Document Type: 
Document Number (FOIA) /ESDN (CREST): 
Release Decision: 
Original Classification: 
Document Page Count: 
Document Creation Date: 
November 1, 2016
Sequence Number: 
Case Number: 
Content Type: 
PDF icon CIA-RDP82-00850R000500010010-4.pdf20.48 MB
APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 , FOR OFFICtAL USE ONLY JPRS L/ 10212 23 December 1981 Translation OPTICAL CAVITIES AND THE PROBLEM OF DIVERGENCE OF LASER EMISSION By - Yuriy Alekseyevich Anan'yev FBO~ FOREIGN BROADCAST INFORMATION SERVICE FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 NOTE JPRS pub'lications contain information primarily from foreign newspapers, periodicals and books, but also from news agency transmissions and broadcasts. Materials from foreign-language sources are translated; those from English-language sources are transcribed or reprinted, with the original phrasing and other characteristics retained. Headlines, editorial reports, and material enclosed in brackets are supplied by JPRS. Processing indicators such as [Text] or [Excerpt] in the first line of each item, or following the last line of a brief, indicate how the original information was processed. Where no processing indicator is given, the infor- ; mation was summarized or extracted. Unfamiliar names rendered phonetically or transliterated are enclosed in parentheses. Words or names preceded by a ques- tion mark and enclosed in parentheses were not clear in the original but have been supplied as appropriate in context. Other unattributed parenthetical notes with in the body of an item originate with the source. Times with in items are as given by source. The contents of this publication in no way represent the poli- ~ cies, views or at.titudes of the U.S. Government. COPYRIGHT LAWS AND REGULATIONS GOVERNING OWNE-RSHIP OF MATERIALS REPRODUCED HEREIN REQUIRE THAT DISSEMINATION - OF xfiIS PUBLICATION BE RESTRICTED FOR OFFICIAL USE ONLY. APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY JPP.S L/J J212 23 De::ember 1981 OPTICAL CAVITIES AND THE PROBLEM OF DIVERGENCE OF LASER EMISSION MosCOw OPTICHESKIYE REZONA'rORY I PROBLEMA RASKHODIMOSTI LAZERNOGO IZLUCHENIYA in Russian 1979 (signed to press 29 Oct 79) pp 2-328 [B4ok by Yuriy Alekseyevich Anan'yev, Izdatel'Btvo, 4,000 copies, 328 pages, UbC 539.11 CONTENTS Annotation Pxeface Introduction. Development of Concepts of the Optical Cavity as a Device for Producing Narrowly Directional Emission CHAPTER 1. GENERAL INFORMATION � 1.1. Laws of Propagation of Light Beams and Angular Divergence of Radiation The Huygens-Fresnel Principle (12). I?istribution in the Far Zone (14). The Ideal Emitter (24). Arbitrary Monochromatic Emitter (19). Nonmonochromatic Emitter (25). Some Conclusions. Measurement of Divergence (26). � 1.2. Optical Cavities and Classification of Them. Initial Information. A Little History (28). Passage ot Light Beams Throsgh Optical Systems. The Beam Matrix (30). Classificatton of Resonators by the Properties of Their Beam Matrices (35). CAndi- tious of Resonator Equivalence (40). � 1.3. Modes of an Empty Ideal Resonator and Their Use for Describing the Laser Situation Classif ication of Nstural Oscillations (42). IntegraL Equation and Natural Oscillations of an Arbitrary Empty Resonator (46). Resonator With Active Layer (48). Suitability of the Standard Model of an Open Optical Cavity for Describing Real Lasers (50). � 1.4. Efficiency of Excitation Energy Conversion in Laser Resonatprs Efficiency of Energy Conversion in an Element of Volume c;f the Medium (53). Accounting for Nonuni=ormity of the Distribution of Laser Radiation Lengthwise of the Cavity (56). General Excitation Energy and Stimulated Emissfon Balance (58). The Meaning and Possibilities of Applying the Derived Relatiflns (61). - a - FOR OFFICIP-L USE ONLY 1 2 4 12 12 28 42 52 [I - USSR - I� FOUO] APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00854R000500010010-4 FOR OFFICIAL USE ONLY CHAPTER 2. RADIATION DIVERGENCE OF LASERS WITH STABLE AND FLAT CAVITIES 64 - � 2.1. Modes of Oscillations of Empty Stable Resonators , 64 Eigenfunctions and Frequencies of a Stable Reaonator with Infinite Mirrors (64). Spacial Structurp of Natural Oscillations (67). Stable Resonators with Mirrors of Fiuite Dimensions (71) $ 2.2. Edge Diff-action and Modes of Oscillations of an Empty Flat Optical Cavity 74 Auxiliary Diffraction Problem (74). Reflection From the Open Edge of _ a Waveguide. Natural Oscillations of a Resonator Made up of Strip or Rectangular Mirrars (78). F'lat Cavity Made of Circular Mirrors (82). Polari2ation of Radiation of Natural Modes (85). ' �2.3. Some Experimental Research Results 87 Early Observations of Stimulated Emission of Solid-State Lasers (88). Divergence of Radiation of Solid-State Lasers (90). �2.4. Multimode Lasing in Ideal Optical Cavities 92 Mechanism of Multimode Lasing (92). Procedure and Some Results of Calculations of the Multimode Lasing Regime (96). Competition of Transverse Modes in Lasers with Flat Cavities (99). Def iciencies of the Model arid the Possibilities of Improving It. (101). � 2.5. Influence of Resonator Deformations 4n Field Configuration of Individual Modes 103 Some General Remarks. Perturbation Theory (103). Flat Cavities . with Minor Aberrations (105). Flat Cavities with Aberrations of - Significant Magnitude (109). � 2.6. Methods of Angle Selection of Emission 112 Attempts to Solve the Problem of Divergence on the Basis of Resonators with Small Diffraction Losses (112). Lasers with Flat Cavities and Angle Selectors (115). Angle Selection of Emisaion of Lasers with Flat Cavities by Reducing the Number of Fresnel Zones _ (119). Flat Cavities of Large Effective Length (121). Multistage Lasers (124). _ CHAPTER 3. ELEMENTS OF THE THFORY OF UNSTABLE RESONATORS 127 � 3.1. Some Initial Information 127 Brief Historical Survey (127). Elementary Examination of the Ideal Unstable Resonator (12$). Properties of Convergent Waves (133). � 3.2. Resonators with Slightly Inhomogeneous Medium 137 Simplest Method of Accounting for Inhomogeneities of the Medium (137). Aberrational Coefficients (139). Some Comments on the Possibilities of the Geometric Optics Approximation (142). � 3.3. Edge Effects and Specrrum of Natural Oscillations 145 Equivalence of Unstable Rssonators and Interrelation af the Solutions for Different Types (145). Unstabie Cavities with Completely "Smoothed" Edge (147). Unstable Resonators with Sharp Edge (150). Specifics of Edge Effects Under Real Conditions (154). � 3.4. Unstable Resonators with Central Coupling Aperture 160 Initial Premises. Oscillations of aTao-Dimensional Resonator that Have a Caustic (160). Z~ao-Dimensional Resonator with Central Aperture (163). Three-Tiimension Resonator with Coupling Aperture. Discuasion of Results (166). , b FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-00850R440500010010-4 - k%* Wk[tIAL USE 0,14LY - � 3.5. Some Problems of Multiple-Mirror and Prism UngtMble Resonatora 170 - Problem of Monodirectional Laaing Mode (170). Stabilizing the Direction of Radiation in Prism Resonators (175). � 3.6. Unstable ReSOnators with Field Reversal 179 Operation of Cross Section Reversal and Polarization Characteristics of Radiation (179). Aberration Characteristics of Unstable Resonators with Field Reversal (182). Resoaators with Compact Outpui Aperture (].PS) . CHAPTEK 4. APPLICATIONS OF UNSTABLE RESONATOkS 189 � 4.1. Unstable Resonators in Pulsed Free-Running Tf^sexs 189 Selection of the Type and Parameters af the R.esonator (189). - Results of Fxperiments with Neodymium Glass ]'.asers (195). Gas Pulsed Lasets with Unstable Raeonators. Froblem of Steadying Oscillations (201). � 4.2. Unstable Resonators in Continuous Lasers 204 Survey of Experimental Work (204). Methods of Calculating the - Efficiency of Flow Laser.s (206). Simplest Model of a Gas Dynamic Laser Medium (210). Problem of Forming Uniform Field nistribution Over the Cavity of a Flow-Through Laser (215). � 4.3. Unsteady Resonators in Laswera with Controlled Spectral-7'emporal Emission Characteristics 218 Simplest Types of Lasers with Control Elements (218). Lasers with Thiee-�Mirror Optical Cavity (221). External Signal-Controlled Lasers (224). Multipass Amplifiers (226). CHAPTER S. OPTICAL INHOMOGFNEITY aF ACTIVE MEDIA AND METHODS OF CORRECTING WAVE FRONTS 230 � 5.1. Thermal Deformations of Solid-State Laser Cavities 231 Origin and t4agnitude of Thermal Aberrations in the Case of Circular Active Rods (231). Consequences of Aberrations and Attempts to Correct Them (235). Various Methods of Reducing Cavity Deformations (238). Lasers Using Active Elements with Elongated Rectangular Cross Section (241). � 5.2. Phase Correction of Wave Fronts. Dynamic Holography and Stimulated _ Scattering 245 Optical-Mechanical Correction Systems (245). Holographic Correction Principles (247). Conditions of Realizing the ProcesG of Holographic "Transfer" and its Energy Efficien;:y. "Transfer" on Thermal Gratings (250). Relation of the Idea of Dynamic Holography to the Phenomena of Stimulated Scattering. Lasers Based on Various Forms of Stimulated Scattering (255). � 5.3. Method of Wave Front Reversal 258 Tha Idea and Theoretical Possibilities of the Method (258). "Reversal" in Stimulated Backscattering (261). "Reversal" by Methods of Classical Optics and Holography (264). "Reveisal" in Parametric Amplification of Light (267). BIBLIOGRAPHY 269 c FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500014410-4 F'OR AFFICIAI. USE ONI.V UDC 539.1 OPTICAL CAVITIES AND THE PROBLEM 0F DI.~'ERGENCE 0F LASER EMISSION ' Moacow OPTICHESKIYE REZONATORY I PROBLEMA icASKHODIMOSTI LAZERNOGO IZLUCHENIYA in Russian 1979 (signed to press 29 Oct 79) pp 2-328 [13oolc by Yuriy Alekseyevich Anan'yev, Izdatel'atvo, 4000 copies, 328 pages] [Text] The book gives the basics of the theory of optical cavities, and examines - the factors that determine divergence of laser emission. Major emphasis is given to the problem of producing narrowly directed radiation; various methods are outlined for reducing angular divergence. The properties of lasers with so-called unstable optical cavities are considered in greatest detail. Methods of calcu- lating and optimizing them are outlined, and the particulare of designs iused in - a variety of laser devices are discussed. The book also gives some informaticn on factors giving rise to optical inhomogeneities in an activ'e medium, the nature af ~ wuch inhomogeneities, and methods by which their influenee can be reduced. Figures- 112, tables 3, references 343. 1 FOR OFFICIAL USE OPILY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 FOR CFFICIAL USE O1VLY PREFACE The work of Nobel Prize winnexs N. G. Basov, A, M. Prokhorov and C. Townds(United _ States) has Ied to the creation of cohexent light generators - lasers, Operating lasers appeared in 1960. Among the astonishing characteristics that attracted general attention, first of all it is necessary to meation divergence of the emission the degree of directionality of a coherent light beam is theoretically limited only by diffraction at the exit aperture of the laser, and many people have held the opinion that there is no obstacle to achieving this maximum directionality. The literature of that time was full of optimistic predictiona with respect to the posaibilities for using lasers in superrlong communication lines (including space), fox long-distance power transmission, and go on. All of these predictions were, as a rule, based on simple manipulations with the known formula for the diffraction limit of divergence of emission ~0 m 1.2 a/D, where a is the emisaion wawelength and D is the coherent light besm diametex. Actually, for ruby lasers (a = 0.694 micron) which appeared first, the value of 0 is 4�10-5 radian or 8 angular seconda for an easily attainable crystal diamezer! pHowever, the firat observations of - stimulated emisgion of such lasers, the active media of which were 6 to 10 mm in diameter, demonstrated that the angular divergence of their emission was appreciably greater than expected (usually by oae or two ordere). _ Analysis of the causes of such behavior on the part of lasers and influenciag them in the require3 direction turned out to be far from a aimple matter. Countless eacperiments, the development of the theary of real reaonators with active medium which, to a great extent, is based on the theory of ideal empty resonators developed in 1961-1966, and so on were required for this purpose. Al1 of this spilled into the overall area of quantum electronics comb{.ning a great variety of inethods of = controllfng tha spatial characteristics of laser emission. Many of these methods are of great theoretical interest; as for pract'Lcal importanee ofwork aimed at de- _ creasing the angular divergence of emiasion, it can hardly be over-estimatRd. Therefore it is no surprise that an enormous number of publications are devoted to ' the problem of divergence. Horaever, up to the present time there has Leen no serious effort to discuss the fundamental concepts of the given problem as a whnle. In all - of the books on laser theory and laser engineering, in the best case the reaults of the theory of ideal resonatora with optically homogeneous active medium (or empty resonators) are presented. The several surveys which have investigated individun2, special problems have nat filled the gap. The preaent book would appear ta he useful in this respect, and to what degree it has served this purpose can be best determined by ite readers, whose suggestians will be apprecisted and receive - immeaiate attention. 2 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE nNLY We have presented only the most general information from the theory of empty ideal resonators whtch is needed tio underetand the following material. A somewhat morE detailed study has been made of the problem of the divergence of emission 'at - resonators with small diffraction losses, primarily in the most wideapread resonator with flat mirrors. The discub,:ion of this problem doas not claim to be especially ~ complete: a laser with a flat cavity is in essence a highly complicated oscilla- tory system with an enormous number of resonance frequencies subjected to the power- ful influence of the most minute disturbance. Therefore the probi2m of the emission " divergence of lasers with flat cavities, which is closely related to a still more complicated and general groblem of their kinetics is truly inexhaustible. The book contains only information which suffices to explain the general situation and , which is needed to read specialized literature. Lasers with so-called unstable resonators are then discussed. The buJlding of these - lasers is the result of an area of research, representatives of which tried not so much to provide an exhaustive description of systems with an enormous number of degrees of freedom as to limit the number of degrees of freedom. Although not everyone has become accustomed to lasers with unstable resonators, they are, in essence, much simpler than trzditional lasers with flat cavities. They do not have such an abundance of Q-similar modes and sensitivity to the most minute disturbances. Therefore the theory of lasers with unatable resonators has already advanced farther - than the theory of other types of lasers that appeared much earlier. The results of this theory will be discussed in conaiderable detail; attention has also been given to the most important experimental papers. In conclusion, there is a brief stsdy of the problem of optical inhomogeneity of act.ive media, and some efforts to solve this problem are discussed. The author is deeply indebted to N. A. Sventaitskaya, V. Ye. 5heratovitov, 0. A. Shorokhov, N. I. Grishmanov, V. P. Kalinin, L. V. Koval'chuk and other of his colleagues for their participation in joinC work with the author, the discussion of the results of which constitutes a significant part of this monograph,and for their assistance in preparing the manuacript for publication. There were useful discus- sions of individual topics with V. V. Lyubimov, M. S. Sflskin. The author 1s also =":idebted to P. V. Zarubin and Ye. N. Sulcharev, whoae suggestions led to many editorial corrections. The author found the constant attantion and active support of Rem Viktorovich Khokhlov in the writing of this monograph especially meaningful. The monograph is dedicated to the memory of Rem Viktorovich Khokhlov. - Leningrad, August 1977 Yu. AnanRyev 3 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY - INTRODUCTION Development of Concepts of tlie Optical Cavity as a device for Producing Narrowly - Directional Emission The first papers on the feasibility of making resonators in the optical band for stimulated emission of coherent radiation were publiahed by Prokhorov, and also by Shawlow and Townes in 1958. These papers to a great exteat predetermined the course of research that led to the development of 1$sers. As.we know, laser action i.s based on the capacit}� of certain media under certain conditions to amplify luminous radiation pasaing through them, Therefore there is no doubt that a major role is played by the properti,es of the active medium itself and the method of stimulating it; however, the properties of the resonant cavity in which this medium is 'ocated alao have an enormoua influence on maay characteristics of stimulated emission. Taking a central place among such characteristics is the angular divergence of emission. Here the resonator plays a truly decisive role: without a resonator, zhe active medium in itself as a rule is capable of amplifying the radiation pasaing through it to equal advantage no mattier what the direction of propagation. Optical oscillators made their appearance much later than rf and microwave oscilla- tora. There�ore the concepts and terminology borrowed from these related f ields are extensively used in describing optical oscillators, and we will continue this praetice. To convert an ascillator to an amplifier, it is neceseary,in the language of elsc- tronics, to close the output of the amp].ifier to ita input, to set up a feedback circuit (Figure 1, a). The essence of the feedbacY circuit ie that part of the amplified radiation goes back into the system, is amplified again and so on, in thia way maintaining a continuous eignal oscillaticn. Oscillators in the opti- cal band amplify a light beam rather than an electric signal. In this connection, the feedback circuit must meet new requirements: after transmission through the feedback circuit the beam its original direction of propagation and atructure. The simplest analog af. Figure 1, a that meets these requirements is the optical_ resonance shown in Figure b. Actually, such resonatora, called ring cavicies, h$ve found some application. However, the first reaonator in optics was a ccriventional Fabry-Perdt interferometer made up of two flat mirrors (Figure 1, c). One mirror is partly transparent, and the stimulated emiasion is coupled out thraugh it. The principal difference o� this resonator from the ring cavity is that the feedback circuit passes through the same medium, and the emission is'repeatedly amplified. 4 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R040500010010-4 FOR OFFICiAL USE ONLY Us eful a) Amplifier hsi 0%11. Feedback .ircul3c_J b) c) - Figure 1. Diagrams of oscillators: a) oscillator in electronics; b) oscillator in optics; c) laser with flat optical cavity. The Fabry-Perot interferometer has been such a successful resonator system that even today it is the most widely u;sed type of laser cavity. Such popularity is - due not only to extreme simplicity, but also to the capability of attaining high energy characteristics of output emiasion (the reasona for this will be taken up in section 1.4). Hawever, the situation with regard to divergence of emiasion etimulated in flat-mirror cavities was trivial. The origia of comparatively - large divergence is explained to a$reat extent by theoretical analysie of the properties of the flat optical cavi.ty, which can be done by methoda developed pre- viously for resonant cavities in t'ae microwave band. Such analysis ahows that transmission through the feedback circuit in a cavity reproduces not 3ust any beam directed along the axis and nearly parallel with it, but only.a laeam with strictly defined distribution of the amplitude :.nd phase of the wave front. This beam is called the fundamental wave form (mode) of the cavity. Wnat i.s more remark able however, is that there are also beame (modes) that are reproduced with somewhat . greater attenuation in which emisaioa propagates at sanall but nonetheless noticeable angles to the axie of the cavity. These beams form a discrete set; the angle of inclination to the axis for modes that are neighbors in classification differs by approximately half the diffraction angle. Thus, a flat cavity in some sense (and, of course, within certain limits) is indif- ferent to the direction of emission propagating through it. The roots of such indiff erence are in the fact that upon passage through the feedback circuit the _ oblique beama, like the axial beam, retain their original directions of progagation; they are prevented from "walking off" by edge diffraction. The mechanism of such diffraction will be i:aken up in section 2.2. Because of differences in attenuation, the different c,ff-axis modes have somewhat different thresholds of excitation; however, because of nonlineari,ty of the medium, theae modes can be present simultaneously in the stimulated emission (Chap- ter 2), which ehould lead to a large angular divergenre of the beam. For this reason, flat cavities with large apesture cannot give small beam divergence even when the medium ia highly homogeneous, , 5 FOR OFFICIAL USE ONII.Y APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 ~ lActive 1 medium I APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 Such is the sit.uation in the idealized'case of an optically hovwgeneous medium. An Even greater diRadvantage of the flat cavity from the standpoint of the direction- ality of radiation is the extremely sltarp dependencp of thF field distribution i.n the cavity on slight distortions of shape (deformation). This can be explained by the sf.mple example of a misaligned cavity, Ilonparallelism of the mirrors is equi- valent to insertion of an optical wedge in the feedback circuit that,changes the beam direction by an angle d that is twice the sngle of misalignment. If we send a Farallel beam into such a cavity, it will be turned through an angle d after the - first pass, through 26 after the secend and so on. Its diaplacement in the trans- verse direction will increase even faster, and a considerable part of the radiation _ will begin to leave the system, missing the mirrors. After a certain number of passes (a fairly large number for small S), the beam shape is distorted so much cha.t it becomes impossible to analyze the process of further propagation without consi- deration of diffractian effects. Therefore it is no simple matter to represent the , steady-state field distribution in graphic form. In essence this distribution is the result of equilibrium between proceases of diffraction and beam rotation. We emphasize that this equilibrium is reached after a process of accumulation of aber- rations over a number of passes. It is also noteworthy, as shown by analysis (sec- tion 2.5), that there is a rapid rise in the number of passes over which aberrations accumulate with an increase in the cross sectional dimensions of the optical cavity. - Therefore flat cavitiea with mirrors of large dimensiona are particularly sensitive to slight aberrations. The situation is similar, but even more complicated in the case of irregular defor�- mations of the cavity. In general, the sources of such deformattons are extremely varied: they include errors of manufacture and alignment af mirrora, initial opti- cal inhomogeneity of the active mediura, inhomogeneity indured during pumping due to _ nonuniform excitation and heating of the medium, scattering of light by microscopic inclusions, mechanical vibrations of the active element, turbulence o.f gas flow, and the list could be extended. It is no wonder that a great many papers have dealt with the problem of a nonideal flat resonator, most of them publiehed in 1965- 1969. The results of research dealing with the moat comlnon principles of behavi.or of lasers with nonideal optical cavities will be given in section 2.5. 0f course, none of this xesearch could eldaninate the fundamental flaws of the flat optical cavity, but it has been very useful for an understanding of the processes that take place in a real cavity. In these same years, as much .research was done to f ind some new solution of the laser beam divergence problem. Two areas of such research can be distinguished. Representatives of one nf these areas tried without givino up the flat cavity to put into the feedback circuit so-called angle selectors; filters that transmit radiation only in a narraw range of angles. Such filtera can be made by uifferent methods: on the basis of total internal reflection, by additianal Fabry-Perot etalons, by a combination of lensea and irises. A1l theae methoda wei:e tried; however, subsequently because of considerable complications and a number of funda- mental difficulties that will be taken up in section 2.6, they found only a few _ apecial applications. Proponents of the other area of research have attempted to solve the prohlem of - divergence (or at least single,-mode laRing) by altering the s,hape of the mirrors rather than by complications in the design of the optical cavity. In particular, there has been a thorough investigation of lasers with so-called stable resonators, 6 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R040500010010-4 FOR OFFICIAL USE ONLY in which one or bAth rairrors are sli,ghtly curved (the equivalent of placement of a weak positive leng in the feedback circuit). Ohviously in such a system a stsady- state beam xith approximately uniformlv :iistributed amplitude and phase and a front cloae to planar ahould have such a small crose section tha.t the focusing effect of the mirrors is compensated by the defocusing of the beam due to diffraction. This is the source of the main disadvantage of stable resonators: they are capable of single-Luode lasing only with very small volumes of the active medium. And it is in lasers with small volumes of the working medium that stable resonators are still being used; an example is the ordinary helium-neon laser. If the volume of the medium is large, wide beams are produced in atable resonators with a complic4ted beam structure corrasponding to an angular divergence larger than in flat cavities under the same conditions (sections 2.3 and 2.4). Some other forms of optical cavities with small diffraction iosses have also heen investigated (section 2.6), but these were also failures. , This sicuation led to the general opinion that the problem of directionality of radiation cuuld not be solved by iu:proving the optical cavity, and that the only possiblP way was to make multiatage systems of a mascer laser and amp].ifiers (since amplif iers are not troubled with the effect r,f multiple accumulation of aberrations, like the �lat cavity). A way out of this dilemma was suggested by Siegman's re- - search with a simplified analysis of an 'yunstable reaonator (in the accepted clas- sif ication; section 1.2), which is formed by two convex mirrors. This analysis showed that in such a resonator, just as in tlie ideal flat ogtical cavity, the:e is a solution in the geometric approximation, only diverging rather than parallel beams propagate in both directions along the cavity, and part of the radfation passes the mirrors (section 3.1). At first Sieguian's paper did not stir up any particular interest: some researchers were already working with unstahle resonators, and were getting only undesirable side effects rather than encouraging results (see the beginning of section 3,1). And Siegman himself even switched his attenticn to the ~ unusual features of edge effects in unstab:Le resonators, devottng some years to the study of this pro;,lem, which is interesting, but as we Ftill show below is far from decisive. From the standpoint of directir,nalit}r of radiat.ion, the most interest_ing peculiarities of unstable resonators were discoverei ony with analysis of the in- fluence of aberratious on steady-state field diRtribution. It was as a result of suc;i an analysis that it became clear that the resonator considered by Siegman was ~ only the f irst oi an extensive class of cavities in which feedback con�orms to a totally new algorithm that has a number of fundamental advantages.. Figure 2, Diagxam of a laser with unstahle resonator, , 7 FOR OFRCIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400540010010-4 - FOR OFFICIAL USE ONLY ~ The essence of this new algorithui i.s clati..�ied hy Figure 2, In the flat cavity, = part of the radiation from the entire cross section of the output beam was introduced into the f eedback loop (aee Figure 1, b) . In the unstable resonator as a rule a12 the radiation is fed back, but only from part af the cross section. For reproduci- bili*yo uf the process, the feedback circuit obvi.ous3.y ha.s to be made in such a way that the beam is widened. In Figure 2, the wave front is planar and the beam is expanded by a telescopic system, but quite different version.s are possible. In par.ticular, the dimensions flf- the cross section of a parallel beam are also altered when it is reflected from a diffrac:tion grating, and when it passes oblique' y ' through the interfaces of inedia with different indices of refraction. Th.... leases and spherical mirrors are not at all obligatory equipment for an unstable resonator. - iVow let us examine what advantages could be realized by what would seem to be such a minor change in the algorithm. As in the case of a f lat cavity, let an optical d % d a) i b) ~ . 's.l %d -1-'- -+j--1"" I I ~ . c) ; i~ i . . Figure 3. Accumulation of wave aberrationa in an unstable reaonator when an optical wedge is _ inserted. - wedge be inserted within the system, leading to rotaton of the beam, and let this - rotation correspond to a magnitude of wave aberrations A (Figure 3,a), Part of the beam goes to the feedback loop; in the case shown in the f igure, its transverse dimension is half the size of the entire beam. On this part of the cross section, the magnitude of the wave aberrations is of course equal to 1/2 A. When the cross _ section of the beam is stretched out by the methcds enumerated above, the magnitude of the wave aberrationa does not change; therefore oy the beginning of the next - cycle a beam arrives at the input of the system with aberration oF 1/2 d, rather than A as in the ca3e of a flat cavity. Upon passage through the system, another A is added (see Figure 3,b) and by the end of the second cycle the ma.gnitude of the - wave aberrations becomes 3I2 A. sy the beginning of the third cycle it is 3/4 A, ; by the end -T 7/4 A (Figure 3,c) and so on. It can be seen that this quantity at the limit approaches 26. Thus, the wave aherrations in the steady state are found = to be only twice the level in a sin,gle pass, In this aituation, shifting of the beam in the transverse direction doea not generally play any particular part; it 8 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-0085QR000500010010-4 FOR OFF[CIAL USE ONLY r, leads only to sli,ght redistriliutton of the fractions of the beam passing to differ- ent sides of tche output mirror, but the feedback loog remains completely filled. In the case of arbitrarily distributed aources of aberrations, of course there are changes in th.e quantitative factors, but the qualitative patterns remain the same: the accumulation of wave aberrations of the beam place over the calculated number of passes (see section 3. 2). As the fracti4n of the beam cross section directed into the feedback loop decreases, this numbe.r approaches unity, a quantity typical of single-stage amplifiers. lie can also conclude from the example given in Figure 3 that in the unstable reso- nator there is only one form of wave tront that is retained after the beam passes through the feedback loop. Actually the direction of the light beam admitted to the optical cavity. rapidly approaches the unique equilibrium direction (which in ~ the case of Figure 3 corresponds to steady--state wave aberrations 2A). Thus in con- trast to the flat optical cavity, the unstable resonator displays rigorous selectiv- ity with respect to the direction of emission; it would seein that here the effects of multimode lasing should have no influence on the magnitude of angular divergence. These encouraging deliberations concerning tlie properties of unstable resonators, - implied by the simplest geometr ic approximation, are conf irmed as well by more thorough analysis in the diffraction approximation (section 3.3). Moreover, de- tailed acquaintance with the properties of unstable resonatora allows us to formu- late the following statement: if the operation of a laser in which a large volume of comparatively homogeneous active medium has been csed is strongly influenced by diffraction, this means that errors have been made in its creation. At first - glance this atatement seems to be paradoxical; in alZ previously used optical cavi- ties the beam structure has been determined in the final analysis by diffraction. - However, this statemen4 has a f irm logical basis. The final goal is to produce a laser with minimum angular beam divergence; this goal is fully attainable in a homogeneous wedium. Beams with minimum divergence are those for which the wave front is f lat or spherical; it is well known that the propagation of such beams to short distances is beautifully described by the geometric approximation, wi.thout resorting to the concept of d3f f raction. This covers the conceptual asp ect, and we turn now to practice. The scheme with two convex mirr ors considered in Siegman's f irst paper was never used in that form, remaining a f avorite subj ect of zesea.rch for theoreticians in - view of its symmetry. But then, a number of specia.lized schemes enabling solu- t:Lr3n oi some problems in laser technology by much simpler means than before became popular. Among these problems, of course, is.the very problem of constructing simple laser emitters with high efficiency and low angular divergence of radiation. In most - cases of practical importance, the so-called telescopic cavity (section 4.1) is suitable for solving this problera, being an asymmetric confocal system made up of a convex and a concave mirror. In this system the path of the rays differs from that of Figure 2 only in the spatial congruence of two oppositely directed beams, tha.t is, in L= same way that the scheme of Figure l,c differs from tha.t of Figure l,b. Also of interest is the ring ve rsion o� the telescopic cavity, wherein conditions are readily brought about such that the radiation �lux propagating along the ring 9 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-0085QR000500010010-4 FOR OFF[CIAL USE ONLY Co one side, without the use o� nonrecipxocal devi.ces' atwuld be many times greater than the �lux �ollowing in the reverse direction (section 3,5). There are a number of new possibilities associated with the idea of utilizing the particular role that is associated with the paraxial region in the unstable re8o- - nator. This idea is as follows. If we trace the path of the rays in an unstable resonator over several cycles, we can see that radiation "spreads out" over the en- tire cross section from a small area that is contiguous with the line tha.t is the axis oE the resonator. By introducing emission into the region with predetermined temporal and spectral characteristics, we cgn obviously accomplish effective control of the radiation of the entire laser. When the "seed" radiation is taken from an external source and coupled in through an aperture in the mirror on the axis of tha - optical cavity, our device becomes simpl.y a multipass amplifier with very Izigh�gain (section 4.3). As the emission intensif ies, the cross section of the beam expands, which facilitates attainment of the limiting possible energy characteristics. CQn- Crol of radiation properties on the basis of this principle can also be realized by making single lasers that havE the valuable property tinat the cross sections of the control elements that are used in them (shutters, spectral selectors) are many times le3s than the cross sections of the output apertures. Some special problems can be successfully solved by a slight modif ication of the feedback algorithm. As an example, we carL cite the problem of stabilizing the direction of radiation; this problem ariaes in the widely encountered case where there is an optical wedge inside the resonator with a size that changes during the lasing pulse (that is,because of inechanical vibrations of the active element). To strongly attenuate the influence of the wedge, it is sufficient to construct a feedback loop such that the beam is reversed in addition to the expansion of the _ cross section, If this change is made in the version shown in Fi,gure 3, by the beginning of th:e second cycle the magnitude o� wave ahexrations is equal to --0.5A, - by the end of the cycle it is -0,5Q + Q; hy the beginnizig of the thixd cycle it is --0.25A, and by the end it is +0.75A and so on. In analysis afront is set up at the output of the system with aberration of 0.67A, that is, one*-thirdof the value in the case of Figure 3. Let us note that both Figure 3 and the example now being considered are essentially different types of telescopic ring cavities. In the case of non-ring systems where the beam in following the �eedhack loop passes through the same wedge, designs can be develaped with even better stability of the clirection of the output beam (section 3.5). Finally, we muet not fail to mention problems that arise with the appearance of a number of new types of lasers. For example in the so-called fast-flow lasers the active medium passes through the optical cavity in the transverse direc*_ion, of ten with excitation beforehand rather than in the cavity. This leads to radiation field distribution patterns that are totally different from those of przviously existing lasers; in particular, when flat mirrors are used, lasing can be localized in a narrow zone close to the edge of the cavity where the active medi-xm enters. The density of the stimulated emission there is naturally extremely hish. A com- pletely analogous situation occurs in the transversely pumped Raman laser:1. Diffi- cultiea of this kind are easily overcome by using unstable resonators: a kind of self-balanced mode arises, and the radiation flux is distributed more or less uniformly aver the cross section of the cavity (section 4.2). All these possibilities were quite quickly realized in specific devices (see Chapter 4), Thus within only a f ew years af ter the publication of the first report on experimental observation of ang.le selection ia an unstable resonator it-became io FOR OFFICIAL USE ONLY 1 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FUR UHFICIAL USE ONLY completely clear that this was the optical cavity that is opti.mum fer the most diverse forms of lasers with narrow~beam Qmission, - All the aspects mentioned above, as well as some other facets of the problem of uiversence will be dealt with more rigorously and consistently later. Before we go into this, let us note one interesting feature of the history of this problem. We have just spoken of the extent to which solution of the problem of divergence is obstructed by appreciable inhomogeneities of the active medium. The results of considerable research in the earl}r sixties (section 2.3) showed that difficulties also pile up rapidly with increasing diameters of the lasing beams: the diffraction limit of divergence is difficult to attain only in the case where the diameter of the active element is large, and the limit is correspondingly small. If beam diame- ter is reduced (that is, by irising the cavity) to a stize on' the order of 1 mm or smaller, the beam divergence as a rule is on the diffraction 1eve1 even when no special steps are taken. - The first gas lasers, typif ied by the pre-eminent helium-neon laser, operated on an � active medium confined in a rather slender discharge tube, and produced a light - beam of sma11 diameter. The mixture itself was quite rarefied, and a negligible amount of power (by present-day standards of laser technology) was expended on excitation; therefore the active medium was nearly ideally homogeneous. Due to _ this confluence of cixcumstances, the developers of gas lasers did not come up - against the problem of divergence in earnest for many years. It mi.ght seem to the superficial observmr that this problem is specif ic in general to solid-state lasers, - and arose primarily because of the imperfectness of solid active media. However, in recent years the intense development of gas laser physics has led to new types of lasers such 3s chemical, gasdynamic and other types, with powo.7 many orders of i-aagnitude greater than that of the familiar helium-neon lasers. The rise in radia- tion power is, being achieved by an increase in the pressure of the working medium, tihe dimensions of the space that it occupies, and specif ic pnergy inputs on excita- tion. All this entails cor.siderable optical inhomogeneities of the medium. "ihus, the problem of beam divergence has turned out to be a problem of all lasers with tiigh emission power, not just solid-state lasers. The developers of gas lasers are now coming up against juat about the same difficulties aa werE - previously encountered by the derelopers of the solid-state laser, and they are using just about the same methods of.solving them. For this reason, research. in the f ield of gas lasers has not added much new information on methods of angle selection. - Most of the research that has played a large part in solution of the problem of divergence, and that will be used in Chis book, has been on solid-state lasers. 11 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00854R000500010010-4 FOR OFF[CIAL USE ON1.V CHAPTER l. GENERAL INFORMATION g' 1.1. Laws of Propagation of Light Beams and Angular Divergence of Radiation Huygens-I'resnel Principle. We shall use the scalar theory of diffraction to study the propagation laws of laser-generated coherent light beams, The easence of this approximation consists in the fact tha.t the di�ferent transtrerse components of an electric or magnetic f ield are considered to be indepeadent of each other, and they are considered separately. The conditions of applicability of the acalar theory of diffraction are discussed in detail in [6]. In the previously investigated situations where the transverse dimensions of light beamg and the psths traveled by them are many times (frequently even several orders) greater than the wavelength, these conditions are automatically satisfied. The most important expression of scalar diffraction theory, which is one of the possible mathematical formulations of the Hupgens-Fresnel principle, follaws; u~xs, ys) = Zi J J ezp ltkri2) cos (n) r1:)l u(xiI Ui) dxldlll� it Here, xl, yl, x2, y2 are the transverse�coordinates of points in the source plane and the observation plane parallel to it (Figure 1.1), r12 is the radius vector joining these points, n is the common normal to the planes, kre 2ff/a; finally, the function o� the transverse coordinates u(x, y) which varies slightly at distances af _X is the complex field amplitude. Expression (1) allows calculation of the field u(x2, y2) in the second plane by the given field distribution in the �irst plane u(xl, ylthe medium is conaidered to . be homogeneous, the sourc~e is conaidered to be strictly monochromatic with time- dependence of the type exp(-iwt) such that u(x. y, t) = u(x, y)exp(-iwt), and the actual value of the field intensity is equal to Re[u(x, y)exp(-iwt)]. Let us note . that the convzrsion. from the field intensity u(x, y, t) to the eanission intensity I(x, y) will be made hereafter by the formula I(x, y) a, where the angular brackets denote averaging over an infinite time interval. For a monochro- matic source I(x, y) _ Ju(x, y)12. The transverse dimensions of the region in which the field is nonzero will alatost always be conaidered limited and not too large. In cases where an opaque scr.een with a hole in it is located in the first plane, and the field sources themselves 12 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-44850R000500010010-4 N'r1R oFFi('tAt, t M AN1,Y are located to the left of this screen, we ahall adhere to I:irchhoff`s principle. . According to Kirchhoff`'s principle, when calculating the field in the second plane, integration in (1) must be performed only over the area of the hole. It is assumed that the presence of the screen has no influence on the f ield distribution in the fiole vi�inity (which is valid when the hole dimensions are signif icantlq greater than . As a rule, in the theory of resonatora it is necessary to consider so-called paraxial beams, the angles of inclination of which to the system axis are small. . This allows replacement of cos (n, r12) in expresaion (1) by one. rhen, in the overwhelmino majority of cases of practical importance, it is possible to replace the exact value of r1,= z' .}.(x9 zl)' (ys - yl)z in the exPonent by the approximate value r12 = z-}- (x2 - zi)212z (y2 - yi)1/2a, and it is possible to replace ' r12 in the denominator of the expression under the integral sign simply by the I, spacing between the planes z. Thus, u eX i) (=kZ) J J exp tik I(s9 2sxi)~ (Y: 2syi)9J1 u(xi, Ji) dx,db,� (2) ~ 1, J ) Formula (2) is usually called the Fresnel approximation. A detailed investigation of its limits of applicability appears in many optics handbooks (see, for example, [7]). We shall discuss only one important item here. Although the aseumption of limited transverae dimensions of the source was uaed in deriving (2), formulas of this type will sometimes be used hereafter for infinite integration limits. In order to understand how theae formulas apply in this case, let us use (2) to derive the propagation laws of an inf inite plane wave. Let this wave be directed along the z-axis so that the plane xy is its equiphasal surface: u(xl, yl) = C= const. Then (2) acquires the form 13 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 Fi,gure 1.1, Huygena.4resnel principle. APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 FOR OF FICIAL USE ONLY � T u(xz1 ys~ = C ez ~=1ks) J eXp tik [(X2 ZsZi)1,.~ (bz 2:Yi)s dyldyl. _ao ..,e The improper interval in the form in which it appears in the right-hand side, does not converge in general. Actually, we shall break down the integration surface into so-called Fresnel zones, inside the first of which the phase of the expression under the integral sign varies from 0 to 7r, and inside the second, from Tr to 27r, the _ third, from 7r to 3w, and so on. As is easy to see, the Fresnel zone boundaries - form a system of concentric circles with the center at the point 0 located opposite the observation point (see Figure 1.1). Their radii rn are defined in the given case by the relations (k/2z)rn = n7r or rn =Vz-. Representing our surface integral in the form of a sum E of aIl integrals over individual Fresnel zones and perform- ing the integration (for which it is more convenient to convert from cartesian co- ordinates to polar coordinates with the center at th2 point 0), in the final analy- sis we obtain the following expression: E_(2i/7r) (sl - sZ - s4 where sl, _ s2, s3, are areas of the corresponding Fresnel zonea. Inasmuch aa sl = s2 snXz, the sum of this series doea not approach a limit; hence, we have the corre- sponding statement with respect to the initial integral. In order to overcome this diff iculty, it is suff icient to consider that formula (1), which is more exact than (2), contains a factor in its expression under the integral sign which decreases slowly as the direction of the emission diverges from the normal to the source plane. Therefore in reality, in increasing order the terms - of the series must decrease slowly, but regularly. In this case the sum of the series approaches a defined limit which dtlgs not in practice dependon thp, law by which the terms of the series decrease, and is equal to the half of the first of them (see, for exainple, [6], � 8, 2). Thus, E= isl/~r. In the final analyais we obtain the correct result: u(x2, y2) = C exp(ikz). The presented arguments indicate that the operation of intermediate integration in infinite limits is entirely regular. This operation is necessary, for example, if the field in the plane of the source depends on only one of the cartesian coordi- nates. Then integration over the othar coordinate leads to the following analog _ of formula (2) for the two-�dimensional problem: - u(x2) = (iks) (s' 2:u(xl) dxl. (2a) Distribution in the Far Zone. Now let us proceed to the primary topic of this sec- tion the problem of angular divergence of emission. Using (2), let us investigate the field distribution at large dietancea from the source z such that the inequality Z ~ 2 (zi -f- Yi~ma: (3) is satisfied. Then it is possihle to neglect the terms with xi and i in the ex- pression undex the integral sign. Also taking the teriqs with x2, 4 out to the common phase factor in fzont of the i,ntegral, we obtain 14 FOR OFF ICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500014410-4 FOR OFFICIAL USE ONLY u (x21 ys) = ezP [tk (a xs/2z + Y22/2z~~ X ~ exp =s (xlxa + yiJt),.u (xi, bi) dxldyl� W J (4) Finally, if we characterize the position of the observation point not by its carte- sian coordinates but h}r the slope angles aX = x2/z, ay a y2/z and the distance of z from the source, we obtain the especially descriptive formulas: : u(ax, ay, z) = exP [dkz (1-- a2 XI2 -i- a2y/2) ] x . . X~11: exp ik (axxl amYl)] u(xi, yi) dxidyit (5) _ j(ax, au, Z) F(ax, aY), (6) I(a, a = 1 ex ik a x x s ' x a) 17j~ P f- ~ x i-I- ayUi)1 u ~i, !li) dx,dyl I. - From theae formulas it follows that the forma of emission distrihution in the planes - at different distances sufficiently removed from the source coincide, except the distribution scale increases proportionally to the distance z, and the density varies, correspondingly, N 1/z2, flere, according to (5), the angular amplitude distri- _ bution is the Fourier transfortn of the amplitude distribution over the source cross section calculated for spatial frequencies fX s aX/a, fy = aY/X with accuracy to the factor in �ront of the integral (for more details see [7]). The value of F(aX, a y ) introduced in (6) is none other than the radiation flux per unit sa1:Ld angle. In classical photometry, this value was usually called the luminous intensity in the given direction; adherfng to the presently more popular terminology, we shall call the function F(a , a) the angular distribution of the radiation intensity (or the intensity disfrib&ion in the far zone), and we sha.ll keep the term axial luminous intensity only for F(0, 0). We shall��use the same notation for the angu- lar distribution of the emission and nonmonochromatic sources, although formulas (5), (6) cease to be applicable here. 1 The intensity digtribution width in the far zone is also the angular divergence of ~ the light beam. Most frequently, when discussing divergence, we mean the so-called "divergence with respect tu the 0.5 intensity level"; this is the taidth of the range - of angles in which the intensity is r..o less than 0.5 of the maximum value. However, real emitters frequently liave broad distribution "winge," for which a signif icant 'part of the power is required. Therefore the magnitude of the divergence with re- spect to the 0.5 intenaity level, that is, in esaence, the width of the central distribution maximum, is not very tndicative unleae the fraction of the total power contained in this maxi.mum ie known. The value usually called the divergence with respect to the 0.5 energy level has greater practical signif icance; this is the angular diameter of a circle in the far zone which encompasses half of the total radiation flux. ~ Before we proceed with analysis of apecific examples, let us note the following characteristic features of the angular distribution of a monochromatic emitter which are a consequence of the Fourier transformation characteristica: 1) if the field diatribution over the emitter aperture u(xl I yl) ca.n be represented in the form uX(zl)u (yl), then the expression for the anguIar distribution breaks down into the produL of two cofactors 15 FOR OFFICIAL USE ONj.Y APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY F(a=, &hr) F,'(06) FY (uv) y (7) 2) if one of the emitter dimensions is increased by K times wtiile maintaining the form of the distribution over the aperture, the divergence of the emission in this directi4n decreaees by K times. - Now let us proceed with the inveatigation of aome important cases which will be encountered in the following sections. . The Ideal Emitter. We ahall consider the emitter ideal if its complex'field amplitude is constant in the exit cross section (aperture). It is possible to use only a hole in an opaque screen ligY~~ted by a point light aource arranged so that the beam in the vicinity of the hole will be sufficiently uniform with re- spect to density and have plane wave front, as the optical system of such an emit- ter. Therefore, previously we talked about diffraction at the corresponding hole instead of an emitter of one form or another. NoW the semitransparezit end of a laser rod frequently plays the role of the 3.dea1 emitter. It is sasy to see that the maximum angular intensity distribution for an ideal emit- ter is always in the ar.ial direction. An extraordinarily simple formula for the axial luminous intensity follAws directly from (6): 2 ~ - (0' 0) I~ ss uodxidul I= u~~_ S, where uo is the field amplitude at the exit aperture of the emittEr, S is the aper- ture (hole) area. If we consider that the value of u0 S is none other than the radiation flux through the entire exit aperture or the emitter power P, then _ F (0, 0) PS/X2. (8) - Thus, for giuen power of the ideal emitter, its axial luminous intensity is di.xectly - proportiona] to thi! area of the exit aperture, aad it does not depencl on the shapt� of th(- exit aperture. - For emitters havin;; identical power and aperture shape, but different transverse dimensions, ~:he pro~portionality F(0, 0) of the area is trivial, and it follaws dir.ectly froin the ,ilready noted decrease in divergence with increase in transverse dimensions. Altholigh the form of the emitter does not count in (8), it is poasible to.use (8) to deve.lop the critierion for estiwating the form of the emitter begin- ning with requiremente imposed on the emitter when combining it with an external - emission shaping system [8]. The fact is that in a11 cases where it is necessary to obtain maxitaum-range (light ranging and detection, distance measurement, and so on), the basic element of the shaping system is the telescope turned with its eye- piece in the direction of the emitter (Figure 1.2). The telescope enlarges the transverse dimensiona of a narrow].y directional light beam by K= f i /fl times, where fl and f2 are the focal lengths of the objective and the eyepece, K is the power of the telescope; the form of the complex amplitude di,atribution o� the field in the first approximation is maintained. As was poiiited out above, with this procedure the divergence of the emission decreases by K times. The axial luminous 16 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 ~ FOR OFFICIAL USE ONLY fntenaity increases hy K2 times, respecti,vel}*; in the case uf the ideal emitter, even after the telescope, the luminous intensitp continues to bQ definad by expres- eion (8) (the emission wavc:. front remains plane), except now S acquires atgnifidance � ;i of the beam cross sectional area at the exit from the telescope. J: a Fi.gure 1.2. Galileo telescope. If the objective dimensions are given and the vignetting losses are not permitted, the maximum axisl l.uminous intensity is reached for the telescope power where the light beam cross sectfon is exactly "inacribed" in the objective cross section, which is usually a circle. This situation is illustrated by Figure 1.3 where the cross sections of two emitters (a, b) and light beams at the telescope exit (c, d) are depicted on the same scale. Let us note that the forms of the emitters presented in the figure correspond to the same laser with different resonator systems (see � 4.2). From the figure it is clear that the axial luminous intensity in such devices is independent of the dimensions of ihe ideal emitter and can he def ined by the f ormula F (0, 0) _ ~ 1'Y, (9) where SD is the area of the telescopic objective, P is as before the emitter pawer (af ter subtracting the losses in the telescope), Y is the f illing f actor which de-- pends only on the form of the emitter and is equal to the ratio of the emitter area to the area of the described circle (this circle is none other than the required cross section of the telescope eyepiece). When comparing ideal emitters of equal power, preference must be given to the emitters having larger y(in our case, the emitter depicted in Figure 1.3,b). Sometimea for the sake of increasing axial . ~ a) b) c) d) e) Figure 1.3. Influence of the form of the ideal emitter on the axial luminous intensity in the presence of an externa.l shaping system: a), b) ideal emitters of identical area but different form; c), d) light beam cross sections at the exit of the shaping system; e) selection of shaping s}rstem for an emitter of irregular shape, 17 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/42109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY ~ luminous intensity, it is even possible to rel4nquJ$b, part of the radiation parser if Y can be increased significantlp as a result. Tbus, in the case illustrated in Figure 1.3,e, it is expedient to use only that pai�f' o� the beam cross section lo- rrated inside the dotted curve. ' Let us conclude the investigation of ideal emitters by presenting data on the emis- sion divergeace with the simplest aperture shapes rectangle, circle, ring. In tfie case of a 2a X 2b rectangle, performing the integration in (6) within the corre- sponding limits (-a < x< a, -b < y< b), we obtain Ftaz, ay) _ rsin (kaax~]2 [sin (kba)l2. p(p, 0) kaak~y ~ Thus, in accordance with (7), the expression for the intensity distribution in the far zone breaks dowm into the product of two cofactors here. Each of them depends on only one angular coordinate, and it is the dietribution for a source in the form of an inf inite strip (slot) of width h, equal to 2a or 2b, respectively; the fozm of this dtstr_ibution is presented in Figure 1.4. This f igure does not require special comment; let us only note that the w�Ldth of the maximum with reapect to the 0.5 intensity level is ~0.9 a/h. For a circular aperture the distribution in the far zone is naturally axisymmetric, and it is described by the formula F (a) 4 (Jl (kaa) I I F (0) _ ~ kaa , where a= a ay is the angle between the observation direction and the axial X direction, a is the radius of a circle, J1 is the Bessel function. The form of the distribution is presented in Figure 1.5 (curve I). Th.e angular dimensions of the far-field pattern, as always, are inveraely proportional ta the emitter dimensions. The pattern itaelf consiats of a central light apot surrounded by a system of con- centric light circles (distribution peaks) with dark intervals (minima) separating them. The intensity of the light circles diminishes rapidly with an increase in radius (in order to reproduce two more peak.s, except the central peak, it is neces- sary to increase the scale on the pattern, beginning with a=X/2a, by tenfold). In contrasC to the case of a rectangle, the maxima and minima are not equidistarct; on going away from the center, the apacing between adjacent rings asymg.totically _ approaches the value of X/2a. The angular radius of the firat peak is 1.22X/2a; thus, the centxal spot occupies a region with angular width of 2.44a/2a. This region requires 84% of the total power of the circular 2mitter, the f iret ring - (more precisely, the region between the f irst and second minima) requirea 7%, and . the second, 3%. In the case of an emitter in the form of a ring with the same ouGside radius a and inside radius ca p(a) _ 4 JL (kau) - EZ Jl (ekaa) a - F (U) kax ekaa 18 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 w'Oft UFFICIA1. U5E UNLY The distribution graphs for two such ring emitters with ring width a(1 - e) differ- ing by a factor of two, are presented in the same Figure 1.5 (curve II, 0.6, and curve III, e= 0.8). For convenience of comparison, all the curves are shown on the same scale, the powers of all three emitters are identical; the axial luminous intensity of the first (circular) emitter is taken as the unit luminous intensitiy. ~ Figure 1.4. Intensity distribution in the f ar zone f or a source in the form of an infinite strip with con- stant amplitude. , l,pmx.e~f (8~ 1,0 0,6 qs 0,4 4rt a,,~r 410 0, OB qv6 4N 402 ~ Figure 1.5. Intensity diatribution in the far zone for ideal emitters with exit cross section in the form of a circle (I) and rings with ratios of the inside diameters to uutside diameters of 0.6 (II) and 0.8 (III). Key; a. I, relative units In accordance with (8), the axial luminous intenaity of ri-ao ?mitters decreases with - the area occupied by them, and in the adopted units it is equal to 1 T e2, which is 0.64 for the second emitter and 0.36 for the third. Thus, the height of the central peak drops significantly; the width of this peak also is somewhat less than �or a . circular emitter (it can be shown that the width of this peak is determined by the average distance between different sections of the emitting surface which is greater for the ring than for the circle). Therefore the fraction of the total power re- quired for the central peak drops sharply, amounting to 37% for the second emitter and only 17% for the third emitter. The radiation intenaity at large angles to the axis increases correspondingly: in the seeond emitter 35 and 15% of the total pcwer - is required for the first two distribution rings in the far zone (inetead of 7 and 3% for the first emitter), and for the third emitter, 20 and 18% of the tdtal power are required. It is worth while to note the following: for our three emitters the angular diver- gence measured with respect to the 0.5 intenaity level is 1.03 X/2a, 0.87 X/2a and 0.79 a/2a, respectively; with respect to 0.5 power level (see above) the angular - divergence is 1.06 X/2a, 2.9 X/2a and 5.25 X/2a. With this instructive example, we conclude our 3nalysis of the emission divergence of systems with a plane wave front and uniform intensity distribution over the exit aperture. Arbitr.ary Monochromatic Emitter. The axial luminous intensity of an arhitrary mono- chromatic emitter, iA, in accordance with (6) equal to F(0, 0) = I~ f f u(xl, y,) dxldyi Ig� 19 - FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 0 ~ z a a " !a APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR cFFIctAt I ISE ONLY Introducing the total emitter power P = S S I u(xl+ yl) 1' dxldyl and the area of the exit aperture S, we can represent this expression as follows: F(0, 0) = pg where ~(I /S) f f u(_l, yl) dxidYl I_ I u IZ (10) (!/S) f f ~ u(zl' V3) 11 dx~dy, ~ u . As is known, for an arbitrary complex fuaction u, the relation p2/ju,2 < 1 is valid. Equality is achieved only when u- const. Hence, it tollows directly that out of all possible emitters having identical powers and areas of exit aperture, the one in which the complex field amplitude at the exit cross section is constant, that is, the ideal emitter, has Che greateat axial luminous intensity (this is why we call it the ideal emitter). This is, perhaps, the only general conclusion of in- terest to us which can be formulated with respect to the properties of the angular distribution for arhitray emittess (not considering the above-+mentioned still more general law that the distribution in the far zone is a Fourier transform of the distribution with respect to the exit aperture or in the near zone). Therefore let us proceed with investigation of the most important special casea, The emitters in which only the intensity distribution in the near zone is nonuniform and the phase ia constant (the wave front is plane) differ little from ideal emitters - with respect to their properties. For the tnentioned nonideal emitters u(x , yl) - ei'OA(xl, y), where A is not a complex coordinate function, but a real, nonnegative - coordinate ~unction. From (6) it is esey to see that in this case, just as for ideal emitters, the maximum of the intensity distritiution in the far zoae is in the axial direction. The dimensionless coefficient Y' which enters into the expression for the axial luminous intensity as a factor is equal to y, _(A)2/AZ < 1, and it acquires the meaning of the fract{.on of the aperture area effectively filled with emission. If the a.perture over which averaging is carried out when calculating y`, ie circular, this parameter becomes entirely analogous to the paraateter Y introduced �or real emitters. In particular, uader the condition of replacing Y by Y`, formula (9) remains valid for the axial luminous intensity after the external shaping system. It is important that the coefficient Y' becomea significantly leas than one only for very large nonuniformity of the field distribution with respect to aperture. Let us present the following numerical example. Let the exit aperturs of the emitter consist of two zones of identical area, over one of which the fraction c of the total radiation flux is distributed uniformly, and over the other, the remaining flux is also uniformly distributed. For such an emitter (10) leada to the formula y' = 0.5 + e(1 - e); for E= 0.2 the parameter Y' is 0.9; for E a 0.1, Y' -0.8. Let us conaider another example. Let the radiation denaity at the exit aperture assume all values from 0 to some maximwn with equal probability. In this case 1 ' 1 - -1 =frd.e ,f x dx =0,89. From these examplea it obviously follows that the nonuni- o 'o formity of the field amplitude distribution ia felt comparatively weakly on the magnitude of the axial luminous intenaity (and the emisaion divergence along with it). 20 FOR OFF[C[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 - FOR O"r FICIAL USE ONLY Phase inconstancy in the exi.t cross section lias much greater ;i.nfluence on the dis- tribution in the far zone. We sfiall investigate the frequently encountered case of quadratic dependence of the phase on the transverse coo5dinates in the greatest detail. This coxresponds (if the coeff icients_on x12 and yl are equal) to a spheri- cal wave front. '1'hus, u(xl, yl) = A(xl, yl) exp ~ 2P (xi yi~,~ here A(xi, yl) is a real, nonnegative coordinate function, p is .the radius of curvature of the wave fron.t. For p> 0 the wave front has convex shape (a diverging beam), and for p< 0, concave (the beam constricts at first). Subetituting this expression for u(xl, yl) in (6), iae obtain F~a:~ a~) = I k,~,~ A(xl, Ui) exP [ 2p txi yi) - - ik (axxl -f- ayy01 dx,dyll 2 _ exp I ik [(=i 2P ax)= ~yi 2P a~)I ~q (xi, bi) dxidyi I2� From a comparison of this formula with (2) it followa that for a wave front with a radius of curvature of p> 0, the form of the angular distribution coincides Grith the ~ density distrfbution pattern I(x2, p2) on the observation plane removed a distance z= p from the source with the same amplitude A(xl, y), but plane wave front (the conversion from a linear scale to angular is made by iormulas aX = x2/p, ay = y2/0. If we change the sign on p, the similarity of these patterns is retained except one of them turns out to be inverted with respect to the othero The results of beam diffraction with plane wave fronts have been investigated in sufficient detail in optics. At short distances (z � a2/a, where a is the charac- teristic size of the source), the shape and sizes of the spot quite precisely repro- duce the shape and sizes of the source. For z� a2/X the spot, on the contrary, can "blur" completely, and we have a far zone the shape of the spot repeats the angular distribution of the emission. As applied to the case of a spherical wa e front of interest to us, this information is interpreted as follows. For I pi � a~/a or a2/aI pI >y 1, the width of the angu- lar distribution is simply equal to the source dimensions divided by (pi; Chis is - none other than the angle between the edge beams normal to the wave front. Thus, the divergence of the emission here has a purely "geometric" origin. For a2/alpl 0 the beam has an imaginary neck, the location of which is uniquely deter- mined by its parameters. If the general phase lead (see g 2.1) is neglected~ the propagation of gauasian beams in free space ia exhaustively described by the following two expressiona, an illustration of which is presented in Figure 1.7; - w' (a) ~ wo + (Az/Awo)'], P (a) = z [i -I- On wo/Xz) z]. (12) 22 FOR UFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FQit OFFiCIAL U&L ONLY Here w is the beam radi,us at a distance zfrom the neck, w0 is its value at the neck, p is the radius of curvature of the wave front. By uaing these expressions it is easy to find the radiation divexgence of a gaussian ' beam with spherical wave front o� interest to ua, Knowing p and w, it iS possible to reproduce the position of the neck and the radius of the beam in it. However, it is still simpler to use the fact that, as was noted above, the far fisld pattern for a beam with radius of curvature of the wave front p is similar to the emission distribution pattern at a distance p from an analogous source with plane front. In accordance with (11), at a distance p from the source of the gaussian beam of radius w and with plane front the beam radius is wA + (ap/'ttta );proceeding from the linear dimensidn to angular, we find that the angular radius of gaussion distribution in the far zone is WP) i/1 -f- 0,Plrcw2)2 = -Pf (w/p)2 -I' (%lnw)'. Hence, it follows tY1at angular divergence with respect to the 0.5 v (In 2. P)2-- (ln2 w)e. It is easy intensity level is In 2- -1 ' (w/p)' (%Inw)a = to see that 1n2(w/p) is "geometri.c" '.divergence, and (ln2/7r) (a/w) is "diffraction" divergence. Thus, in the given case the square of the resultant divergence is equal to the sum of the squares of the geometric and - diffraction distribution widths. Let us considar another example of a beam with uniform intensity distributi.on with respect to a rectangular aperture. In this case F(ax , ay ) decays into the product _ of Lwo like co-factors, each of which is the angular distribution of the source emission in the form of an infinite strip with cylindrical shape of the wave front. Figure 1.8 shows the results of numerical calculations of the intensity distribution in the far zone for such a source for different rElations between the geometric di- vergence h/p and the diffraction divergence a/p (h is the atrip width). The radia- tion density in the f ar zone is taken the same in all caaes, and the axial luminous intensity of an ideal "strip" source with plane wave front (p is tak.en as the unit for measuring the intensity in the far zone. Here the sha.pes of the geometric and diffraction distributions, in contrast to the preceding case, do not coincide, and the resultant pattern is much more complicated than for a gaussian beam. Let us emphasize the following. As is eaey to aee, the geometric and diffraction - divergences are equal to each other when the deflection of the wave front is a total of a/8. For deflection X, the geometric divergence is eight times the diffraction , divergence and completely predominates over it. As follows from a comparison of curves 1 and 4 in Figure 1.8, the presence of this def lection wfth respect to only one of the coordinates causes a decrease in the axial luminous intensity by more than 11 times as compared to an ideal source; if analogous deflection occurred also in the other direction, the axial luminous intensity would decrease by as ma.ny times! This again demonstrates the enormoua influence on the distribution in the far zone f elt �rom even such small phase disturbances. Now let us proceed to another type of source which we must deal with hereafter. - We are again talking about beama with plane equiphasal surface, but the real ampli- tude of the field on this surface will be a sign-variable value. Let us limit ourselves to the simplest case of unifornt distribution of the complex amplitude of - the type 23 ' FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAI. USE ONLY cos(m-f2a)a:~ m=0, 2,4l . um (x) _ - a < x`G a. (13) sin ~m 2a m =1, 3, 51 . . I(a) a A7h- Figure 1.8, Angular emission distributiona of aources with uniformly distributed amplitude and spherical wave fronts 1- "geometric" di- vergence ia misaing (the plane wave front ia the same curve se in Figure 1.4); 2-- h/p = 2a/h (the deflection of the wave front ia a/4); 3--- h/p = 4.5a/h (the deflectinn of the wave front 9a/16); 4-- h/p ~ 8X/h (the wave front deflection is X). These functions assume. zero values at the ends of the interval, and they each have - m zeros inside; thus, all of them, except the first, are signwariable. Let us write them as follows: lm(,Z)=C08~2 (n,u1)z~.m,l~~ . L ] � ~ 2 uap tin 1(M a ~ rn]} 2 eXp `2 [(m a1) x -I- m,}� (14) The field with complex amplitude of the type C exp(ilcxa) for ci � 1 is a plane. wave inclined at an angle a with respect ro the axis. xherefore u(x) is a superposition of two rw1tually coherent plane waves, the directions of propagation of which make . a the angles +(m + 1) /2) ( /2a). Calculating f exp ikaxx) um (x) clx (cm. (6) we obtain _a ~ r / \1 . 1'' a Isill lkula~, m2 i 2 11 -1 msialkala-{- am2 1 a 2 ~ ~ ~ m (x)N + - I I l;a(ux-ml i la) ka(aa-{-M~ 1 2 ) 24 FOR OFFICL'+L USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 r5 -4 -,T Z y U ~ G u ~ APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY a -,l -J -2 -1 0 1 Z 3 at a: ,~~a A%2a Figure 1.9, Distributiona in the far zone for coherent and noncoherent superpositions of two plane waves. Solid curvea source with ampli- tude distr3bution of the type (13); dotted curves incoherent sum of the eame waves; curves 1, 2, 3, 4--r for m- 0, 1, 2, 3, respectively. ' The graphs of Fm (aX) for the firat four functions are presented in Figure 1.9 (solid curves). We shall return to this figure soon, but let us note meanwhile that the sign-variable nature of the amplitude leads not only to growth of the divergence, but also to decay of the angular distribution into individual apots with comparable intensity (in our example when considering the analogoua distribution with respect - to the second coordinate yF(a,X, a y ) there would be four peaks of equal magnitude). The angular distances between the spota axe _X/A, where L1 is the diatance between , the:amplitude diatribution nodes in the near zone. Nonmonochromatic Emitter. Let our source emit a set of monochromatic waves different frequencies simultaneously: u(xi, yi, t) i= ul(xi, U)eXP(-iwid) -I- us(xI y0eYP(-iw2t)-}-... ; Thanks to the superposition principle the f ield at any diatance from the source can be written in analogous form: u (Xz, Jz, t) uI (xz, Ys) exp (-tca lt) -l-us (xz, ys) exp (-i(02t) . where the complex amplitude ui(x2, y2) o� each component fs calculated by substitu- tion of ui(xl, yl) in the standard expreasions presented at the beginning of this item. In accordance with the formula there, let us calculate the emiasion intensity averaged over a large time interval (the indices on x and y are omitted, the * will ! denote complex conjugation): I(X, y) CI u(x, y, t) 12> = Cu (x, y, t) u+ (x, y, t)) _ ( (ul exp iwit) ua exp iwat) + . . . J X ' x fu; exp (iwlt) + u= exp (iWSt) -f- . . . - lul (x, J)I'--Ius(x, v)I' -f- 25 FOR OFF[CIAL U5E ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY All of the crass ternts oscillate arS,th dEfference fxeq,; therefore when averag- ing over a large time interval they do not contribute to the general sum. Thus, the total field intensity turns out tv be equal to the sum of the intensities of indivi- dual monochromatic waves, Let us again consider Figure 1.9. Graphs of the funGtions ! [Yea ~ sin ka I ax - m 2 1 2~~~ 8 ain (ax m 2 1 2 `LaJ Fm (ax) = l, ~ kara _m+ i 1 I kara ,+,m+i 7~ 1 2 2aJ l x 2 2af are plotted on it as dotted curves for the same m as the solid curves. Thus, if the - solid curves correspond to the case of coherent addition of two plane waves figuring in (14), the dotted curves correspond Co incoherent addition of rhe same waves which - could occur at different frequencies. From this example it is obvious tnat the coherent and incoherent additions can lead to significantly different resulta. On the other hand, it is possible to see that the differences here are small except in cases where both added waves have noticeable intensity. This is understandable inasmuch as coherent addition diff ers from inco- - herent by consideration of the cross terms which include the amplitude cofactors of - uifferent waves. - We have paid definite attention to what would appear to be a quite elementary prob- - lem, for an unclear distinction between the concepts of coherent and incoherent additions is also encountered in the literature on optical resonators. Piost fre- quently the confusion arises when using a standard procedure consisting in expansion of the field distribution of a real device in a series with respect to the eigen- functions of an ideal system (see, for exataple, g 2.5). Such a series frequently contains high-order terms which, if taken separately, correspon,d to very great beam divergence. This appears to be a sign that the total divergence is large, but in reality it can be quite the oppositz. Let us present an instructive example. Let the amplitude distribution u= C- const which ie rectangular in the interval -a < - 3-- x< a be expanded in a series with respect to the functions of (13) ; u= 4n (uo u 5- Iiere, as a result of coherent addition auccessive consideration of terms of higher and higher order does not lead to an increase in the total diver- _ gence, but to a decrease (to be convinced o� this, it is suff icient to compare the solid curve 1 in Figure 1.9 which pertaina to the first term of the sum with Figure 1.4 which correaponds to the f inal resul.t of summation). _ Some Conclusions. Measurement of Divergence. The above investigation of the angu- lar divergenCe of the emission of different sources is not at all exhaustive, but it should make it easier to underatand how certain factors considered hereafter in- fluence the divergence. The primary conclusion which can be drawn ia that for sys- tems with narrowly directional emission it is necessary first of all to strive for maximum uniform phase distribution of the emiesion with respect to the exit aperture. Nonuniformity of tlie amplitude distribution is far from so terrible (the case of aign-variable amplLtude which would aeem to be exempt from this rule can be inter- preted aa the case where there are phase jumps of Tr). Even multifrequency is dar.gerous only when a significant portion of the total energy goes to the components = Qither with sreat divergence or with different directions of propagation. 26 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL fJSF. ONL.Y - In concluaion, let ua consider the pxobeltq og weaauxing the angular divergence. On analysis of expression (2),we saar tbat the txansition to the far zone where the form of the distribution ceases to depend on the distance is related, to tlze possibility of neglecting the terms containing xi and yi in the phase factor of the expression under the integral sign. Under ordinary conditions, for thia purpose it is neces- sary that the inequality (3) be satisfied, which, as a rule, requires significant removal from the emission source. Thus, for a= 0.5 micron (green light) and a source diameter of 3-5 cm, the far zone is completely formed only at a distance on the order of several kilometers. However, it is easy to achieve complete absence of the mentioned terms also in direct proximity to the emission source. For this purpose, it is necessary to install the so-called quadratic phase corrector at the source exit; the simplest device of this type is an ordinary thin lens with spheri- cal surfaces. The fact is that with small thicknesa of the lens, the field ampli- tude digtribution doea not change on passing through the lena. The curvature of the wave front only changes discontinuously, which is equivalent tn the addition of a phase factor of the type of exp[iY(x2 + y2)], It is easy to calculate the coeff i- cient Y knowing the index of refraction of the lens material and radius of curvature of its surfaces; however, it is possible to proceed more simply. Actually, it is known of a positive lens (f > 0) that it converts a spherical wave front from apoint source at its focal point to a plane front. The radius of curvature of a spherical front at the approach �to the lens will be f in thia case; the compiex amplitude of the field with such a wave front has, as we already know, the phase factor exp [(ik/2f)(x2 + y2)]. The phase is constant at the plane wave front; thus, passage through a lens with a focal length f leads to multiplication of the complex field amplitude by exp[(-ik/2f)(x2 + y2)J (a small total phase lead plays no role in this investigation). Now let us return to expression (2). Multiplication of u(xls yl) by exp [-(ik/2f) 2 2 2 + yi)] leads to mutual reduction of the terms containing xi and yi if the dis- (xi tance to the observation plane z is equal to f. Hence, we have the simplest formula of distribution observation in the far zone which everything follaws: at the source exit a lens (or more complex optical syetem) is installed with focal length f> 0. The pattern in the focal plane is entirely like the distribution in the f ar zone ;for transition to the angular scale it is nececsary to divide the linear scale by fj. Inasmuch as the angular distribution of the emmisi,on at any diatance from the source remains the same, the distance from the source to th,e measuring lens plays no special role; it is also necessary to see that the lens always "encompasses" the entire light beam. When there are grounds to assume that the wave front has a spherical companent, the compensation of which causes a decrease in the divergence, it is Worth while to measure the beam width d not only in the focal plane, but also at other distances k from the lens. In order to understand what information such measurements give, let us mentally divide the lens into two components with focal lengths f' _(1/f - 1/k)-1 and f" a!C, respectively. Inasmuch as tha observation plane is the focal plane for the second of these components, it becomes clear that the ratio d/Q is none other lA similar situation has already been encountered when analyzing beam divergence with a spherical wave front: transition frrna a sphetical front with p? 0 to a plane front is equivalent to passage of the beam through a leas with f p. 27 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY than the divergence o� the source eaission wi,th the fixst component as the exit correcr.or. Finding the distance R,p at ~rFich the ratio d/Q passes tnrough a min.imum, we at the same time determine the focal power of th.e optimal corrector 1/f - 1/ko and the divergence achieved when using it. � 1.2. Optical Cavities and :'!nasification of Them , Initial Information. A Littic History. Now lpt us proceed with the investigation of the most important sources of higtly coherent narrowly directional emission- lasers. One of the basic laser elements, along with the active medium and the syatem for stimulating it, is the resonator. The standard requirement imposed on the resonator is the presence of comparatively high-Q (slowly damping) characteris- tic oscillations, the frequencies of which fall into the amplification band of the active medium. On the other hand, it ie desirable that there not be too many oscillations with approximately the same Q-factor aimiltaneous stimulation of too large a number of them can lead to almost incoherent emission. - Optical-band cavities are characterized by the fact that their dimensions usually are several orders greater than the operating wavelength. This excludes the uae of closed resonators in the form of a closed cavity with reflecting walls: the number of high-Q oscillations on optical frequencies would be extremely large in such cavities. Therefore open resQnators which do not have side walls are used. The simpleat of such resonators consist of two mirrors inatalled opposite to each other with the active medium between them. The placement of the mirrors leads to isola- tion of the predominant direction of pxopagation of the emission which must aharply decrease the number of high-Q modes by comparison with closed resonatore. Actually, light beams traveling along the axis of the syatem (the common normal to the mirrors) are reflected alternately from the mirrors and obviously damp more slowly than others; inclined beams exit from the system, and thia occursfastex the larger the . angle they make with the axis. Similar argumenta led Prokhorov [1] to the basic idea of the prospectiveness of us- ing open resonators. On the other hand, it is still impossible ta extract informa- tion from these arguments as to whether such resonators have natural-oscillations. It is possible only to conclude that theseoscillations must be found in the form of a superposition of light beams directed along the resonator axis or slightly in- clined to it. Only a careful analysis such as will be performed in this book can answer the questions of what sortof oscillations these are, how they are influenced by the presence of an active medium, how many of them will be present in the stimu- lated emission.and, finally, what the resultant divergence of the emission will be. This analysis is for the most part based on the results of the theory of empty (tlzat is, containing no active medium), open resonators. The primary period of development of the theory of empty, open resonators extends from 1961 to 1966. The famous paper by Fox and Li [9] marked the beginning. The problem of the existence and properties of the lowest (that ia, the highest-Q) modes of an empty open resonator was stated and solved numerically for several special examples for the firet time in the paper by Fox and Li. The concept of diffraction losses was introduced for the f irst time. Diffraction losses can be defined as the fraction of the total radiation flux scattered as a result of dif- fraction and paseing by the mirrors. For a number of reasons, this concept is much more use�ul as applied to cavities than the Q4actor concept, 28 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOIt OFFICIAL USE ONLY and at thia tixne it has alinost completelp displaced the Q�-factor concept. _ After reference [9] came the articles by Boyd, Gordon and Kogelnik [10, 11], in which a more general study was made of open resonators consisting of two spherical mirrors with arbitrary radii of curvature, aud a classification of such resonatiorg with respect to magnitude of the diffraction losses was presented. It was discove- . red that in a defined range of geometric parameters of the resonator (radii of curvature df the mirro�:s and spacing between them), the diffraction losses are very ' small, but on going outside this range, these losses increase extremely sharply. During those years it appeared that atable lasing could,be achieved only in resona- tors with small diffraction lossea; therefore such resonatora were called stable; as we shall soon see, a system of two slightly concave mirrora can serve as a characteristic example of such resonators. For this reason, resonators with large losses which include, for example, a syatem of convex mirrors,were called unstable, and researchere did not consider them for several years. A further significant step forward was made when r.eferences [12, 13] appeared. The general methods of analysis of complex devices conaisting of an arbitrary number of optical elements with plane and spherical surfaces were developed in these papers. As a result, it turned out to be possible to reduce almost any complex resonator to the equivalent two-mirror system with the same f ield distributions of the natural os- cillations on the mirrors and the same diffraction loases. Papers by Soviet scientists V. p. Bylcov, V. I. Talanov, et 31. an impor- tant contribution to the development of the given field of science. Amor_g them a special role is played by the brilliant research. cycle of L. A. Vaynshteyn which was summed up in the monograph [3] which was already mentioned in the introduction. Previously, he developed powerful, universal methods of analyzing microwave systems. The application of these methods to open optical-band cavities permitted Vaynsh- teyn to obtain simple analytical expressions for field distributions and losses in many cases where other researchera were forced to resort to machine calculations. His formulas were also suitable for high-order modea and for large-mirrer resonators, that is, they even encompassed cases where machine methods were useless in view of the extraordinary awkwardness of the calculations. , All of the above-mentioned studies pertained to flat and "stable" resonators which had small diffraction losses. The reaults of theae papers, that is, the theory of - empty resonators with small losaes, have also been reflected quite completely in almost all monographs on quaatum electronics. Accordingly, we shall not go into a detailed reatatement of the methods of solving certain problems or the results of calculations performed for a great variety of systems (a significant part of whicr did not find application subsequently). We ahall only present a minimum amount of information which we need for the follawing discussion. We shall try to systematize this information as much as possible and avoid numerous "historical features" which would only complicate the discussion. Open resonators are first of all divided into ring and linear cavities (see Figure V. 1). Ring cavities are made up of at least three (moat frequently four) mirrors or other elements which change the direction of a light beam; they are used in a small ~number of types of special laser devicea (laser gyroacopes, and so on). Ring reso:~ators are almoat never used in ],asers with narrowly directional emission: they _ do uot have any special advantages from the point of view o� divetgence of the 29 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500014010-4 FOR OFFlCIAL USE ONLY - emissi,on, and they are complicated to adjust; at the same time, the requirements on adjuetment precision increase as tlie emission divergence decreases. Therefore linear cavities will be the primary subJect of otir analpsis although the results of the analysis can be extended to the case of ring systems Grithout difficulty. Although a linear cavity contains fewer mirrors than a ring,cavity, as a rule the - linear cavity is an integral optical system. The mirrors themselves are frequently not plane, but spherical; an active element made of a material with an index of re- fraction different from one (but still variable with reapect to cross section) is placed between the mirrors; if necessary polarizera, shutters, lenses, and so on are also placed between the mirrors. Therefare f irst of all we must discuss the laws of transmission of light beams through optical systems made of elements with - spherical or plane surfaces. ~ (r,.s , yr) Figure 1.10. Thin lens between aource and obser- vation planes: 1-- source plane; 2-- observation plane. Passage of Light Beams Through Optical Systems. Beam Matrix. Eixst let us consider the simplest example where there is only one thin lens with focal length f be- tween a monochromatic source and the observaCion plane; the distance from thi,s thin lens to the source is R, , and the distance to the observation plane ia RZ (Figure ~ 1.10). Acco?:ding to (2~', the field at the point with transverse coordinates x, y in the rePerence plane located directly in front of the lens is lik _ 9 u~x~, Y') ezpi , J kl1) r exp 211 11' `y 21 a yt~ 1 u(xl, !/i) ds,; ~ here dSl = dxldyl, u(xl, yl) is as before the complex amplitude of the f ield on the reference plane of the source. After passage through the lena, the complex amplitude ia multiplied by exp 9 / 9 [-ik(~~ ~ 2j(v) finally, applying (2) again, we obtain tihe expression for the field diatribution u(x2, y2) in the observation plane: e:p [ik (Ii + la)] ~ s ~ - 1)2 u(xa, ys) ~ ~1~~li I exP tk 211 211 - . _ (x I ; (Y )a (s' 2isX,)s (v 21sV~~ 11 u (xs, Yi) dS1dS'. J 30 FOR OFFICiAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY In order to expresa u(x2, y21 directly ~n textaa of u(txl, yl), i.t is ner.essary to change the order of integration and take the integral over the surface of the lens S`; here we shall consider that tTie Ieus more tfian "encotnpasses" the eatire light beam, and integration can be performed within infintte limits. Thus, our problem reduced to calculation of the integral ~ esp ik ( 21i + 2l1 (s') _ ~x )Z 2/(y )2 d_ `ral,~x )2 T `Yr2lzy J~,1 ds' _ t~~-I�9 zj-F!/ 12c- 1 -f- - ( j + -L - j ~ (xo yo) X i i s X f cap { i ( T ,t - ; ) [(X, - Xa)s + (y~ - JXds', W ) 1 z i f i`-1 r�~ 1 ~ r! i i 1-1 rU y2 I. are the coordi- where xo ;~1 -F- ~t 11 ~2 Yo - l l~ ~Q - j l I l i Z l-- l ~ \ nates of a point on the surface of the lens at which the beam should be directed from tne point (xl, yl) so that after refraction in the lens it will hit the point (x2, y2) (see the f igure) . The last integral is similar to the ones investigated by us when discussing formula _ (1.2) at the beginning of g 1.1 and also is equal to half the integral over the - first Fresnel zone. The center of a system o� concentric circles which form the boundaries of the Fesnel zones (see Figure 1.1) is in the R1._en_�case_ lg.cated at the point (xo, yO the radii of the zones are equal to r� nX/(1/1, 1/12 - 1/f After all of the transformatione,we arrive at the formula: exp (ik (1i + jz)) ik u(.T i~ c- 11 c~f~ exp t, (c t- t i X (11 z z , z ,s~f) (X i yi) (xs -I- yi~ (15) - ? (xizs -f- y,ys)1} u (x~, yi) ds'i� Let us know that the sum 1C1 + Q2 in the exponent in front of the phase factor inte- gral must be replaceu by the optical distance L between the source and observation planes measured along the system axis wh.en conaiderin; finite thickness of the lens. The exponent in the expression under the integral aign has the form ik.A(xl, yl; x2, y2), where A is the difference between the optical distance from (xl, yl) to (x2, y2) measured along the beam passing through these points and L0. In the investigated example we have not considered partial reflection of light from the lens surface. Consideration o� thia phenomenon would in the given situation only lead to a small decrease in the i,ntensityr of the emiasi.on passing through the 31 FOR OFFICIAL USE ONLy APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 FOR OFFICIAL USE ONLY lens (if we neglect the slight influence of multiple reflection). However, inside an optical cavity light beams originating at the iaterfaces are superposed on the primary beam reflected from the end mirror. This can lead to consequences which cannot be neglected in spite of the apparently negligible intensity of the _ scattered light. Let us present a simple numerical example. Let a coherent beam - with intensity 0.04 IO (the reflection coefficient equal to 4% haa an interface be- tween the air and glass with index of refraction of 1.5 for normal incidence) be added to the primary beam having intensity I and_, consequently, field amplitude u0 The amplitude of this beam is 0. 0410 m 0.2u0; at ti~e peak formed as a result of superposition of the interference pattern beams, the intensity is equal _ to (1.2 u0 )2 = 1.44 IO, at the minimum, (0.8 u0)2 = 0.64 1 0' It ie unnecessary to comment on these figures. Therefore in the following analysis, in addition to the specially stipulated cases, the interfaces will always be coagidered coated. In order to get away from the influence of the interfaces in the experiments they are frequently not coated, but incliaed slightly. Then the ref lected emission leaves the resonator immediately and has no influence on the structure of the output beam, - n;tly slightlydecreasing its intensity. Now let us proceed with investigation of more complex optical systems. It would be possible, of courae, simply to perform an intermediate integration aver every inter- face or thin lens inside the resonator; that is how Collins proceeded in reference E121, which was one of the first articles in the given field. Thia method is highly _ universal, but awkward. In the case of paraxial (sli6::tly inclined) beams that are 4f interest in our syetems with spherical and plzne interfaces, the beam matrix method is more suitable. Koge]iiEik . and Li [13, 14] developed this method 3,n detail as applied to resonator theory. The beam matrix is especially convenient in that it is calculated in the geometric approximation, but as we sball see later, knowing its elements makes it possible, if necessary, to describe the transmission of the light beam through the optical system also in the diffraction approximation. The beam or ABCD-matrix relates the values of the transverse coordinate x, y and slopes aX, ay of the light beams at the entrance and exit of tlte optical system: yi ~1 B x U 1 1 zs ~ aax �L+u ~ T a C D~� aix aiY 1 (16) where the index 1 corresponds to the input plane, 2 corxeaponds to the output plane (Figure 1.11). The matrix product in (16) and hereafter is calculated in accord- ance with its standard definition. Being written for any one of two coordinate paira, this formula acquires the form a? J.JA D q� I~1 I, or x, = As1 Ba alx - Cx Da Ix+' 1 ~ls� lx The beam matrix eleinenta are uniquely related to the focal length of the optical system F and the position of its principal planes. In particular, C- --1/F. 32 FOR OFFICiAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R040500010010-4 FOR OFFICIAL USE OriLY ~ ais> U4 I ~ I - ~ Xj>p - I ~ i I a~r 0 the magnitude of the aresine will be within the limits from 0 to +7t/2; for AB < 0 it will be within the limits from +/27rto +7r. - If the same integral equation is written not in a cartesian coordinate system, but cylindrical and we f ind the solution by the method of separation of variables in the form u(r, = u(r) cos Q~, we arrive at the following set of eigenfunctions and frequencies: rrt ~P) (r�/tu)l I.1, (2rzliv9) eXr r510) cos lT, (3) c 2Lo ~~JtQ f1pl~+ (4) ~ where 1,;, v'~'`I' (u) ~luv I ( xl) v) ~'1'i 11 are the generalized Laguerre polynomials, 8ri =(2p l. 1) nrcc�.os j,' AU, the parameter w retains ita former value (2,1rc) 1/2 (--A.B/CD) 1/4. The form of the radial functions u(r) resembles to a known degree the form of the functions presented in Figuxe 2.1. In particular, u ~(r) has p roots; for !C # 0 another root is added at the point rm 0; inErease in the radial index p, the reoion of noticeable intensity shi,fts in the direction of larger, r,and so on. 65 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 hl3li llFM.llt. ~s,%c, t,iM A very good idea about the structure of the heams described hy formula (3) is pro- vided by Figure 2.2 which we borrowed from the book [33],t ffA~,, TEMaI Tf:1.;,, I EAf", Tf.M� TF.M,< TfM10 TEM1t TE~Wia Figure 2.2. The �ield configuration of natural.modes of oscillations of a stable resonator described by formula (3). On the solid closed lines the amplitude is 0.2, 0.5 and 0.8 of the maximum value; on the dotted lines the amplitude is equal to zero; the dash-dot line bounds the region inside which 86.5% of the total radiation flux is included. The ambiguity of the choice of the syatems of eigenfunctions, on the one hand, and the similarity of the behavior of these functions on the other, have a common basis. The fact is that a11 the functions of system (1), except the lowest (m = n= 0) are degenerate an infinite. set of times; this means that there ars as many differ- ent functions as one might like with the same natural frequency. Actually, it is little that the frequencies coincide for oscillations with different combinations of transverse indexes with identical sum of them m+ n(see (2)); it is possible to rotate the x and y axes as many times as one might lik.e around the resonator axis. The same thing also pertains to functions of system (3): oscillations not only with equal 2p + R, (see (4)), but also oacillations differing simply by the origin of reckoning of the azimuthal angle have identical frequency. The superposition of any number of natural oscillations with identical complex fre- quencies (in the given case there is no damping and the frequencies are real) de- scribes a strictly periodic process with the same frequency, that is, it is also a natural oscillation. Tlius, for degenerate eigenfunctions it is possible to com- bine them arbitrarily, from which ambiguity of the choice of the systems of these functions arises. If desired it is possible to find how it is necessary to combine functions of the type of (1) in order to obtain (3) and vice versa. - 66 FOR OFFiCIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFFICiAL USE ONLY Spatial Structure of Vatural Oacillations.. In oxder to understand the meaning of the obtained solutions, it is necessary to consider the propagation laws of the light beams making up these solutions. Inasmuch as the eigenfunctions (1) and (3) are real, the mirrors are equiphasal surfaces, and therefore the beams have spheri- cal wave fronts near the mirrors. The lowest functions of the sets (1) (m = n y 0) and (3) (p � Q= 0) coincide; this is the gaussian beam already known to us; its transverse radius, the def inition of which is presentea in g 1.1, is equal to w. - For other beams the effective spot size, as is obvious from Figure 2.1, exceeds w the more, the larger their transverse indices; therefore we shall call w the linear distribution parameter in the general case and not the beam radius. Using the Iluygens-Fresn=l principle (1.2), it is possible to see that the beams described by formulas (1), (3) have the same propagation laws as gaussian beams. In each such beam at any distance from the source the initia.l form of the amplitude distribution is reproduced; the wave front remains spherical; only the radius of curvature of the wave front p and the parameter w vary; this occurs in all of the beams identically in accordance with formulas(1.11), (1.12). Thus, Figure 1.7 can pertain to an equal degree to any beam from the families (1), (3) accept the dis- tribution picture For wtiich w gives the scale, and each beam has its own. Inasmucli as on the exit mirror w and 'p are identical for all of the beams, they coincide in any segment. of the length of the resonator. Figure 2.3 shows the law of evolution of w for such a family of beams. Any other characteristic dimen- sion of the cross section of each beam varies exactly the same, in particular, the transverse dimensions of the region of high intensities outside which the field decreases exponentially. The surface bounding this region is called caustic; for beams of the type of (1), it has a rectanguiar cross section, (3) has a circular cross section. The volume included inside the caustic surface (or the caustic) naturally increases with the transverse mode indices. As an example, in the figure the dash-dot line depicts the caustic surface of the beam f.or which the field dis- tribution with respect to the direction perpendicular to the axis lying in the ' plane of the figure is described by the function u with j= 12 (Figure 2.1, d). In the same figure 2.3, the dotted lines show the aquiphasal surfaces which are common to a1.1 beams of the family; there is an infinite set of such surf aces. Installing the spherical mirrors with the same radii of curvature at the location of any two of these surf aces, we obtain un empty stable resonator, the solutions for which will be all of the beams of the given family. Using (1.11) and (1.12) to relate the resonator parameters and the families of beams, it is possible to obtatn the Lollowing expression for the linear distribution param.eter on the right-hand mirror of tiie resonator: 2 u~ r'i ri2 t tr~- (all of the notation is the same as bef re). This expression is a special case of the more general expression w2 =(a/~)192(-AB/CD)114. The parameter wl on the left-liand mirror is defined by the same formulas, in which it is necessary only to exchange places witti R1 and RZ or in the general case, A and D; wl =WT)1/2(-BD/ AC)1/4; hence, it Lollows that wlw2 =(x/'T) -B C. 67 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY Finally, from tlte same Huygens--Fresnel pXi,nciple, ure halie the followi,ng expressions for the phase lag AY which occurs as a result of fiaving the phase velocity exceed the speed of light (see � 1.3) on a path of length z, beginning with the "neck": for type (1) beams A '1f _(m -1- ra t) arcig o for type (3) beams A'1' _(2p l-i- 1) arc:t.g (%z/nw2,); here, just as in (1.11), (1.12), wn is the linear parameter at the "neck." If we use these relations to calculate tTie phase correction in an empty two-mirror reson- ator, we again arrive at the formulas which are special cases of the morw general expressions (2), (4). As was already noted in � 1.3, the phase correction is larger, the smaller the characteristic dimension of the transverse beam structure. In the given case with an increase in the transverse indices, the distances between ad- jacent peaks of the amplitude distribution decrease (see Figure 2.1), and the phase correction increases correspondingly. - ,-s ^ ~ t ~ Figure 2.3. L�'volution of the linear distribution parameter w for type (1) and (3) beams. In order to trace the behavior of gaussian beams and, with them, the more complex beams type (1) and (3) inside an arbitrary resonator, it is possible to use the following procedure. The complex amplitude of a gaussian be:im on an arbitrary � r- reference plane is exI) z2w, d2I exI [ N(xa ys)] (see � 1.1). It can be given the same form exp I lp ('cI-{- y$),~ as for the complex amplitude of an ordinary spherical - wave with uniformly distributed intensity. For this purpose it is suff icient to introduce the complex radius of curvature p, defining it using the expression ik ik 1 1 1 ih 2p 2P u,2 ' or p P-~ n~u2 ' This form of notation, of course, would not be of special interest if the conversion laws for p in the optical systems did not compare exactly with the conversion laws of the radii of curvature of ordinary spherical waves, Actually, on passage through a lens with focal length f, the complex amplitude of any beams turns out to be mul- tiplied by exp[-(ik/2f)(x2 + y2)] (see � 1.1). The value of 1/p in this case has -1/f ;idded to it; the curvature of ordinary spherical waves is transformed in the same way. Using (1.11), (1.12), it is also possible to see that as a result of traveling the distnnces z in empty space, z is added to p, just as to the radius of curvature of an ordinary wave. Thus, all of the relations which can be derived using geometric optics are valid for p; in particular,_thP cnnversion law of p in an arbitrary optica2 systeiahas the form (�1.2)j7)2=(Api -}-B)/(Cpi -{-D). Hence, in turn, it follows that equation (1.21) and its solution (1.22) retain their meaning even Yor ABCD t 0. The radii of curvature i,n this case turn out to be purely imaginary 68 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-00850R440500010010-4 FOR OFF'[CIAL USE ONLY and equal to � AB CD. Waves with such radii qf cuxvature have the following amplitude distribution; uI, (x, eXp I (tlr/?P')(xE + ys)] - exp [ t (n1A.) p'- CD/AB (x' 01. The amplitude of one of the waves increases without litnit on going away from the axis; this ave ia of interest to us. The second is a gaussian beam with radius w=W( AB/CD)1~4. Inasmuch as the phase factor in the amplitude distribution is absent, the mirrors of the equivalent and, therefore, the initial �resonator are equiphasal surf aces; thus, we have arrived at the solution already known to us. Tntroduction of the complex radius of curvature permits the parameters of the gaussian beam to be traced without diff iculty (and any beams of the type of (1), (3)) along the entire length of the arbitrary resonator. For maximum simplif ication of this procedure in its time special nomograms were compiled ("maps"); anyone who wishes to bPCOme familiar with them is referred to [12], [13];.we shall proceed to what goes on outside the resonator. In the case of stable resonators, the zadiation most frequently is output through the partially txansparent exit mirror. As was demonstrated in g 1.3, the trans- verse structuxe of the natural oscillations does not depend on whether the reflec- tion coefficient of the exit mirror is equal to one oz is not equal to one (only that it be the same on the entire mirror surface). Therefore the semitransparent exit mirror of large transverse dimension originates beams with the amplitude of the type of (1), (3) for which it is an equiphasal surface. For a gaussian beam with radius of curvature of the wave front R2, the amplitude distribution in the far zone also has the form of a gaussian function with angular radius p V(ivl1?.Ay -1- (XInw)' 1.1). Inasmuch as the beams of the type of (1), (3) behave similarly to gaussian beams, in all of them the amplitude distribution in the far zone is described by the formulas (1) (3) in which x, y and r must be replaced by the slope angles aX, y and a= etX + ay , and the linear distributiofl parameter w must be replaced by the angular parameter S. In the expression for (3, the first term under the square root sign has a"geometric" origin, and the second term, "diffraction" origin (see � 1.1). The geometric term in stable resonators does not have theoretical significance, f or it can easily be reduced to zero. I'or this purpose it is sufficient to install a lens with f= R directly behind the exit mirror; this lens converts the spherical wave fronts ofe all t;ie beams to plane fronts; the common angular parameter 5 decreases to the value 5f)- a/7rw, and the emission divergence decreases correspondingly. Hereafter, we shall always consider that this simplest measure with respect to decreasing the emission divergence at the output of a stable resonator has been adopted, and the geometric divergence is absent. For this reason it is necessary to note the follow- ing. It is easy to avoid geometric divergence so easily in this case only because the exit mirror of a stable resonator is clearly designated a spherical equiphasal surface common to all beams. The position of this equiphasal surface remains f ixed even if a tliermal "lens" occurs inside the resonator or there is some other disturb- ance of similar type even if the resonator continues to remain �stahle." Ln- fortunately, the given property is almost the only advantage of stable resonators 69 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFlCIAL USE ONL1' from the point of view of beam divergence1 fox laxge apertwces lasing takes place in them on high-order modes wliich correspond to ver}r large diffraction divergence 2.4). In all of the resonators whicfi provide small diffraction divergence and are actually used in lasers with quite large aperture, the mirrors are not equi- phasal surfaces; the shape of the wave front of the stimulated emission, as a rule, differs noticeably from an ideal sphere, and still worse, it frequently varies with time. Under these conditions the control of "geometric" divergence becomes, on the one hand, extraordinarily difficult, and on the other hand, it acquires primary significance. We shall discuss the methods of this control in the last chapter of the book. As for diffraction divergence ~ of any of the beams (1), (3), it is possible to obtain a very useful formula for it if we use the fact that the relation between the angular dimension of a spot in the far zone and the angular pazameter a 0 is equal to the ratio of analogous values in the near zone. Hence, we have the expression V - - cniae m, (5) where (D is the size of the spot in the near zone (at the exit mirror). Now let us trace how the beam dimensions vary in the near and far zones on variation of the parameters of a stable resonator. Let us do this in the example of an empty two-mirror synmietric resonator with R1 � R2 a R, wr{ch is "stable" for R> k/2;.the spacing between the mirrors SC will be cons3dered fixed. For a symmetric two mirror resonator -AB/CD - R21C(R - k)/[2R - R.)(R - SC)]. For R� SC, the resonator is close to flat, -AM/CD z RQ,/2. Together with R, the cross sec- tional dimensi ns of all the beams in the near zone are comparatively large: w2z (i ~/2.~2)194. The angular distribution parameter in the far zone S0 ~(2J~2/ ~ R!C) ~ is correspondingly small. As R decreases and the resonator approaches confocal (R = Q) - AB/CD decreases, approaching 0; w also decreases, approaching XQ/ff; the angular parameter ~0 increases correspondingly and approaches a/'~l~. The beam dimensions behave the same way if R approaches R from the other side: for R?, Q/2, that is, when the resonator is "stable," but close to concentric, w is large, and (3d is small. As R increases and approaches !t, the garameter w ak/~r, So- A. Thus, the beam dimensions of resonatora with R12 < R< k and R> k pass through the same value; why this can oncur-t:from is explained by Figure 2.4, a, b. A symmetric confocal resonator is isolated frcm the others: the value of the ratio -AB/CD for R= k will become undefined; therefore the beam cross sections on the exit mirror can have any transverse dimensions. Only the product of the linear parameters wl and w2 on the left and right mirrora turns out to be f ixed: wlw2 = aQ/�R (Figure 2.4, c). It is necessary to note that not only the above-investi- gated symmetric resonators, but any other two-mirror gtable resonators have a large value of the given prc,duct. Actually, in the general case B 7l 1 ?,l ! wlws n Y C` n 11R1 i/Rs - l/R,Ita glg!' 70 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500014010-4 FOR UMFI( . % . a) ; ; . b) ' 'lAL USE ONLY c) , ^d) Figure 2.4. Natural oscillations of stable resonators. a), b) reson- ators with identical spot dimensions on the mirrors: a) resonator with Z/2 < R< k, b) resonator with R>k; c), d) confocal resonator; c) two different beams with infinite mirrors, d) type of resonator ~ oscillations with mirrors of finite dimensions. where g1,2 = 1-Z/R1,2; let us remember that only for a symmetric confocal resona- tor is gig, = 0; for the remaining stable resonators 0< glg2 < 1. From the subse- quent investigation it will be obvious that as a result of the indicatad peculiari- ties a symaaetric confocal resonator has the amalleat diffraction losses among all possible empty two-mirror resonators with given transverse dimensions of the mirrors and distance between them. Stable Resonators with Mirrors of Finite Dimensions. When considering finite dimen- sions of mirrors, degeneration of the eigenfunctions is removed to a signif icant ' degree; the "arbitrariness" of the selection disappears in the case of rectangu- lar mirrors the solutions are only functions of the type of (1), and for circular mirrors,(3). The shape of the functions undergoas some changes; the equiphasal ' surfaces of a11 resonators except a symmetric confocal resonator cease; to compare exactly with the surfaces of the finite mirrors. The spectrum of the eigenvalues - also is subject to adjustment. The frequencies become complex: diffraction losses appear, and with them, damping of the natural oscillations. This adjustme~nt to - a significant degree involves various types of oscillations: the scales and nature of the variations are determined primarily by how the transverse dimensions of the mirrors relate to tite dimensions of the region encompassing the caustic surface of given mode. ' For sucli modes, the width of the caustic of which is noticeably less than the width - of the mirrors, the field distribution remains almost the same as for infinite mirrors. The magnitude of the diffraction losses of the resonators not too close ' to confocal can in this case be calculated directly using formulas (1), (3): it is ' sufficient to calculate what portion of the total emission flux goes tothe distri- bLtion "tails" which are beyond the boundaries of the mirror surface. Inasmuch as - beyond the boundaries of the caustic the field decreases very sharply, as the i caustic approaches the mirror dimensions, the losses increase rapidly, remaining nevertheless, very small with respect to ahsolute magnitude. HEnce, it follows, that under othex equal conditione the higher~-order modea have gxeater losses they have b.roader caustics. 71 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00854R000500010010-4 FOR OFFICIAL USE ONLY If ttie resonator is asymznetric, the hastc cantribution to the total losses is made by reflection from a mirror, the diatensivns of which are closeT to the dimen- sions of the corresponding caustic. This pertains to all stable resonators except a confocal one with R1 = R2 =!C: inasmuch as for such a resonator it is not Che spot dimensions on each mirror that are f ixed, but only the magnitude of their ' product, these dimensions are redistributed for f inite mirrors proportionally to _ the mirror dimensions (see Figure 2.4, d). As a result, the two mirrors make an identical and small - contribution to the diffraction losses; tYie total losses turn out to be less than they would be for any other relation between the spot dimensions on the mirrors. If we consider that the product of the spot dimensions itself is minimal in the resonator of the given type, it becomes clear why it has record-small diffraction losses. Now let us f ix the conf iguration of the resonator and proceed successively to the modes with greater and greater transverse indices. Here the caustic sooner or later must approach closely to the edge of one or both mirrors. Beginning with this event, it is impossible to neglect the influence of the edge effects on the mode structure, and formulas (1)-(5) derived in the approximation of infinite mirrors become inapplicable. At the same time, the types of oscillations of still higher order still exist; their classification can be based as before on the number of distribution zeros with respect to the carresponding directions. In order to understand the nature of the variations of the mode structure under the influence of the edge effects, it is best of all to trace the behavior of a speci- fic type of oscillations in detail as the stable resonator gradually approaches a flat (or concentr{c) cavity. This analysis can be performed by the Vaynshteyn method; those desiring to become familiar in more detail with the mathematical aspect of the problem are referred to [34]; we shall only describe a qualitative picture of the phenomena. Let us do this in the example of a completely symmatric resonator consisting of two mirrors with R= R > k and with identical transverse dimensions. The given dimensions and the spacing between the mirrors Q will be considered f ixed; the initial curvature of the mirrora will be selected so large that the width of the caustic of the oscillations of the type of interest to us will be significantly less than the width of the mirrors (Figure 2.5,a)., Now let us begin slowly to decrease the mirror curvature. The approach of the resonator to a flat cavity is, as we already know, accompani,ed by an increase in the linear parameter w; the transverse beam cross section increases with it. This process lasts until the beam begins to fill the mirrors entirely (Figure 2.5, b). A further decrease in curvature of the mirrors causes other consequences: the beam cross section remains almost unchanged, only the distribution peaks gradually become equidistant, and their heigiits equalize there is a gradual transition to the corresponding type of oscillations of the flat cavity (see Figure 2.5, c). All of this can be given the following graphic interpretation. For the given "se1f-imaging" (type (1), (3)) structure of the light beam its _:oss sectional dimensions in a resonator with quite large mirrors (Figure 2.5,a) are established so that dynamic equilibrium is achieved hetween the processes of expansion of the beam as a result of diffraction divergence and focusing of it on reflection from concave f inite mi;rrors. When the curvatuxe of the mirrors decreases their focus- ing effect is weakened, and the state of equilihxium shifts in the direction of the larger cxoss sections which correspond to smaller di��raction divergence. This is what happens until the situation depicted in Figure 2.5, b. Then, in 72 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY spite of the continuing attenuation of the focusing as a result of cuxvature of the mirrors, the beam cross section dimenslons cease ta grow, and the deflection - divergence almost does not cfiange. This can be given a unique explanation: a new factor is added that counteracts the divergence of the light beam; edge diffrac- tion is a factor of this type. As the resonator subsequently approaches a flat cavity, the f ield at the mirror edge increases somewhat; the role of edge diffrac- _ tion increases with it. Finally, in a flat r.avity edge diffraction remains the only reason that the beam "does not walk out" of the system and does not have such large losses. Adhering to Vaynshteyn's terminology, it is possible to state that the field in a flat cavity is fixed not by the caustic as in stable resonators, but by the mirror edges (g 2.2). Edge diffraction still "contains" the field less eff iciently than the caustic; - therefore the diffraction losses of all types of oscillations on transition from a confocal resonator to a flat cavity (or concentric cavity) increaae monotonically. Figure 2.6 can be used as a graphic illustration in which the relations are pre- sented for the losses of two lowest types of oscillations of a symmetric resonator - with circular mirrors as a function of the Fresnel number N= a2/Xk (2a is the mirror diameter) for different values of lgl which vary from zero (the confocal resonator) co one (a flat or concentric cavity). We conclude the investigation of stable resonators with this. Let us only state that much more detailed informa- tion about the properties of empty, stable resonators, including the magnitudes of the diffraction losses, the phase corrections as a result of finiteness of the mirrors, and so on, can be found in the monograph on submillimeter wave engineering [33]. This is no accident: submillimeter resonators are excited by elementary dipoles or throLgh an aperture in one of the mirrors; then the Q-factor of the system plays the decisive role, and reao.nators of the stable type fall into first place. On f illing of the resonator with active medium as is done in the optical band, the initial Q-factor is not so important (g 1.3), and the stable resonators 11119 a ) n - - - � 2 .r. b) ~l r,~ n ,�r.I'. ~ L� i - - u~- - - n ..i. Figure 2.5. Evolution of a natural os,cillation on tzansition from the symmetric stable resonator with tranaverse mirror dimensions _ of 2a to a flat cavity: a) stahle resonator; b) stable resonator witli smaller curvature of the mirrora;c) flat cavity. 73 FOR OFFICIAL US" 'LY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 FAit Ub'!'il'1A1 lN-F' UN1,1 lose their exclusiveneas. In addition to everythi,ng elae, i.t will hecome obvious that for achievetRent of- small angular divergence of the exit emission of powerful lasers, in general they are not suitahle; in order to arrive at this conclusion, the inforuiation which we already have at our disposal is entirely sufficient. ,nn - bo . ~a - - TO - - - - - ~ ~ 4 - - - 191=11) . - - - F 1-- 0,95 ~ - - . U9 . ~a~q - - fl] 0,' p c, ~ C !l40,6 1 Z 4 6 10 ..'0 O6UA !0/1 L L (a70d � _ IE Mai ,;71 - - - - - y 4 g~ ~�I,D . . � o " 0,9,f _ 116 114 Q5 - UP . . : _ ~ l Q 0, 2 1Q6' 1 4 h' A ' ,'J N Figure 2.6. Losses of the two lowest types of oscillations of a symmetric resonator with cir- cular mirrors as a function of the Fresnel number N for varioua values of lgl (the magnitude of the losses is presented for passage through the resonator in one direction). Key: a. power losses, % ~ 2.2. Edge Diffraction and Modes of Oecillationa of an Empty Flat Optical Cavity Auxiliary Diffraction Problem. Let us consider plane wave diffraction in a perio- dic structure made up of semiinfinite absorbing screena (Figure 2.7). The period of the structure is equal to R; the direction of propagation of the wave incident - on it makes a small angle a with the plane P passing through the edges of all of the screens. Let us successively construct (beginning, of course, at the bottom) the field dis- tributions on the r.eferenca planes normal to the direction of propagation of our wave and passing tlirougll the edgea of the scr.eens as shown in the figure. For this purpose let us use tiie F:-esnel approximation (1.2a) for the two-dimensional case - of interest to us; 74 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500014010-4 FOR OFRICIAL USE ONLY exp (lks ( s - 1 s u(x$) _ s exp Iik Z 2ss , u~x1~ ~i� IP I ~ Zrt ~ . ~ 4 01 X4 :r ~ 9 0 i i it , � Figure 2.7. Plane wave diffraction on a periodic structure made of screens: 1-- periodic structure made of absorbing screens, 2-- initial plane wave, 3-- reflected wave, 4-- reference planes. Let us remember that here 2 is the distar.ce between the observation and source planes, the difference x2 - xl has the meaning of the length of the pro3ection - of the radius-vector connecting the observation and source points on any of these planes. In the investigated problem the transverse coordinate x, as illustrated in the fig- ure, will be reckaned not from the common normal to the reference surf aces, but fzom the edge of the corresponding screen. Thus, the origin of the coordinates wht^ making the transition to the next plane shifts in the transverse direction by aSC, and x2 - xl must be replaced by x2 - ak - xl. The distance between reference surfaces is z= R,cos az R, - ka2/2. It is impossible to neglect the smal.?. correc- tion term 0/2 only in the rapidly varying phase factor exp(ikz) (see also � 1.1). In the final analysis we obtain the following recurrent formula relating the field distribution on two adjacent reference surfaces: e u ni�1 8"` likl 0 - a2/2)1 exp 1ik (X - x2~ al)'un (x') dx'. y e,l - Introducing the dimensionless coordinate R k/kx and the comaon phase factor exp[i,1R(1 - a2/2)1, we axrive at the final form of the recurrent forntula for tlie functions U(T) = u( R/kT): C\1I (t - S)2/21 Un l'[~> dT', (6) _ :ro 75 - FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/42109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY where s=ayrk_T; here and hereafter n wtll fie the nwmlaer of screens on which the wave Itas alreadybeen dif fracted. The lower reference plane is an equiphasal surface of an initial plane wave; thuag UD = const. In combination with formula (6) this means that the entire set of functions U('r) (and with them, the complete picture of the ditfraction in a perio- dic structure) is uniquely defined by the magnitude of the parameter s. The most interesting case is the one where the �lopC angl.?s of the plane wave a are so small that the condition s� 1 is satisfied. In Figure 2.8 res~1 lts are presented from numerical calculations of the radiation intensity IUn(T)IL on several reference surfaces for s= 0.3.1 The function JU1;T)I2 - (Figure 2.8, a) has the form well known in optics: it describes the diffraction pattern of the plane wave on a halfplane (aee, for example, [6], Figure 8.37). The boundary of the geometric shadow is located at the point T= s= 0.3; although - the edge of the second screen (T = 0) is located to the left of this point, the field amplitude and, with it, the radiation intensity turn out to be smaller near this edge than f or the initial wave. Therefore near the edge of the next screen = the intensity turns out to be even less (see Figure 2.8,b). With it the radiation energy in the region T> 0 absorbed in the screen decreases (the area under the curve decreases); Yere this area turns out to be less than it should be from geo- - metric optics (the area of a rectangle). On making the transftion to the subse-' quent mirrors, the absorbed energy continues to decrease, and in the final analysis it is ;-~cablished at a very low level. Hence, it follows that as a result of dif - fraccion, the periodic structure of the screens basically does not absorb, but scar.ters the emission ir.cident on it at small angles. In the vicinity of T< 0 deep modulation of the amplitude develops with time; the modulation period gradually increases, approaching a def ined limit (Figure 2.8,c, d, e; on the last graph, the field distrib+ition was almost steady-state - varia- tions on transition to subsequent screens are already small). This indicates that in scattered emission a discrete wave is formed, the interf erence of which with the initial wave leads to amplitude modulation; the great depth of modulation indicates that the intensity of this wave is comparable to the intensity of the initial wave. ' The property of a plane (or spherical) wave undergoing diffraction on an opaque screen known in theoretical optics helps to understand the structure of scattered - emission. This property consists in the fact that such a wave can be represented in the form of the sum ef a wave which, being completely absent in the geometric shadow region, is not distorted by diffraction in the remaining space, and a wave, the fictitious source of which is the edge of the screen (sez,.for example, [6], g 8.9). Actually, in our example, after simple transformations the following expres- sions can be obtained for the field ul which arises as a result of diffraction on the first screen: u, (.r.) c-(c~s,'irc) Fix- al x C al, (~~p/'in) F I i/n/Xl ~ x-- ai 11, x> al, 1The dutlior takes, this opportunity to express his appreci,ation to L. V. Koval`chuk who performed these calculations on a computer specially for thi,s publication. 76 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY it) b) d) Os z e) Figure 2.8. Result of successive diffraction of - a plane wave on periodically arranged absorbing screens for s = 0.3. ~ where c is the amplitude of the initial glane wave, F (a) = f oxp (il') d (a > 0) is _ tlie Fresnel integral. For the Fresnel integral it is possible to propose the Eollowing approximate expression giving a correct reprPSentatiQn of the behavior of F(a) in the entire range of variation of the argument; F(a) --{Ii/(2 (a +y;T/-n)]) exp (ia2). Substitution of this expression in the last formulas for u~(x) indicates that tne f:[.eld in ttie geometric shadow zone has the form A(x)exp[ik(x-al)2/21], where A(x) is a comparatively slowly varying function of the coordinate: A comparison with the expressions for spherical waves from g 1.1, indicaCes that the given relation describes a cylindrical wave that diverges from the edge of the first screen. The amplitude of this wave A(x) decreases on going away from the edge of the geometric + shadow; the nature of the decrease is clear both from the above-presented expressian for the Fresnel integral and fram the upper graph in Figure 2.8. In the vicinity of x< at, in addition to the ent3.rely analogous wave, there is also an undistorted plane wave with.initial amplitude c. It is the interference of a plane wave with a cylindrical wave that leads to the appearance of characteristic 77 FOR OFFiC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 c) � - -10 -B -6 -4 -Z 95 2 z APPROVED FOR RELEASE: 2007/42109: CIA-RDP82-00850R000500010010-4 FUR 4FFICIAL USE ONLY amplitude modulation, the depth of which decxeaSeS qn going away fzom the edge of the shadow together with the cyIindrical wave amplitude, Diffracting on the second screen, the undistorted plane wave generates a new cylin- drical wave, the fictitious source of which is the edge of the second screen. In addition to the cylindrical wave, the resultant f ield contains, as before, an un- _ distorted plane wave, diffraction of which on the next screen leads to the appear- ance of the next cylindrical wave, and so on. Thus, the field o:` emission scattered as a result of edge diffraction is a super- position of cylindrical waves, the sources of which are the edges of all of the screens, and the amplitudes decrease identically as the direction of the emission - deviates from the direction of the initial plane wave. As a result of the inter- ference of many cylindrical waves succeeding in getting suff iciently far from the point of their generation, a number of discrete directions ar e isolated in which the amplitudes of these waves are summed (ent�irely similar to how this haFpens on exposure o:. a diffraction grating to a monochromatic plane wave, see [6], g 8.6). The angles 0 which these ::irec~ions make with the plane P sa isfy the obvious con- _ dition kQ(1 - a2/2) - kR(1 - 0n/2) = 2n~' (An � 1) or 6n . a~ + 2nJ~/Q, n= 0, 1, 2, Let us note that these formulas can be obtained directly'from the known rela- tion of diffraction grating theory sin 6- sin 8=+ nX/d, where 6 and 6 are. angles which the incident and reflected wave make with the normal to the plane of the grating, and d is the grating period (in our cass its role is played by k). in the intermediate direction, mutual interference "extinguishing" of the cylindri- cal waves takes place. As; for plane waves formed as a result of interference addi- tion and prop3gated in the isolated directians, inasmuch as the scattered light in- tensity increases rapidly with an increase in A, in the scattered emission the wave with the least A predominates; this is the wave mirror "reflected" from the plane P(60 = a). Its presence also causes deep amplitude modulation of the established field distribution with spatial period of X/2a (in the dimensionless coordinates Tr/s). The remaining waves have significantly smaller amplitudes and lead only to small, fine-scale amplitude modulation. - Let us sum the results of the investigation of our auxiliary problein. If the angle of inciclence of the plane wave on a periodic structure made of absorbing screens is stifficiently small, the greater part of the energy of the emission is not absorbed, - but scattered as a result of diffraction. In the scattered emission, the primary role _ ie played by the "reflected" wave, to which the greater part of the total intensity of the scattered emission goes; thus, the amplitude of the reflected plane wave is comparable to tite ampZitude of the initial incident wave, approaching it as a de- creases. Reflection from the Open Edge of a Wave Guide, Natural Oscillations ot a Resonator made up oF SCrip or Rectangular Mirrors. The above-describert phenomena are tre basis for the mechanism of diff raction "containment" of the f ield inside the reson- ator that we.,mentioned in the preceding section. Actually, the problem of diffrac- tion on a periodic structure of screens is eritlrely similar to the problem of dif- _ ~raction on tlie edge of an open semiinfinite wave guic.e formed by two flat, totally veflecting *-,irrors installed at a distance R from each other (Figure 2.9). The radiation ;:bsoxbed in tlie screens here corresponds to the radiation leaving from the open edge of the wave guidet Of course, in hoth cases the nature of the setup processes depends on the initial conditi:ons and can be quite diff erent, but 78 FOR OFFICIAL USE ONI.Y APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL'USE ONLY the steady f ield distrihutions must that in a wave guide not one, but a the open edge, converting into each 2.9). Inasmuch as the nodes of the the mirrors, an additional conditioi where q is a large (on the order of coincide. It is only necessary to consider set of two plane waves go in the direction of other on reflection from the mirrors (Figure total field distribution must be located in 1 kR(1 - a/2) = qTr is imposed on the frequency, 2Q/A) integer. i a I � I - Z I & I , . . . IP Figure 2.9. Set of two plane waves ("wave guide" - wave) propagated in the direction of the open edge (P) of a flat, semiinfinite wave guide. lising the wave guide approach, Vaynshteyn and his followers were able to obtain many valuable theoretical results not only for plane, but other open resonators; there- fore we shall discuss the laws ohtained above in the language of wave guide theory. The set of two waves that convert into each other on ref lection from the wave guide walls, the frequency and angles of inclination of which satisfy the above-indicated condition, is the so-called wave-gui3e wave. It is propagated through a plane wave guide with ideally conducting (ref lecting) walls as large a distance as one might like taithout damping. The waves f or which the gi.ven condition is not satisf ied can not be propagated through the wave guide and damp quickly. The initial wave guide wave with q= qQ, being propagated from lef t to right, approaches the open edge of the wave guide; diffraction on the open edge leads to the fact that the emission almost does not exit to the outside and "is reflected" back. This reflected (or scattered) radiation, in turn, breaks down into a set of wave-guide waves, the sngles of inclination of which a satisfy the same condition ~ with q= qo, q0 - l, qo - 2, A wave with q= q0 is similar to the initial wave and differs from it only by the opposite direction of propagation; it is called re- - flected, and the ratio of its amplitude to the amplitude of the initial wave is - called the reflection coefficient from the edge; for small a the modulus of the reflection coeff icient approaches one. Waves with q0 qo are calied transformed; in the case of interest to us for very small a, their amplitudes are smzll, and liereafter we shall not be especially interest in the transformed waves. The field of the wave-guide wave propagated from left to right with constant ampli- tude in time has the form exp iwl) oxI) (ilrYx)( sig) ytqz, - i,o 2 0,5 rlQ >,0 19 Figure 2.11. Radial eigenfunction factor of a flat optical cavity made of circular mirrors: a) f00 (r); b) f0l(z); 1- N= 2, 2-- N=5, 3- N=10. Key: A. phase, degrees B. amplitude 84 . FOR OFF'IC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE UNLY The corresponding calculations lead to a discrete set k r = vmn /a(1 + E), where 2a - is the mirror diameter, e has the same value as in (9), vmn is the (m + 1)st . positive root of the equation .Tn(v) = 0(Table 2.1). Iience, we have the follaooitig formulas for the eigenfunctions and eigenvalues of a resonator made of circular flat mirrors: cos n(pvMnr 1 (,os n~ u'mn (f') (sin n~) a [t -f- P(i i)sin n(p x amn /Vmn M (bt 2~i) jMn gVmn p (M -i- 0) I- 0112 [(M p)' where M= 8~r/aQa. The diffraction lossea for large M are equal to 1fp1~mnA1-e = A v?,,,N-312; let us remember that N= a2Ak is the numher of Fresnel zones, R= ' n 1/2n 0.824. The graphs of the functions f00(r) and f0l(r) are presented in Figure 2.11, a, b. The nature of the difference between these functions and the exact solutions of the integral equation i.s the same as in the case of a strip resonator (see Figure 2.10,a). - Polarization of Ra3iation of ilatural Modes. Wit}out considering the systems investigated in � 3.6 with field reversal, the situation with polarization in all empty, apen resonators is identical; the information presented helow pertains to ' an equal degree both to the flat cavities investigated in this section and to resonators of other types (in particular, stable resonators which were discussed in the preceding section). Inasmuch as from the very beginning we used scalar diffraction theory, there is no dependence on polarization direction in any of the dexived formulas; in particular, the natural frequencies for oscillations with any linear polarization are identi- _ cal. Hence, it follows automatically that the natural oscillations can not only have arbitrary linear polarization, but also be,.a superposition of oscillations with different liiiear polarization, that is, have circular or elliptic polariza- tion. Hereafter, for more descriptivenesa we shall represent the oscillations as linearly polarized; let us begin with the case of rectangular mirrors. Figure 2.12 contains a schematic description of the field d�'stributions of TEM00' TrMl0 and TEM01 oscillations with respect to their linear cross section with rectangular mirrors; on the dotted lines, the f ield will be equal to zero. In the general case of unequal sides of the rectangle, the frequencies of the TEM10 and TF.M01 oscillations are different; fur square mirrors they coincide addi- tional degeneration occurs. By superposition of such degenerate oscillations, oscillations with more complex f ield topology can be obtained as illustrated in Figure 2.13. 85 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047/02/09: CIA-RDP82-00850R400504010010-4 FUR OMFIGIAL USE ONLY - 7F. Ala~ - - ~ ---r I 7 EM1G - ~ - _ Tf Mo~' I rfh,ol;, ~ I I ' TT~1f t~ TEMo6, k'igure 2.12. Tield distribution with respect to resonator crass section with.rectangular mirrora. I ~ I T i ~ + - - ~ Figure 2.13. Formation of ascillations with comp3.ex fi,ald topology in a resonator with square mirrors [9]. 86 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 NUR OMFIC'IAI. USE ONI.Y . C) - ~ 1 y ~l a) , d) ~ 9 y y . y f) ~ � Z' ~ ~ @ h> Figure 2.14. Field distribution with respect to resonator cross section x ~ds with circular mirrors j31: a) TEM00~; b) TEM0~ c) TEM01 , u- - d) TEM~i), u- sin 0; e) TEM~i), u- sin 0; f) TEMDi), u- sin 0; g) TEM~r) ; h) TEM01~. Finally, Figure 2.14 shows TEM0Q and TEMQ1 oscillations of a resonator with.circular mirrurs; superposition of the oscillations depicted in Figure 2.14, c-f, leads to the natural oscillations TFM01 with radial (g) and azimuthal (h) directions of polarization. -.i With this, we complete the investigation of empty resonators with small diffraction losses. � 2.3. Some ExperimPntal ResLarch Resulta It was already mentioned in tiie izitroduction that the first gas lasers had highly uniform and extremely weakly amplifying active medium; in addition, the active medium was included in long narrow cells which led to a small number of Fresnel zones N on the resonator mirrors. For small N, as will be demonstrated hereafter, the optical nonuniformity of the medium, :Imperf ection of the resonator elements and bain saturat�ion effects are most weakly manifested; f or these reasons, the experimental data of the transverse f ield structure of gas lasers at that time did not of,.ar too much to consider. It is possible to note, it is true, that when using "stable" resonators distribution patterns similar to the patterns predicted by the theory of ideal empty resonators were observed quite frequently. In the 87 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY case of f lat mi.rrars. (which were used comparatively xarely f or gas lasers), small deviations from ttie ideal configuration led to sharp distortion of the form of the _ field distrihution. Thus, in one of the fi.rst papers on investigation of continuous gas lasers written byHerriott [36] such distortions were ohserved for deviations from ideal planeness of the mirrors of nX/100. In the 1970's when powerful gas lasers with large voli:mes of optically nonuniform active medium began to be more and more widespread, the problem of the divergence of their emission was solved on the basis of unstab]e resonators rather than stable or flat cavities (see Chapter 4). Therefore the basic sour:e of information on the transver se field structure in nonideal resonators with smail diffraction j losses remain the results of the studies of solid-state lasers performed in the 1960's. Early Observations of Stimula*ed Emission of Solid-State Lasers. The first experi- ments demonstr ated that the atL;;ular divergence of the emission of solid-state lusers was significantly greater ( by one or two orders) than the expected value. In order to discover the causes of the comparatively large angular divergence of the emission, various studies were made of the spatial structure of the stimulated beam as a functiur, of the degree of uniformity of the active medium, the pumping inten- sity, and so on. The greater part of this research pertains to the case of a re- sonator with f lat mirrors. When comparing experimental data with the results of the theory of empt open reson- ators, it was discovered very quicklv that the re6. lar spot distributi ir: the near or far zones corresponding to def ined o3cillation modes of an ideal r_sonator is almost never encountered. Even approximaL2 similarity of the f ield diatributidn pattcrns with the oscillation modes of an :deal resonator is observed, as a rule, only when the lasing threshold is slightly exceeded, the medium is homogeneous to the maximum and under other exceptionally favorable conditions [37-40]. With an increase in pumping intensity, the number of oscillation modes present in the stimu- lated emission increases. Modes appear with more and more complicated structure and - greater and gr eater divergence of the emission [37, 38, 40, 41]. In addition to everything else, the spatial distribution of the emission during a lasing pulse does not remain constant at all. Thus, if the stimulated emission pulse lasts more than 10-5 seconds, as a rule, it decays into a large number of already mentioned emlssion "spikes" randomly distributed in time; the field structure varies almost randomly from spike to spike. Therefore if we are able to observe individual modes of oscil- lations, then it is possible only with the help of high-speed pltotography, recording the f ield distribution for individual spikes. Even with high-speed photography, recognition among the other modes of the single, lowest mode which has diffraction divergence of the emission, is faY from a frequent event. When using active rods with noticeable optical iahomogeneitiea, identification of the individual modes of oscillations becrnnes impossible [42, 43, 39, 40], In many experiments it was discovered that very small resonator aberrations with respect to magnitude are sufficient for complete distortion of the field distribut:ton cf the lowest-order modes. Thus, for example, in one case [44] such distortion was caused by elastic def ormations of a ruby rod (as a result of especially applied forces), in another case it was caused by tranaition from samples with optical length variations AL - 0.1X to samples with QL - 0.25X [39]. Accordingly, in subsequent papers an efFOrt was made to analyze the mechanism of the effect ot optical inhomogeneities on the field structure. 88 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY The papers hy, Eytutiov ^nd Neeland 138, 45] and, iz1 paxti.cular, Leontovich and Veduta [46.] wexe a significant step forwazd in understanding the essence of this mechanism. In these papers it was demonstrateri that if a nonuniform active rod aerves as the source of the second--order wave aberrations (of the positive lens type), then the resonator wtth flat mirrors is made equivalent to an empty reson- ator with concave mirrors. The observed f ield structure in such cases corresponds to the predictions of stable resonator theory. Thus, the concept of the equivalent _ resonator was used for the first time to interpret the experimental data. In ref erence 1461 an explanaticn was also presented for a number of laws observed when using stable resonators. When the threshold is slightly exceeded, only the lowest types of oscillations with least diffraction losses located near the - axis of the sample are stimulated. The appearance of stimulated emission first in the central zone is promoted by the f act that the pumping emission density and, with it, the s:nverse population frequently reach the maximum value on the axis of the sample j47, 48J. With an increase in pumping power, the threshold of stimula- ted emission of higher-order oscillation modes is reached, and the region encom~ passed by the stimulated emission expands. Finally, when the threshold is exceeded by a great deal, tlie modes of such high order that the stimulated emission fills the entire active medium appear. ~ The above-described picture of replacement of the oscillation modes was actually _ observed in many papers (for example, [49, 50]); the argument that the maximum transverse index of the modes present in the stimulated emission is determined by the condition of filling the er..tire resonator cross section also found complete conf irmation [51-53]. Let us note that resonator "stability" was insured in different experiments both by using concave mirrors [51, 53] or introducing posi- tive lenses into the resonator with flat mirrors 152, 53] and by "lenticularity" of the sample itself [54]. It is also interesting tha.t on appearance of modes with high transverse indices in the stimulated emission, the lowest oscillation modes of the stable resonator ar2 "displaced" they cease to be observed in the stimulated emission (for example, [49]). The following is especially important for us. Inasmuch as with an increase in order of the modes, the emis~~ion divergence also increases, the radiation pattern of lasers with "stable" resonators not only does not become more constricted as the active zone diameter increases, but it even becomes broader. Thus, the situation here turns out to be quite clear, and it is extzemely unfavorable from the point of view of achievin6 small emission divergence. .10 89 J FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY With a decrease in sphericalness of the mirrors and as the resonator approaches a flat cavity, information about the angular divergence of the emission becomes less and less systematic. The modes of oscillations observed in stimulated emis- sion frequently have such an irregular structure that whatever it is, classifica- tion of them becomes impossible; frequently it is not even possible to trace the correlation between the width of the radiation pattern and the perfection of the resonator. This is all the more true, as numerous studies have demonstrated, in that in the case of a flat cavity the field structure turns out to be especially sensitive to intraresonator wave aberrations, the sources of which are highly varied. In spite of this fact, for large diameters of the active elements, the _ emission divergence, although it exceeds the diffraction limit significantly, is far from that in the case of stable resonators (for example [551). Therefore lasers with flat mirrors remained the basic type of solid-state lasers for many years. Divergence of the Radiation of Solid-State Lasers. Comparatively small emission di- vergence and, together with this, a variety of factors influencing it, have insured constant interest on the part of researchers in lasers with flat cavities. Let j us present the basic results from numerous studies in which the causes limiting the axial luminous intensity of these lasers was investigated. A signif icant part of the studies pertain to ruby lasers. Being the typical crpstal- line active medium, the ruby has an enormous number of randomly distributed micro- and macro inhomogeneities leading to significant scattering of light. The chara~- - teristic magnitude of the losses to low-angular light scattering is 0.01-0.1 cm [56] for this material (just as also for many other cryatalline active media [17]). In the case of noticeable light scattering, the width of the central core in the angular distributio,, usuall.y is 5-20'. The core is surrounded by a system of com- paratively intense rings [57-60, 39, 46] with angular radii equal to the radii of the rings in the Fabry-Perot etalon for light with a wavelength equal to the emis- sion wavelength in the central core (for example, [60]; let us note that the magni- tudes of these angular radii form the same sequence as the slope angle s of the transformed waves A in the preceding section). According to some observations, the ring width in tRe angular distribution is determined by the same expressions as in the passive etalon, and it depends on the quality of the sample, respectively [39]. Finally, in the paper by Vanyukov, et al. [61] the relation of the indicated rings to the light dispersion was finnaly prnved by introducing an additional scat- tering element into the resonator. Observations of the spatial coherence [57-59] and emission spectra [62] led to the conclusion that the emission pertaining to individual oscillation modes is in prac- tice distributed over the entire spot of the far-f ield pattern and the shape of the wave fxont at the exit from thF� laser has an extraordinarily complicated and irre- _ gular structure. All of this does not fit within the framework of the concepts developed by the theory of ideal empty resonators. The interpretation of the observed phenomena is also complicated by the fact that in almost every specif ic case the influence of a great variety of types of resonator aberrations is exhibited. Nevertheless, in many of the experiments it was possible to trace a clear correlation between opti- ca1 pPrf ection of the active medium and the degree of directionality of the emis- sion: with a decrease in uniformity of the sample,the angular divergence of both 90 FOR OFFICIAT. USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFF[CIAL USE ONLY the amplifiers [63] and lasers [63-68] increases. It is worth while especially to note the fact that with large angular divergence, the magnitude turns out to be close to the magnitude of the angular divergence of a high-parallel beam from an extprnal source af ter single passage of it through the same sample [67-69]. In references [66-68], an experimental study was made of the pecuiiarities of the influence of such specific sources of light dispersion as the sliding surfaces, interfaces between modules, and so on. From the presented data it follows that for macroinhomooeneities, basically the light dispersion on the microinhomogeneities on the distribution "wings" is felt in the magnitude of the central angular distribution core. The angular divergence and power of the emission as a function of length of a flat cavity for the case of a laser using a macroscopically uniform medium but having noticeable dispersion on microinhomogeneities (CaF2:Sm2+) was studied in [26 ; it was demonstrated that the divergence of the emission is proportional to L-1/ ~ in a very br,oad range of variation of L. It is noteworthy that with an increase in the resonator length, a decrease in divergence until it approaches the diffraction limit was accompanied only by insignif icant reduction of the lasing power. As has already been pointed out, the light acattering sources in crystalline active media are distributecl more or less randomly with reapect to volume. The aberra- tions caused by nonuniform heating of the active rod (so-called thermal deforma- tions of the resonator) have an entirely different nature. Their source is the thermal variations of the index of refraction and the phenomenon of photoelasticity caused by the presence of thermal stresses [70, 71]. The given eff ect was detected in 1963 [72, 73] and was studied in detail in many subsequent papers. On the basis of their origin itself, in contrast to light dis- persion, the thermal deformations reduce to the presence of an index of refraction gradient which varies slowly with respect to the cross section of the The thermal deformations are especially large in active elements of lasers operating i:i the periodic [74, 75] or continuous [54] mode. Iiowever, even when the laser operates in the single spike mode, thermal deformations at the end of the pulse frequently reach noticeable magnitude and have a signif icant influence on the angular divergence of the emission. Thus, in one of the early papers by Vanyukov, et al. [76] pertaining to a comparatively small neodymium glass laser with �umpin6 trat is not very intense, a two-fold increase in the divergence at the end 3f the lasing pulse was observed which was correctly ascribed to thermal effects. Here- after, as the size and power of solid state lasers increase, the control of the thermal deformations has become one of the most important problems of laser engineer- ing. We shall discuss the methods of solving this problem and other aspects of the problem of optical nonuniformity of active media in more detail in Cha.pter 5. Vow let us only note that analogous phenomena are also observed in powerful gas lasers: microinhomogeneities occurring during the process of excitation of the gas medium or during its turbulent motion frequently lead to noticeable light scattering; in- stead of the thermal deformations of the active element, here thermal deformations occur in the resonator mirrors. In addition to the enumerated factors, the spatial structure of the beam is in- fluenced by a number of other factors. Thus, great significance is attached to the 91 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY accuracy of altgnment of the flat mirrors: if they are not para11e1, the spot in the far zone is elongated in the misalignment direction, and in the case of suffi- ciently misalignments, it breaks down into a number of individual spots [77- 80]. Resonators with unifor.m medium and correspondingly amall emission divergence are especially sensitive to nonparallelness of the mirrors [80]. The relation was also noted for the angular divergence of the emission as a function of the nature of the distribution of the inverse population with respect to the cross section of the sample [81]. - The set of many causes in the final analysis also leads to the fact that the magni- tude of the angular divergence when using resonators with sma.11 diffraction losses usually is several angular minutes and greatly exceeds the diffraction limit even in the case of lasers with highly uniform active media. From the point of view oF the beam structure itself, the great magnitude of its - angular divergence can be the fault af both the presence of a large number of oscil- lation modes in the stimulated emission and the fact that in systems with noticeable - aberrations the emission wave front belonging to the highest Q-nodes, can differ sharply from planar. The latter phenomenon is frequently called mode deformation; although this term is not entirely precise, it is convenient to use in many studies. From the data presented ir, many articles [39-43, 57-59, 62, 77-80], it is possible - to draw the qualitative conclusion that in the case of lasers with flat mirrors in the presence of noticeable aberratior.s the mode deformations are especially impor- - tant. The decrease in aberrations is accompanied by an increase in the role of the effects connected with multimodality (for example 140]); finally, if special csre is exercised in conducting the experiment and a uniform active mediuua is used, the angular divergence of the emission turns out to be primarily caused�by multimodality of lasing. Now let us proceed with a more consistent analysis of the phenomena. Let us begin - with multimodal la:;ing. Its mechanism is so complex that for lasers with noticeable optical nonuniformities, noone has seriously studied it. Only the theory of multi- mode lasing in ideal resonators is more or less advanced, 3udging by the experi- mentall.y observed trends, which we discussed above, this case is also of interQSt, and we shall investigate it in Che following section, _ v 2.4. Multimode Lasing in Ideal Optical Cavities Mechanism of Multiinode Lasinfi. At the beginning of the 1960's, the approximation where the inverse population is considered uniformly distributed through the vol- ume of the resonator was used to describe the operation of a laser. Within the framework of this approximation, multimode lasing is totally unexplainable, Actually, in the case of steady-state lasing, as we saw at the beginning of � 1.4, under the joint effect of pumping and stimulated emission, the inverse population is establisFied at a level such that the gain in accuracy compensates for losses in the resonator. If such compensation occurs t or the mode of oscillations with the lowest excitation threshold (the frequency is closest to the frequency of the maximum ampliFication band, the diffraction losses are minimal), then for the 92 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R400540010010-4 FOR OFFICIAL USE ONLY rernaining inocJeK tiitt baJ.n turnx out� to be leas than ttie lossea, and the modea ahould not be exc.ited.1 _ AS we have seen, the results of experimental studies of the transverse field struc- ture do not correspond to such representations. The data on the emission spectrum are, perhaps, more indicative: even for soiid-state lasers based on media with uni- formly broadened line, the lasing spectrum width frequently is comparable to the uain band width [17]. Thus, in Figure 2.15, the x's are used to plot the results of the measurement of the intensity distribution between oscillations with different axial indices q performed in [62] (the corrections to the frequency as a result of the transverse f ield structure were quite small in the givzr_ case); dotted lines are used to plot the luminescence line. From the f igure it is obvious that the lasing spectrum has components so removed from the center of the line that the gain of their frequencies is noticeably less than ma.ximum. n�e1 9 >,0 i ~ . ~ : ,.I . . . _ . x _ ~ I+ -I 0 I z j = 6,7 1.0 - . x x ~ Uf " x x x -S-f J Z-> 0 1 2 d 4 5 nm~: - ~5 d.5 - x L-JL U ~-:'lll1~,~ ~n - U,,S K n) n,~,t=1,B ,:J x x x X C  101245 /Imc' _ ~ ~3 .:;7 - Ax Figure 2.15. Lasing emission energy distribution among spectral bands ~ with different axial indices (the value of (v - v0 ))c/2n0 9,)'1 is plotted on the x-axis, where v0 is the frequency at the center o� the luminescence line). Temperature of the specimen: a) 15� K, b) 40 K:nmax the threshold is cxceeded with respect to pumping intensity; x-- experi.mental data (energy or the emission belonging to individual spectral bands, in relative un.its); calculated data; the dotted curve is the curve. for the luminescence line. ; 1'L'he etfects expanding the lasing spectrum in the case of a nonuni,fpxtaly broadened _ operating transition band or in the presence o� factors causing mode synchronization will not be discussed here, 93 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 FOR OFF[CIAL USE UNLY Tt,;e data on the emibsion spectra also led to the idea that the primary cause of multimodality is the nonunifarmity of the inverse population distribution with re- spect to the resonator volume arising under the effect of the stimulated emission field. The first paper in which it was demonstrated that in spite of the elementary theoretical predictions, the single-frequency lasing mode is unstable, was writttn by Kuznetsova and Rautian [82]. Soon Tang and Statz [83, 84] proposed a simple model of multimode lasing based on the proposition that with simultaneous excitation - of several modes of oscillations, their frequency differences are sufficiently large that the inverse population does not change noticeably during the intermode beat period. When calculating ttte inverse population this permits summation not of the field amplitudes of individual modes, but their intensities directly (see � 1.1 uti the addition of fialds of several sources with different-frequencies). On the basis of the given model which will be discussed below, Tang and Statz were able to find the steady state distribution of the lasing emission between modes with pumping in- _ tensities somewhat exceeding the threshold. As a result of this, the solution was extended to the case of exceeding the iasing thresho'Ld by large amounts [62, 85, 86]. Another fact is very important. The excitation thresholds of individual modes do not coincide as a result of two factors the presence of differences in the mag- nitudes of the diffraction losses and the fact that the gain depends on the position of the frequency inside the luminescence band. As for the diffraction losses, as is known, they depend on the axial index. When investigating angular divergence, this permits us to do away with the problem of the emission apectrwn [84, 86]. As a result, it is possible to calculate the set of valties of A. =2AmqI where mq q - is the power scattered in the mode with transverse index m and axial index q, and ' the summation is carried out with respect to all possible q. The set of values of Am usually is called the intensity distribution between transcerse modes, although in essence we are not talking about individual modes, but groups of modes, each of which contains all the oscillations having identical transverse, but different axial ~ indices. In the case of a flat cavity, on the contrary, the oscillation frequency is basi- _ cally determined by the axial index; the tiorrection as a result of the transverte structure is usually small. In turn, this permits us to get away from the presence of the transverse structure of the field when calculating the emission spectrum; the calculation result is a set of values Aca>L-= Vmq+ which usually is called m the distribution between the axial modes. For this reason, it is possible to make the remark made about the values of Am. Now let us explain the multimode lasing mechanism in the example of spectral distri- bution of the emission in a flat optical cavity. Here is it necessary to consider the field structure along the resonator axis; the transverse structure, as was demonstrated above, can be ignored. Let us conside.r Figure 2.16. The curve in Figure 2.16,a provisionally depiets the - intensity distribution of one of the types of oscillati.ons along the resonator _ axis (in reality, of course, an immeasurably larger number of modulation periods equal to X/2 are f itted into the length of the resonator than on the figure). We shall consider that the frequency of this oscillation mode fits the center of the li!minescence line, and thus, it has the lowest excitation threshold. 94 FQR OFFdC[AL USE UNLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047/02/09: CIA-RDP82-00850R000504010010-4 FOR OFFICIAL USE ONLY ~ . i i a) ~ I I . . ~ L z , i ' b) ~ - - - - r - - " i i ~ I L 'z ~ ',c) i i U L/Z L z Figure 2.16. Induced spatial modulation of the inverse population as j the cause of instability of the unimodal conditions; a), c) oscilla- tion intensity distribution with different axial indices with respect ; to resonator length (provisionally depicted); b) induced spatial modu- lation of the inverse population during lasing un the first oscillation, the dotted line denotes the threshold level of the inverse population. ; Let the pumping intensity be distributed uniformly along the length of the resonator. - While it exceeds the threshold very little, the lasing intensity is negligibly small, the inverse population is distributed almost uniformZy and is equal to'the . threshold (the dotted line in Figure 2.16, b). When the threshold is noticeably exceeded, the situation changes: the lasing power increases, stf.mulated transitions begin to play a significant role in the inverse population balance, anzi the dis- tribution of the latter acquires the form represented by the solid curve in Figure 2.16,b. At the f ield distribution nodes, a large number of excited atoms accumulate which do not participate in the lasing process in the given oscillation mode. As a result, the inverse population averaged with respect to length turns out to be appreciably greater than the threshold value (this fact also leads to the situation j where the laser efficiency during unimodal lasing is less than the efficiency for : multimode lasing; see � 1,4, Figure 1.20). It is obvions that sooner or later lasing will begin on modes with adjacent axial ; indices: although their frequencies do not fall in the center of the ].ine, which decreases the gain, part of the intensity peaks are not at the inverse population distribution minima as in the f irst mode, but at the maxima (see Figure 2.16,c). With still higher pumping intensity the lasing power in these modes increases so , much that they, similarly to the first, create unfavorable inverse papulation - distribution for themselves; as a result, the lasing threshold can be reached on - subsequent side frequencies, and so on. As the number of modes of oscillations present in the stimulated emission increases, the total emission intensity distribution becomes more and more uniform. The in- verse population averaged over the rdsonator length gradually approaches the threshold value. Obviously, 3ooner or later the time must come when the amount that the threshold poptilation exceeds the threahold value cannot compensate for the decrease in gain on transition to the next side frequencies; there�ore the 95 - FOR OFFICIAL USE ONLY i APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00854R000540010014-4 FOR OFF[CIAL USE ONLY number of modes on which lasing is realized still remains f inite and not too large with limited increase in the pumping intensity. The mechanism of spatial competition of the transversed oscillation modes is entirely analogous. If the pumping intensity is distributed uniformly over the resonatior cross section, the mod.e with least diffraction lossss is axcited first. As the lasing power increases, the inverse population distribution writh respect to cross section becomes more and more nonunform; the averagQ inverse population with respect to cross section increases until rhe excitation threshuld of the next transverse mode is reached, and so on. Thus, the multimodality arises f irst of all as a result of the fact that the emission intensity in indivi3ual modes of oscillstiorLs is distributed nonuniformly with re- - spect to the resonator volume. Thia argument, clearly formulated in the r.entioned - articles by Tang and Statz, quickly found experimental confirmation. In a number of papers [87-89], various procedures were tested which ma3e it possible to avoid - modulation of the radiation intensity taith respect to the res.onatior lengtY! (the simplest of them consists in using a traveling wave ring cavity [85]); in all cases a sharp decrease in width of the lasing spectrum was acLUally observed by compari- son with ordinary type lasers. Procedure and Some Results of Calculations of the Multimode Lasing Regime. ln order to calculate the steady-state multimode lasing regime within the scope of the model discussed above, it is necessary to solve the system of equations, each of which is _ the condition of keeping the amplitude of one of the oscillatiun modes present in the stimulated emission constant in time. In the case of a four-level medium with unpopulated lower operating level, this system described as applied to the problem - of f inding the emission spectrUm has the form t - d ln Ai ky�,,xj - coa 2q,tts1T) dt XpAk (1 C09 2q1ns/1) 0 ~ 0, 0, t1t f2,...,tr~ ~ -vp1-1n~4f r+ (12) here ql = qo + i is the axial i*dex, i is the mode numher (reckoning from the center _ of the luminescence line where q= q0); k0 g3in is the gain on the central frequency _ in the absence of lasing; X4 = X(vi), X(9) is a functiocz which describes the shape - of the luminescence line (in the center line X= 1; for the remainir.g frequencies _ X < 1); Ai is ttie energy dissipation powcr in 2he itr, mode (including the emission power leavinb the resonator) in the corresponding dimansionless units; 60Q are the tuactive losses, R' is �he reflectfon coefficient of the exit mirror. Lasing ts considered to Ue realized on 2r + 1 frequencies arranged symmetrically with respect to the center of the line. In f.ormulas (12), the factor 1-cos2qiTrz/k= 2 sin2qiTlz/k describes the intensity distribution of the ithoscillation along the resonator length (RI is considered O 1-" !J ~ChAh 0 - COS Z9a76Z/,`) i close to one); the expression kgg~ ] is the gain distri- bution on the central frequency established under the overall effect of the entire 96 FOR OFFIeIAL i)5E ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR UhFICIAt. USE UNI.Y lasing emission field, Hence, it is obvious that the integral in (12) i,-:k the gain averaged over the resonator length on the ith mode frequency calculated with weight equal to the intensity distribution function of the given mode. Thus, the idea of ~ conditions (12) is quite obvious: for modes present in the stimulated emission the gain must be equal to the losses; for th,e remainder, less than the losses ( the regime will be unstable). Let us also note that the method of averaging the gain used here is based an calculating the number of induced emission processes caused by the given mode field, Thus, formulas (12) are in essence the balance con- ditions: of the number of photons in the individual modes. - As an example of how the results of the solution of system (12) look, let us present the figure 2.15 which is already familiar to us and in which the cixcles are used for the data on the emissiun spectrum calculated in [62] for the same conditionsunder wrich the experimental measurements were taken. It is obviotKs that the forms of - the calculated and the experimentally obtained distributions are quite similar. The system of conditions analogous to (12) can be isritten also as applied to the problem of finding the intensity distribution between the transverse modes of' oscillations. Depending on the frequency, the factor X can in this case be omitted; then it is necessary to add the diffractiori losses which depend on the transverse index to the losses to inactive absorpt3on and transimission of the exit mirror already considered in (12). The msin change, of course, is that the expressions 1- cos2q.7rz/Q describing the intensity distributions along the resonator axis must be replaced by the distribution functions of the intensities of the different trans- verse modes with respect to the resonator cross section. Here we encounter one of the primary difficulties of the theory of filled resonators for the f irst time. The f act is that any nonuniformity of distribution of the gain with respect to the resonator volume (including that arising under the effect of the lasing field itself) must, generally speaking, cause a change in the individual mode f ield structure by - comparicon with the structure in the empty resanator. While the problem of spectral composition of the emi.ssion was considered, this could be ignored: the spatial modulation period of the gain along the resonator axis i:s sd small (-a/2) that no noticeable change in the flux densities in opposite directions talces place along _ its length. 'rience, it follows that the result of interference of these fluxes is described with extraordinarily high accuracy as before by the expression 1- cos2qwz/R,. On making the transition to investigation of the angular divergence of the emission, the situation changes sharply: the spatial modulation period in the transverse - direction of both the lowest mode fields and occurring in the presence of quite in- tense gain distribution fields will be compared with the mirror dimensions. Thus, ' the width of the zones inside which the gain predominates over the losses or vice- versa, is less, is not so small. Inasmuch as the light waves entering into the transverse modes are inclined very slightly with respect to the resonator axis, transmission across this zone corresponds to an incomparably longer path along the reLonator axis. As a result, the light waves in the corresponding segments can acquire signif icant additional gain or attenuation by comparison with the case of an empty resonator. This, in turn, must cause significant changes in both the form of the f ield uistribution and the magnitude of the diffraction losses. iJevertheless, in a significant oart of the gapers devoted to specif ic calculations o� the competition of transverse modes, the same valuas of the diffraction losses - 97 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400540010010-4 FOR OFFICIAL USE ONLY and field distributions were used as izx an empty resonator, This is most justified in the case of stable resonatorst the characteris.tic dimenaions o� the transverse field in them arc less than the characteristic dimenaions in flat cavities (the causes of lower sensitivity of stable resonators to various typea of excitations will also be discussed in the f ollowing sectiofi). Therefore the results of the papers on stable resonators theoretically will be reliable: let us begin with them. - Let us first note that in the case of stable resonators the natural oscilla.tion - frequency depends sharply not only on theaxial, but the transverse indices (see g 2.1; the values of dmn and bPl usually are significantly superior to the phase corrections in a f lat resonator). Therefore the problem of finding the emission spectrum will become more complicated; during lasing on a very large number o` types of oscillations, the spectral width turns out to be appreciably less than wculd be expected from the solution of (12). As for the angular divergence of the emission, the basic argument about the possibility of estimating it without considering the presence of the f ield structure along the resonator axis remains valid. The results of the corresponding papers [90, 91, 53] turned out to correspond com- pletely to the experimental data. The first of these papers belongs to Fox and Li, and was done by exact numerical methods providing for automatic consideration also of the mode deformations as a result of nonuniform gain distribution and :pos- _ sible instability of the kinetic regime; the subsequent results were calculated by the above, less strict method permitting, however, estimates to be nade in an - immeasurably larger range of variation of the -esonator parameters. In all casea, a gradual increase in rhe pumping intensity was accompanied by the process of suc- cessive "displacement" of the lower--oXder modes by the oscillation modes with higher transversQ indices. This process is explained by the fact that in thF_ case of stable resonators the emission of the highrorder modes is distributed over greater vQlume than the emission oL the lowest modes (see Figure 2.1, 2.2). As - a ree+ilt, the situation arises which is illustrated in Figure.2.17. Curve 2 in Figure 2.17 provisionally depicts the lowest mode field intensity distribution; curve 1 provisionally depicts the higher-order modes. If these two modes were excited simultaneously, as a result of the totai effect of their fields, the gain distribution would assume the form of curve 3. It is clear that the mean "effect- ive" value of the gain for the lowest mode will in this case turn out to be less than for the high-order mode which indicates imposeibility of simultaneous excita- tion of them with comparable intensities. 3 I \V0 - J~. ~ _ � Fibure 2.17. Competi.tion of modes of different volume; 1, 2-- f ield distribution of different transverse modes with respect to the resonator cross section (depicted provisionally) ; 3- gain distribution simultaneous excitatian of both modes. 98 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R400540010010-4 FOR OFFICIAL USE ONLY _ Competition of ttie Transverse Modes i.n I,asers with Flat Cavities. Now let us pro- ceed to the case of lasers with flat cavities which we shall discuss in somewhat more detail. There are no special grounds for assuming that the emission diver- _ gence with circular and square m:Lrrors of the same area must differ significantly. As for square or, in the general case, arbitrary rect-angular mirrors, with- in the framework of the above-discussed simplest model, the angular divergence with . respec*_ to any two mutually perpendicular di.rections can be calculated independently of the divergence with respect to the other [84, 86] (similarly to how the divergence and emission spectrum can be determined independently). The solution of the system of equat3ons of the type of (12) using the values of the diffraction losses, calculated for,.a,t ~empty reso%atoL and the approximate inten- sity distribution f anctions with respect to cruss section of the type of I Ilnt (x) ~2 c4:s2)(m I t) n: (See (11) ) leads to the following xesults [86]. If the pumping is sin2n uniform, the power A dissipated in individual transverse modes of oscillations of a strip resonator (i~ the case of rectangular mirrors Am = n A~) will turn out to distributed among ttiem by the law Am =A0 - Bdm, where 2dm are the diffraction losses (in the case at zectangular mirrc;rs, the part of the diffraction losses which is connected with the f ield structure in the given direction, see 9 2.2), A a�nd B are parameters whicli depend on the properties of the medium of the resonato~ and also the pumping intensity. The greatest power goes to the lowest mode with minimum diffraction losses. Tr�,As, in flat resonators the excitation of highrorder modes does not lead tU displacement of the lower ones (as a result of the fact that the volumes ci a11 the modes are approxima.tely identical). The number of modes A in a two-dimensional resonator and, consp4ueLit?:y, the magnitude of the angular divergence with respect to one of the directjons 0_IU/2a are saturated extremel.y quir.kly as 3 the pumping intensity above the threshold increases (K - ( where x is a parameter which is equal to the ratio of the pumping intensity to its threshold value for a four-level medium). With uniform distribution of sufficiently intense pump- ihg and K;-~ 3, the following simple formulas xre valid: K~ 1,5 V(t.l In ai 11"TI~,~~~} 0 0,7 ~/U�1. ln (i/~''~{'~ I~ T/1,3~;~.a Key: a. equiv where Lequiv - L- 1C(1 - 1/n0), L is the distance between the mirrors, Q is the _ length of tiie model with flat ends made of a medium with index of refraction n0 (sPe the comnents to Figur.e 1.18, g 1.3). From these fornwlas it is obvious that inasmuch as K, a, with an increase in cross- _ section of the active elements the angular divergence of the emission does not de- crease and is farther and farther from the diffraction limits; the correctness of this conclusion was confirmed, in particular, by direct experimental checking �or tlie case of a neodymium glass laser 126]. 99 FOR OFF!"CIAL USE ONL3l APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 hUK utMICiaa, uSL un,.r Within the �ramework of the same model, it i.s also simple to consider the influence of the nonunif ormity of the pumping distribution and the losses with respect to the resonator cross section [86]. Here, much depends on the specif ic �orm of these d;gtributions. Without going into details, let us note the following trend: in a laser, generally speaking, an erfort is made to establish the lasing emission f ield such that the resultant gain distribution will apprflach the loss distribution to the maximum (that is, the threshold condition will be approximately satisfied over the entire cross section). If this emission f ield is representable in the form of the sum of f ields of several lowest modes, lasing will be realized in them. This in- dicates not only comparatively small emission divergence, but also that the calcu- latea model itself is not so rough in the given case: on satisfaction of the thresh- old condition in the ent;!.re resonator cross section mode deformation is absent, and the use of the same f ormuias as in an empty resonator to describe the mode structure becomes valid. A simple analysis shows that this situation must be ob-served most frequently in cases where the ratio of the pumping intensity to the losses decreases - smoothly from the center to the edge of the resonator; the lowest mode intensi:.y in this case increases noticeably by comparison with the case of uniformly distributed pumping and losses.l If the indicated ratio, on the contrary, increases on going away from the axis, this must be a..,ompanied by noticeable mode deformations and must, as a rule, lead to large angular divergence. Finally, with sharply asymnetric pumping distribution (or loss distrihution), both the r.ode deformations and the angular divergence of the emission must be especially lurge. Actually, the proposition that in the given case lasing can be realized on nodes � which are similar with respect , to structure to the modes of an ideal empty resonator leads to logical contradic- - tion. Indeed, the sum of the intensities of any number of * iftodes of an ideal reson- ator is always distributed symmetrically with respect to cross section and cannot "correct" the asymmetry of the gain distribution caused by nonuniform pumping. ' This, in turn, can lead to significant mode deformation which contradicts the initial assumption. ' All of the enumeraCed laws are actually observed when investigating lasers if ineas- ures are taken to reduce the phase aberrations of the resonator to a minimum. Thus, i.n reference 192], the greater pumping concentration on the axis of a cylindrical rod caused a decrease in angular divergence. The results of a carefully made comparison of the cases of symmetric and asymmetric pumping distributions with respect to resonator cross section are also of interest [81]; they can serve as a good illustration of the presented arguments (Figure 2.18). The object of inves- tigation was a laser using an active element made of neodymium glass of rectangu- � lar cross section. The pumping distribution with respect to one of the directions _ along the large cross sectional dimension - was uniform, and with respect to the other dimension, it varied (Figur e 2.18, a). This caused variationa of the angular divergence of the emission with respect to the corresponding direction (Figure 2.18, h). Sharp asymmetry of the angular distribution in the case of � asymmetric pumping distribution indicates high deformation of the modes, for the modes of an ideal resonator have symmetric far-field pattern. lIn real lasers such effects can be manifested only in complete absence of any reson- ator deformati.ons, including thermal. 10.0 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R040500010010-4 FOR OFFICIAL USE ONLY Def iciencies of the Mode1 and the PCS,wtbatt~,es of ImpzOVin$ It. In spite of the fact that the ahove�dtscus.sed sitaplest wodel of multiYaode lasing leads to plausible explanation of a number of experimentall}r observed laws, the possibility of using it for quantitative estintates of the angular divergence is far from obvious. In- deed, it is possible to use the f ield distributions of an empty resonator, as we have seen, anly in certain special cases; in the remaining r,.ases, it is impossible to neglect the mode deformations as a result of nonuaiform distribution of the gain (or losses), generally speaking. The proposition that the frequency differences of the transverse modes participating in lasing are quite large in the case of a large number of Fresnel zones N also is not entirely justified. Finally, the steady intensity distribution of the emission between the individual modes predicted within the framework of ti:e -givem approximation was never ob*served in practice, most frequently, as a result of the "spike" lasing regime. It is true that it was explained in � 1.4 why the energy characteristics of the laser must on averaging with respect to a sufficiently large number of "spikes" cor�respond to the calculation data in the steady-state approximation; the same arguments can be stated also in favor of correctness of the discussed model, but ita deficiencies axe still obvious. ~ (A) 4 ~ o. . ~ ~ 1 i� ~ ~ _ / J e o 0,6 r. - ~ 0~/I~ i i t 1 Figure 2.18. Angular distribution of laser emission during pumping that is nonuniform with respect to one of the directions in accord- ance with the pumping distribution [811. ' _�Key: A. pumping emisaion denaity, relative units B. IV relative units . C. angular minutea Numerous efforts have been made at more complete consideration of all of the pro- - cesses occurring in real lasers. timong them it is worthwhile to r.ote the numerical calcu?ations of the processes of establistunent of the multimode lasing regime af ter inclusion of the previously absent pumping, performed in the previously mentioned papers by Tang and Statz [83, 84] Expressions of the tn2 of (12) were used in the calculations. Considering the dependence o� A on time, theae expressions are a system of differential equations of the type ot d ln Ai/dt = Fi(Ao, A1, It turned out that although the steady intenaity distribution taetween the individual modes is also established in the fi:nal analysis, the setup process lasts for an extremely long period of time, and the oscillations ~,i total intensity occuring in this case are of an irregular nature and are similar. to the experimentally observed 141 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 ~ ~-1 U 1 Z 34f 'ygf(t) C APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY random "peak$," Tt taecatqe posetble to undeicataad that the atpalleFtt "f ailures" in operation of the luaer cauaed br pumplng instabi,l.i,ty$ variation of the resonator parameters as a xesult of vibratfons, ttierntal ef�ects, and so qn are sufficient for the intensity oscillations to last indefinitely. Subsequently, numerous - experimental studies (for example, 193, 94]) demonstrated that the causes of the "spike" mode are indeed such. We shall not discuss the results of other papers devoted to detailed study of multi- mode regimes and the emission kinetics af lasera --ith flat cavities in general. Although among these.papers there are very interesting one--, either special cases are investigated everywhere or models are used which are only a little more per- fected than the one discussed above. In addition to everything else, the behavior of real lasers with flat cavities depends to'a high degree on such a large number of factors, frequently not controllable during the course of even the most careful expertments, that it is inconceivable to fully take them into account. Let us present the following instructive example to confirm this argument. In any model of a laser with a flat cavity, boundary conditions are induced in one form or another at the edge of the resonator.or diffraction losses defined by edge effects are giver.. In particular, when deriving the above-present-ed formulas for a number of simultaneously excited transverae modes and 3ngular divergence, the values of ' the diffraction losses. found in g 2.2 were used: 28~, yn� x( 4N,/s~ = 0,26 (m !)=a-' When the losses themselves were considered, it-was assumed, that the resona- tor mirzors are installed exactly oppoaite each other and are.equa.l in size; if these cond3,tiona ate not satisfied, all of the foYmulas hecome, in general, invalid. In order to understand what deviations from ideal resonator configuration can lead to such an effect, let ut return to Figures 2.7, 2.9. If the edge of one of the mirrors is shifted in the transverae direction with respect to the edge of the other by Aa z ak, one of the mirrors begins to cover the second- reflected wave generated by the edge, and the entire edge diffraction pattern is distorted signi- ficantly. The angles of inclination of the waves entering into the mode with trana- verse index m are aZ ((m + 1)/2)(X/2a); thus, for noticeable variation of the losses of this mode it is sufficient that inaccL.racy in the mutual arrangement of the mirrors or inequality of their dimQnsions reached Aa i 1. lQ ! a. In par- ticular, if the halfwidth of one of the mirrors exceeda the halfwidth of other by this amount, the given mirror can be coneidered infinitely large in the first .approxi*nation. Transmi.ssion of the.light beam through this.resonator in both di- rections is equivalent-to.trasmiasion in one..direction through.a symmetric flat reaonator of double length.; the diffraction losses reduced to a unit length increase in this case by Y72- times. , Thus, for N� 1 very small,deviations �rom resonatar symutetry are sufficient for the.diffraction losses of the lowest modes to become different. The magnitude of the losses can feel the frequently existing scattering of the light on the lateral surfaces of the active element or cell walls, and so on. It*is txue that in the case of large N usually many transverse modes are excited; the properties of the high-order modes are less subject to the effect of random causes (indeed the criti- - cal magnitude of the deviations from ideal configur.ation Aa -m+ 1); therefore ttieoretical estimates of the total number of modes are still meaningful. 102 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400540010010-4 FOR OFFICIAL USE ONLY - The efforts tq create tqqdels af multigode la&i1i$ moxe adequate to. the actual situa- tion in a la&er with aflat resanator are continuing. Howev'exi they can obvi.ously lead only to deeper suhstantiation of the fact already long obvious to experimenters that flat cavities with a large nwaber o� Freanel zones cannot tnsure minimum di- vergence of the emission even witfi an ideally uniform medium. Therefore it is best to procPed to an investigation of the effect of intraresonator aberrations. g 2.5. Influence of Resonator Def ormationa on Field Configuratfon of Individual Modes In this section a study will be made of the problem of how the form of the field distribution of individual modes vartes under the effectof various types of intra- resonator aberrations. Inasmuch as in the final analysis we are interested in the possibility of achieving small divergencc, special attention will be given to the case of comparatively small optical inhomogeneities. Some General Remarks. Perturbarion Theory. The anal.ysis of mode deformations with small diffraction losses has been the subject of a hroad literature. The ma.jority of reasonable results were obtained either by a numerical iterative method proposed in the paper by Fox and Li [9] or by direct solution of differential equations for an electromagnetic field or, finally, uaing the expaneion in a series with respect to eigenfuncti.ons ef an idQal resonator. The latter method is very deacriptive and permits unique consideration of the influence of the most varied factora; we shall pay special attention to it. The expansion of the arbitrary field distribution in a series with respect to eigen- - functions u of the operator of an idea1 resonator P(see g 1.3) is possible, - strictly speaking, only in the case where these functior.s forai a complete system. Only sta.ble resonators with infinite mirrors have a complete system of eigenfunc- tions; perturbation theory has been used to calculate them since 1964. [95, 961. With f inite dimensions of the mirrors, the eigenfunctions of open resona.- tors, as Vaynshteyn demonstrated [3], do not form a complete system. This is all the more the case in that for modes of very high order even the scalar formulation of the Huygens-Fresnel principle on which all of our discussion is based, becomes unexceptable. However, the eigenfunctions corresponding to modes with sma1.l trans- iTerse indices coincide {n the case of stable resonators with the functions of reson- =-_tors made of inf inite mirrors, and with flat mirrora they are very similar to the igenfunctions of a closed resonator with flat mirrors which, as is known, form a ~.amplete set. In addition, the coefficients of the corresponding series usually riecrease rapidly, and it is possible to getby several of the firat terms. .l:erefore the expansion in a seriea with respect to eigenfunctions of both stable a:.:d flat resonators is still valid (a more detailed discussion for the case of a f.-4at cavity appears in 197]), and it leads to results which almost coincide with t::e data of much more tedious exact -calculations. In the case of small aberrations, the methods of perturbation theory are applicable. In the f irst approximation of perturbation theory the solutions o.f the equation (P + P"_~ = R m um corresponding to a resonator with aberrations are described by the f or:r.ulas , PA'm 14m = iLm +YwQmA1Lkr QmA - ~ _ P + m Jh ~m = 1'm + pmm� (13) 103 FOR OFF:CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 FOR OFFICIAL USE ONLY Here P' i,s the pertuxhatipn apexatpr, u~ and , axe e4genfunctions and eigenvalues . of a resonator with aberrationsy Pk,� = f uhl'umda � axe matrix elemeate of the pertur- - bation operator. The mathematical aspect of perturhation theory is explaiaed in any text on quantum mechanics. The only difference of the above presented relations from the standard ones is that in the expression under the integral siga for P~ the operatioa of complex conjugation of one of the eigenfunctioas is absent (the substantiation of the exped:iency of this notation appears in r95, 96]). These relations have, in the given case, the following meaning. The matrix elements of the perturbation opera- tor are none other than the relative amplitudes of light waves scattered as a re- sult of perturbation from certain modes -)f an ideal resonator to others (it is possible to understand this when considering specif ic forms of the perturbation operator). The values of a are the ,amplitudea of the induced oscil- lations; it is natural that they are inv~sely proportional to the frequency dif- ferences of the "inducing force" and the free oscillations of the system B . In the case of stable resonators usually these differences are many times greater than the differencea for the case o� flat mirrors (see also the semark in the pre- ceding section on the emission spectrum of lasers with atable reaonatora). The given fact is the basic cause for comparativeiy weak dependeu'.,Z of the form of the field distribution in stable resonatora not only on misalignments of the m-irrors, leading, primarily, to a shift of the resonator axis), but also on aherrations of other types. As a result, the angular divergence of the laser emission with a stable resonator, as a rule, is determined not bp the influence of the aberrations, but the presence in the stimulated emission of higher order oscillation modes (see gg 1:3, 2.!). Tharefore those who desire to become familiar in more detail with the behavior of stable resonators in the presence of aberrations are referred to the available broad literature cn the subject (for example [95, 96,'98, 99]), and we shall limit ourselves to further investigation of lasers with flat mirrors. ~ 1 I I ~ 1 I ~ I I i1 Ill fl! l.'I It Figure 2.19. Equivalent diagram*of a reeonator with flat mirrors: _ T, II mirrars, ITI perturbation zone. In the case of flat mirrors with a large number of Fresnel zones, the effect of the _ operaLOr P reduces to a significant degree to parallel tranafer o� the wave front, the diffraction "displacement" is small. As follows from the commenta preaented in g 1.1 for formula (1.2), the field source at an arbitrary point on the surface of the resonator mirror (aee the equivalent diagram in Figure 2.19) ie, in essence, the initial wave field in the region Q around the same point encompassing aeveral Fresnel zones. The given fact greatly facilitates calculation of the matrix elements of the perturbation operator. i 104 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R040500010010-4 FOR OFFtC1AL USE ONLY Actually, let the perturhati.on souxce 6e concentrated im the zone accupyi,ng a small part of the length af the reaonator ('pigure 2.19)i on paasage, the wave amplitude is mulciplied tiy, the afierration tactor f(r) wFiicfi varies slowly with respect to the resonator cross section (In the general case this factor also includes the amplitude and phase corrections). Tf the variations of f(r) in the dimensions o� the region Q are small, the field distribution of the wave passing 2 through the entire resonator forward and backward turns out to be multiplied by f(r) independently of the segment of resonator length the perturbation zone is located in. Therefore all such sources of aberrations (including the nonuniformly excited active medium) can be considered concentrated in narrow zones near the mirrors, which has been done in many papers (for example, [100, 101, 90, 86]). Thus, the aperator of a resonator with aberrations can be represented i.n the form of PP, where k' is a factor describing the total influence of all sources of aberra- tions; the perturbation operator p' - FI' - P=(F -1)P ~ hence, it follows that ppm uh (r) u. (r) (F (r) -1J d8 (14) (we used ~m 1). When calculating the expanaion coefficients amk by formulaa (13), (14), it is expedient to.introduce another gimplification, replacing the eigen�unctions and the eigenvalues by the corresponding values for a closed resonator. Indeed, inasmuch as when N� 1 the value of UM in formulas (11) ia a ama11 parameter, the eigen- ~ functions of the open strip resonator are close to the functions of a closed one cos n(m-}-i)i u,�� (x) _(sin Za As for the difference of the eigenvalues, for an open resonator ~ it is determined by the magnitude of the phase corrections 6m with accuracy to terms of the same order ef smallnesa. Phase corrections, in contlast to diffrac- tion losses, in practice do not depend on random parallel shifts or inequality of the size of the mirrors (see the end of the preceding section) and coincide with the corrections for a closed resonator. In the two-dimensional case - Sm ~ (m -4- i) g1v ~ pm (ks - m's) SN' Hence, it follows that the zesult of the effect of the perturbations on the form of the field distribution depends little on random causes. Therefore information - ohtained using the above-discussed approximation can serve as an objective charac- teristic of the emission field of real lasera; calculation of the influence of the perturbations on the magnitude of the diffraction losses requirea a much more com- - plex analysis jfor example, 1102]). Flat Cavities with Minor Aberrations. Let us proceed with the investigation of specific types of perturbations, beginning with purely phase aherrations. The re- sults of this investigation pertain to some degree not only to empty resonators, but to the case of multimode lasing where the gain in the entire resonator cross section is close to threshold (see g 2.4) and the amplitude perturbations are thexefore small. 105 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 _ FOR OFF[CIAL USE QNLY The most wideripxead 4ouxce qf f ~atToxd~ ~dYe abexxat~:Qn~, (the o~t3,ca1 wedge) is _ nonparallelnea& of the mi;xxoxa�. I,h tfits case., for tb.e. two diqengianal (&trtp) resonator F (x) - 1= exp (i 2 2exl.- 1 x i2A 2ex - (e is the angle Uetween the mirrors). The perturbation opexator is antisymmetric; accordingly, amk with odd im-- k l are not equai to zero. A simple analysis shocrs that with an increase in the misalignment angle E, the center of gravity of the - field distribution shifts monotonically in the direation of more remote edges of the mirrors (the opposite conclusion in the monograph.j3] is based on inaccuracy in the discussion). In particular, the expression tor the eigenfunction of the - lowest mode has the form uo uo /4 ~ Nul ([97]; (Figure 2.20, a). In accordance with this expression, the fundamental mode turns out to be noticeably deformed even for extremely small misalignment angles. When e reaches a value of X/4aN, the _ angular divergence of the emission of the fundamental mode is approximately doubled [103]; the perturbation theory itself simultaneously ceasea to be applicable for description of this mode.. Inasmuch as dm - (m + 1)2, with a decrease ira the traneverse index the deformations of the modes of oscillations decrease aharply. Therefore, in the ussal multimode regime, the total magnitude of the angular divergence turns out to be appreciably less sensitive to misalignments than the fundamental mode fie1d configuration. More detailed information about the field structure and the diffraction losses in _ r:!sonators wi,th flat misali,gned mirrors can be found, for example, in [79, 80, 99, 102, 104--1061. In the last of the above-enumerated articles, the corresponding results were obtained by the Vaynahteyn method in analytical form. It it note- worthy that in accordance with the data of [102], with an in the angle of misalignment, the diffraction losses increase most rapidly for the lowest order modes (that is, with the least number of f ield distribution "nod es"). As a result, - the curves on the graph repreaenting the losaes of different modes as a function of _ the 'misalignment angle intersect ,(Figure 2,21), Analogous intersection of the curvea was also obsexved in xeference 1107] when per- floxming the calculations (by the iteration method) for a flat cavity with large irregular aberrations cauaed by thermal deformation of the active element in imper- fect illumination systems. This fact demons::rated that in the case of reaonators with large diffraction losses efforts' to classify the modes by the magnitude of these losses can lead to complete confusion; we shall discuss an exaraple of such - confusion in g 3.3. The properties of a flat cavity are alro of interest in the px'eSence of second- order aberrations (slightl}r concave or convex mirrors, swall thermai lens, and so on) . In the case of a stri,p resonatox the perturbation operator with second-order ~ aberrations has the forin I' (x) - 1,~ i~ 2h(x/a)', wfiere h. is the difference in the distances between mixrors at the edge and in the center of the resonator (Figure 2.20, h, c). The operator is symntetxic* u0 r, u0 , 0.6(h/a)Nu2 1971. 106 FOR OFFICIAL USE ONLY � APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00854R000540010014-4 FOR OFFICIAL USE ONLY , h , I I uo uo ~~o . -1 >x/a d01 uf �o-, a) ~OP uY -1 1 x/a b.} 1402 U2 c) Figure 2.20. Effect of phase aberrations on the lowest mode field distribution: a) mirror misaligtimerit; b) concave mirrors; c) convex mirrors. Por h< Q(th.e mi.rrors are concave) tbLe lowest mode f ield natuxally i,s concentrated ' near the resonator axis (aQ2 A 0, see Pigure 2.20 b), The d3.ffraction losses de-- crease in this case 1102, 104]. For deflections 1h.1 ~ X/10N in order to find the distribution in the fundamental mode usually it is quite possible to use the re- sults of stable resonator theory. For h> 0(convex mirrors), the field distrihuti,on wi,th respect to resonator cross section becomes more uniform (Figure 2.20, c); the diffxacti,on losses increase - sharply. Naw we see how the lowest mode fi.elds v'ary i.n the presence of amplitude aberrations. The source of amplitude aberrations is nonuniform distri,bution of the inverse popu- lation or.loases with respect to the resonator cross section. Kuznetsova [108] dt. nstrated that for amplitude aberrations a diffraction "makeup" of the zones with small inversian (or large losses)with radiatioa frrnn tfie region in which the gain predominates over the losses takes place. As a result, the equiphasal sur- face begins to be distorted and even for the lowest mode the eanission divergence can differ noticeably from the diffraction limit, 107 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFF'IC[A4 USE ONLY tt" 10-j 10''r >0-if 1~ Figure 2.21. Diffraction losaes in the reaonator with circular flat mirrora of radius a as a function of the misalignment angle * 110]]; 1 T"' TEM00' 2~ TEM01, 3 r-- TEM11, 4-- TEM21, 5--- TEM31. It is of interest that the nonuniformi,ty of disttribution of the inverae population can lead to the presence not only o� amplitude.but also phase aherratione.. Thus, if lasing is realized on frequenci,es noticeablyr distant from the center of the - luminescent line, the phenomenon of anomalous dieperaion begins to be �elt (see, for example, I109]). As a result, the index of refraction turns out to depend on _ the magnitude of the inverse population; for frequencies in different directions from the center of the line, the variations of the index of refrsction are opposite _ with respect to sign. The same perturbation theory helps to understand the mechanism of the influence of amplitude aberrations, just as phase aherrations. The value of F- 1 defining the - matrix eleaoaents of the perturbation operator is in the general case equal to 1971 oxp (h;C j (1- p) exp [i (,qk,,l !-nAL)] -1 ,^s ( ~ kyel - P-I- i C191'r:l + ~ OL Key: a, gai,n where p is the total losses (including losses to passage thxough the uiirrors), OL is the variations in optical length of the xesonatoz as a result of phase distor- tions not connected wi,th inverse population, n gain SC axe the phase distortions as 108 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 FOR OFFICIAL USE ONLY 1 a xesult of the pxesience qf anoaalon% dtape~stan of the actiye mediuta. k'or uniform Lorenttan bxoa,dening tx " 2(v T~ VaZNVR, where v and vQ are. tize. qpexattig i'requency and the i,'requency in the center of the Iuminescence liaies, respectivelp, OvR is the halfwidth of the luminescence line (see, for ezample, j109]). If the phase aberrations are absent and the only source of perturbations is nonuni- formity of distributfon of kgain and p with respect to the resonator cross section (p = 0), the matrix elements of the perturbation operator are real. Here the coeff i- cients amk, Just as am - Rk, turn out to be almost purely imaginary values (see (13)). For this reason the amplitude aberrations lead f irst of all not to amplitude, but to phase distortions of the eigenfunctions. Within the framework of the �irst approximation of perturbation theory it is also possible to find the self-consistent solutions for the lasing regime ueing one lowest mode 197], The reaeon for the nonuniformity of distribution of the inverse population in this case is the presence of a nonuniform lasing field itself. The corresponding analysis ahows that for small phase aberrations of the type of a negative lens (convex mirrors), the range of pumping intensities for which the uni- modal regime is stable expands by comparison with the case where the aberrations - are absent (or, all the more, have opposite sign). This is also understandable: the valume of the lowest mode increasea (see Figure 2.20), and it "burns up" the inverse population more uniformly with respect to the resonator cross section. - Flat Cavities with Aberrations of Signf,ficant Magnitude. Let us proceed to the cases where the modes of oscillati.ons o� a resonat= with aberrations have so liztle in common with the modes of oscillations of an ideal empty resonator that it is necessary to forget perturbation theorp. Thus, especially for small amounts of excess over the lasing threshold, the situstion is widespread where emission of individual modes is not distributed over the entire resonator cross section but localized in a small zone af it. In the already mentioned papers by Kuznetsova [108], a study is made of the case where this situation is caused by sharp nonuni- formity of distribution of the gain which for the greater part of the cross section is appreciably less than the threshold value and exceeds the latter only in a small segment where the lasing f ield is located. With suff iciently removed edges of the mirrors from this section, the diffraction on these edges can be neglected, which was done in 1108]. Let us note that on the bhsis of the given model in 1112} it was possible to explain some of the experimentally observed peculiarities of the energ}r characteristics of lasers with.nonuniform pumping. - Soskin and KravchenLO 1113, 80], using a model of a resonator with "stepped" mirrors, investigated the case where such localization was caused not by amplitude aberra- tions, but phase aberrations. For phase distortions of a random nature, the emission can turn out to be concentrated in several separate segments of the cross section (like "local stable resonators"); as a result of the presence of 1Generally speakitg, the index of refxacti.qn depends on the degxee pf excitation of the active medium not only as a result o� anomalous-dtsQerston, Ih particular, for neodymium glass it is more izapoxtant tfiat the atoms in the ground and excited states have differQnt polarizabilit}r 1110]; the same effect is desacriTied for the case of a ruby, Eor example, in jlllJ. 1Q9. FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 FQR OFF[CIAL USE ONLY exponentially decxec'tsi.x1$ dtatx~~tton k"tatU" aveak- dif fxaction xelatton i.s realized hetFteen the segments. Us#a& the Siven models it is poasa:hle to explain a number o� pfienomena observed near th+e lasin$ tfire$fiold. Iirnwever, wfien the - threshold is exceeded litcle, the laser emissioa iatensity distribution inevitably becomes much more unif orm (as a result of the "equalizing" ef f ect of the nonunif or- mity of the gain distribution induced bp the lasing f ield) The experimental data mentioned in g 2.3 158, 59, 62] also indicate that orith intense pumping the emis-- sion of individual modes , although it has small-scale amplitude modulation, is distributed in practice over the entire operating cross section (in the sometimes observed 'intensity spikes filaments a negligible fraction of the total emission flux is concentrated). Therefore the models, the basis for wrich are the concepts of localization'of the field in iadividual segments of.the cross section, are unsuitabZe as a rule for estimating the angular divergence of the ' emission in ordinary lasing modes. Efforts at strict investigation of the general case of arbitraxy phase aberrations when considering the effecC of the laser inverse population on each other lead to ext Y awk~r Therefore it turns out to be expedient to do away with the details of.the inverse - population distribution and only consider that, to a signif icant degree as a resuit of the effect of this distribution on the emission field the lasing is realized predominately in modes with maximum volume. In this way it is possible to obtain comparatively simple formulas which correspond to the experimentally observed laws to estimate the angular divergence of the laser emission as a whole. Let us pre- sent such formulas for two extreme :ases where the source of the phase aberra- tions is dispersion of light on micxoinlwmogeneities and where a slowly varying gradient of the index of refraction with respect to cross sectiAn exiats. The most successful method of considering the ef�ect of the randomly arranged'.small scal.e inhomogeneities was proposed in 1968 by Lyubimov 1114], who made a large contribution to the development of the theory of empty resonators with phrse aberrations. This method is based on the concept of the mode of oscillations of a real system as a complicated complex characterized by a aingle frequency which is a superposition of many modes of an ideal resonator with randomly distributed amvli- tudes and phases. The relation between individual modes is realized as a result of light dispersion; inasmuch as it is quite weak, the complex can in.clude only modes with initial �requencies that do not differ too much. The estimates made bv - Lyubimov for the spectral width of the range in which these frequencies 1ie lead to the folluwing resultant formula for the angulax aperture A of the emission of such complexes; ez pd1/4 isp xlLequiv I1151. The experi.mental data of 126, 1151 pertaining to a fLuorite and sama.rium laser where the macroinhomogeneities are absent and there is "pure� light dispersion on the micr4inhomogeneities (perceived as a halo when ohserving a point aource of light through a crystal), at least do not contradict this formula. It is, however, - necessary to exercise known caution with regard to the formula inasmuch as certain assumptions which are far from always Justifiable were used in its derivation. In particular, it was assumed that the width of the light dispersion index is less than the width of the angular distribution of the lasing beam formulated in the presence of light disperaion, The following is also doubt�u13 the presence of a noticeable light dispersion leads to a sharp increase in angular divergence of the radiation, already not very small, in flat cavities with.large N. The cause of 114 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 FOR OFF[CIAL USE ONLY the light dispersion can be not onl}r microinhamogsneities. Thus, sometimes an effort is made to use metal reflectors with a lgrge numbe;e of small apertures as ~ semitransparent mirrors for lasere in the far infrared range (see g 4.2). Diffrac- tion in the aperture system leads to the appearance of a signif icant amount of - scattered light, which in turn, causes the most undesirable consequances from the point of view of divergence of th.e emission 1116]. Now lez us proceed to the most isn;3ortant and widespread case where the optical length of the resonator varies slowly with respect to its croas section. 'The cause of this can be macroinhomogeneities of the active elemeat, thermal deforma- tions, errors in making the mirrors, and so on. This problean was investigated most comprehensively for arbitrary deformations equal to the wavelength with respect to order of magnitude in ref erence [117]. The authors use the following approach, the " mathematical aspect of which wili not bE discussed. By solving the corresponding - dif f erential equation of the f ield f or each of the waves traveling to the edge of the - resonator, superposi.tion of which forms a transverse mode, an'.equiphasal surface is constructed. According to the optical-geometric approximation,the normsl to this sufface at each point determines the direction of the emission originating from the given point (in g 1.1 in the example of a spherical wave front we saw that for deflections of -X the emission divergence can be completely estimated by this method). The resultant formula for the maximum angle between the direction of emission and the resonator axis has the form . amex {[21t/l -I- (ph)'/92)ma: - [2h/1 -I- (Vh)a%12)m1n}1/9, where h are the variations of optical length of the xesonator caused by wirror dis- tortions (the index of refraction of the medium was considezed equal to one). If h varies within the limits of the cross section so slowly that it is ossible to neglect the terms (Oh)2/12, the formula acquires the form %.X 2AY./Q , where Ak is the total variation of the optical length in the working cross section, In the same paper [117], it was pointed out that terms with (Oh)2.vanish pletely when the mirrors are flat and parallel, and variations of the optical length are caused by dependence of the index of refraction on the transverse coordinate (hence, it is clear what type of errors can occur as a result of "diaplacement" of ' the optical inhomogeneities along the resonator length), In this case the formula for the divergence can be derived bp a very simple method. Actually, let us trace the trajectory of a beam multiply ref lected from flat mirrors and gradually inter- secting layers with different index of refraction as it moves acrass the resonator axis. Variations of the angle a' between the beam and the axis in this case are caused exclusively b}r the refraction at the boundaries between the layers and can be calculated directly by the sine theorem sin(r/2 q,�)/sia(7t/2 ^ a2) n n2/nl, . where cci and o4z ar.e values of a' in layers with index of refraction ril and n2, respectively. Using a` � 1, wte obta~ (9t~2~2 ~(q,1~2 m 2(n2tn1 ~ 11. No~t i;t re.~ mains onl}r to consider that the mode encompaasing the eatire las$,ng region coxze: - sponds to a family, of beams intersecting this enti'xe reg3,on. Hence, it fo1loFts� directly that in one of the s ents of the cross section the angle a,t' xeacfies at least a value oP am~ = 2~n n, whexe n is the total,on o� tfie index of refraction within the limits of the lasting zone. inasmuch as the angle in outer space a is related to d' by the expression d'= n G(',.we arrive at the final formula qmax -~An. FOR 07FICIAL,USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 FOR OFFICIAL USE ONLY ~ Let us note that five years later another method of deriving the same formula based on similar arguments was published in [118] (instead of the sine theorem, the standard equation of a beam trajectory in a nonuniform medium borrowed from [119] was used). - It is time to swa up our investigation. The following fact is most remarlcable. Ia flat cavities, as a result of the effect of multiple passage of the light through the same large--scale optical inhomogeneity, the resultant distortions of the wave front of the steady-state oscillations usually greatly exceed the distortions acquired on one pass. In particular, this pertains to cases where the magnitude of the aberra- tions in one pass is small, and the characteristic transverse dimensions of the op- tical inhomogeneities, on the contrary, axe large. Actually, the variation of the optical length NZ in the characteristic dimenaion b in the final analysis causes, as we have seen, deviation of the direction of the emission from axial by the amount ~ 2p9,1Z. At the same time, the deviation of the beam direction in one ass (that is, in the amplifying mode) is 0t/b; the ratio of these two values is 2b /ZAZ - As a result, the "sensitivity" of a flat cavity to small phase aberrationa turns out to be extremely high. Comparison of the formulae presented in this and the preced- ing section shows that for U rX the divergeace of the emiasion of individ'ual modes exceeds the divergence which would occur in the absence of aberrations as a result of multimodality of the lasing. Thus, the theoretical elements, just as the set of experimental data (g2.3), indicate that in the overwhelming majority of cases the primary cause of cowpar atively ,large divergence of the emission of lasers with flat cavitiea is not multimodality of las- ing, but deformation of the modes utder the effect of intraresonator wave aberrations. Section 2.6. Methods of Angle Selection of Emiasion In this section an anal.ysis will be made of the efforta to decrease the divergence of laser radiation with planar or stable resouators (or resonators similar to them; see be'_ow) which have been built at different times and with varying degrees of _ succc-s. The majority of these methoda are now only of historic interest; hnwever, some o' them are used even today. Attempts to Solve the Problem of Divergence on the Basis of Reaonatora with Small Diffraction Losaes. In a number of papers a atudy has been toade of the poasibility of creating resonators from mirrors with-aspherical surface, the ahape of which is selected in such a way that the diffraction losses in the lowest mode are just as small as for stable resonators, but they increase with the transverse index faster than for stable resonators. This theoretically facilitates the achievement of un:t- modal oscillation. Some systems of this type are presented in Figure 2.22. � The resonators depicted in Figura 2.22,a,b, were made from dihedral ref lectors, the angle between the f lat faces of which is w - a in the former case, and 7r/2 - a in the latter case (a � 1). The effect of the first type of reflector on a narrow light beam to a certain extent is similar to the effect of a concave mirror; some "fine focusing of the beam is realized. The reflector of the second type only adds "in- version" of the beam cross section to this effect; therefore resonators af these two types are equivalent to each in thQ abseace of aberrations. It is possible to see that the tranaverse dimensions of the light beams corresponding to individual transverse modea increase in them with the transverse index faster than in stable - 112 FOR OFFiCIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY reaonatora,l which ensuxes greater sel.ect~tity% fox xeflectaxS, o~ ftnite dimme.nsions. The properties of these resonators are described in more detafl in 1124, 121J. - a - - - - - c - � ' . 90_K - d - - b - - Figure 2.22. Some types of resonators with aspherical mirrors: ' a, b--resona- tors with dihedxal reflectors [120, 121]; c--resonator with cen- tral "indentatioa" [122]; d--resoaator proposed in [123]. _ Figure 2.22, c, depicts a resonator in which the eigenvalue.spectrum is still more radically rarefie,d. The dimeasioas of the central section of the left-hand reflector--the "indeatation =-caa be talcen ao that they will be equal to the di- mensions of the basic mode spot formed by this section and the right-haad mirror of a stable resonator. Then the loeses of this mode will be small; broader beams corresponding to other modes will go beyond the limi.�s of the ceatral sec- tioa aad scatter quickly in the peripheral part of the resoaator. This must . lead to signif icant increase in the losses. At f irst glance it may seem that an analogous effect is achieved by simple irising of the stable resonatorf Aowever, in'the latter case as we ilave seen in Sections 2.1, 2.2, the high-ordar mode f ield begins to be conf iaed inside the resonator by edge dif fraction (the edge of the hae the same effect on the light beam as the edge of a mirror); in the resonator depicted in Figure 2.22, c; the diffraction on the edge of the "indeatatioa" turns out to be significantly attenuated as a result of the presence of halation. Vaynshteyn (3] indicaCed the possibility of using a similar procedure; the effort at practical implemen- tation in the optical range (ia the example of a helium-neon laser) is described in [122]. Sometimes proposals for a different plan are eacouatered in the literature. Thus, in [123] the properties of the reaonator depicted in Figure 2.22, d, were analyzed. With a strictly defined form of the componenCs of its-reflectors, the equation for finding the natural oscil.latioas has only one solution. However, in [123] a study was made only of the equation of an mPtY resonator made up of infinize mirrors; consideration of the edge effects and the introduction of an active medium should change the situation sharply--nothing may remain of "uni- modality." In addition to everything elae it is uaclear how such reflectors would be made with the precision required in the optical raage. In contrast to the Last-mentioned version, the possibility of using the resona- tors depicted in Figure 2.22, a-c, is unquestioned. 'However, they are all char- acterizee by the same deficiency as the ordinary atable resonator itself: under 113 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 FoR oFr-Ic;IA,. u5L VINLY conditions of unimodal oscillation it is possible to make effective use of only a very small volume of the optically uniform medium (it is aufficient to poiat out that ia the mentioned experimental work [122] trie depth of the "inden- tatioa" was �A/10). Therefore the devices with aspherical reflectors are not able to compete with the stable resonators which are simple to make and aliga. It must be noted that for cross sectioa dimensions af the active medium not ex- ceeding several millimeters, unimodal oacillation can be achieved using both stable and planar resonators. la particular, it is appropriate to note the successful method of empirical selection of the optimal parameters of the reso- nator near the "stability" boundarq used for the first time in [124] aud also used successfully in a number of subsequent experiments. The method is based on the application of a combination of plane aad coacave spherical mirrors, the distaace between which L varies near a value equal to the radius of curvature of the concave mirror R. For L- R(or in the case of the preseace of an active elemeat of leagth k with an iadex of refraction no, for Lequ{v - L-z(1 - lln R) a so-called semiconcentric resonator is realized which is equivaleat to the planar resonator aad thus is at the "stability" limit (Section 1.2). For shorter lengths, the resonator is stable; it is importaat that for Lz R small variations ia the distance between the mirrors lead, as it is easy to see, to significaat simultaneous variation of the diffractioa losses and spot dimensioms of the fuadamental and otlier transverse modea. TEiis makes it eas}i to select the optimel cambfaation of them from the point of view of the output characteristics of the laber. Obvionaly, witen thia cboice is made empirically, the "'1enticvlar- ness" of the sample is automatically taken into account if it exista, aad so cn. p/pi.r.s O,N Qel 41 n," Figure 2.23. Results of experiments in oacillation selection in a resonator cloae eo a semiconceatric resoaator [128]. Figure 2.23 shows a standard graph for use of the indicated method borrowed from (28), where this method is diacussed ia great detail. The graph was obtained in experiments with a helium-neon laser; the ratio L/R is plotted along the x-axis, the line L/R - 1 is the "stability" limit.* Oa approaching it from-the left, the diffraction lasses increase, the total oacillation power P decreases slowly, then the number of transverye mpdes present in the oscillatious are reduced. At the point noted by the arrow, only the lowest mode TEM,0 remaias. The resonator length corresponciing to this point can also be considexed optimal: further mavement toward tha stability ?imit aud transitioa to the region of instability 114 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R400540010010-4 FOR OFFtCIAL USE ONLY - located to the right causes only a sharp decrease in power aad then cessation of oscillation. - EL1.1 of this is good, however, only in the case of lasers with amall cross sec- tions of the active medium for whiah the problem of radiation divergence in gen- eral is not acute. As far the lasera of primary intereat to us which have a large exit aperture, for them the application of stable resonators leads, as has already been noted more thaa once, to oscillatian on higher-order modes with a broad radiatioa pattern. Nevertheless, efforts have been renewed many times to solve the problem of divergence even in this case. In Chapter 5 there is a _ brief discussion of the conversion of light beams corresponding to the high- order modes to narrowly directional light beams by the methods of holographic correctioa. A search was also conducted for simpler metitods; from the work in ' this direction there are some interesttiag papers (for exataple, [125]), but their _ practical sigaificance is low. Therefore hereafter we shall limit ourselves to the investigation of the methods of angular selection (decreasing the angular di�vergence of the radiation) in plauar resonators which has received signi�i- cmat development in its time. La.sers With Planar Resonators and Aag1e Selectors. For constriction of the radiation pattern of a laser with plaaar resonator it ie necessary ia the gea- eral case also to deczease the aumber o� modes ia which lasf~ig-=--~ is realized and, what is usually even more important, the deformatfons of these modes. The number of modes ia determined primarily by the ratio between the diffraction aad nonselective losaPS. Therefore for angc}e:- selection in the hypothetical case of aa ideally uniform medium where the mode deformations are small, it is neces- sary to try to increase the differences of the diffraction losses. In the presence of aberrations of any tqpe, the'most important problem turas out to be decreasing the mode defdrmations; in accordance with perturbation theorq (Sect3:on 2.5), for this purpose it is aecessary to increase the differences of the eigenvalues of the operator includiag the phase correctic-ns. Making this remark of a general nature, let us proceed with investigation of the specific methods of angrear selection. . In order to obtain the desired effect usually additional elements called angle selectors are introduced into the resonator. They are essentially filters, the transmission of which depends sharply on the direction of P:apagation ot the radiation. Historically, the first type of an81e selector was a system of two confocal lenses and an iris with ,small apert.ure _ placed at their commou focal point [126, 127]. A concentric resonator with an iria, �in the central plane [128, 1291 (Figure 2.24, b) is entirely ideatical to a planar reaonator with such a selector (Figure 2.24, a). The operating principle of suzh a selec- tor is cbvious. Znstead of an .iris,�, a passive ahutter can be used: part that cleara firat then acta as an iris apertura 1130J. The effect of the selector based on the Fabry and Perot etalon [131, 1321 is based on the fact that the transmiasion of the etaion depends not only on the 1.15 FOR OFFICiAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R040500010010-4 _ VVK vtr 1%.U%a. vou, vV.... waveleagth, but also the directioa of p-ropagation of the radiation. Inasmuch as for inclined incidence of the beam, this relation becomes sharper, the etalon is installed at an angle to the resonator axis (Figure 2.24, c). For realization of angular selection in both directions it ia necessary to use two etalons. Probably the method of selection based on using the dependence of the reflection coefficient at the iaterface of two media on the angle of incidence had the greatest popularity. Near the critical aagle of total internal ref-lection., the iadicated relation is especially sharp; therefore these angles of iacidence are used. Ia order to elimiaate the selective effect it is possible to make the light undergo multiple reflections (Figure 2.24, d). In the 1960's a large num- ber of versions of selectors of this type [133-140] grouped under the general head- ing of total internal reflection aeZectora were proposed. Let us consider the mechanism of the effect of the selectors oa the angular d3- vergence. From very general argumeats it is clear that the presence of a fil- ~ ter, the transmission of which dependa on the direction of propagation of the radiation is primarily felt in the magaitude of the losses of individual r.rans- verse modes. Z'he phase corrections are determiaed for fixed configuration of the resonator ia practice only by the numbex of aagles of distributioa of the amplitude with respect to cross section (that is, the transverse mode indea.), and ia the presence of a selector, they must vary inaigaificaatly (aee Section 2.2 for the similarity of phase correttions ia open aud closed resonators). The reaults of strict calculations [141] confirm this obvious conclusion. Let us preaeat the data for the idealized case of a Gaussiaa selector, the shape. of the passbaad of which is Intermediate between the shapee of the baads of real selectors p2esented in Figure 2.24, and it is deecribed by the formula g2(~) - eXp[-(O/p~) ](Figure 2.25; see [241]); 0 ia the angle between the directioa of propagation of the radiation and the resonator axis; Ao is the pasabaad width, g2 is the transmission with respect to iatensity. If ure consider that the transverse modes with the index m correspond to values-of 0 's t(m'+ 1)9diff/2 (see Sections 2.2, 1.1; Adiff `X/Za), the magaitude of the.loases introduced-by (m.+ 1)~(9diff/2DO)2 (this same result was the selector follow directly A(2dm) o obtained in [141] by a stricter procedure). Let us now trace how the-magnitude of the angular divergeace of the radiation - must vary with the passband width of the sal2ctor. . . . In the absence of aberrations the role of;the~selector reduces to variation of the conditions of competition of ths modes (S~ction 2,4) by.,inereasiag ;the loss differences. The losses introduced by the Gaussian eelector tura out:to be greater than the diffraction losses an ideal emptY' resonator oa satisfactioa of the condition 20~/9diff < (a2/U)3 4� Iwwmuch as for lasers with large cross section.a2/J1L a N � l, the aagular divergeace in the given idealized case caa decrease aharply even for comparatively greater width of the selector pass- baad. Estimatioa showa that to achieve the uaimodal conditions with an ideal active medium it is sufficient to use a selector with Ao several times greater � than 9diff/2� 116 FOR OMCIAL USE ONLY ~ APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047/02109: CIA-RDP82-00850R000504010010-4 2 ~ S I s Z , . a � I ' , - 4 1 b ' . I Z ~ z 1 , c � ' ' Z ~ d _ f i ' . ~ - 2 . - Figura 2.24. Diagrams of lasers with an61e selectors: 1--active sample; 2--plane mirror; 3--epherical mirror;'4-- iris wfth aperturef,~-- ` lens; 6--Fabry and Perot etalon; 7--plane-parallel plate. - Figure 2.25. Shape of the passbaad of different anSle selectors; Gaussian - selector (dash-dot line), the aelector based on the Fabrq and ~ Perot etalon (dotted line) and the ideal selector (solid liae). This possibility is based on the fact that R2, the resonator is also unstab le; one of the centers is located inside the resonator (Figure 3.3, b). _ The asymmetric confocal systems (R1+R2=2k, R1OR2 are interesting versions of un- stab le resonators; for determinaey we shall consider R1>R Both the confocal resonator made of concave mirrors (R1>R2>0; Figure 3.39 c~ and the resonator made of concave and convex mirrors usually called telescopic [153] (R1>0, R21, Figure 3.7, a) all terms of the series (6) decrease monotonically. The nonuniformity of the type of an optical wedge is manifested comnaratively strongly (k=1; first-order waVe aberrations): for values of M=2 to 5 which are characteristic of many types of lasers (see Chapter 4), the beam at the exit from the resonator is deflected from its initial direction by an angle exceeding the angle of inclination by 3.5 to 2 times on a single pass through the introduced wedge. The effect of a nonuniformity of the lens type (k=2, secoiid-order wave aberrations) in the same variation range of Tt turns out to be greater than in the case of a single-pass amplifier by only 2-1.5 times; higher-order wave aberrations are manifested still more wea?cly. The confocal. r.esonator made of concave mirrors (Ms aa :/a 6) (15a) Figure 3.10. Field distributions of the lowest modes ia a resonator with Gaussian mirrors (M - 2, nNe uiv ' 50): a--amplitude dis tribu- - tion; b--phase distributioa gthe origia is the apherical wave of _ the geometric approximation). Then it will be obvious that these arguments formulated in [151] are basically tr.ue; however, it is still necessary to become familiar with the properties of resonators, the mirrors of which have regular geometric shape with sharply out- lined edge. 144 FOR 0MCIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 a,s I'o is Zo rid APPROVED FOR RELEASE: 2047/02/09: CIA-RDP82-00850R000504010010-4 Unstable Resonators With Sharp Edge. As we have already mentioned in Section _ 3.1, ia Siegman~sfirst paper [4] he stated the argument that in unstable reso- nators with large losses the edge diffractioa must influence only the peripheral part of the beam immediately exiting from the resonator. Hence, it follows that the field distribution on the mirrors (or with one-sided output, on the output mirror) aad the magnitude of the ?osses must not noticeably depend on tiie zdge effects; the analogous conclusion regarding the properties of unstable resona- tors with large losses can also be found in Vaynshteyn ([3], problem PIo 8 for Chapter 4). However, precise machine calculations parformed by Sigmen and Ar- ratun (157) by the iterative method demonstrated that the picture of the prop- erties of unstable resonators with totally reflecting mirrors of fiaite dimen- sions is far from simple. It was discovered that the field distribution for - oscillations with the least loases does not differ too strongly, but neverthe- less, noticeably, fram the predictions of the geometric approximation (Figure - 3.11). It turned out that the nature of this distribution and the magnitude cf the losses depead in a complpx way on the traasverse dimensions of the mirrars, revealing explicit periadic depeadeace wita variation of Nequiv for fixed M. _ The typical form of the relation calculated ia [157] for the losses as a fuac- tion of Nequiv is Presented in Figure 3.12, a(the procedure used made it pos- sible simultaneously to find the losses of the two highest Q modes). ZO N . ~ ~ a 7 L.1 rl i~ �-1 tb ~ Figure 3.11. Amplitude distribution of modes with least losses near the degen- eration point in the resonator with sharp edge (M = 1.86, Nequiv t 5) [157]. Curve I--Nequiv S S; curve II--NeQutv ~ 5� The work by Sicgman and Arrat:hoon was a major contribution to the theory of unstable resonators with a sharp edge; ia particular, the parameter Nequiv Was iatroduced here. However, the physical meaniag.of this parameter reatained un- clear; in addition, when iaterpreting the calculated dara the authors of [157] erroneously considered that the lower "wavy" line GHJ corresponds to one mode of the lowest order, and the V-type branches AGB, CHD and EJF another _ symmetric mode. In reality, as was indicated in (152) and confirmed by the re- sults of the latest machine calculations [160, 161] the apparent periodicity of the variation of the losses is caused by the fact that as Nequiv 8rows, the types of oscillations having the highest Q-factor alternately exchange places (similarly to how this occurs in Figure 2.21). This exchange takes place near 1"SQ FOR aFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00854R000540010014-4 the integral values of IVequiv for which the modes turn out to be doubly degen- - erate with respect to losses (but not with respect to frequencies). Let us note that the conf igurations of the fields of two adjacent modes near the degenera- tion point are presented in Figure 3.11. In Figure 3.12, b, c, a more complete picture of the behavior of the natural oscillations af a two-dimensional resonator [160J and a three-dimensioaal reso- nator with circular mirrors [161] is depicted. It is obvious that in the fina.l aaalysis there are a small number of modes, the 3.osses of which vary with Nequiv , quasiperiodically so that these oscillationa~alteraately become the highest Q. In the three-dimensional case these laws are also maiatained for large Nequiv at the same as in the two-dimeasional case, beginning with a defiaed value of Nequiv, the curves cease to intersect--the degeneration of the modea with re- ~ spect to the losses is removed. Without going into a detailed analysis of these phenomena, it is possible di- - rectly to draw the conclusion that the edge effects in unstable resonators are still manifested although only the ceatral and it would appear, almost undis- turbed part of the beam as a result of diffraction is incident on the exit mir- ror. In order to understand the cause of this, it is necessary to consider that as a result of diffraction, in addition to the undistorted reflected wave, an additional wave also appears, the fictitious source of which is the edge of the mirror (see Section 2.2). Although the amplitude of the additional wave de- creases sharply with removal from the directioa of the reflected wave (Figure 3.13), some part of the radiation scatters alao at large angles, including in the direction opposite to the direction of the incident wave (aoted in the fig- ure by the dotted arrows). This radiatioa givea the beginning of the converging wave, the properties of which were investigated in detail in Section 3.1. Let us remember the basic characteristic feature of the converging wave: at the same time as the intensity of the basic wave decreases on a single pass through = the resonator by M or M2 times, the radiation pertaining to the converging wave remains entirely inside the resonator for many passes. As a result, the con- verging wave, in spite of its negligible intensity near the edge of the system where it is formed, is amplified as it approaches the resonator axis to such a degree that it has a significant influence on the eatire field structure. The discussed picture, although primitive, aevertheless permits uaderstanding of - the role of the parameter Nequiv [154]. For this purpose let us return to Fig- ure 3.9. In Figure 3.9 the dotted line depicts the equiphasal surface of the diverging wave moving in the direction of the mirror for the presently iavesti- gated case of an ideal resonator (the medium is abaent or uniform). It is easy to see that this surface is equiphasal also for a converging wave moving away from the mirror. Therefore the radiation of the diverging wave incident on the edge of the mirror and then forming the converging wave passes between the equi- phasal surfaces of these waves a total distance NequivX� Thus, on variation of Nequiv by one, the phase difference between the diverging wave and the converg- ing wave occurring as a result of edge diffraction varies by 21r, which also leads to quasiperiodicity of the properties uf unstab le resonators. iSi FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00854R000540010014-4 ~ w C1 o BJ ~ ~ 6G w ~ I~'I� q~2 q2~ R>s q08 BO ~ - gC.D E f ~60 . A. ; v~� `uJ ~ ~ - - - 0~~ -~+v - ~ C20 d c~ . . . . . p . . I . . g � � ,f- - 4 6'Ne quiv a 0, ;1 8 JA S 6. 7/YeQ4jy b . ..,.M . d" 1P tY c Figure 3.12. Losses and eigeavalues as a function of Nequiv: a--losses in the two-dimensional resonator, M= 1.86 [157]; b--losses in the two- dimensioual resoaator, M a 3.3 [160] (the dotted line is used to plot the losses for the lowest resonator mode with smoothed edge); c--the eigenvalues in the three-dimensional resonator with spheri- cal mirrors, M g 5, the azimuthal mode index ia equal to zero (there is no dependence on the azimuthal angle) [161]. Let us note that it is possible always to determine Nequiv by the distance be- _ tween the equiphasal surfaces of the diverging aad converging waves near the element boundiag the beam cross section in the resonator, whereas the def inition of N quiy given in Figure 3.9 someCimes makes no aenee. This occurs primarily in ring cavities and also in systene ia which the beam croea section is limited not only by the exit m3rror, but by tfie irie placed at a noticeable dietance from - it. A more careful investigation of the properties of uastable resonators with a sharp edge caa be made by the analytical methods of Vaynshteyn similarly to how this was done ia Section 2.2 for the case of a planar resonator. We shall 152 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R400540010010-4 FOR OFFICIAL USE ONLY becoma somewhat familiar with the mathematicaZ aspect of the question in the following section where there is a discussioa of the properties of resonators with central beam hole: the solution in the geometric approximation is in general absent there, and it is not possible to get along without the Vaynshteyn apgroach. Now we only note that as a reault of considering the interference ef- fects the radiation scattered by edge diffraction inside the resoaator turns out to be distributed not with respect to a].]., but with respect to a number of dis- crete directiona. Juet as in Section 2.2, one of them corresponds to the wave "reflected" from the edge, and the rest, to the "tranaformed" waves; only inas- much as the angle of incidence of the initial wave an the edge of the mirror a (see Figure 2.7) is very large here, among the "transformed" waves there are waves which are scattered not onlp at anglea larger tfian the "reflected" one (that) is, with A> a), but also at smaller aagles (9 < a). A comparison of Figures 3.13 and 2.7 shows that in the case of unstable resonatora, when a divergent wave is incident on the edge, the wave that is "reflected" from the edge is indeed convergent. The approximate calculatioas made by Lyubimov et al. [162, 1631 demonstrated that the above-described complex laws actually are explained by interaction of the fundamental wave with the "reflected" wave and the one or two "transformed" waves closest to it. The radiation scattered in otYier directions quickly exits from the reaonator, and therefore plays no significant role. Figure 3.13. 7.'he formatiola of scattered waves during edge diffraction. Subsequent foreign publications in which the Vaynshteyn methods were still used (see, for example, [164]) do not add anything theoretically new to the results obtained in [162, 163]. It must be noted Chat in a case of unstable resonators such calculations, especially consideriag at leaat one "transformed" wave, re- quire extraordinarily awkward calculations aad in the fiaal analysis it is not possible to get around a machins solution of the systems of traasceadeatal equa- tions. Therefore their results are very difficult to see and are not clear. In addition, later it will be obvious that the problem of classificatioa of the na- tural oscillations in unstable resonators, in general, aad in those with a sharp edge, in particular, is primarily of academic infierest. In practice only the nature of the behavior of the modes with the least losses turas out to be impor- tant, and it is entirely clear from the results of references [157, 152, 160, - 1611 in which the numerical iterative method was used (see Figure 3.12). There- fore we shall consider this problem in more detail and proceed to a discussion of the properties of real systems. 153 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 Specifics of Edge Effects Under Keal Coaditions. The resonators with significant crose sectioaal dimeneions and not too small IMI - 1 are of the greateat intereat from the point of view of the problems of divergence of the radiation; obviously for them INequivi 1 is satisfied. The specific nature of the diffraction effects in such resonators primarily coasists in the fact that the fundamental diverging wave is incident on the edge of the mirror at a com= paratively large angle a defined by the geometxy of the system (see Figure 3.13). From the radiation scattered as a result of diffractioa only that part remains in the resonator which is deflected from the reflected beam at aaglea exceeding - a. We have already seen that even in this part of the radiation the primary role is played by the converging wave corresponding to the aagle of deviation 2a. It is easy to show that the values of a aad Nequiv are related by the expression a` NequivX/a. Ia real reaonators of powerful lasers I3equiv uaually varies from several tens to many thousands (see Chapter 4); her.e the aagles 2a reach several degrees. At the same time it is known that the intensitq of the diffraction scattering of light at large angles depends atrongly on the aature of th@ edge of the mirrors [1651. The above-enumerated theoretical papers in which the effects of the mode degeneration with respect to loases, and so on were obaerved easentially pertaia to the hypothetical case where the edge of the mirrors is ideally sharp aad pre- cisely outlined along a straight line or curve with center on the axis of the reaonator. Such a model is eatlrely admissible when we are talking, for exam- ple, about the lowest niodea of planar resonators which correepond to small an- gles of incideace of radiation on the edge of the mirror. For unstable resona- tors it is impossible to neglect the natural. "blurriag" of the edge and imper- fection of the mirror outlines; both of these factors lead to attenuation of the - diffraction scattering of the light at large aagles aad, consequently, to a de- _ crease in the iatensity of the convergiag wave. If this decrease is signifi- canC, the degeneration with respect to loases is removed aad the efgenvalues ap- proach the values predicted by formula (15). Let us estimate what the deviati-ons, from the ideal conditions shauld be ia order that the intensity of the con.verging wave decrease sharply.. This is done most frequently for the case where the mirror reflection coefficient decreases from one to zero aot discontinuously, but over the exteat of a zoae of finite width d (a similar situation occurs, in particular, whea using mirrors with multilayered interference coatings). Actually, significaat variation of the amplitude of the beam reflected from the mirror at the characteristic dimensioa d indicates that in the expaasion of the amplitude in a Fourier series there are componeats pres- ent with spatial frequencies of -1/d. Inasmuch as the aagular distributioa of the radiation is the Fourier type of distribution in the near zane (Section 1.1), these components correspond to the anglea of inclination of �a/d. Hence it fol- lows that in order that the light be scattered primarily at angles less than 2a = 2NequivX/a, it is necessary to satisfy the condition d> dQ a/2NeQuiv [1511. Let us note that the situation here is eatirely the same as in rad:Lo engineering where a decrease in the steepness of the pulse front is accompanied by the corresponding constriction of the signal spectrum. 154 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 A more careful calculation of the decrease in intensity of the scattered waves can be made as follows. Let us break dowa the entire "smoothing" zone in a series of sectioas Ax wide each. Replaciag the "smooth" fuaction p(x) by a _ step function, we obtain the set of "steps" of height -(dp/dx)Ax. Each "step" obviously generates edge waves with relative amplitude -(dp/dx)Ax (see Section 2.2). Adding them in the far zone considering the phase relations and replacing the summation with respect to all "steps" by integration, we find that the in- tensity of the edge waves scattered at an aagle ~ is proportional to + D(~p) oxp ~~hs) X ds ds [ 1(38J ; a- ~a let us note that B(~) is the angular spectrum of a source of width d with ampli- tude distribution -(dp/dx). If we coasider that for d-* 0 the value of B(*) 1, it becomes clear that replacement of the sharp edge by a smoothed edge causes a decrease in the amplitude of the radiatioa scattered at an angle * by 1/IB(*)l times. The estimates made using these relations in [166] demonstrated Ehat the parame- ter d0 introduced above is actually the critical width of the "smoothing" zone, on achievement of which the diffractioa "reflection" from the edge turns out to be significantly attenuated. With a further iacrease in zone width, the inten- sity of the convergiag wave contiaues to decrease rapidlq (in some cases experi- encing pulsations; the specific form of the depeadence of 1/IB(2a)l on d/dp na- turally is determined by the law of decrease of p inside the zone). In the same paper [166], Sherstobitov and Viaokurov performed the corresponding calculations for two-dimensional resonators made of cylindrical mirrors with comparatively small Nequiv. The calculations were performed by the aumerical method of Fox-L i; the "amooth3ng" of the edge was introduced by direct assign- ment of the form of p(x). It was found that smoothing does entail elimination of degeneratioa of the lowest wodes with reepect to loases. As aa illustration Eigure 3.14 ahows the losses of the two lowest symmetric modes as a fuaction of Nequiv in the resonator in which p decreasea to zero in a regioa of width -dp by a linear - law. It is obvious that the degenPration is completely removed, and the losses are close to the losses in a resonator with ideally smoothed edge (dotted lines). The field distribution of the basic mode uo in this case is excellently de- scribed by the formula (5); the distribution of the second mode, although not so _ good, still satisfactorily coincides with the reaults of the calculations by - formula (14) with substitutioa in it of the same up (see Figure 3.15). . Let us note that in general on making the traasition to the higher-order modes, tr,e effect of the converging wave regularly inereases and for elimination of it, a greater and greater degree of amooChiag of the edge is required. This is easy to understand if we begin with the field distribution in the resonator with idzally smoothed edge and introduce the partially smoothed edge of a disturbance leading to the formation of a coavergiag wave. The initial field of the high- order modes is comparatively high on the periphery of the resonator and small on its axis (see Figure 3.10). Therefore with an iacrease in ti:a transverse index, 155 = FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/42/09: CIA-RDP82-40850R000500410010-4 on the one side, the initial iatensity of the converging wave increases, and on the other, ita iaflueace on the field structure in the central part of the reso- nator increases. Hereafter we shall nat touch on the problem of higher modes, but limit ourselves to analysis of behavior of modes with the least losses. >00 F iv~~P _ ~ Ms; ~ JD w ~ ~ N oB0- f-4 t w 70 m`~ . ~ ;6 i8 ~0 ~S3 0Nequiv - Figure 3.14. Losses of two lowest symmetric m:+des as a fuaction of Nequiy in a resonator with partially amoothed edge [166]; the dotted lines are the values of the losses in the resoa-ator with completely smoothed edge; M = 3.3. -W O~ .-cci O,f D 0 ~ ,a _ �ti w T d a - ~ a) d Figure 3.15. Field distributions of the loweat aymmetric modes in a reaonator - with partially smoothed edge for M- 3.3, Nequiv = 4[166]: a-- amplitude distribution; b--phase distributioa (the origin is the _ spherical wave of the geomatric approximation). The solid lines represent the machine calculations, and the dotted lines, the cal- - culation by formulas (14), (5). Thus, for removal of the c}egeneration of the lowest modes of the two-dimensional unstable resonators with small Nequiv slight smoothiag of the edge is sufficient (let us remember that far large Nequiv the degeaeraCion in the two-dimensional resonators is absent even without any smoothing). This conclusion can be 156 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 0,5 1,0 ,V/R APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 directly generalized to the case of a three-dimeasional resonator with spherical rectangular mirrors, for in such resonators, as we have seen many times in Chap- ter 2, the variables are easily separated. In resonators with Gircular spherical mirrors the edge effects are manifested significantly more strongly: Here, in contrast to the three-dimensional resona- tors with cyliadrical mirrors, a simple increase in Nequiv in the preseace of a sharp edge does not lead to removal of the degeneration of the lowest modes (see Figure 3.12). The reasons for this coasist in the fact that the density of the converging wave increases as it approaches the center more aharply for spherical mirrors than for cylindrical mirrors (it is known that the diffraction structure at the center of the pattern is expressed more strongly with diffraction on a circular hole than on a slit of the same transverse dimension). In references [163, 3381, the Caynshteyn method was used to show that for removal of the de- generatioa in resonators with circular spherical mirrors it is necessary to de- crease the amplitude of the corrverging wave by ccmparison with the case of a sharp edge by approximately e la(2trNequiv)/ln M times. This correspoads ta a not so small width of the smoothing zoae: for the most favorable decreasing law P (P~r) - Z-1/ ~ ~�cP i 1 d~ [1GG] ~ i 0 it will be (dp/n) 1+ ln ln(2nNequiv) ln M] , in the case of a linear law and laws close to it [166] it reaches -O.Sda ln(2nNequiv)/ln M. Nevertheless, for large Nequiv this is only a small portion of the total size of the mirrors. Finally, tiie time has come to see what relation the case of the resonator with partially smooched edge discusaed by us has to the properties of the real sys- tems. The parameters of solid-state lasers deecribed in the following chapter correspond to the widths of thE smoothing zone required for removal of the de- generation of the lowest modes by losses 0.1-1 mm; this blurring of the edge oc- curs quite frequeatly. There are other factors which can lead to attenuation of the influeace of the edge effects. Thus, for the smallest misalignmeats of the resonator the values of Nequiv measured from different sides of the system axis begin to differ from each other; these differeaces also arise as a result of the iafluences of aber- - rations (see Figure 3.9). Finally, the edge of the mirrors caa be inexactly outlined. Al1 of this leads to the fact that the waves beginaing ia different sections of the mirror loop arrive at the axis with different phases and there- fore are mutually extinguishing (it is knawn that usiag the diaphragm with un- even edge, it is possible completely to b lur the diffraction pattern far from the boundary of the geometric shadow). The meaaing of the parameter dp here be- comes entirely clear: For variatioa of the distaace from the axis to the edge within the limits from a- dp/2 to a+ dp/2 the value of Nequiv varies by one, the phase difference between the diverging and converging waves, by 21r. 1.57 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 . . _ Everything that has been stated together with certain other results of the above-enumerated experiments suffices for drawing the following basic conclu- sion: The degeneration of the lowest modes -of real resonators with Nequiv 1, as a rule, is absent, aad if necessary it can be reliably prevented by usiag such simple measures as the applicatian of a bevel to the edge of the mirror (for more information about the suppressiaa of edge effects by this method see (3]), the applicatioa of a serrated iris, aad so on. Of courae, we are talking only about cases where the degeaeration is cauaed by diffraction on the edge of the mirror aad not light dissipation in the active medium or os interfaces (the effect of the light disaipation will be briefly coasidered in Section 4.1). - The removal of the degeueration with respect to losses is accompanied by the fac t that the field distr.ibution begins to be described with a high degree ofaccuracy by the formulas of the opticogeometric approximation (it is only neces- sary to consider that manifestations of diffraction remain in the peripheral part of the beam leaving the resonator with small smoothing of the edge). The _ applicability of the opticogeometric approximatioa in turn indicates that the operation of the laser will be reliably realized on one transverse mode. Actu- ally, the condition of steady-state laeing on tfie fundamental mode has - the form y a K(O)p(O)/IMIP - 1(see (11), (15a)). Hence it follows that the am- plif ication coefficient of the medium on the system axis for any intensity and distribution of the pumping in the free lasing mode is equal to IMIP/p(0), which in accordance with the same formula (15a) is appreciably lesa than the am- plif ication coefficient required for excitation of the other modes. Thus, the lasing regime caa be unimodal only in the presence of such large dis- turbances that the fo naulas (14), (15) become inapplicable even for tb-R lowest modes. A special case of such disturbaaces is, as we have seen, the presence of an ideally sharp and exactly outlined edge of the mirrors. Al1 of these problems pertaining to edge effects in wnstable resonators are of unquestioaEd cogaitive interest. However, from the gractical point of view in its elf the problem of degeneration of the modes witbL reepect to losaes is aot so im.- portant as caa be demonetrated. Fram the numerical calculatioas performed ia j157, 153, 1621 and other papers, it follows that for ama11 Nequiv tfie degenerate modes correspond ia practice to the same angular radiaticn distribution, especially in the case of noatransparent mirrors, although only the outer part of the beam will exit from the resonator (see'also Figure 3.11). With an increase in Nequiv, the fields of the highest Q oscillations, independently of the presence or ab- sence of degeneration, differ less and lesa from the spherical wave field of the = geometric approximation undergoing single diffraction on the mirrlr aperture. In addition, even for aa ideally eharp edge if NQquiv is sufficieatly large these differencea are of an almost irregular nature (Figure 3.16). Thus, degen- eration cannot be essentially felt fa the angular divergence of the radiation, and it is undesirable only when constructing siagle-frequencq lasers. If the transverse dimensions of the cross section of the active medium are so small that Nequiy cannot exceed a few units, for achievement of the single- frequency lasing it is possible to select, in accordance with Ste,gman's.�recommen- dations [167], reaonator parameters auch that Nequiv will be close to the a half-integer. In this case the difference of the losses of the two highest 1,58 " FOR OFFICIAL USE ONE.Y APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 Q modes turns out to be comparatively large even for a sharp edge (aee Figure 3.12). It is only necessary to consider that the field distribution in the resonators with small Nequiv. as arule, is highly nonuaiform, which preveats tha achievement of the maximum high output parameters of the lasers (for more information see Sections 1.4, 4.1). ~ 04 r4oz r ~ ~ ~ y i 0 0,2 Q~ 46 x/a 0 2 4 6 d X12y w q a H Figure 3.16. Amplitude and phase distribution of the basic mode of a two-dimen- sional resonator for Nequiv '30, M= 2.38 [166]. In conclusion let us aote that the above-discussed arguments were basically stated ia [151, 154], and with minimum corrections, they are confirmed bq the results of the numerical calculations in [166, 163, 338]. Thus, the pro,blem of suppression of nonuniformity of the field distribution occurring as a result of the edge diffraction, first acqnired urgency and found theoretical solution as applied to the problem of opticsl resonators. Some time later the same problem arose for the creators of powerful laser amplifiers. The fact is that whea the density of the amplified radiation approachea the self-focusing threshold occur- ring under the effect of any factors the noauniformity of the distribution can lead to the fact that at any point this threshold will be exceeded, and the ac- tive element damaged. One such factor caa be edge diffraction; for suppressioa of its effect, the same procedures in essence were used as were discussed in [151] (the so-called aaodizing of the aperture). -9 159- FOR OFFICLAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFIC[AL USE ONLY �3.4. Unstable Resonators with Central Coupling Aperture Initial Premises. Oscillations of a Zfao-Dimensional Resonator That Have a Caustic. An 'important property of unstable resanators is the fact that in them the eiaission somehow "spreads" from the central segment of the cross section (see Figure 3.6). As a result of this, the possibility appears for controlling the operation of the laser as a whole by influencing the indicated segment. In the last chapter there will be a discussion of amplifiers and lasers based on the given principle. Ti.air characteristics are determined to a great extent by how efficiently control is realized; the latter, in turn, depends on to what degree the self- excitation th reshold of the laser with an unstable resonator rises with shielding of the central segment of its cross section. , This section of the book is devoted to a theoretical analysis of the properties of unstable resonators with shielded central segment of the cross section or aper- ture in one of the mirrors which uaually is called the coupling aperture. This analysis not only leads to useful practical conclusians, but also forces us to discuss a number af important items of the di�fraction theory of unstable resanators in more detail which were discussed inadequately in the preceding section of the book. Some information which we shall use later can theoretically be extracted from the already quoted article [163]. However, in our discussian we shall fully adhere to reference [168] which was specially devoted to resonators with coupling aperture and was pub lished almoat simultaneous ly. In reference [168] the question was investigated quite completely and consistently. It is impossible in practice to find exact solutions for azi unstable resonator with coupling aperture in analytical form; therefore we shall limit ourselves to - approximate estimates. The performance of tliese estimates can to a significant - degree be based on analyzing the behavior of the eigenfunctians of the resonator without an aperture. The fact is that, as will be obvious latex, among these func- tions there are those for which the field emplitude in some region adjacsnt to the resonator axis is negligibly small. With an increase in dimensions of the indi- cated regian, the eigenvalues decrease (that is, the loases of the corresponding modes increase). Inasmuch as the presence of a disturb ance in the zone o� a negligibly small field must not have significant influence on the field distribution as a whole, it is possible to talk aUout the existence of a class of functians which almost coincide in resonators with and without an aperture. Among them, the function in which the dimensions of the small-field region approximately coincide with the dimensions of the introduced aperture has the maximum eigenvalue. It is natural to assume that the given function also corresponds to the highest-Q mode of the resontitor with aperture; then it will be obvious th at this assumption is entirely justified. Let us proceed with an analysis, which as we have already mentioned, can be made by the Vaynshteyn method. First lefi us consider }he case of a symmetric two- dimensional resonator formed by two convex cylindrical mirrors. In accordance with the Vaynshteyn method, for constructian of the solution in a resonator with finite mirrors, the eigenfunctions of a resonator made up of un- limited mirrars are used. As was demonstrated in [162], the eigenfunctions and 16Q FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R400540010010-4 FOR OFFICIAL USE ONLY eigenvalues of the integral equation of a resonator made of convex cylindrical mtrrors of infinite dimensions have the form _ Fv ~~l) = D, ( t e-tn/a~~z -~V+ 21 , Yv = M , (16) (17) where r1= 4TNx/a is a dimensionless coordinate, D. is the Weber function, and the index v can assume an arbitrary complex value. It is known that functions of the type of (16) satisfy the differential equation a2_ F� 1+1) pt _(y + 21 ij Fv (~1) = 0 (18) ~ / and in the general case they describe a set of three waves. OnE of them, converging, is propagated in the direction of the resonator axis (depending on the sign of the argument D. in (16), from the larger positive or negative r1 direction); then it partially travels through the central region, partially is reflected from it and at the same time starts two expanding waves which spread in different directions from the resonator axis. For Im v>0, the greater part of the emission passes through the central zone of the resonator; for Im v0.5, ya0 I M0 I DS>l (for calculation of Nequiv the distance from the 0 axis to the edge of the sectional plate BC must be taken as the - characteristic transverse dimension a). With this, we end the investigation of the most general problems of the theory of unstable resonators. A number of more special theoretical problems important for certain specific applications, will be considered in the following.chapter when discussing these applications. 188 FOR OFFICIAL USE ONY,Y APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R400540010010-4 CHAPTER 4. AFPLICATIONS OF UNSTABLE RESONATORS � 4.1. Unstable Resonators in Pulsed Free-running Lasers The basic result of using unstable resonators usually is expected to ba tlze achieve- ment of small angular divergence while maintaining the same energy characteristica of the laser as in the case of using planar or unetable resonators. It is perhaps simpTest of all to do this for pulsed lasers using large volumes of active medium that are not too nonuniform. Actualiy, in the case of pulsed excitation usually high amplification coefficients are achieved; as will be obvious later, this permits the application of resonators with quite large magnif ication M, which without any of the cantrivances of the type described at tlie end of the preceding chapter have low sensitivity to the effect of optical nonunifQrmities. If the operating cross section of the active zone is large, the values of Nequiv also turn out to be large; therefore the edge eff ects almost have no influence on the mode structure and have no negative effect on ttre directionalitq of the radiation. Selection of the Type and Paranetera of the Resonator. The most important factors which detemaine the angular divergeace of the radiation were investigated in rhe preceding chapter. T'herefore now we shall primarilp dietcuss the problems pertain- ing to the energy characteristics of lasers witt unstable resoaators: in the case of large volumes of the active medium, the problem of efficiency acquires pri- mary significance. As for the efficiency of the laeers during their operation on a single transverse mode, unstable resonators here have explicit advantages over resonators of other types. Actually, it is known that the maximum efficiency of a laser is achieved usually when the laser radiation distribution in general - features repeats the pumping distribution 1.4). In planar resonators and, especially in resonators with stable conf iguration the field of one mode cannot unj- formly fill a suff iciently large croas secti4n, the correlatien between the forms of distribution of the lasing and pumpiag fields is verq weak (thus, in � 2.5 it was noted that the amplitude aberrations in a planar resonator primarily cause not amplitude, but phase distortiona of the wave fronta). Accordiagly, in lasers with such resonators usually the multimode oscillation mode is realized with large angular divergence of the radiatioa (H 2.3, 2.4). Unstable resonators provide incomparabl.r more f avorable conditions for achievement of high efficieacq in the presence of single-mode oscillation. With a uniform active medium and mirrors with reflection coeff icient coastant with respect to cross section, *he f ield distiribution of the lowest mode is close to II-type. If the inverse population is noauniform, then, as was noted in [152], in accordance with formula (S) the lasf~.g field acquires a eimilar nature of distribution. 189 FOR OFFTCIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-40850R040500014010-4 iiowever, for reali4ation of the indicated prerequisi,te it is necessary tlzat a num- ber of conditions be satisfied. First of all, of course, it is neceasary that the lasing radiation fill the cross section of the active zone well over its entire length. If this is not so, the output power of the laser decreases, frequently more sharply than in accordance with the proportion of the used volume: the region in which the medium is excited, and the lasing radiatioa is absent, isa powerful _ source of luminescence and, as a result of amplif ication of it (so-called super- luminesceuce [32]) it can signif icantly decrease the inverse population in the re- maiaing volume. On the other hand, it is necessary to see that the lasing rad:ation does not hit the sidewalls of the active element or,tiYe eell in which the lasing is realiaed. In addition to decreasing the eff iciency of the system, this can lead to an increase in angular divergence of the radiation as a result of the influence o� the scattered light. = Thus, it is neceseary that the coaf iguration of the beam propagated to the exit mirror exactly coincide with the conf iguration of the activ.e element (the beam propagated in the opposite directioa has less volume for one-way output). This. imposes restric:tions on the type of resonator used.. Most frequently, the active zone haa a cylindrical ahape. In thia case the beam going to the exit mirror must be at leaat close to parallel. If M exceeds one by very little, the given condition is more or less observed even when using tfie simplest resonator made up of a planar and convex mirrors (see Figure 3.3,a), the beam cross section in which on the path from the plane mirror to the convex mirror - increases by ZM/(M + 1) times. However, with large losses to radiation, it becomes - necessary to use.the asymmetric coafocal systems depicted ia Figure 3.3,c,d. The transverse dimension of the exit mirror must be IM+: times less than the cross sec- _ tion of the active element. Here, its shape should obviously be similar to the shape of the cross sectioa of the active zone (from reading the theoretical napers it appaars that it must be circular or square; in reality, this ia not so at all). = A confocal resonator made of concave mirrors is lesa sensitive to aberrations (�3.2); in addition, for the same I Mi and distance between the mirrors it has (I MI + 1) / (IMi - 1) times larger value of INequivI than the teleacopic one. On the other hand, it also has a very large def iciency: the ceater of the spherical wave re- ~ flected from the exit mirror is inaide the resonator (Figure 3.3,c). As a result, the beam following from the exit mirror fills the resonator cross sectian worse than in the telescopic one; however, it is still more important that the beam den- _ sity at the focal point reaches an extremely large value and can easily exceed not only the rupture threshold of the solid active medium, but also the threshold , af formation of breakdown in the gas. For the indicated reasaas, in the majority of practical applications the theoretical resonator proposed in [185, 152] is used. Then the question arises of aelecting the optimal magnif ication M. From the point of view of diminishing the eff ect of large-scale optical nonunif ormities it is desirable tha4 M be as large as possibls. For this purpose it is expedient to destroy other sources of losses to radiation and use completely reflecting mirrors. _ Then the admissible value of M is determiaed by purely energy arguments. In order = to understand what these arguments are, let us consider the dependence of the output power of a laser with teleacopic resonator on M. W'ust as in g 1.4, we - lga FOR OFFICIAI. USE ONI.Y APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 FOR OFEICIAL USE ONLY shall ] ouraelves to the analysis af the simplest case where th,e pumpiag is uniformly distributed with-respect tc cEie volume. of the active medium having a cylindrical shape, the interference pfieaomena between the radiation fluxes following in "opposite directions do not apgear, and the relatioa betWeen the amplification cGeff icient and the total radiation lias the form of (1.33). Introducing the radia- tion deasity in dimensionless units p= aI, ler us rewrite (1.33) in tlie f orm . kyc = ky,I('1-F P+ -F- P'), (a) Key: a. amplif ication (1) where p+ and p are the flux densities to the exit mirror and in the opposite di- - rection, respectively (Figure 4.1). Y ff Figure 4.1. Radiation fluxes in a laser with telescopic resonator. - It is easy to aee that inside the region I which is f illed by the fluxes in both directiong, the inverse population and the flwc deasity itself vary only along the length z(0 < z< L), and these valuzs are constant with respect to cross section. Then for p+ and p in the indicated region the obvious relations are valid: _ p+ (z) = p+ (0) exp ~[k,.(a') - v;j ds' ~ ~ Z : . p- (z) = p- (0) eip - ~ [kyc (a')- a.] az' s~-s ' (z) 0 where 60 is the c:oefficient of inactive losses; the factor [,a0 /(zQ - z)]2 describes the decrease in the density of the diverging apherical wave on going away from an imaginary center z0 coinciding with the common focal point of the mirrors. Equations (1) and (2) must be solved jointly with respect to p+(z) an.d p(z) con- sidering the boundary conditions p+(0) - p(0) and p+(L) = p'(L). Finding the exact solution in analytical form is impoasible. For approximate solution it is convenient to use the fact that according to (2) p}(z)p"(z) = p(0)z0 /(z0 - z). Inasmuch as the aum p+ + p is with an accuracy to several percentages equal to 2p47r- up to values of p+ and p differiag by two or three times (larger differ- ences in practice do not exist) it is posaible by the correspoading substitution ' to convert equation (1) as follows: 191 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 cVal vrrta,ana. VJG VI~L� kyo a kio -F- 2P (U) s�I(:. - s)1'1 � (la) The substitutioa of (la) in (2) with. subsequeat integration-leads to the equation for determination of p(U): 2M P(0) 1n IZP 0) !jM~ i_ In M-}- a~L . ~ 2p (0) M + ! 1;0L ' where M- z0 / (z0 - L), as always, is thz magnif ication of the resoaator. Finding p(0), it is posaible by using (2) to calculate the diatribution p+(z) ia the region I. 'When the part of the radiation flux remote from the axis (r�. > a/M, 2a is the diarneter of the active elemeat) intersects the bouadary bstween the regions aad goea into zone II where p a 0, f urther amplif ication of the f lux is easily consi- dered bp the correspondiag formula for the aatplifying awde [186], which is a direct consequence of the radation transport equation: ~s (b) In tPP x-~k~�,~ - Qe) 1-{- k~ ln 0kI i) " iP.wc ~ Key: a. out b. emplificatioa where p is the flwc density at the eatrance to regioa II; puut is the flwc density - at the exit from the resonator as a function of the distance k traveled by the flux in this regiun (see Figure 4.1). It is obvious that the value of R,, aad with it also pout, increase with an increase in r. Figure 4.2 shows the graphs of the deasity distributions of the radiation leaving the resonator calculated in this way. As is obvious from Figure 4:2, with aa ia- crease ia M the radiatioa density at the laser exit decreases, but the width of the radiating zone (a - a/M) naturally iacreases. For some value Mopt the power of the _ outgoicag radiation will be maximal. - p; amX e;l. - (a) 4d F ~I Z wv 1,Ll a q2 qs qs qe rya Figure 4.2. Density diatribution of outgoing radiatisn with respect to cross section of tha active element for the caae k L= 3.0, QQL = 0.12 and different values of MaMp Key: a. relative uaits 192 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY The effectiveaesa of the eaergy coaversioa and the resonator % introduced in g 1.4 caa be calculated as the ratio of the number of lasing piLotons leaviag tfie resona- or f I dS (iategration is performed over the area of the exit aperture) to the (S) total number of acts of filling out the invexse population ia the volume of the _ oacillator ; PpUMP dv. I*_+asmuch as in the case of a four-level medium with uaiformly (V) broadened line and unpopulated lower level of the operating transition PpUMP = k~/a (see $ 1.4), we obtain X= ( I P+dsljr r k�y,dvl. In particular, for the case of a ~cs> J 1 cv) J circular cylindrical element with unif orm pumping X k� La' PD~'a(r) r dr. af (b ) /Key: a. out b. amplif ication Figure 4.3 $hows the efficiency of the resonator g calculated using the last formula as a function of the value of M for a number of values of Q0L and kampL (let us remember that in the case of a four-level medium the ratio k~p/cto is the amount the lasing threshold is exceeded in the abseace of losses to radiatior:, that is, for M 9 1). The data for a telescopic resonator are compared with the data pertain- ing to a planar resonator with the same total losaes and lasing tbreshold (the = reflection coef�icient of the output mirror R' is 1/M2). = Eron Figure 4.3 it follows that the eff iciency of the energy conversion in a tele- scopic resonator in the given case is somewhat lesa, but it is very close to the eff iciency in the corresponding planar resona.tor. Thus, for a laser with a tele- scopic resonator, the formulas of � 1.4 can be used under the condition of replace- ment of R' in them by 1/M2. In particular, for QQL that is aot too large, the value of X can be determined using the expression Xln M i_ ln M+ asL n 34L ( kyaL j' (3) - the optimal magnif ication of the resonator, by the-formula ln M ra) z os L/cy�,o7vp i~r (4) . Key: a. opt finally, thc maximum value of X, just as in the case of a plane resonator is approximately equal to Xma: ~ (1 - YQolkj~~~. (5) The above-discussed analysis was performed ia ref ereace [187], being the f irst example of calculation of the energy characteristics of lasers with unstable reson- - ators. Later, more complex cases were iavestigated wbich require very awkward machine calcuLations (we shall touch on the methods of performing these calculations 193 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY in g 4.2). Thus, in I1881 calculati,ong wese made of the efficiency for nonuniform distribution of the pumpiag with re.spect ta cross sectioa and large excesses over the threshold; it turned out that the formalae (3)-(5) rEmain in force under the crnndition of aubstitution of the value of kamp averaged over the cross sectioa in them. Analogou$ laws occur, as is kaown, also in the case of plaaar reson- ators. X �4 O7 ~ )=Z , Q6 ; ' ~ / nf ~ / QZD q~ q� a X kizLa~ 47 _ ' If / I~I \ + ~Z 1, O,IB q3 - Z 1 N B' ~ O,Z ' ~ , ; S QZ Q~ C,J II,Y, R Figure 4.3. Eff iciency of the resonator with telescopic (solici curves) and planai (dotted curves) resonators ~s a function of the values of 1/M and R', respectively, for k~L a 2 and 3. Key: a. amplification This similarity of behavior of the telescopic and planar resonatora of coursp is no accident. Its causes consist in the fact that the aature af filliag of the cylin- drical active element with lasing radiatioa in these two types of resonators with identical losses to radiation is aot so stroagly distiaguished as it appears at first glaace. Actually, with equality of the losses in the medium, identical meaa valuea of the amplif ication coefficient are establishEd; consequently, the mean radiation deasities approximatelp coincide. Then, along the path from the "blind" mirror to the exit mirror in a planar resonator, the radiatton deasity iacreases by 11T timea (4 1.4), in the telescopic reaonator, by M times, that is, in the same ratio. Hence it is obvious that p+ in these two resonators has sinailar values in the entire volume of the active medium. As for the radiation traveliag in the opposite directioa (p�), in the telescopic reaonator its distribution is lesa favorable: although the total radiation fluM is approximately the same as in the planar resonator it is distributed not with reapect to the entire cross section, but only with respect to.part of it; wors~ of all is the filling near the exit mirror. Aa a result, the total density p+ p is distributed over the active vol- ume in the case of a telescopic resonator aomewhat more.nanuniformly, which leads to ineignif icant decrease in the eff iciency. Thus, the telescopic resonator inaures an eff icieacq which is close to ita maximum value defined by formula (S) if we aelect the radiation losses 1- 1/M2 such as the optimal planar resonator would have. Moreover, the value of X in the vicinity of its maximum varies very slowly with M, and the variation of M within kaown limits is not related to a signif icant reduction in the eff iciency of the eystem. This can be used so that when selecting M the arguments connected with divergence ' of the radiation are considered. For reduction of the divergence, it is, as a rule, expedient to use resoaators with the largest poseible M. Here the sensiti- _ vity af the resonator to the aberratioes decreases 3.2); ia addition, the ring 194 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-40850R040500014010-4 at the exit will become less narrow. However, It is necessary to proceed to values of M signif icantljr exceeding 2 with great caution. Firat o� all, from 9 3,3 it follows that single-rmode lasing with uniform field distribution is achieced _ most reliably for lar$e Nequiv' If tfie tran.bwerse dimeasions of the active element - are given, Nequiv reaches a maximum for M- 2. A futher increase in M, in spite ~ of some increase in curvature of the mirxors causes a decrease in Nequiv as a re- sult of a fast decrease in the transverse dimensions of the exit mirror. ~ Extremely large values of M can be disadvantageous also in the presence of light dispersion, especially at angles close to 180� (usually all poasible interfaces are sources of this light dispersion). It is easy to understand the reason for this if we consider that with an increase in M the proportion of the radiation participating in the "regular" feedback channel decreases, and the light intensity ~ mixed wich it as a result of dispereion of the basic flux remaias unchanged; thus, the role of the light dispersion increases. . The proper choice of the resonator parameters even in the case of-unifoxm active medi,um still does not guarantee that a high axial luminous intenaity wi11 be obtained. - From tl-.e iaformation preseuted in the precediag chapter it is clear tha.t for this to occur, it ia necessary to exclude the formation of converging waves with notice- able initial intensity. In the case cf a telascopic resonator, the purely converg- ing wave is formed, as ia easp to see, with partial reflection of the basic wave from the plane interf aces perpendicular ta the resonator axis; therefore the inter- faces existing in the laser (for example, the end surfaces of the rod) must be in- clined noticeably. Results of Experiments with Neodymium Glass Lasex s. The above-discussed arguments about the choice of the type and th,e parameters of an uastable resoaator and also a signficant part of the concepts developed in Chapter 3 regarding the properties - of unstable reaonators were develaped during the course of experimental studies of neodymium glass resonators [5, 152, 189, 153, 190, 191, 1968, 192-197], aad they . were confirmed by the results of these studiea. In the example of lasers of the given type a most detailed comparison was made between the characteristics of the iasers with planar and unstable resonators; ia gractice all of the new version,s of the systems based on unstable resonators were tested anai studied f or the f irst time. Let us discuas the basic results of the experiments pertai,ning to the lasers with the simplest two-mirror resonators diacussed in this section. For a dia.meter of the neodymium rod of 10 mm and length of 120 mm, the application of an unstable resanator led only to a twofold gain in axial luminous fnteneity by comparison with the case of a planar resonator 1152]. Ia the greater part of the subsequent experiments, a highly eff lcient laser based on a rod 45 mm in diameter _ and 600 mm long which was described in [198] was used. It so-rved as a prototype - for the sPries manufactured GOS-1001 lasers and various versious of them. Here the axial luminous intensity on replacement of th,e pleaar reaonatar by an unstable one_.inr- creased by tens of times. The angular divergence of the radiatioa measured by the half intensity level decreased frvm 2' to 15-20"; with respect to the half eaergy level, from 5' to 40" (Figure 4.4, cume I) [152]. Let us note that this situation is quite characteristic: the larger the laser. the greater the effect from us- - ing an unstable resonatior in it. Th,e achieved gaia in divergence also iacreases ' with an increase in optical cnif ormity of the active medium; in this respect the _ investigated laser was entirely satisfactory. 195 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 rvic vrc~a,iwa, ua~, t,~t,Y ~ ~ ; N _ i (8) J a z ~ 6 e ft 1P AP,JZR.rrix (b) Figure 4.4. Angular distribution of the emissicn of a solid-state laser with telescopic resonator-1152]: I-- resonator withcut iater- _ faces perpendicular ta the axis; II a coated glass plate was in- - etalled inaide the resonator perpendicular to the axis. The propor- tion of the energy included in tlze cone with apex aagle 0 is plotted oa the y-azis. Key: a. enF.xgy, relative uaita b. angular miautes In the experiments with roda 45 mm in diameter the mirrora were totally ref lecting. The losses tio emiesion were equal to the optimal transmiseioa factor for the case of planar mirrors, and they amouated to r75% (M - 2). Th,e use of a telescopic resonator insured approximately the same radiatioa energy as in the caae of a plaaar rasonator. In the case of a planoconvex eystem of mirrors, the output energy dropped by 1.5-2 times as a result of worae fil?iag of the active elemeat with the lasing radiation. After these experiments with large losses ta emission, only telescopic reaonators begaa to be used everywhere. Subsequently, the output energy of an emissioa of a laser of the given type was brought to 4500 joules, and with series installation cf two active elemente in one = resonator with M a 5, to 8000 joulea [194]. The angular divergeace of the radiation was: with rESpect to 0.5 iatensity level, -40"; with respect to half eaergy level, - about 1'30". Let us note the fallowing 3mportant fact. In order to realize amall angular diver- geace, in the case of a telescopic resonator it turaed out to be aecessary to in- cline the ends of the active element by 2-3� with respect to the resonator axis, which made it possible to avoid the couvergiag wavea generated by Fresnel reflection. The necessity for taking such measures was proved by the following demonstration experiment: a glass plate with coated surfacee aTas inatalled strictly perpendicular to the axis in a telescopic resonator e.lemeat, the ends of which were inclined; the residual reflection 9f the coated surfacas did not exceed 0.3%. This turned out to be sufficieat that the lasing pattern changed strikiagly, and the angular divergence of the radiation iacreased so much that it approached the value characteristic of a planar resonator (curve II in Figure 4.4) [153]. The corre- sponding ghctographs are prettented in Figure 4.5, a, b [photos not reproduced]. 196 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 FOR OFFICIAL USE ONLY The mQcbanism responeible for poor directionality of radiation in auch - cases was stu.iie3 in [197, 336]. It turned out that on imtroduction of a tlzird planar mirror into the telescopic resonator, "spurious" modes appear which corre- spond to closed beam trajectories. Many passes through the active medium go with - one reflection from this mirror. Therefore the "spurious" modes even for the smallest reflection coeff iciente of the planar mirror have lower excitation thresh- olds than the fundamental mode of a two-mirror resonator. Inasmuch as these modes,in addition, are characterized by high nonuniformity of the field distributioa, some of them are excited immediately with all of the sad coasequenees following from this. Aaid this is no surprise: in 9 3.3 we encountered the situation where the presence of even a negligibly weak converging wave generated by edge diffraction leads to (iegeneration with respect to losses. Therefore the efforts sometime made to influence the lasing mode (in particular, lower its threshoid) by artif icial ~ initiation of converging waves obviously always must lead to an iacrease in diver- gence of the radiation [336]. . However, let us continue the discussion c;f the propertiea of lasers with "normal" unstable resonators not having sources of converging waves. Among the discovered peculiarities of such lasers, high atability of their output parameters is remark- able, including the form of the aagular distributioa of the radiation. Such phenomena characteristic of lasers with p3.anar resonators as variation of the an- gular distribution from pulse to pulse, a gradual increase in angular divergence 3uring aging of the active element, and so on were not observed. This property of lasers with unstable resonators is to oue degree or another inherent in all systems with spherical mirrors, and it is frequently connected with their small critical- ness with respect to the alignment grecision.As the experiments demonstrated, sma11 rotations and shif ts of the mirrors in the transverse direction cause only small changes ia the beam directioa. The magnttude of these variations corresponds completely to the predictions of the geometric approximation. The form of angular _ distribution is essentially distarted only for such large rotations of the mirrors that rhe axis of the resonator tightly approaches the f lat surface of the sample [152] (aa analogous cycle of studies for the case of a C02 laser was performed - later in the paper by Krupke and Suya [199]). Also in accordance with the geometric approxi.mation, the diaplacemeats of one of tha mirrors in the longitudinal direction cause.variation of the curvature ni ~he wave leaving the resonator. In the case of a telescopic resonator it is possiblL to use tbis means of focusiag the beam at a givea diatance d� L from the laser, increas::azg the distance tietween the mirrors by colparison Grith the distance L for confocal location of them by the amouat _(M + 1)Y. /(M - 1)d (focusing at the dis- tance d- L ta also possible, but it is accompaaied by a decrease in the output power as a reault of "tapering" of the light beam in the active element). In reference [152], a study waa also made of the spectral and time characteristics ef the emisaion of a neodymium glass laaer with unstable resonators. There were no special differences from the characteristics of 3.aaers with plaaar mirrors: the same random apikes, approximately the same integral width of the apectrum; only the duration of f�xrh spike turned out to be somewhat less, and the average time = interval between them increased. I The reduction in duration of the spikes arose from the fact that the oscillations in unstable resonators are set up somewhat faster than in planar resonators. The = 197 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 maia reason for this is the preaeace of some mechanism of forced "spreadin.g" of the emission over the cro&s sectioa. We sball discuss tliis question in more detail at the end of thie section when we are taiking abaut lasers for which the rate of establiahment of the oacillations plays the decisive roleo As for the spectral diatribution of the radiation, for planar and unstable resonators it is essentially - the distribution of the radiatioa intensity between modes witlx different anial in- - dicee (see � 2.4). Near the axis of an unstable resonator the same interference of two counterf lows and ehe f ormation of standing waves occur as in a planar resoaator. _ Therefore the mechanism of the spatial competition of the axial modes in resonators of toth type3 is identical in spite of the fact that in the unatable resonator the - peripheral part of the active elQment is f illed with radiatioa propagat-ed ia only one direction (see also the discussion of thp problem of spectral selectioti in ring cavities in � 3.5). The observations of the time expansion of the spatial distributioa of the emission [152, 153, 192]demonatrated that during lasing, iasigaif tcant shifts of the center - of symmetry of the angular disribution of the emission talae place. In addition, the aagular distribution also in individual spikes differed from the distribution _ for the ideal emitter. As a result, the iategral width of the angular distribution with respect to time for the investigated lasers noticeably exceeded the diffrac- tion limit. Obviously, this was a consequence of thermal deformations of the reaonator, the vibrationa of the samples, and so on. Takiag measures agaiast the eff ect of these f actors 1ed to a decrease in the radiation divergeace. In particular, the results of experimenta with a laser based oa an active element of great length aad with rectaagular cross sectioa are indicative [152]. During the pumping pulse the sample underwent aoticeable mechanical vibratioas along the - small dimeasion of the cross section. Accordingly, the center of gravity of the _ angular distribution completed complex oacillatory movement in the same direction; the divergence of the radiation with respect to this direction was four times greater than the diffractioa limit, aad it amounted to -2120". The replacement of the concave mirror by a dihedral prism with convea eur�ace turned into the resona- tor and an edge parallel to the large dimension of the cross section led to cam- � plete coYrespondence to Che ideas developed ia 9 3.5, almost total ^ stabilization of the direction of radiation. T,he aagular divergeace decreased in -this case to 1'. If a telescopic resoaator made of two prisms was wsed, the de- gree of stabil3zation of the direction was somewhat leas, and the divergence was -1'10". Extraordinarilg high selective propertiea of unatable reaonators with large Fresnel _ numbers were fullp manifested in tize test experiaents performed in [1901. The manif estations of optical nonuniformity of the medium and other similar cauaes were completely eliminated here. A two-dimeasional uastable resonator with M= 2 made of totally reflecting mirrors, one of which was planar and the other, convex cylin- drical (Figure 4.6), was used. The active element, ,juat as in the preceding case, was a rectangular parallelepiped, the location o� the flashlaatps insured high uniformity of distribution of pumping in the direction of the large digension of the cross section. The'curvature of tbe wave front emitting from the resonator was compensated for by an additional lens. 198 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 FOR OFFICIAL USF ONLY ~ rt) 6 Figure 4.6. Diagram of a laeer with rectaagular active rod and two- dimensional unstable resonator: a) sqaometric, b) asymmetric exit of the radiation (the crosshatched rectangles on the right depict the exit beam cross section). Two methods of exiting thi:: radiation from the laser were tested which are illustra- ~ ted in Figure 4.6, a, b; Nequiv Was ^'1700 and ..7000, respectively. In the former ~case the integral angular distribution of the radiatioa with respect to pulse time ~ along the direction in the plane of the �igure.coiacided almost with the Fraunhofer diffraction pattern on two rectangular openings (P'igure 4.5, c jphoto not reproduced]; ~ the long extent of the pattern with respect to the eecond direction is connected with the fact that ia this direction the.Yesonator was equivalent to a planar resonator; ia addition, the compensatiag lens was not cylindrical as should be, - but spherical). T`ue baad contrast is w*y close to one, which iadicates higr. ' spatial coherence of the radiation (let us note tbat the distance hetween two beams leaving the resonator was --120 mm). ' Noticeable deviationa from the reaults of the diffraction aC the eait aperture of ~ the laser and when using the system depicted in Figure 4,69 b with exit of th,e radiation in the form of a single beam ia one direct{on from the axis of the resfl- nator were not detected. As a result Qf iacreasing Cfle aperture width, in this , case the divergence was less than in the first case, and it was 2" or 1�10"5 rad _ (Figure 4.5, d jphoto not reproduced]). : Af ter performing the$e experiments the thesis that unstable resonators with large = Nequiv with uniform medium provide single-mode lasiag with diffraction angle o� ' divergence of the radiation could be considered proved. It was only Ie`t to estab- lish whether departure of the divergance of the radiation of real las-re t:om the diffraction limit is a consequence of such prosaic cauaes as imperfectiou of the elements of the resonator, optical nonuuigormitp of the mediua4, or the "hiAh-order modes" sometimes mentioned ia the Iiterature are at fault here. According to � 3.2, ~ the wave front o� a light beam which is the fundamental mode of an unstable rpsonator is formed in it just as oa transmisaicn of the beam through a multistage amplifier - (Figure 3.6). Tberef ore the best answer to the stated problem can be givea by direct experimental comparison of the divergence of the rgdiation ui a real l.aser i with an unstable resonator with divergence at th6 exit of the mul�tistage amplifier ~ 199 ' FOR OFFICYAL USE ONLY ( APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 0 (see � 2.6) constructed from analogous active elemeats. This comparison was uruier- taken in [195]. The basic results of [195] are presented in Figure 4.7. Curve 1 corresp:nds to the angu],ar distribution r-t the iaput of the final stage of amplification, curve 2, at the output of the mu.'.tistage system. The data far the laser with telescopic resonatcr are plotted uaing x's; it is obvious that the multistage system in its "pure" form insures somewhat leas divergence of the emissior.. However, if we cover thE central part of the exit aperture of the multistage system so that the bea.m acquires the same circular cross section ss in the case�of a':elescopic resanator the divergence of the radiatioa of the devices of both type� cumes close to coinciding (see curve 3). I F r tw' Q ~ c~ (a) r (b) Figure 4.7. Angular distribution of the radiation eaergq (the propor- tion of the energy in a cone with apex aagle 0 is plotted on the y- axis): 1- at the entraace nf the f iaal stage of a multistage system; 2-- at the exit of a multiatage system witb. cixcular aperture; 3- annular exit of the radiatioa; s-- telescopic resoaator, multi- stage system with annular diaphragm. Key: ,s. energy, relative units ~ b, aagular minutes Attention is attracted by the fact that the dif f erences between curves 2 gnei 3 cannot be explained by purely diffractioa ph,enomeas: the halfwidth of the angular energy distribution as a result of diffractioa of the plane front in a circular opening was 5'.' more under the cenditions of [195],, and in the annular openiag, a total of 4" more. Obv iously these diff erences were caused primarily by the fact that the optical uniformity of the ceatral part of the pumped necdymium glass rod is higher thaa its peripherq (information about the nonuniformity induced in - such cases is presented in [200]); therefore the addition of the central part to the annular exit aperture also cauaes a noticeable decrease ia the divergence. From Figurs 4.7 it is obvious that preliminasy stages and the final amplifier in the given case make similar contributions ta the total radiation divergeace with re- , spect to magnitude. From the materials of � 3.2 it followa that for a telescopic resonator used in [195] with M- 2 the distortioae of the wave front coming as a- result of large-scale nonuniformities before the laat pass through the active rod and directly in this last pass are also similar with respect to magrritude. The analogy between the multistage syatem and the telescopic resonator is quite ebvious here. 200 _ FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 FOR OFFICIAL USE ONI,Y Thus, carefully made unstable resonators are actually capable of insuring the _ same radiation divergence a.s multistage circuits (considering obvious correctioas connected with the differen.ces ia the exit apertures). As will be obvious in 94,3, by using these resonators amplifiers can be constructed which provide enormqus ampli- fication of the westk signal in one etandard active elaaent (considerable work is required, in particular, ln devices consisting of many channels syachroaized by a single master oscillator). At the same time, the experieace of [195] demoastrated that tlie probZems pertaining to energy eff iciency are solved on the basis of un- atable resonators much more simply chan when usiag multistage systems (as a result of a signif icantly smaller number of elements in the optical system af the 3evice). Therefore the sphere of application of the awkraard multistage systems primarily re- mains the rare cases where for the sake of achieving record radiation parameters any means are considered j ustifiable. Tha entire research cycle discussed above which was performed in the example of pulsed neodymium glass lasers with free lasing permitted suff iciently complete _ discovery of the possibilities connected with the application of unstable resonators in their simplest version. This cycle was preceeded by a total of two articles with reports on experiments with similar resonators. The f irst of them is the initial paper by Siegman[4] which we have meationed many times aad in which, in addition :to the formulation of a number of the most important thsoxetical argument.a (9 3.1), the preliminary results of studying a ruby laser with unstable resonator aYe also discussed; the second is the paper [201] on experimeats with a pulsed argon laser having a discharge tube 7 mm in diameter. Ia both cases it was impossible to ob- - tain any positive effect from transition to the unstable resonator. ~ Gas Pulsed Lasers'with Unstable Resonatnrs. Problean of Steadyins Oscillations. Subseqaent foreign publications about the application of unstable resonators in pulsed lasers began to appear c+nly ia 1972 [202, 203], aad they pertaiued to the case of C02 lasers. No aew ir.formation about the properties of unstable resonators was contaiaed in theae papers, and nothiag special was preseated on a purely tech- nical level. Actually, among several of the f irst experiments the highest output parametera were achie ed in [203]: for a pulse energy o� 3.5 oules, th radiation divergence was -2�10 radians, and the brightness was ..2�10 l~watts/(cm~-steradian). For comparison we mention that in Soviet laaers with telescopic resonator based on neQdymium glass long before this the smaller angular divergence was "mastered" with an output eaergy of n103 jaules, and in 1971-1972 there were alreadp lasers with ' brightness of ~1017 watts/cm2-steradian) [204, 2051. Beginning with that time, unstable resonators begaa to be uaed also with invariant succESS in lasers of almost all types. It is sufficient to mention, for example, the creation of an electroionization laser with a pu.lse energy of 7500 joules [206]. However, the results of experiments with metal vapor lasers are of the greatest cogni- tive interest [207]: the specific peculiarities of theae lasers forced a new look at some theoreLically known properties of uastable reaonators. The fact is that the amplification coeff icieat of the medium is extraordinarily high here; on the - other hand, the population inversion here exists for such a short time that during this time the light travels through the resonator onJ.y a f ew times. Since low diver- gence is desirable at all costs, {t is of primar3r importance for the ;,ntical. cavity to be capeble o!' rapidlq isolating the fundamental mode from noise radiation. 201 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 The problem of establishiag oscillationa in optical resonators was first encoua- _ tered in 1961 bp Fox and Li [9]: tIie iterative calculatidns that they performed - to asfgnificant degree simulate the processes oceurring in real lasers; the iaitial field distribution written inta thp iterative prncedure plays the role of the "nucleating center," the source of which in real ]asers is apontaneous emission. Thea Fox and Li noted the obvious relation between the rate of establishment of steady oscillatioas and the differences in the losses of the iadividual naodes: actually, during the time of passage o� the light through an empty resoaator the intensity ratio of any two%modes varies bq (1 )/(1 - A.)) times, where A1 and A 2 are their diffractioa losses. Obviously the inteasity ratio will vary in the same way also in the presence oi a medium with uniformly distributed amplificatian coef- ficient kamp with respect to volume (the eigenvalues of the integral equation of the resonator turn out to be multiplied by the same coeff icient exp(kamp SC) in this case, ($ee � 1.3). If the medium was uniformly excited at the time af beginnin8 of deve- lopment of lasing from the noiae "nucleating ceater," the situation will be main- tained until the lasing power builds up to such a degree that saturation level is reached. Hence it follows that the more the losses of the individual modes differ, the faster - the generation of cartain modes againet the backgrouad of the others will occur; in particular, the osciliations in uastsblp resanators must be set up much faster than in p1ane, and eapecially stable cavities. For Chese reasons the rate of establisizment of the osci].lations in the moat unstable resonators increases with an - increase in iMi. ' Althuugh all of thess facts were weYl-known, the problem of calculating the time - required for the diffraction :irected beam to form in an unstable resoaator from spontaneous emission was clearly stated only in [207], and it was gradually solved in [207-209]. It is true that the authors of the indicated papers for some reason considering the given situation exceptional tried to get around uaiag the kaown results and methods of the theory ot optical resonators, even such generally used . ones as the~iatroduction of the equivalent resonator. Accordingly, for this quite simple problem they obtained a very complex solutioa, making, in the course of the matter, aa entire seriea of erroneous etatementa (for more details see [210]). In , particular, their advice to select the resonator parameters so that the converging beam will expand to the former crosa sectional dimensions invariably in an integral ' number o� pasaes is meaningless. Theretore it is better to discuss a signif icantly simpler solution to the same problem preaented in [210]. Let the active medium have Che crosa sectioa 2a X 2a and be placed inside a -ale- - scopic resa.ator (Figure 4.8). Let us follow, f or pxample, the fate of the nucle- ated radiation which at the initial point in time is emitted near the convex mirror ~ in the direction of a concave mirror, We shall take the spherical equiphasal sur- face of the diverging wave located on the convex mirrox, the center of curvature of which is at the common focal point of the mirrora as the reference aurface. The complex amplitude of the nucleation fisld of one of the polaxizations on refer- ence surface can be represented in the form of the series . , u(Z, !1) unl .~xp ~knt a etp Inl a l, k, t= 0, f 1, f 2,.... 242 FOR OFF[CIAi USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFF3CUL USE ONLY i o! i t-- : _ - - bj . . ~'--`-t~~~---.__ lE- l_- I . CJ Figure 4.8. Traasmission of spherical waves t'hrough a telescopic . resonato:: a) one of the "diverging" waves; b), c) traasmissioa ofdiverging waves with centers of curvature near the zone 2a wide 00 and outaide this zoae (c) over an equivalent line. Thia series corresponds to the set of apherical waves wi�th randomly distributed . amplitudes ukl, the mean value of which can be calculated beginaing with the power of the spontaneous radiation for a solid angle (1/2a)2. The cmters of curvature of these waves lie in the focal plane aad are shifted from the co~on focal point of the mirrora ia two directions a diatance bXf2/2a, iaf2/2a, where f2 is the focal ~ length of the Goavex mirror, k aad Z are the wave indices. The centers of curva- ture of the waves, the radiation of vhich completelq covers the concave mirrar, " fills a zonr: 2a x 2a in ize (see the figure); the numbeL of such wavea, conse- quently, is equal to (4a~/Xf 2) 2. Ae the f igure shoon, some of the radiation of waves Iying near this zone also reaches the ranse of the laser; Iwwever, it very ~ quickly leaves the confinee of the optical cavity entirely. Theretore, eacept ; for the very earliest ata~es, the number of wavea taking part in the process of ' lasing onset is (4a2/at2) , i.e. the epontaneoue emisaion falling ia a aolid an,gle of (2a/f 2) 2. . ' Vow let us trace the behavior of the waves; this is entirelp possible within the framework of the geometric approximation uatil the divergeace tightlp approaches the diffraction limit. Obviously af ter the first reflectioa fram the concave mir- ror part of the cross section of each wmve equal to 1/M2 remaias iaside the reson- - axor; the waves themselvea go from aph6Li=al to planar with propagation directidns -inclined with reapect Co the axis at aagles of (k/M)(A/2a); (I/M)(X/2g). The width of the entire range of angles, that is; the total divergence of the emission is - 4tt� x 7s- - 2a/Af/.. Af ter each subsequent pa$sage through the resonator, the amouat ~ of radiation remaining in it decreases by M2 times. The slopes of all the beams, and with them, ,a].so the total angle of divergence, decrease by M times. Thus, af ter n passes through the reaonator, the proportion of te radiatiion of all of the waves entering inte the "nucleating center" equal to 1/Mln remains ia it, 203 FOR OFFIC7AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 _ and the total 19geometric" divergeace is 2ajef2. The aumber of paeses nQ during which the "geometric" divergence decreases to X/2a and thus the formation of the diffraction diverging beam the ba~ic mode) is completed, is define~d by the expres- = sion 2a/ (Mn0f 2) = 7~/ 2a or ~0 = 4a /(Xf 2) , By tliis time the proportion of pr3mary - "nucleating" radiation equal to 1/M2no s(Xf /4a?)2 remains inf,ide the resonator. It is easy to see trat this proportion corresponds to the radiation intensity which initially pertained on the average to one wave. In spite of the primativeaess, the given analysis leads to the same quan*itative = results which were obtained in [207-209] using somewbat more complex manipulations with the "compressing wa.ves." Using these results, it is entirely possible to un- dertake specif ic calculations of the kinetics of auch lasers. Fiowever, the most important conclusion can also be drawn without performiag such calculations. From _ the above-presented relations it followa that the time of formatioa of a diffrac-. tion-directional beam with fixed dimensions of the resoaator slowly and monotonically with an iacrease in t4 (inasmuch ae nD` ln [4(M - 1)a2/aL]/ln M). Thus, the thesis that to obtain the smallest possible divergeace it is necessary, considering the atipulations made on page 53 to use-unstable resonators with the largest possible magnif icationa, has received another, quite weighty substantiatioa. This entire concept was checked experimentally in [207]. When using a telescopic resonator with M- b in a capper vapor pulse laser, the calculated time required for -fundamental mode laeing turned out to be greater than the time during which ' the radiation density inside the given laser could grow from the noise density _ to Ciie saturation level. As a result, the integral divergeace with respect to the pulae duratioa exceeded the diffraction almost an order. For M= 200 (1) this limit was reached, it is true, at the price of a sharp drop in radiation power as a r.esult of the extraordiaary rise ia the lasing threshold. Probably for inter- mediate M it would be posaible to achieve both small divergeace and suff icient = radiation energyl). � 4.2. Unstable Resonators inContinuous Lasers Survey of Experimental Work. . So lewhat later than in pulsed neodymium glass lasers, unstable resonators begaa also to be used in continuous gas lasers. Among the first publications only one short, but exceptionally interesting article [211] _ stands out which to a great extent antictpated the future development of the reson- ators of gas dynamic lasers; we shall retura to this paper later. As�for other . studies in 1969-1973, their subjecta were low-presaure electric discharge C02- lasers [199, 212, 213] and chemical lasere 1214, 215]. The early paper by Krupk e and Sooy [199] is isolated here. In this paper a telescopic resonator was - used in practice for the first time; as a reault of the carefulness of the experi- ment and the high optical uniformity of the actiye medium the authors were able to observe the diffraction structure of the distribut on in the far zoae with full width oF the central peak on the order of 1' (3.5�10 radians)2. -This was done in the recent experiment [332]. zIn this paper an analysis was made for the tirst time of the consequences of mis- al.ignment and variation of the curvature of the mirrors in two types of confocal unstable resonators. 204 FOR aFFICIAt USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY - It must be noted that in all of the enumesated studies except 12111 not only thE goals, but also the coaditions differed sharply from tie conditions of the experi- ments wita neodymiuffi lasers, primarily by the fact that in view of the modest di- mensions of the resonator cross sections aad the long wavalength, the Fresnel numbers - N were quite small. The theoretical limit of the angular divergence of these lasers, on the other hand. was not snaall. Thus, in the same paper by Krupke, and Sooy, the width of the radiation ring at the exit from the laser was -1 nm; the angle into which half the energy went, although it was not measured, must have been equal to 5'10-3 radiaas under these conditions. This divergence could be completely achieved also using a planar resonator. Th,exefore the basic positive result of the men= - tioned experiments is not the achievements in thP: f ield of angular selection, but experimental testing of a number of conclusions of the theory of unstable resonators pertaining to the magnitude of the losses, the nature of their dependence on N uiv' and so on. A detailed summa.ry ot the results of testing the theory is availabg~ in the highly substantial survey by Siegman [216]. It is of interest that although the majority of researchers, in accordance with the recommeadations of Siegman [167], _ have carefully selected the resonator parameters so tha.t Nequiv will be close to a - half integer, nc article contains data indicating that this choice actually is useful from the point of view of angular divergeace of the emission. 1 t J r' Figure 4.9. Schematic representation of the active zone and resonator ~ of a powerful gas dynamic laser [2111: 1- planar mirrors, 2- con- _ vex exit mirror, 3-- external concave mirror, 4- exit beam. Key: a. flow Now let us more carefully consider the results of [211]. This article is devoted to experiments with a powerful gas dynamic laser; tlie schematic representation of the active zone with the resonator borrowed from [211] is presented in Figure 4.9. As is obvious from the figure, the resonator is installed so that the lasing radia- tion will pass through the gas flow perpendicularly to the direction of motion of the flow itself. This arrangement of the cavity, which is called transverae, is adopted because the high average powers require large flow rates of the gas mix, and organization of uniform gae flows with large flow rates is possible only in the casE here the resonator elements are beyond the limits of the flow cross sec- tion (in eiectric diacharge lasers the same thing occurs alsa with the electrode assemblies). 205 FOR OFFICIAL U5E ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 Teata wexe run on a gas dynami,c laser in 1211] wi,th resonatora of tLro types. The = f irst type waa a stahle resonator made of t4ro large metal mirrora, in oae of which a system of many hales 1.5 mm were made, th;e total area of whic1L was on the order of 6% of the total area of the mirror. procedure does not witb.stand any criti- cism from the point of view of divergence of the radiation 2.5), but it permits sealization of a resonator with given small losses to emission, aad, as the _ experience of [211] shows, it can be used for experimental optimization of the enersq characteristics of the laser. With this resonator the output power was 55 kilowatts. The second type resonator is depicted in Figure 4.9. It was formed by four mirrors and was already unstable. One of the mirrors was convex, it served as the output mirror, and accordingly was smaller in diameter than the others. The beam passing near it fell on the additional concave mir.ror and was focused on the hole in the wall of the cavity through which it was taken outside. The use of a sqstem with three passes through the active medium permitted, in spite of small amplif ication per pass, operation of the laser with magnif ication of the resonator M- 1.6; the output power was 30 kilowatts. An almost twof old reduction in lasing power by comparison with the case of a stable resonator indicates that the achievemeat of the maximum eff iciency of eaergy con- veraion in the gas dynamic laser resonator is not such a trivial problem as for pulsed lasers. It is necessary to consider that such a resouator consists, as a rule, of several carefully aligned (or even equipped with an automatic alignment system, see [217]) cooled mirrors and is, together with the f astenings, the align- ing slides, and so on, a quite complicated device. It is part of a still more complicated engineering structure which is the fast-flow laser as a whole. Inasmuch as the purely empirical selection of the optimal resonator under such conditions becomes too thaakless, the theoretical methoda of analysis have been widely de- veloped, the iavestigation of which we shall proceed with. . Methods of Calculating the Efficiency of Flow Lasers. It is necessary to distin-- guish two cases immediately. Whereas the fast-flow laser operates in the frequency mode where ttie excitation of the medium is realized bp indiv idual, periodically re- peating pulses, the procedure for calculating its eaergy characteristics does not differ ia aay way from the procedure for calculating the eff iciency of the re9 onator of Vinary pulsed lasers. Actually, one pumping pulse lasts, as a rule, 10- to 10 seconds; the medium during this time can travel such a short distaace that it can be considered stationary during the pulse (the flow velocity in the frequency lasers usually does not exceed 100 m/sec. In pulsed lasers, as we kaow from g1.4, the state of the active medium in any eleffientary volume of it under the given pumping conditions is uniquelq defined by the density of the lasing radiation pass- ing through this volume. This makes it poasible to talk not only about the eff iciency of the.laser ae a whole, but also about the efficiency of eaergy con- version in any part of its volume, which sigaificaatly facilitates the understand- ing of the basic laws 1.4). The local nature of the deppadence of the amplifi- cation coefficient on the radiation.deasity leads to signif icant simplif ication also of the quantitative calculations of lasers with unstable resonators. First, the equations describing the atate of the medium themselves are s3mple in this case. Secondly, it is very easy by using the threshold condition to f ind the radiation density on the axis of the resoriator before f inding Che distribution in the entire volume (see � 4.1), which greatly 3ccelerates the convergence of the ordinarily used iterative procedure. 20.6 - FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY In continuous- Wave f low laaexa tha si,tuation i,a quite dif�erexLt, The concept of local efficieacy here cannbt exigt in general: tiLe previously excited active me- dium floers entire beam of generated radiation, and the numbex af atoms reacting with the beam can be calculated only beginning with kaowledge of the radi- ation diatribution as a wh,ole. It is also more difficult to calculate the radiation density in advance on the system axis inaemuch as the iaverse populatioa on the axis depends not on this density, but oa the entire history of the medium reaching here, in particular, the f ield density on its entire path. SiLiilar laws occur also in the case of side optical pumping of iniuced Raman scat- tering lasers frequently called Raman [combination] lasers or VKR�-converters[218]. On the other hand, the amplification coefficieat of the lasing radiatiion cn the combination frequency is proportional to the pumping radiation deasity on the in- itial frequency. On the other hand, the attenuation of the pumping radiation on passage through the medium is almost wholly determined by its interaction with the generated radiation - without lasing, the primary pumping radiation is poorly absorbed by the medium. It is easy to see that the pumping Iight density in Raman lasers is, from the point of view of resonator theory, the complete analog of the inverse population density in flaw lasers; the residual absorption ia the Raman laser medium corresponds to a decreaee in the inverse popula'tion down stream with respect to the gas as a result of spontaneous relaxation of the active medium of the flow laser. In 1968 Alekseyev and Sobel'man [219] pointed out that the application of a planar resonator in a Raman laser with side pwaping is fraught with highly unpleasant consequences. Inasmuch as the amplif ication coeff icient near the edge of the resonator from the side of which the pumping is realized, usvally noticeably pre- dominates over the losses on the combination frequency in the mirrors and the me- dium, the lasing radiation density here turns out to be extraordinarily high (if. ' its growth does not prevent the beginning of lasing on subsequent combination fre- quencies, for which the radiation on the first cambination frequeacy is itself pumping). At the same time, with aa increase ia the density of the converted (Raman) radiation, the attenuation coeff icient of the primary radiatioa increases - the region of powerful lasing screeas the remainiag volume fzom the pumping ra- diation. Thus a tendency ehows up for the lasing region to contract into a very narrow zone, and tbis ia difPicult to eliminate. Analogous phenomena must occur also in f low lasers although uaually not in such sharply expressed form (we shall discuss the causes of this somewhat later). - In the case of an unstable resonator, independent developmeat of lasing on the peri- phery of the converter or the flow laser is impossible, for the lasing radiation must come from the central section of the crosa section. Extraordiaarp growth of the lasing radiation density on the a.N:ts of the system is impossible, for it causes rapid growth of the density also on the periphery, which.leads to a decrease in the amplification on the axis. As a result, the regime turns out to be self- balanced; the lasing radiation fills the resonator cross sectioa, its density is established on a level such that the number of pumping quanta reaching the axis in the Raman laser or excited atoms in the flow laser iaeures exact satiafaction of steady state conditions. ihe first letters of the Russian words that meaa induced Raman scattering]. 207 EOIt OFFICUL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 - These arguments are formulated in 11541, and tfiey, wpse confirmed by tbe experimental data of the above-quoted re.ference 1211] (th.e aoauniformity of the intensity distri- bution observed there with respect to cross sectioa of the unstable resonator did not exceed +SOx). In all of the subsequent tiieoretical papers devoted to estimation of the efficiency of the continuous lsaers, a study is made of the self-balanced lasing regisne on the lowest transverse mode. IC is true that in the experiment bf [220] with def ined resonance, a mathematical model of a flow laser was proposed from which it follows that the lasing regime must be not steady, but autooscillatory with an oscillation period on the order of the drif t time of the gas flow througti the resoaator zone. Zf this model were correct, the autoosci,llations would be chaxacteristic of even a broader class of lasers than = the authors of [220] themselves pZoposedl. Sowever, the medium was considered _ single-component in [220], the speed of light was conaidered inf inite, and the geometric approximation, valid, even under the conditions where the properties of the medium vary sharply over the extent of small sections of the resonator cross section near its axis. Judging by the results of [221], it is suff icient to do away with the proposition of single-component mixture so that the trend toward the autc;- oscillations will decrease sharply (and in real lasers, as a rule, mixtures are used which contain more than oae component). Consideration of the fact that for the light to pass through the resonator takes a f iaite time is extremely fmportant and must also lead to dampiag of the oscillatioa$ (from the papers at the beginning of the 1960's devoted to the kinetics of eolid-state lasers it is known that neglect- ing this fact usually leads to absurd results). Thus, the deep autooscillation mode is hardly widespread. As for the ordinary aad unavoidable oscillations of - intensity caused by fluctuations of the resonator Q-f actor, and so on, theq can hardly lead to signif icant departure of the- energy parameters of the laser averaged over a large time interval from the calculation results in the quasistationary approximation (see � 1.4). ' In itself the calculation of the efficiency of a laser operating in the gteady state mode in the general case reduces to finding a self-consisteat combination of distributions of the amplification coefficient and the lasing field. The equation:. describing the dependence of the amplification coeff icient distribution oa the ex- citation conditions and the lasing f ield depend on the peculiarities of the medium and are quite diff erent. As for the lasing field distribution, for it it turns out most frequently to be suff icient to use the geometric approximation. Actually, we have already diacuased the causes by which it is poasible to neglect the effect of the edge diffraction in lasers with large Mequiv 0 3.3). The consideration of large-scale nonuniformities of the active medium does not require diffractioa _ approximation (g 3.2). Moreover, if the medium is not too nonuniform, it is possible also to take the path of the beams the same as would occur in an ideal resonator. Hence, it follows that weak optical nonuniformity of the active medium, just as the edge diffraction, can ia getteral not be considered in the energy calculation. On the other hand, inasmuch as the angular divergence of the radiation depends primar- ily on the phase distribution at the laser output. and the nonuniformity of lAt the end of the indicated article, aa additiaaal factor is erroaeously introduced to describe th,e phenomeaa of amplif icatioa saturdtion, at the same time as canside- ration of these phenomen.a is already built into tb.e initial equations. 208 FOIt OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 the inteneity diatributioa influences it weakly, the amplification saturation - phenomenon must be felt little in its magaitude; this conclusion follows from th.e - materials of "s 1.1 and is conf irmed by the results of sometimes uadertaken specif ic calcula+tions (for example, j222]). Therefore the width af the radiatioa pattern can be estimated in the first approximation without calculation of the energy characteristics of the laser. ' In order to find the self-consistent solutioa most frequen.tly an iterative method is used. The simplest iterative p.:ocedure used in the 1960's to study the pheno- - mena of amplif ication saturation in planar and stable resonators j100, 90, 1091 conststs of the following. An initial field distribution (usually uniform) is taken; this distribution is.substituted in the equations of the medium; the latter are solved, and the spatial distribution of the amplif ication coeff icient is found. Then the new f ield distribution is calculated as a result of single pa.asage of the initial beam through the r.esonator with the active medium. The ne;;?y obtained f ield distribution is substituted in the equations of the medium, and so on. As applied to the calculations of lasers with unstable resonators in a weakly nonuniform medium performed in the geometric approximation, the given procedure can be written as follows: F.+i(r)�/[iL�(r)], *ta.+i(r)=F�(r)u�(r/M), where F is the aberration function - calculated considering the distribution of the amplification coefficient (see g - 3.2), and the index denotes the number of the itezation. The calculations are quite tedious, for although only the f ield distribution at the exit mirror u(r) figures f.n the above formulas, for the calculatiou of F(r) it is necessary to deal with the amplification coeff icient distributions (and, c.nsequently, the f ield dis- tribution) in the entire volume of the laser. As was noted above, the convergence of the iterative procedure can be accelerated signif icantly by preliminary calculation.of the radiation density and inverse population on the resonator axis. In the case of lasers considered in the preced- ing section, it was suff icient for this purpose to consider the conditions on the axis itself. Later it will be obvious that this problem is also solved for flow lasers, it is true, using more complex calculationa. The data obtained on the magnitude of the f ield under the state of the medium cn the resonator axis are used in the subsequent calculatioiis; here the condition of stationarity of the regime turns out to be automatically satisfied in all phases of the iterative procedure. Finding the amplif ication 4oeff icient distribution next, this makes it possible for us to not limit ourselves to single "transmission" of the beam through the reson- ator with the medium and to f ind the steady state f ield distribution corresponding to the given picture of the state of the medium. The iterative procedure acquires the form p.+i(r) /[u.(r)j, ~+i~r) = u(0) F. (r)!'. (r/M)F�(r/A!2)..., where u(0) is thE previously found field amplitude an the axis. As a result, the volume of the cal- culations is reduced noticeably, eapecially for complex mathematical models of tne medium. The calculations of pulsed lasers Grith noauaif orm optical pumping mentioned ~ in � 4.1 (18$] anci the calculations of gasdynamic lasers discussed below were performed by approximately the same method. Other procedures are also used which permit reduction of the volume of the calcu- lations both when fi-nding the field distribution by the given parameters of the resonator and state of the medium and for calculatioas of lasers as a whole. Although the authors o� of these procedures pr"ant very convinciag arguments in its favor, it is now diff icult to determine which of them is actually more 209 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 effective. Therefore thcse wEio deai,re to Pamil:Urize tflemselvea ia more detail with the methnd of the calculaticns are referred to numeraus coXresponding arttclas [222-227, 188, 156], and so on); at thE same time we shall limit ourselves to the fact that wa shall Qxplain snme peculiarities of the calculations and the behavior of flow lasers in th.e example of gas dynamic lasers with two-mirror telescopic resonator imvestigated ia [188, 228]. All of the basic laws her.e are the same as in the casQ of resonators of the type depicted ia Figure 4.9 (if, of course, we compare lasers with identical amplif icati.on not oa the beam width, but oa the entire path from vae terminal mirror of the resonator to anothex). Lasers with resonators, the axis of which is "broken" not across, but the direction of the gas flow behave diff ereatly; we shall not coneider them. Simplest Model of a Gas Dynamic Laser Medium. Methods and Results of Eaergy Cal- culations of Gas Dynamic Lasers with Tw o-Mirror Resonator. In the overwhelming ma,joxity of gas dynamic lasers the inverae population is created, by the proposal of Konyukhov and Prokhorov [229], by adiabatic ex,pansion of a gas mixture consist- ing primarily of C02 and NZ. These two components also play the primary role in the lasing process. C02 molecules are characterized by a comparatively short _ oscillatory relaxation time; the laser tranaition is also realized in them. Fast loss of the oscillatory energy reaerves in this component is frequently compeasated _ for by resonance transmission'of excitatioa from the N2 molecules on collision with them. The relative molar conceatration 1- c of the aecond component of the mixture is larger than the f.irst (c) and the major,ity of the total oscillatory energy reserves is concentrated in the second componeat, N29 which is, therefare, a type of energy "reservoir." As a result of the long natural oscillatory relaxation time of the molecules of the second component, the energy from this reservoir is consumed pri- marily for excitation of molecules of the first component. _ Although the atoms located on a quite large number of oscillatory levels of both components participate in the operation of the laser in one way or aaother, Konyu- khov [230] proposed limiting ourselves to a system of a total of two equations for deacription of the relaxation proceases of the medium occurriag in the resonator zone. These equatioas, being reduced to the maximum canvenient f orm for calcula- tions of the resonators, have the form [188] dkl (s, t) eJes - e) k1 kt ds a ~N'-""'- _ -r - (P+ P ) kir dkl e) cks -(1 - Q) k1 _ k' _'3x- � - he . Here kl is the amplif ication coeff ieient of the laser radiation; it is proportional to the oscillatory energy reserves in the firat component (the population of the lower laser level is considered equal to zero, whicfL usually does not lead to large errors); k2 is the value having dimensionality (but not meaning) of the amplif ica- tion coeff 3cient and characterizing the oacillatary energy reserves in N2; p+ and p, just as in the preceding aection, are the densities of the lasing radiation fluxes directed in opposite directions ia the correspondiag units; the designations of the coordinates are presented in Figure 4.10; h1, h2 are the distances during the time of passage of which by the flow of mixture oscillatory relaxatioa of the 210 FOR OFFICIAL LJ5E ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-40850R040500014010-4 FOR OFF'ICiAL USE ONLY individual componeats will accur in the abseace of resonan.ce eaergy exchange; finally, h is the distance during th,e time of passage of which. the processes of resonance energy exchange, neglectiag processes of oth,er types (oscillatory re- laxation, induced emission), cause relaxation of the ratio of the energy stored in the components Icl/k2 to the equilibrium value of ~c/(1 - c) (it is coasidered that on collision of two diff erent molecules the probability of exchange of excitation does not depend on which of the molecules is excited before collisioa). Let us - nota that as a result of the fact that the euergy exchaage between the componsnts is realized with finite velocity (h 0 0), th� total oscillatory energy reserve in the medium cannot decrease too rapidly with as high a density of the generated radiation as one might like. This should attenuate the manifestations of the mentioned trend toward "constriction" of the radiation in the case of a planar resonator. ~ ~ ~ c ~ ~ (a}-- ~ JL_ Figure 4.10. Diagxam of a telescopic resonator in a flow laser. Key: a. flow It is possible to demonstrate that the transition from the deasity of the output radiation p+ in the adopted units to the resonator eff iciency X def ined as the ratio of the number of quanta of output radiatioa to the total number of excited molecules at the entrance to the resonator ahould take p1aEe according to the formula .r = S P+ (a) dt (S [kl (a) + k, (i)1 da'}-1, where the first integral is talcen with respect to the exit optical aperture o~ the resonator, and the second with respect to the f low crosa section surface at tts antrance to tha resonator. In spite of simplicity of the given model, its application iasures satisfactoxq precision of the energy calculations of the gae dynamic laser in a def ined range of variation of composition and parameters of the gas mixture 1231]. In the greater part of the specif ic calculationa performed with its help, the results o� which will be discussed below, the folloviag iaitial data staadard for gas dynamic lasers were used: mixture composition 15% C02, 83.5% N2, 1.5% 820; gas flow 211 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047/02/09: CIA-RDP82-00850R400504010010-4 velocity 1500 m/sec; gas pressure :Li the vicin.t,tr of the xesonatar 0.1 atm, tempe- rature in the flow 354 K. These data correspond to the �cliowitlg va].ues of the paraaeters which figure in the equatioas o� the mediumt c= 0.15, b.= 0.1 cm, hl = - 4 cm, h2 = 250 cm. Let us also preseat the value of sucll an interesting charac- - tertstic of the active medium of the flow laser as the length H on wliich independent relaxatian of the exaitation is realized in the absence of the lasiag field. As is - kaowa, any multicomronent mixture, the energy exchange rate between the components of which signif icatitly exceeds the rates of the remainiag proceases, has such a - unique relaxation leagth. For our medium this condition is satisfied (h � hl, h2); the total relaxation length is H=[(c/%) +(1 - c)/h2]'1 z 25 cm. At the entrance to the resonator the active medium w s considered excited unif ormly over the flow croas section and in equilibrium (ki/k~ - c/(1 - c)), the total magni- tude of the applicatfon coefiicient of the entraace to the resonator k~~L varied from 0.3 to 0.6 (L is the f low width of the medium). For maximum energy pickup for the location of the mirrors depicted in Figure 4.10, thair shape obviously must be rectaagular, which was assumed in the calculations. The reflection cveff icient of each mirror was considered to be 98%. . The calculations were performed for resoaators both made of cylindrical mirrors and made of spherical mirrors. In t,:s f irst case the caZculations were aot too compli- cated inasmuch as all of the diatributions are esseatially two-dimensional. How- ever, the resonator made of cylindrical mirrors caa insure aagular selection er1y with respect to one direction. In the case of apherical mirrors which in prac- tice is more important the aalculations are greatly complicated: the gas flow alternately intersects the p:aaes passing through the axis of the resouator at di:ferent a.ngles to the direction uf mvtion of the flow, and the solutions in these planes turn out to be dependent on each other. The_refore when ccnetructing the - solution in the entire volume it is aecessary in all steps of the iterations to manipulate the total three-dimenaional f ield distributions and amplif ication co- efficient distribution. The only exception is one plane pasaing through the resou- ator axis on which the solution caa be found without constructing distributions in the rr.maining volume the plaae in Figure 4.10. The solution of the correspond- ing two-dimensional problem permits the f ield on the resonator axis ro be found immediately, which greatly facilitates subsequent calculations. It also turned aut to be convenient to divide the resonator by the axial plane x= 0 into two parts having in the general cas,e diff erent lengths I 1, R2, (see Fig- ure 4.10) and to proceed with calculations of the distributions in the right- hand side only after the solution on the lef t-b,and side has been completely con- stxucted. For optimization of the reaonator, its magnification M, the widtha of the left and right-hand sides kl and t 2 were varied. Ia addition to the effectiveness of the resonator X, for each version the values of the relative losaes characteristic of flow lasers were also calculated relaxation in the volume X rel' removal from the resonator Xrem and abaorption in the mirrors Xinir - 1- X- Xrel - xrem' Let us remember that X is the ratio of the number of quanta leaviag the laser in the iorm of useful radiation to the total number of eaccited molecules eatering the resonator at the eame time; the relative relaxation and removal losses are 212 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 FOR OFFiCIAL USE ONLY fractions of thi,s total nuaaber lost as a result af the processes of deactivation by collisions and as a result af reaoval of excite.d melecules the.r have incom- - pletely interacted with the radiation froiu tha resonatar; the meaning of X.m-Lr is entirely clear. ! ~d as (a) - 0,2 ~b~�` _ X.. ~ 0 d 6 9 12 15 1~, c.r~ ~ Figure 4.11. Efficiencq of the left-hand side of the resonator Xm, the rilative relaxation losses ia its volums, XP' and the removal from it Xr~ as a function o� the length of the lef~~d side kl for ~ M= 1.33: curves 1, 2, 4-- for kiL . 0.6, 0.5 and 0.4, reapectively. ; Keq: a. XTem b. xrel ' Let us proceed with the investigation of the resulta of the calculations pertaining to the case of the rasonator with spherical mirrors. The relationa are presented in Figure 4.11 fo the efficiency of the left-hand side of the resonator RR and relative losses X~~, Xrel as a function of the width of thia part with respect to the flow Z1. As is obvious from the figure, with a decrease in R1 the losses to ; relaxation are reduced approx3mately proportionally to kl at the same time as the loeses to removal vary slowly; as a result, the efficiency RR iacreases. The meaning of these Iaws becomes clear if we consider the followiug peculiarity of the solution for the lef t-hand side. On the axial plane x- 0 separating the lef t and zight sides of the resonator, the distributions and densities of the f ield and, what is especially important, the amplification coefficient turned out to be quite uniform in all af the calculated cases. Thus, over the extent of the lef t hand side of the resonator the amplificatioa coeff icient with respect to th.. entire flow cross section decreases from the initial value of k.l approximately to the threshold value ki =(In M- In R')/L (R' is the mirror ref?ection coeff3.cient). It is necessary to add to this that for the selected mixture the energy exchange be- tween components was so fast that the present radiation field could not signif i- cantly disturb the equilibrium ratio between the number of excited molecules C02 and N. Hence it follows _hat the losses to removal from the le;t-hand side of the resonator far any width must in the given case be cloae to lc.l/ki. On the other hand, the less the width t 1 becomea, the greater the fields must be - insuring such a decrease in the amplification coefficieat. This leads-to an ia- crease in the role of stimulated� radiation proceases by i omparison with the role of the relatation processes, that is, to an increase in X as a result of a de- crease in Xrel' 213 FOR OFFICUL lJSE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 cva~ va�as~a^a. vuc. v*.Ad. Now 1et us proceed to the r. Ight*hand stde, Aa we bave seen., the state of the medium at tFie eatrance to it can be considered in tfie fixst approximation uniquely defined by the resoaator losses. TFiExefore the only parameter wgiich depends on the conditions in the lef t--iiand paxt of the resonator subj ect to variation wh.ea = investigating the right--hand part remains the radiation flwc deasity on the axis ~ p(0, 0); this makes it possible eigai.ficantly to reduce tlLe aumbex of variants of the calculations of the right-hand side. - The construction of the solution in the right-hand side offers the posaibility of calculating its efficiency Xlr defined ae the proportian of the eaccited molecules "proaessed" into outgoing radiation lef t after passage through the lef t-hand side. The results of the calculations Iff for M= 1.33 aad three values of p(0, 0) are presented in Figure 4.12. It is abvious that the dependence of e on Z 2 differs with respect to nature from the similar depeadence for the left-hand side: with aa increase in k2, the eff iciencp first increases rapidly, then more and more alowly and, f inally, it reaches an extraordinarily gently elopiag maximum. This is explained "by the fact that on passage of the active medium through the right- tiand side ;,f the resonator the reserve af the molecules is completely exhausted; as a r.esult of a'decrease ia field denBity dowa stream this - process takes place more and more slowly. The maximum eff iciency is achieved when the amplif ication in the medium becomes so small that it compares with the - losses on the mirrors. Of cours, it is possible in practice to limit ourselves to a signif icantly 3maller width of the right-hand side, losiag very little in efficiericy, and on the other hand gaining sigaificantly in size of the mirrors. For the above-iadicated parameters of the medium, the value of k- 15 cm (see [228] and also Figure 4.12) can be used which is also used in til subsequent cal- culations. X" gi p(4oI,~pl QpZS ~8 ~ -------O,A75 ' 0 10 ZO J9 !f, c.+v . Figure 4.12. Eff iciency of the right-hand - side of the resonator as a functioa of its length k2 for M- 1.33 and three different P(Q' 0)� Now it is possible to proceed with investigation of the properties of the eaonator as a whole. In Figure 4.13 we have the relations for its efficiency X=%~ + Xr~%~ as a function of the magnif ication M aad width of the le�t-hand part for k2 = 15 cm. It is obvioua that the given relation coiacidgs with respect to nature with the analogous f unction for the lef t-side onlp (see Figure 4.11). This is also under- , standable: for small M and t1, p(0, 0) increases, and as a result the efficiency not only in the lef t-hand side, but also the right-hand side increasea, and to- gether with them, the efficiency of the resonator as a whole. The growth of X with a decrease in M and kl continues in th:e entire investigated range of parame- - ters, alowing only in the case of 1c~ - 0.6 for R1 3 cm and M- 1.3; at this time 214 FOtt OFFICY,PL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-40850R040500014010-4 FOR QFFICIAL USE ONiY X alread}r rea.chea 60%. I.t makes no sease- to iuvestigate the region of still smaller values of M and Z li Eor ordinary sizes of the active zoae oi a gas dynaffi.ic laser the geometric approximation ceases to be valid tFere, aad therefore the direction of the radiation must become worse. Thus, in order to achieve high efficiency of eaergy conversion it turns out to be sufficient to select a total resonator width which is not too much less than the _ relaxation length of the medium (let us remember that fi- 25 cm), the width of the left-hand side is selected approximately an order less, and the resonator losses such that the threshold value of the amplificatioa coeff icient is approximately half the value at the entrance to the resonator. This choice has quite clear mean- ing and maq fail to give the desired reault only in case of inadequate energy exchange rate between the compoaents of the mixture. Actually, then the total oscillatory eaergy reserve will not be able to keep up with the gain reduction in the presence of the field, and the losses to removal must increase. In [228] a series of the correaponding calculatinns were made for mixtures with the compositions 5% C02, 90% N2, SX H20 and 8% C021 90% NZ, 2% H2O; in both cases the value of k~L was 0.8, the pressure was taken equal to 0.05 atm, and the remaining parameters were the same as in the above-inveszigated case. The energy exchange rate between the componeats for these mixtures was noticeably less (primarily as a result of reduced pressure). The calculations demonstrated that for_ analogous selection of the position of the axis and the dimensions of the mir:ors, the eff�aciency on the order of 0.6-0.7 was aGhieved only when the threshold value of the amplif ication coefficient was less than the input value by 3 or 4 times. From the procedural point of view the following is netwQrthy. Alttwugh the ampli- fication coeff icient in the plane x- 0 is as before close to threshold, the con- tent of the excited molecules N2 here also depends on the f ield density. Inasmuch as, in turn, in the entire indicated plane the field density is close to p(0, 0), the value of p(0, 0) even for mixtures with 8ma11 energy exchange rate remains the oaly parameter subject to variatioa for calculations of the right-hand side. Problem of Forining Uniform Field Distribution ov-er the Cavity of a Flow-Through Laser. The above-investigated example is quite typical; _ it is clear fram it what arguments must be and are being talcen iato account when selecting the resonator for a fast-flow laser. It is also obvious that even the X X ~ 04 ~ \ k; L.Qs a! 0 3 6' 9 1Z /5 11, cn !t �T (S M Figure 4.13. Eff iciency X of a resonator as a whole as a functioa of its parameters with a length o� the right-hand side k2 = 15 cm: a) dependence of the magnitude of X on the length of the left-hand side for M a 1.33; b) the value of R as a function of parameter M f or Ql s 3 cm. 215 . FOA OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 si;aplest unstable laser made up of spfiexical mixxars with.accurate selection of - its parameters can entirely insure satisfactory efficiencq of the energy conver-- - sion. iiowever, for practical applicatioas it is almost as important to have more ur less uniform distribution of the radiatioa intensitp witb, xespect to zpc- - tion (otherwise the local thermal loads of the mirrors and angular divergence of the radiation wi11 iacrease). Alth,ough with uniformity o� field distsibution in =the unstable resonators of this type tltings g,o better than in plaaar resonators, they are not suff3.ciently good. Thus, ia Figure 4.14 we have the intensity distri- bution on a coucave spherical mirror (curve 1) fos the very case where eff3ciencq of 60% (k~I. - 0.6, k= 3 cm;- M = 1.3) was achieved for a mixture with a pressure of 0.1 atmosphere. iere the graph is also plotted for a two-dimensional resonator (curve 2) for =he same parameteTS of the medium, losses and position of the axis (let us note that under these conditions it has approximately the same eff iciency). From the figure it is obvious that the f ield distribution is quite nonuniform, especially in the more important case of spherical mirrors (ia tae diffraction approximation a sharp distribution "peak" caa smooth off somewhat, but ths total _ nor.uniformity will still be clearly noticeaUle). A more favorable form of dis- tributioa is observed only in vPrsioas with low efffciencyL primarily with exces- sively large M when the f ield indide the resonator turns out to bo� insuf f i.ciently intense. This cype of situation obviously occurred in the above-described experi- mants using a gas dynamic laser [ 211] , which is iadicated. by the 1QTe ef f iciency of the resonator with satief actory uniformity of distriUution. qZ3 gt 0,>5 0 B 9 lt 13 J~d ~ u' Figure 4.14. Field density distributioa in a concave mirror; 1-- for spherical mirrors, M- 1.3; 2-- for cylindrical mirrors, M- 1.69. The use of multipass systems of the type depicted in Figure 4.9 permits a sharp in- crease in M. As a result of the compli^itioa of the system, a aoticeable positivs effect is achieved, in particular, the ~actness o� the exit aperture increases, and its f illing factor y increases (see L.1, 3.6; Figure 1.3, a b corresponds to the single pass and one of the possible versiona of multipass systems). Sow- ever, it is still not possible to achieve a combinatioa of high eff iciency with high uniformity of the field distribution here. In order to solve this most important problem it is possible to consider the prism resonatora investigated ia g 3.5 in w`.ich th.e effect of kave aberrations of odd order was significantly atte.n.uated by comparison witiz the ordinarp telescopic reson- ater. The effect of the amplification coeff icieat gradient directed acsoss the 216 FOR OFFICIAL USL ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-40850R040500014010-4 ax�e ie ideatical to the ttlf luence oP the gr$dieat of the isu:ex of ref raction kwedge) and the wave phase, and it must be attenuated to the degxeet Tt is true that in the far infrared range where the greatar part of the flow lasers operate, tzanspa:2nt totsl internal reflection prisms analogaus to those used in aolid state lasers are bardly realizahle. Iiowever, f or low magaif ications M which are a}iaracteristic of the flow lasers with siagle pass rasoaators, the application of $ confocal resonator becomes mandatory inasmuch as satisfactory f illing of the , operating croas section with lasing radiation is also achieved in a resonator made up of planar and slightly convex mirrors (see also � 4.1). Replacing the planar mirro.r by a dihedral 90-degree reflector made ug of two planar mirrors, we obtain the desired resonatqr. For equalizing the intensity distribution, the edge of the reflectcr must be orientEd ohviously perpendicular to th.e plane of Eigt:re 4.10. This possibility was investigated experimentally in the example of the flow-type C02-laser of :-mparatively smaller power in reference [232]. The systems discussed above were cested (Figare 4.15, a, b); for convenience of selecting the optimal parameters of the resonator, the convex mirror was not spherical, but cylindrical with regulated curvature [233] sa that the unstable resonator would be two-dimen- sional. For the same purpoae the radiation was coupled out by two auxiliary mirrors, the position of which could be adjuated. The preliminary recordings of the radia- tion patterns of the amplif ication coefiicieat with respect to the resonator cross section [234] demonstrated that the medium in the given l.aser relaxes quickly, and the conditions are quite typical for continuzus flow systems. � P , 1' ~I /lom~a ~ ~ j ~ II ! I 0 ~ - i ~ pt a) b) . c ~ d> Figure 4.15. Unstable resoaators in the flow-type lasera: a) resonator made of planar and convex mirrors; b), c) resonator made of a corner ref lector and convex mirror; b) symmetric; c) asymmetric radiation output; d) various versions of the type of projection of the exit aperture on the plane perpendicular to the resonator axis; the axis passes through the point 0, the line PP' is perpendicular to the plane of the figuCe; a) to c). Key: 1. f low The results of the experiments turned out to be highly hopeful. When using the system depicted in Figure 4.15, a a tenfold decrease in intensitX downstream was observed insiae the generated beam. The dif ferences between the intensities and the left and right halves of the re.sonator with corner reflector (Figure 4.15, b) did ndt exceed 25%. Another ixLteresting gosaibili,tp is connected witli the appli- cation of the corner reflector: in order to increase the compactness of the exit 217 FOR OFFiCIAd. USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 aperture it is sufficient to resort to aeymatetric output of the radiation. In the caae of a thzee-dimeneional resonator Cfiis is doae using an output mirror, the possible versions of the shape and arraagement of wfiich.are explained by Figure 4.15, d. Iiz the two-dimensional resonator it is poasible simply to remove one of the two output mirrors (Figure 4.15, c). Then the rad iation which was incident earlier on this mirror and immediately left the resonator naw makes an additional pass t: rough the resonator and leave.s from the opposite side. The width of the output zone approximately doubles, the radiation distribution in the far zone be- comes more favorable. From the results of the experiments and rough estimates [232] it is also possible to draw some conclusions relative to the energy character istics of the flow lasers in whieh au Lnstable resonator with corner ref lector is used� If we select magni- fication M and the width of the left-hand side the same as in the case of an ordi- nary resonator, the average radiation density here will remain as before; the officiency of the left--hand side almost does not change. As for the right-haad - side, its width in the systems with beam "inversion" is automatically close to the - width of the left and, as a rule, falls far short of the optimal width of the right-hand side of the ordinary resonators. However, as a result of much greater radiation density the eff iciency of the right-hand side also remaias approximately on the former level. Thus, it is possible to hope that resonators with corner ref lectors will permit insurance of the same eff iciency of energy _ conversion as ordiaary resonators with noticeably lesa width of the operating zone and greater compactness of the exit aperture. - The confocal unstable reeonators made of two concave mirrors must have similar properties (see Figure 3.3, c), where "inversion" of the light beam is also realized. However, their use ia many cases can cause undesirable phenomena at the common - focal point of the mirrors where the rarliation density reaches an enormous value. On the other hand, the great prospecti-veness of the application of unstable reson- ators inveatigated in 9 3.6 with field rotation in continuous-action lasers is , unquestioned. The quite recently published results of the correaponding experi.- ments [235, 3331 completely coaf irm the correctness of the arguments discussed in this regard in � 3.6. The introduction of the operation of rotation of the cross aection actually cardinally equalizes the intensity distribution in the aear zone. Sensitivity to astigmatism decreases so much that the latter is weakly manifested - even when the ccrner reflectors �ozming the linear resonator include spherical mirrors, the angles of incidence of the light on which are, therefore, 7/4. Finally, the use of a sectional output (see Figure 3.28) instead of annular not only increases the compactness of the exit aperture, but also leads to a signif icant de- crease in divergence of the outgoing radiation without reducing its power. It only remains to mention that sometimes reports appear in the literature and resonators, the elements of which have the shape of a eurface differing sharply from spherical conical, toroidal, and so on (see, for example, [236]). The possibility of obtaining a small angular divergence of the emission in this way still is far from obvious and we shall not analyze properties of such systems. � 4.3. Unst eady Zesonators in Lasers with Controlled Spectral Temporal Emission Characteristics Simplest Types of Lasers with Control Elements, For maay practical applications it is necessary that the stimulated emiesion have not only small divergence, but 218 FOR OFFdCIAI. USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAT, USE ONLY also given time-spectral charactexi,stics, Tbeir $cb.ievement usually is insured by the fact that the corresponding control elemente are located inside the resonator. The first publication on building a giant-pulse _ laser with unstable resonator - belongs to the year of 1969 [189]. This was a solid state neodymium-doped glass laser; a passive shutter covered the entire cross section of the active eleaent. The output energy was -20 joules, the peak power was 1.54109 watts, tb,e angular divergence with respect to 0.5 intensity level was -4"(2�10-5 radians), but a sig- nif icant part of the energy went to the "tails" of the aagular distribution as a result of light dissipation in the shutter liquid. As a result of rapid "spread" of the emission over the cross section of the unstable resonators the pulse aura- tion turned out to be noticeably shorter than when using a planar resonator. Subsequently,giant.pulse lasers with unstable resonators have always been constructed by one of two circui*_s depicced in Figure 4.16; here the shutter does not cover the entire cross section of the active element, but only the exit mirror. This leads to signif icant improvement of the output charac*_erietics: the peripheral part of tne beam runaing "to the exit" from the laser, bypassea the shutter and does not undergo additional absorption and dispersion in it [204]. In addition, possi- bilities are created for controlling the radiation flux having larger area using a small shutter. If we consider that the optically improved fast-acting shutters usually have small aperture, the prospectiveness of using such systems to construct spikeless, monopulsed and other controlled-Q lasers becomes obvious. Thus, at the beginning of the 1970's highly improved a giant-pwlse neodymium_., glass- lasers were - built with radiation brightness of _1017 watts/(cm2-steradian) [204, 2051. Let us also note the achievement of the quasisteady lasing mode of an analogous laser in [237]. > ~d 2 ~ f 1 1 ~ t  Figure 4.16. Diagrams of 4-switched lasers; . 1- resonator mirror. 2-- additional mirror, 3-- active sample, 4-- shutter The entire resonator crosa aection or its exit mirror caa be covered, of course, ' not only by the ahutter, but also the spectral selector. It is necessary, however, when installing any controlling elements to deal with the specifics of unstable - resonators which impose restrictions on the types and methods of placement of these elements. First of all it is necessary to take steps to prevent ~ the occurrence of a converging wave. If the resonator is telescopic, the f lat surfaces of the controlling elements must be also iaclined with respect to the rescaator axis ,juat as any other interfaces (aee 1 4,1; in the light of t:iis fact Eigure 4.16 is provisional). _ 219 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 Another important characteriatic of unstatle resonators complicating the problem of controlling their emissian is the f act that the in.troduction of a small optical wedge does not lead to sharply exceeding the lasiag threshold, j ust as in a planar resonator, but it causes only a shift of.the optical axia. As a result, it is neceseary to depart from a number of traditional methode of control based on ths application of an optical wedge which varies in time or depends on the wavelength. - Thus, the placement of a disperaion prism inside aa uastable resonator leads not to spectral selection of the emission, but only to the �act that the radiation epectrum will be expanded in the corresponding direction in the far zone iastead of one spot the spectral distribution image appears. Efforts to modulate the Q- factor usiag a rotatfng prism can ha-rdly lead to good results. , Thua; ia giant-pulseJ.asers with unstable resonatora it is aecessary to use predomi- nately passive or elect.rooptical shuttersl; for spectral selection primarily the Fabry and Perot etalon3 are suitable, on passage through which the value does not change its direction. flowever, even here it is necessary to consider the fact that in any lineaz, unstable resonator, not a plane wave, but a spherical wave will be propagated in at least one of the two opposite directions. Under these conditions the introduction of the etalon will not cause intensity modulation with respect to the resonator cross section except ia the case where the angular width of the maxi- mum transmission of the etalon will exceed the angle of opening of the spherical wave. The angular width of the maximum traasmissioa of the etaloa in turn is equal to the angular distance between adjacent rings divided by the ntmber of iaf erf ering beams N which depeads on the reflectioa coeff icient of the working surfaces of the etalon. As a result, we arrive at the following condition imposed oa the magnitude of the etalon base t[196]: 7l M !lL t~ Nainfp6tp-~!ND da 4p' (6) where D is the diaateter of the traasverse cross section of the active element; M and L are the magnif ication and the equivalent length of the resonator; ~ is tbe angle between the normal to the etalon surface aad the resonator axis; finally, A~ - (M - 1)D/ML is the angle of opening of the divergiag beam in the telescopic resanator (when deriving (6), the inequality ao which usually is satisfied was considered to be valid). Multiple beam ref lection in an inclined etalon also leads to some "blurrin.g" of the position of the resonator axis, which in turn can cause an iacrease ia the radiation divergence. The corresponding"calculations iadicate that the condition of smallness of this increase by comparison with the diffsaction angle again reducea to formula (6). Inasmuch ag the etalon base determiaes the width of the region of its disper- sion (AX _ X2/2t), conaequently the expressioa (6) limita the minimum width of the spectrum reached in the laser with direct placement of the selecting etalon ia the teleacopic resonator. � 1Self--Q switching of unstable resonators described in 1238] and a number of subaequent papers is reAtized only in tlie smallest lasers and leads to output parameters wh3ch are hardly recorded even for thts class of laser: 220 FOR OFF:CiAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFEICIAL USE ONLY Lasera with Three-Mirror Optical Cavity. Fs3tttter improvement of. themethods of con- trolling the emission of lasers with unstable resonators is conaected with the idea formulated and experimentallp substanti.ated in [153] of the effect on the central section of the resonator cross section, from which the radiation "spresds." For normal course of the beams this section is the analog of the master oscillator, and the remaining part of the croes section, a multipass amplif ier. In order to realize this idea, it is sufficient to make an opening at the ceater of one of the mirrors and install an auxiliary mirror behind it; the coatrol elements of very emall size can be conveniently placed in the narrow "appendix" formed in this wap. This three-mirror resonator and the acheme eQuivalent to it without an open- ing in the mirror are depicted ia Figure 4.17. Figure 4.17. Diagrams of resonators with radiation control in the central section of the cross section. The possibility of eff icient radiation control using the given procedure is limited by the fact that, as follows from the materials of g 3.4, a laser with unstable resonator is capable of lasing even with a completely shielded central section of the cross section. Therefore, if in the "appendix," for example, the shutter and the axial section are "blocked" at a given point in time and the lasing threshold increases inaignificantly, independent lasiag which is not controlled, will de- velop in the remaining volume. Hence, it is clear what important practical sig- nificance the nature of the dependence of the lasing threshold on the dimensions of the covered central segment of the cross section has. The results of the measurements of this relation performed in [191, 168, 193, 1961 turned out to be entirely in correspondence to the theoretical represeatation developed in g 3.4. In the case of two-dimensional resonators the tEreshold ia- creases with an increase in hole sizes (more precisely, the slit width) extraordi- narily sharply. Thus, the shielding of the central section -3 mm wide has approxi- mately tripled the threshold intensity of the pumping of the laser described in � 4.1 based on a rectangular large active elemeat with resonator made of planar and cylindrical mirrors (Figure 4.6) [168]. Therefore the control of the radiation characteristics of lasers with two-dimensional unstable resonators is realized without special diff iculty. In particular, the use of the simplest disc modulator has made it possible to convert the mentioned laser to the regular "spike" mode - with repetition of these spikes from 25 to 50 kilohertz [168]. In reference [193] special selection of the radiation of the same laser without a noticeable decrease in its output power was successf ully produced by the introduction of the Fabry and Perot etalon into the "appendix." - Lsers with resonators made of spherical mirrors behave entirely differently. Their lasing threshold increases very slowly with the size of the central circular 221 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 - operaing. Thus, in a laser with active neodymium doped glass element 45 mm ia dia- meter and a telescopic re;saaator with magaification M= 2 for a hcle diampter ia the concave mirror of 4 mm, the pumping threshold inteasity iacreased bp 1.3 times; for a diameter of 8 umi, it approximately doubled by comparison with the case where the hole was absent. Even the presence of a hole with a diameter of 20 mm and almoeti - reaching the diameter of the convex mirror (22 mm) caused onl i triple imcrease in the self-excitation threshold [191]. ? . ! ~ y 90 . t 60 (a)~ 79 -#7 411 Z 5 10 ZO JO 40 50 N, ;(b ) Figure 4.18. Losses of the lowest modes as a function of the size of the coupling hole in a three-dimensionp.l resanator with circular mirrors (M - 2, N - 60). The dots represeat the experimeatal = data; 1-- the caggui lation for the caae of a sharp mirror edge; 2- analogous calculation in the presence of "smoothiag" of the edge (IRref 1I decreased by four times). . Key: a. energy loasea, % b. Nequiv Ia Figure 4.18, the data on the dependence of the loeses of the lowest mode on the siZe of the coupling hole calculated by the reaults of experimeatal observations and by formulas from � 3.4 in [168] are compared. As ie obvious from the figure, the experimental points lie aoticeably above curve 1 obtained on substitution of the value of the coeff icient of diffraction ref lectioa from the edge Rrefl calcula- ted by the formula for an ideally sharp edge (3.25) in expression (3.24). In order to achieve comparison of the calculated and the empirical data it is necessarq to decrease the value of IRzef 1i substituted in (3.24) by four times (curves 2). For ' this decrease IRreflI in the given case there was sufficient "smoothing" of the edge (see � 3.3) in a zone -0.1 mm wide, which, apparently, are curved in the experiment. Inasmuch as the shielding of the small axial section here only causes an insigni- ficant increase in the self -excitation threshold, th,e posaibilities for controlliag the radiation turn out to be more limited thaa ia the case of two-dimeasional resonators. In particular, it is explicitly impossible, when placing the control- ling elements only in the "appendix" to realize giant pulse.lasiag or a lasing mode cloae to it. If the operating transition liae is broadened�nonuniformly, the spectral selection of the radiation is also complicated. Actually, in the loop 222 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R440500010010-4 FOR OFFICIAL USE ONLY of thia line with iatense single-frequeacy lasing a"trough" arises, aad the amplification coefficient on the side frequeacies became.s noticeably larger tban on the frequency at which lasing occurs. Therefore the isolatioa of one frequency usiag the selector in the "appendix" cannot prevent self-excitation of the lasing on the side frequencies ia the remaining volume. Nevertheless the problem of apectral selection in unstable resonators with spherical mirrors frequently can be solved even in the case of a nonuniformlp broadened line. According to [196] it is aecessary to try to install one etalon having a small base directly betweea the primary mirrors of the resonator so that it covers the eatire ' operating crosa section. The purpose of this etaloa must be to isolate the band, the width of which does not exceed the width of the "trough" ia the line contour. If this can be doae without violating condition (6), for further constriction of the spectrum it is possible to use the "apnendix." The favorable situation actually occurred when performing the experiments described in (196) with a neodymium-doped glass laser with the active element 45 mm in diame- ter and 600 mm long. The telescopic laser had magaif icatiaa of M= 2; the output power without spectral selection was 500 joules per pulse. The preliminary con- striction of the spectrum from the width usual to neodymium lasera equal to several naaometers, to a value of -0.5 nm was carried out using an etalon having a base of only 0.05 mm. Iasmuch as all of the conditions were satisfied which were initially discussed, the introduction of this etalon did not cause any changes in the form of the angular diatribution (its width with respect to the 0.5 intenaity level was 8�10-5 radians). Iiere the output power dropped to 400 joules. Subaequeat introduc- _ tion of the etalona with bases appreciablp larger than in the f irst one iato the ' "appeadix" caused only further decrease in the width of the spectrum, reachiag, ia the final analqsis, 0.003 am; not oaly the radiation divergence, but also the out- put power remained constant. The following is of interest. In these experiments the width of the opening beyond which the "appendix" with additional etalons was located and amounted to a total of 3 mm. The overlap of this small section of the cross section exceeded by only 10% - the self-excitation threshold of the eatire laser. Ia the ceatral section taken separately, the lasing threshold, considering all of the introduced etalons, uacon- ditionally would be highe=. This demonatrates that the above-mentioned division into the "master oscillator" and "amplif ier" is purely arbitxarp inasmuch as the radiation control of the "amplif ier" is realized whea the self-excitation threshold , of the latter was below the threshold in the "master osciliator." The fact is that, as the experimeats demonstrated, with the "oscillator" aad "amplif ier" spatially combined in the three-mirror system, the radiation arrives not oal.y from the first to the second, but also as a result of diffraction, from the second to the f irst. This l.ncreases the radiation density in the "appendix" especially for a small dia- meter of the appendix, and the role of the "appendix" turns out to be quite large. - The unitorm part of the line broadening that determines the wl.dth of the "trough" was rather eztensive in the s111cate neodymium glase used in Ref. 196, which made it easier to pick out tfie precontraction etalon arlthout violating condition (6). It might not be possible to pick out such an etalon in lasers based on media with a narrower uniform broadening band. Wiiat to do in such cases is still unclear; it is possible that it remains only to do away with the linear and resort to the uni- directional ring cavitie5 for which.condition (6) becomes invalid. 223 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 Ending with three-mirror resanator lasera, let us aote that if there is too much absorption in the "appendix," its role can be sigaificantly streagtheaed bp place- ment of an additional small-cross section active elemeat in it. It would appear that it is possible to achieve an aaalogous effect on introducing a special de- flector (or orienting oae of the interfaces perpendicular to tiLe resonator axis) to produce a convergent wave that "overflows" to tfie center, but this leads to a sharp iacrease in the radiatioa divergence (see [196] aad also � 4.1). - External Signal-Controlled Lasera. Now let us proceed to another method permitting - the use of the special role of the central segment of the crosa section of an un- , stable resonator. This method was proposed in 1969 [239J, and it consists in the fact that radiation from an external source ia input to the laser with unstable reaonator through the central coupling hole. The properties of such a laser depend - decisively on the relation between the amplif ication of the medium and the losaes of the resonator with the coupling hole. If this relation is such thafc self- excitation of"the system does aot occur in the absence of aa external signal, we are dealiag with a''pure" amplifier capable of operating in the slave mode and thus, suitable for effective amplif ication of powerful short pulses aad other purposes. If the self-excitation threshold is loW and turns out to be exceeded, it is more correct to talk not about an amplif ier. but about a laser controlled by an external signal.* Let us f irat consider the latter case. � The comtrol of laser radiati.on by an external signal is a we].1-lmowa method of obtaining the given spectral-time characteristics. This method was used success- fully in plaaar resonators (see, f or example [240-243]), where the "aucleating cen- ter" from an extemal source was iatroduced, as a rule, immediately over the entire crosa section di the resonator. Inasmuch as in uastable resonators the radiation quickly "spreads" from the central region, for such sffective control here it is obviously auff icient to introduce a beam from an external source only into the re- $ion which permits us to go on with quite low power of it. Unfortunately, a com- - parison of the results achieved in practice obtained whea using planar and ustable resonators is almost imposaible inasmuch as these results pertaia to eatirely different cases. For planar resouators, a study was prim,arily made of the spectrum control mode'of a giant pulse by injection of the "nucleating center" at the beginning of its development. As for unstable resonators, for them the control of the spectral-time characteristics of the emission wda still realized only under conditions close to quasiatationary although the posaibility of coatrolling mono- - pulse lasers also raises no doubt. In order to obtain the same output power for a controlled laser operating in the quasistationary mode as for an ordinary laser it is obviously suff icient to use aa unstable reaonator for it with the same M and introduce radiation with the same density into the coupling hole which is developed iaside the ataadard lasers with- out the coupling hole. However, for more reliable control it is expedieat either to increase the density of the controlling signal (which both in the pulsed and in the flow lasers leads to a decrease in amplification on the axis) or noticeablp increase M or, f inally, do both. Thea the systmoa with the supplied external signal turns out to be significantly below its self-excitation threshald, which improves its controllability in essence;'the amplification mode occurs. It ia only = necessary, by increasing Mi, to see that this does not lead to a large decrease in the autput power. This may occur especially quickly in flow lasers with low amplification. As for pulsed lasers using highly amplifying media, the parametera - of their resonators can, as a rule, vary entirely "painlessly" within the broad 224 FOR OFFiCiAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R400500010010-4 FOR OFF'iC1AL USE ON] Y limits, especiallg in the presence of a quaaistationazy controlled sigaa]. (we ahal.l ' diacuss the causes for this somewhat later). i Let ua touch an the problem of aelectiag the size of the coupliag hole. The lossea ' of Yesonators made of spherical, mirrors and the radiatioa densitp required from aa ! external source to prevent self-excitation depead very slightly on the dimensions' ' of the hole. iience, it follows that the required control signal power is ia the 'f irat approximation proportional to the area of the hole. Therefore it is advan- ' tageous to use quite small holes; however, they must not be so small that the resoa ' ator axia caa go beyond their-limita as a result of aligament errors, vibrations, ; aad so on. Beginaing with these ar&uments, the parameters of aa unstable resonator were selec- ted for the firat laser of the given claea built in [191]. This laaer, just as in ~ maay experiments described above, was built on the basis of an active element made ~ of neodymium-doped glass 45 mm in diameter aad 600 mm long. The telescopic reson- ~ ator had magnif ication of M- 5 iastead of the usual M -2; the diameter of the , coupling hole was determirned by the precision of the alignmeat and it was 3 mm. The - external master oacillator operated in the mode of raadom radiation "spikes" which is usual for solid-state lasera; its output power was more thaa two orders less than the output power of the basic laser which was several hundreds of joules for a purp- ing pulse duration of -1.5 millisecoada. Although the preseace of intervals between ! the "spikes" does not at all promcte control reliabilitq, complete synchrony of the ; "spi.kes" at the output of the master oscillator and the system as a whole was ob- served. In [191], the requirementa on the required accuracy of mutual alignment of the ex- i ternal oscillator and the control laser were also discovered, aad it was demon- ~ strated that they are aot at all extraordinary: The admissible magnitude of the ~ angle between the direction of the beam from the esternal oscillator and the axis of the powerful laser was in the given case ZW. ! One and a half years later a report appeared on work of a similar nat.ure ~ in the f ield of C02-lasers : under the ef f ect of aweak signal introduced in tlne axial section with f requency corresponding to one of the rotational transitions of the COZ molecule, a powerf ul laser with unstable resonator began to generate on thia - frequency although without the input of the external signal it operated on another i rotational tranaition [2441. In the same paper the argument was atated of the i possibility of feeding the control beam not to the central but to the peripheral ~ section of the croas section so that a converging wave amplif ied as it approached the axis was formed. Uaing this procedure, the required spectral characteristics probably can b2 obtained, but there will hardly be amall divergence of the radiation ; simulataneously with this. ~ Now let us discuss one of the most peculiarities of pulsed lasers with unstable resonators operating under the effect of radiatioa fram an euternal source. We are talking about the extraordinarily weak dependence of their output power on ~ the reaonator parameters and the coatrol signal intensity if the latter was pre- ~ sent. In this respect the resulta of the calculations of the energy characteristics of a controlled laser the parametera of which are typical of powerful neodymium- doped glass lasers performed in 1245] aza indicative. The diameter of the coupling ! hole ia a cancave mirror of a telescopic reaonator was takea equal to the dismeter ' of the convex mirror or less than it by M times so that the beam from the external 225 FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 aource during its expaasion to filling tlze ex1,t aperture completed three or f ive paeaea through the medium, reapectivelp (if the givea laaer was not capable of aelf-eaccitation in the abaence of aa external signal it would 'oe a"pure" three or five-paes amplifier). The value of M varied from 2 to 20 (M - 2 corresponded to the optimal value of the magnification calculated bp formula (4) for an ordiaary laser conatructed on the givea active elemeat). The calculationa demonstrated that as M increases with fixed power of the external signals falling withia reasonable limits, the efficiency of the controlled laser decreased monotonically, but noticeably more slowly tban the eff iciency of ordinary lasers. This is understandable: when M is so large that the lasing in an ordi- nary laser is completely curtailed, the laser with coupling hole operates as a multipass amplif ier, and on supplyiag the external signal it has a fiaite output power frequently commensurate with maximum� The eff iciency of a controlled laser depends atill less on the power of the exter- nal signal. It ia suff icient to presoat the following data: in the entire above-indicated range of variation of M, the variation of the controlling signal power by more than three orders did not lead to a decrease in the eff iciency below the level equal to 40% of ita maximum value. This also has a simple explanatioa: it is known that the relative fluctuationa of the power at the output of any laser amplif ier operating in the quasistationary mode is alwaqs signif icantly less than the relative power fluctuations at its input as a result of amplification satura- tion. Here, as a result of the presence of several radiation passes through the medium, the given effect appears still more rarely. The results of 1245]. permit aome conclusions to be drawn which pertain to the properties not oaly of controlled but also ordlnary lasers with uastable resonators. It happens tbat aay disturbance on the central secticn of the cross section strongly reduces the losses of an empty resoaatar, and with them, the lasing threshold of the corresponding laser. Thus9 the presence of even weak small-scale nonuniform- ities easily leads to the formation of local "stable" resonators near the axis with all of the consequences following from this [246]. It can be demoastrated that a signif icant reduction in the threahold arising from auch causes must cause a signif icant increase in the lasing power. However, in reality the variation of the conditions on the small axial section can aharplp influeace the output power only in the case where the laser was quite close to the lasing threshold before this. If the threshold is greatly exceeded, the output parameters have little sensi- tivity to variations of the radiation density on a small segcnent of tIie cross sec- tion iadepeadently of whether these variations were caused by the iafluence of local disturbancea or the radiation ia supplied, as in a controlled laeer, from an external source. � Multipass Amplif iera. In conclusion, let us brief ly discuss the "pure" multipass - amplif iers, the unstable resonators of which have euch large losses that the self-excitation does not occur even in the absence nf an external signal. Most frequently these amplif iers are constructed oa the basis of a telescopic resonator; then they are called (at least in Soviet literature) telescopic. The possible versions of telescopic amplif iers are presented in Figure 4.19. As a result of the presence of several passes through the medium, one such amplifier is capable of replacing several single-pass amplifisrs with telescopes hetaeen them (see 92.6, - Figure 2.28) [239]. 226 , FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 FOR ( ;IAL I a) . . . . C) Figure 4.19. Telescopic amplifiera: a) multipass, b) three-pass, c) two-pass amplifiers; in the last amplii:ier the mirror ape.rture is replaced by an absorbing�screea. As a rule, amplifiers with uastable reaonators are used in the intermediate stages of powerful giant�-pulse lasers (e. g. 12479 24$]; aimple single-pass amplifiers are often preferred for the final stagea, as they have greater radiation streagth �through eliminatioa of the output mirror�and concomitant radiation in the reverse direction). Tlierefore, the important here is not so much efficiency as it is the weak-aignal gain, which preferablq should be maximized. The amplif ication of the weak signal in the n-paas amplif ier is equal to ICn, wtaere K is the amplif ication on one pass of the active medium. Therefore for the given K it is necessary to strive for the largest possible n; this promotes the achieve- ment not anly of high amplification but aleo batisfactorp energp efficiencp of the device. ' If K is small, multipass sqstems of the type depicted ia Figure 4.19, a can be used; let us begin them. We shall consider that the radiat3on is fed through the coupling hole in a concave mirror as depicted in the~nigt~~~; here a is odd, the beam cross section expands inside the amplifier bq M 1~ timea. Sence it fol- lows that n- 1+ 2 ln (D/d)/ln M, where d and D are tbe beam diameter aC the input and the output oE the amplifier, respectively. In order to obtain a large n it is obviously aecessary to use a resoaator with the lawest possib].e magnif ication M and minimum coupling hole diameter d. Iaasmuch as the self-excitation thrashold of a laser with amall coupling aperture is almost equal to the lasing threshold of an ordinarp la8er which is achieved for K= M, the least admissible value of M is equal to K. As for the diameter of the coup- ling hole, it is limited from below, as we have already poiated Aut, by the pre- cision of the alignment. The latter, it is true, dep4ads not onty oa the ovarall dimensions of the resonator, but also oa M, but for the large M characteristic of the amplifiera this relatioa is very Weak. Therefore we shall conaider tiie minimum aperture � diameter f ixed and equal to do. 227 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 Thus, the maximum n i,a nMaX - 1 t 2 ln(pldQVla K. Hexe ai4plif icatiqa of the weak signal is achieved by Knmax - K(p/dQ~2 times 1249] (ezactl}r the same power traxis- formation coefficient occurs.also ia controlled lasers operating in th.e quasista- tionary mode 1191]). Fram the condition n max ? 5 we obtain the criterion of applicability of multipass systems: K< D 0. For large K three-pass aystems can be used (Figure 4.19 b) insuring a general ampli- �ication coeff icieat K3. Inasmuch as here M - D/d, even without consideriag tiie increase in the threshold as a result of the presence of the coupling hole, self- eaccitation ahould not begin until R- D/d. If we consider that the coupling hole - diameter ie no smaller than the convex mirror diameter and if necessarq can be made as shown in the f igure larger than the convex mirrox diameter (at the price of insig- nificant decrease of output power), the admissible value of K increases as a result of a corresponding increase in the self-excitatioa threshold to values oa the order of 1.5 to 2 (D/d) [249]. Hence it folloysstl~at the criterion of applicability of three- pass :~ystems has the form K 0, P+ Q in this case also turns out to be positive. From formula (4) it follows that ia this case for radiation polarized along y, the index of ref ractioa gradient is absent, and for the other polarization component the element is equivalent to a scattering lens. In planar or telescopic resonators this necessarily leads to an increase in losses; therefore the x-component can be present equally with the y-componenc only at the very beginning of the pumping pulse until a noticeable temperature gradient appears. Spontaneoua polarization of the laser emiasion with planar active elements must occur under certain other combinations of conditions aad be actually observed in practice [284]. In spite of individual hopeful experiments with planar active elements to 40�204�600 mm3 in size [192], lasers of this type still have not become especially widespread. The reason for this is that for now for the majority of solid state lasers an effort is being made to consider the possibility of Q-ewitched operation. The large-cross section planar active elements are not too euitable for monopulsed lasers contiguration is highly uaf avorable from the point of view of superluminescence which plays an extraordinarily important role in the "slave" operating mode. As for glass lasers with high output energy in the free lasing mode, the procedure discussed above for constructing them obviously is optimal [1941. In conclusion, let us briefly touch on the problems of thermal deformations of the liquid laser resonators. The value of dn/dT here is very large and as a result of the absence of static stresses is not comFensated for any way; therefore the varia- tions of the index of refraction turn out to be very large. Th,e always available mixing of different layers of liquid leads to the fact that the temperature f ield is unstable, and the nonuniformities are irregularly distributed with.respect to volume. For these reasons small divergence of the powerful liquid lasers almost unattainable. Z'herefore the liquid media are widelv used only in 243-244 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047/02/09: CIA-RDP82-00850R400504010010-4 small lasers with tunable frequency, on the angular divergeace of the effiission of which as a rule special requirements are not imposed. � 5.2. Phase Correction of Wave Fronts. Dynamic Holography aad Stimulated Scattering Inasmuch as in the optical channels of lasers usually all posaible nonuniformities are present, even an insignif icant decrease ia which frequeatly is extremely ccim- plicated, there is a natural effort to find methods pesmittiag insurance of high _ output characteriatics even when such inhomogeneities are present. A brief dis- cussion of research for this purpose makes up the content of thYs and the follow- iag sectiona. Optical-Mechanical Correction Systems. Let us begin with the possibilities which are related to purely mechanical c:isplacements of certain elements with respect to the channel. First of all, the systema for automatic adjustment of la.ser resonators which are gradually entering into pracCice deaerve attention. Their function is adjustment of the mutual arrangement of the mirrors in order to optimize any laser parameter. Without going iato the technical details, let us only diseuss the most theoretical problem of selecting a parameter, the magnitude of which is subject to optimization. The output parameter of the laser which ia quite "sensitive" to the resonator geometry aad at the same time most simply coatrolled is its power. Theref ore it is easy to create a self-adjustment system which maintaias the maxi- mum high level of output power. However, this operatiag algorithm can be far from always used: the pursuit of an insignif icant gaia in power can easily become enveloped in the large f luctuations in the positioa of the resonator axis, aad, along with it, also the direction of the generated radiation. In addition, cases - are possible where the shift Qf the resonator axia leading to an increase in power is accompanied not only by "drift" of direction, but also other undesirable con- sequences. We encountered one such example in 1 4.2 - as the resonator axis of the f low laser approaches the point of entrance of the flow of inedium into the resonator which is advaatageous fram the poiat of view of eff iciency, the noauni- formity of the radiation distribution with respect to crosa section of the lased beam and, consequently, the divergeace increase aharply. Finally, it is necessary not to forget that the syatems based on measuriag output power can realize adj ust- ment of the resonator only some time af ter lasing has started. Therefore f or the. overwhelming majority of pulsed lasers suCh systems are disadvantageous. It is - also difficult to use them for certain contiauous-action lasers, at the time of startiag of which frequently such large mieali.gnmenta occur that the laser in general does not begin to lase without resonator tuaing. For all these reasons, the most eff icieat operatiag algorithm for the autoadj ust- ment systams ia maintenance of the reaonator axis in the given position. It is possible to deal with this problem by using the radiation of a low-power external - source which will permit adjustment of the laser without begimting o� its lasing. A system for autoadjustment of unatable resoaators investigated in the already- mentioned paper [2171 was constructed by the same principle; the radiatioa of the suxiliary helium-neon laser is iaput to the resonator here along its axis through a small opening in the center of one of the mirrors. Undergoing multiple reflec- tions, th-is radiation then "spreads" over the entire cross section. If the resonator was misaligned, the intenaity distribu;l-ion with respect to cross sec- tion turns out to be asymmetric, which ia uaed for generating the error signal 245 FOR OFFiC7AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 putting the corresponding automation circuite into operation. The sqstem described in [217] per.nita insurance also oE satisfactorp precision of adjustment of the errors (-5") and "capture" of the initial misalignment (5'-7') in a sufficier.tly large range of angles. The application of an auxiliary laser and the receivers of its radiati4n operating on a frequency differing from the frequency of the basic laser permits possible interference caused by the lasing process in the latter to be avoided. The system with introduction of the auxiliary laser through the central opening is also convenient in that this radiation passes through the adjusted reson- ator oa the ssme path as the generated radiation; therefore it can be used so that the radiation will be directed at the measuring devices and so on visually or auto- maticai7.y. A f urther step in i.mpr w ing the methods of inechanical automatic tuning of lasers is the application of the so-called adaptive optical system. Linnik's idea [286] about the possibility of autocompensation of phase distortions of the wave front caused by a turbulent atmosphere and interferina with the observation of remote objects marked the beginning of this area of study. The typical method of solving this prablem by the methods of adaptive optics coasists in the following: one of the elements of the optical channel of the transceiviag system, for example, the mirror objective of the shaping telescope is made up of many more small mirrors having the possibility of insignif icant translational displacements. By using these displacements, campensation of the large-scale phase distortions is also insured. When creating such devices it is most complicated to deal with the generation of error signals. Let, for example, the prvblem of the formation of a beam with di- vergence as small as possible be solved. The recording and analysis of the form of the intensity distribution in the far zone with invariant arrangement of the indi- vidual parts of the composite mirror still do not permit judgma.nt of which of their displacements are needed to improve this forta. Actually, the same distribution in. the far zone can be observed f or different forms of the wave front; thus, in g 1.1 we are dealing with the fact that the beams distinguished by the sign of the cur- vature of the spherical wave Eroat had identical divergences. ~ In order to obtain the required information about the deviation of each individual mirror from its optimal position corresponding to the maximum axial luminous intensity - of the devices as a whole, it is simpleat to use the f ollowing fact. The dependence of total axial luminous intenaity F on the coordinate of one of the mirrors x measured along the axis perpendicular to its surface is depicted in Figure 5.6. If this mirrar undergoes oscillations with of amplitude Fi much siwrter than a wavelength nears its optimal position xp, the axial luminoua intenaity as is obvious in the figure, re- mains almost constant. If the average poaition of the mirror is sfiifted to anJ poiat xl aoticeably remote from xO, the magnitude of the axial luminous intensity begins to fluctuate with the same frequency as the mirror. The relation betweea the phases of these two fluctuations will obviously depead oa the direction from optimal that Che average mirror position is shif ted. Now, leaving only one of the mirrors stationarq, let us force each of the remaining ones to oscillate with its individual frequency aad let us carry out spectral expansion of the time dependence of the axial luminous intensity. Obviously, the presence in this expansion of compoaents with frequencies equal to the frequencies - of the oscillations of individual mirrora will indicate that the positions of the 246 FOR OMCiAI. USL ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/42109: CIA-RDP82-00850R000500010010-4 2h J", Figure 5.6. Axial luminoua intenaity as a function of the position = of one of tbe parts of the compoeite mirror. given mirrors are subject to adjustment. The phase relations provide information about the direction of the required shifts. This principle was also proposed and rea].ized in the knowa papers [287, 2881. In the first of them an adaptive system of seven elemeats was manufactured and success- fully tested; in the second, a system of 18 elements. Later the possibilitiea of adaptive optics were unconditionally expaaded. Iastead o� the set of individual mirrors now siagle flexible mirrors are begianing to be used this corresponds to the transition from "step" approximatioa of the givea phase distribution to more _ improved approximation using continuous fuactioas. The nwnber of indepeadently adjustable parameters will also increase; familiarization with [334] gives a suffi- ciently good idea about the paths of development of adaptive optics. In spite of all of these proapects it ia diff icult to couat on broad applicatiou of adaptive optics directly in the l,aser resonators: the creation of a low-inertia adjuatable mirror capable of operating under high beam load conditions is a highly complicated problem. In addition, the sma11 phase aberrations and unstable reson- ators, as we have seen in Chapters 3, 4, canaot lead to either multimodal lasing or to large reduction in the output power, aad they only cause phase distortions of the wave front of the generated radiation which caa be entirely eompeasated for also outside of the resoaator. Therefore the nptimal location of the complex adjustable mirrora is the exit. of the shaping spstem where the density o� the -radiation usually is many times less thaa the density inside the oscillator. As for resonatars, among their mechanical correctioa devices probably only the simp- lest sutoadjustment systems have become widespread which can be supplemented by very useful and not too complex syatems for automatic compensation of the second- order wave aberrations which are variable in time (lanticularity). Other methods of conversion of the wave front form are also of great interest. The most universal and common of them is obviously holography. Iio].ographic Correction Principles. Uaually two mutually coherent light beams participate in the recordfng of the hologram depicted in Figure 5.7, a. In the - case of staadard use of halograplhy for recording the image of real objecta one of - 247 - FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R040500010010-4 ~ i ~ . a, bi c) a) - Figure 5.7. Problem of holographic corrections: a) recording of a thin hologram; b), c) reproduction of one of the beama on illuminatian - of a hologram by the others; d) beama and three-dimenQional hologram. these beama I-- is the light scattered from the object illtaainated bp the co- herent radiation; its wave front ha.s a compleac shape which carries iaformation about the imaSe of the object. The beam II the reference beam - has a regular, for example, planar sh8pe of the wave front. When the hologram, which is aa ~ imprint of the interference pattera between these two beams is made and theu illuminated by the reference beam itself II, the light scattered from it in the direction of one of the diffraction ordera has, as is knowa, the same structure as - the beam I has during the recordiag, that is, as before, it is the carrier of all ' of the information about the imsge of the object (Figure 5.7, b). Sowever, the ~ light beams of I and II are theeretically entirely equivalent. If the hologram is illuminated not by the ref ereace beam, but by the object beam I the light scattered in the corresponding direction will have the same structure as beam II has during ' the recording, that is, it will have regular (planar) shape of the wave front. The given process is also depicted in Figure 5.7, c. This property of a holograph also provides the basis for holographic methods of correctin.g the wave f roat of the wane emiasion f irst realized b,y Soakin, et al. [2891. The functions of the obj ect beam of I here are performed by the laser ra- diation, the shape of the wave front of which ia subject to ccrrection. The co- herent ref erence beams II with f lat wave front can be formed, for example, fram part of the same emission transmitted aagle selector. When using holography to decrease the convergence of the laser emission, the decisive role is played whexe the density of tlLe 'efficiency af conversion - the fractiou of the power of the beam illuminating the hologram which pertains to the required diffruction order. T'fie lowest effi- ciency not exc?eding a few percentages occura,for thin amplitude holograms which are obtained on ordinary processing of thia-layer photographic material. Bleach- ing converts the metallic silver to a transpareat compound with a diff erent in- dex of refraction than for the emulsioa gelatin, and the hologram is converted from an amplitude hologram to a phase hologram - the peaks and the minima of the recorded interEerence pattern on the hologram made correspond to sections of dif�erent optical thickness. 'I'he eff iciency of thin phase holograms can theore- tically approach 30'6; such hologr.ams were used when investigating the possi.bilities of the given correction method uaed by the same,group of authers in 1969-1972 [289- 291]. � 248 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 _ Among the other results contained in the mentioaed papera, the realization of phase correction of the radiation of the helium-neon laser generated on one trans- verse mode of the stable resonator TEM [290] deservea to be noted. Before cor- rection, the lasing beam had a flat equi phasal surface in the hologram zone and 3uet as the radiation of any nonzero mode of a stable resonator, it had sign- variable amplitude distribution (see � 2.1). It is possible to interpret the _ change in sign of the amplitude as a phase shift by tt; as a result of the holo- _ graphic correction such phase shifta disappear, and the wave front becosnes geauinely planar (although the amplitude distribution naturally remains nonuniform)� ~d he the conditiona of [290] this has led to a fourf old decrease in divergence, - axial luminous intensity increased noticeably althuugh the energy efficiency of the conversion was a total of ..lOx (291]1. No less interesting experiments were performed with a ruby laser operatiag in the ordinary "spike" free lasing mode [291]. Here the gain in the divergence turaed out to be still more signif icant. The possibility of realiziag the correction it- self in the free lasing mode which is more frequently multimodal when using a planar resonator is, generally speaking, connected with two facts. First, in the case of an optically nonuniform active medium the general diatortion of the wave ~fronc caused by aberrations of the resoaator characteristic of diff erent trans- verse modes can prevail over the differences in the shape of the wave fronts of the iadividual modes (which, probably, occurred also in [291]). Secoadly, although the spatial field distributions caa vary from spike to apike, the peaks of these dis- - tributions naturally do not coincide; therefore the overlap of the holograms re- corded by the radiation of the iadividual "spikes" is only partial. flowever, pax- tial superposition of the holograms must lead to a redu.^.tion in energy efficiency o� conversior.; under tha conditions of [291] it waa a total of 3%. The successful performance of the individual expntinherent method of holographic correction of the serious def iciencies or8aaically in it. The use of thin-layer photographic materials alone immediatelq imposes rigid restrictions on thewoxkiig Wavelength and on the admissible radiation f lux densit3es and, finally, the energy efficiency of converaion. It is still more important that the correction using the previously manufactured holograms ia $enerally possible only in rare cases where the laser operates uader exceptionally stable conditions, and the shape of the wave front of the generated radiation remains the same. - All of this dictates the transition to dynamic volumetric phase holography based oa reversible changes in the index of refraction of certain nonline ~dmedia itioainthee radiation f i.eld; as a result of the presence of the known Bra88 light incident on the three-dimensional hologram is scattered predoffii.nately in one diffraction procedure, and the efficiency of conversion can theoretically reach 100%. A].1 of the subsequent studies in the given field have proceeded alung thir. path. The system which is used for correction by the methods of dynamic three-dimensional holography is depicted in Figure 5.7, d. The volumetric interf erence between two 1Having high energy eff iciency, although less elegant, the procedure for realizing the correctioa in such cases with the help of a stationary phase plate was de- scribed after several yeara in 12921. 249- FOR OFFICIAL U8E ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 FOR OFFICiAL USE ONLY coherent light beams leads to the fact that a hclographic phase lattice is formed in - the nonlinear medium conaisting of alternating lapers with diflfereat iadex of re-. f raction. If sucha f ixed in some way and then illuminated by one of the beams participating in the process of its formation, as a result of ref lection from the.larLice-it acquires_the structure of the second beam entirelq similarly to how this occura in the case of a thin hologram (aee Figure 5.7, a-c). Reflection takea place without loss of intensity if the absorption in the medium is quite low, the lattice is suf f iciently thick (the number of its periods along the direction of the incident beam must be significaatly greater thaa oae), aad the coudition An/n � 4 sin2 6 is satisfied [293, 294], where An is the modulation amplitude of the in- dex of refraction, 26 is the angle between the initial beams (see Figure 5.7, d). The meaning of the last condition reduces to the fact that the reflection from one lattice period taken separately must be quite small; only thea does the three- dimensional nature of the lattice acquire a genuinelp important role. Otherwise the light, just as with a thin lattice will be scattered also in other directions. Hereafter we shall consider that the conditions of three dimensionality are satis- f ied, and the scattering of the light in other directiona is absent. Naw let us consider a real situation where both light beams exist simultaneo+lsly together with the three�-dimensional hologram created by them. As a result of the interaction of each of these beams with the hologram, scattered (or, if one likes, reflected fram the hologram) radiation appears. However, inasmuch as the shape of the wave front o� the scattered radiatioa coincides with the shape of the wave front of anotherbeam, the structure of the iaitial two beams remains unchanged; only redistribution of their intensitiee can occur. The idea of the dynamic holographic correction is conaected with this redistribu- tion: if it actually occurs, Che possibilits appeara for transmission of the energy - of a powerful light beam, the shape of the wave front of which is subject to cor- rection, to a reference beam with small initial intensity and plane wave front. This method of obtaining (more precisely, amplifying) narrawly directional irradia- tion is obviously applicable when the shape of the wave front of the powerful beam varies in time; it is only necessary that these changes be suff iciently small during the lattice relaxation time. Conditions of Realizing the Process of Holographic "Transfer" aad Ita Energy Effi- ciency. "Transfer" on Thermal Gratings. In spite of the apperent "transparency" of the above-discussed idea of using dynamic holography, in reality it is not so simple. A more careful analyais indicates that under steady conditions and in an isotropic nonlinear medium redistribution of the energy between beams of identical frequency does not occtir whatever the initial ratio of their intensities [295]. The reason for this is the following. Strictly def imed phase relations exist be- . tween any of the two light beams and the radiation scattered in its direction as a result of interaction of the second beam with the volumetric hologram. These re- lations depend oa how the holographic lattice is arranged relative to the inter- ference pattern between the beama. Under the above indicated conditions the posi- tions of the lattice and the interference pattern coincide - the extrema of the index of rQfraction are matched with the extrema of the total field iatensity. It turns out that in this case the acattered radiation is phase shif ted by 7t/2 orith respect to the radiation of the beam to which it is added, and therefore it does _ not cause a change in its intensity. The phase shift and togecher with it the result of the interaction of the beams with the twlogram, becames different only when the holographic lattice ia shifted for certain reasone by a fraction of a period in the transverse direction with reapect to the interference pattern. 250 FOR OFFIC[AL U5E ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/09: CIA-RDP82-00850R000500010010-4 Initially we sha11 consider the hypothetical case Whexe th,e radiation absorption in a nonliear medium is absent. For derination of the corresponding equazions let us represent the total field of two beams in the form r E(r) = Ai exp (ik,r) + A= exp (tk=r), where A1 and A2 are the complex amplit�~des of beama I aad II, lcl aad lc2 are the wave vectors of these beams where kl a k2 a k0 - 2"n0/71 where n0 is the indeu of refraction of the nonlinear medium in the abaence of a field. The indea of refrac- tion ia the presence of the two beams will be considered equal n= no[i a dAi oxr,(tklr) Az exP(lksr)exp(ta) 1!]. (5) The value of the index of refracti~n averaled with respect to the lattice period obviously-is nmean a n0 [1 + a(IA11+ JA21 axid the modulation amplitude Dn ~ 2laA1 A21 . The introduction of the factor exp (16) in one of the terma in the right-hand side of (5) leads, aa is easy to see, to the fact that the holographic lattice described by this formula is shifted with respect to the iaterfereace pat- tern by a fraction of a period equal to 6/27r; the direction of the shift for d> 0 is shown in Figure 5.1, d by the dotted arrow. If the conditions are steady state, and the medium ia is?tropic, S - 0; thea the formula for n acquires a standard form n � n0(1 + alEl The calcuZation of the interaction of jight ~eams with a nonlinear medium reduces ~ !r~ (1 2a ~ A~ ocp (lkir) -h A, o~p (Jkir) e~p (t~) h�' - to the solution of the wave equation V E+ k Ea 0 conaideriag the dependence of k2 on the radiation f ield. Inasmuch as the terms with a play the role of a small correction in the formulas for a, the following approximate formula is valid for k2 . k~(n/n0)2: Substitnting this f ormula and the expression for E in the wave equation, as a re- - sult of the calculations analogous to those diseussed in the known paper by Kogel'- nik [296] by the theory of three-dimeneional holograms we obtain the following differential equations relating A1 and A 2 : dA ^i,~ -j.jj- = t -Z~n-' (ncp - n.) Ik.a I /1s., 13 esp ld)r (6) where Q1 2 are the distances along the directione of propagation of beams I and II, respectibely. The equations obtained have obvioue meaniag. The fixat terms in their right-hand sides are identical and do not depend on the lattice shif t parameter 8; they describe the phase incursion occurring as a reault of the fact that the average _ index of refraction %ean in the presence of a field differs from no. Tb,e origin of the latter is explicitly related to the reflectioa from the lattice. Tbis is 251 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY indicated primarily by the preaence of tiu "shift" factcr exp (�ib). Ia addition, iaaemuch as An 4 IAlA211 the amplitude of the radiation scattered in the direction of beam I must be proportional to jAlA21, and ia the direction of the second beam ~AiA21; this dependence is also contained in (6). From equations (6) it follows that for 8- 0 the scattered radiatioa leads onlq to additional phase variations which turn out to be greater for the less intense beam. Whea d# 0(the lattice is shif ted) in addition to the imaginary terms real terms also appear in the right-sides of (6). Accordingly, tae intensity of one of the beams begins to decrease as it is propagated and the other, to increase so- called energy "transfer" takes place. The direction of the "transfer" is defiaed only by the sign of the product a sin d and does not depend on the relatioa betweea the intensities of the interacting beams; therefore the energy of one of them can be transferred entirely to the other. We have succeeded in drawing this optim3stic conclusion after the authors of [294] only because, ,just as in [2941, we introduced the lattice shif t purely f orma-lly, not analyzing the natural situations in which it could really occur. Nevertheless, it will be seen later that the "transfer" will be realized ia its classical version (the interacting beams have the same frequency) primarily in the presence of sig- nificant linear abaorption in the mediva. The corresponding theoretical analqsis wae performed in [297]. In the case of linear absorption (that is, when the absorbed power is proportional to the field intensity) the term -6/2 is added to the right-hand side of each equation of syatem (6) where cs is the absorption oeff i- cient. Af ter transition from amplitudes to intensitiea 1 1 s lA112, 1 2 = JA21 2 the syatem acquires the form dl ~ ~ = f 2k.a1,,, min 8 - v. (7) Hereafter, for determinac,y we shall consider that a< 0(which is characteristic of the thermal lattices investigated later) and sin d> 0. In this case the besm I is the donor, and the second is atnplif ied. From (7) it is obvious that -k0 all sin d has the meaning of the amplification coeff icient in a medium f or a aecor.d - beam in the presence of the f irst. The density of the amplif ied beam iacreases while the condition 1 1 > IO 9 Q/Ok0lal sin S) is satisf ied. It follows i.mmediately from this that for maximum use of the energy of the donor beam it is desirable that its density decrease in the zone of interaction of the beams to a value of Iot. In the most favorable case where the densitiea of the two beams at the entranceo the medium I'1, I'2 are constaat with respect to cross section, this condition can be satisfied by using the flat layer of the medium (Fig- ure 5.8), the thickness of which muat be correspoadingly selected. In Figure 5.9, a, b, the data obtained by solution of the system (7) aa the optimal thiclcness of the layer of inedium for the cases depicted in Figure 5.8, a, b whea the beams enter into the layer from one or differeat sides are preseatedl. Although f or the - same I'1. I'2 the optimal thickaess of the layer in these two versions does not ; coincide, the attained density of the amplif ied beam 1 2 max turns out to be the same; 1The calculationa for the second of these cases were performed by V. D. Solov'yev. 257 FOR OFIFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 the data on it are presented in Ftgure 5.9, c. It is obvious that high energy efficieacy of the "tranaf er" (on the order of 50% or more) caa be achieved only a) 0) - Figure 5.8. Geametry of'the interaction of beams during "traasf er" in a flat layer of inedium: a) one-sided incidence of the beams on the layer; b) counterbeams. ca d~""~1~ t0 J~~~~'4~ 4~ > as ;S � i !~=Rl 0 S !0 /6 SSI~/! e 6 � I... /!i e) Figure 5.9. Optimal thiclrnessee of the layers of the medium and maximum attainable denaity of the amplified beam during "tranafer": a), b) dependence of the thickneea Az of the layer of inedium ia the initial values of the beam densitiea I' , I'2 ia the diagram in Figure 5.8, a, b, respectively; c) dependenie of the maximum attainable deasity of the amplified beam 1 2 max I'1, I12� Key: 1. opt when the initial density of the donor beam exceeda 1 0 by at least several timea; the initial density of the amplified beam also must not be too small. Such is the theory of the interaction of beams with the holographic phase lattice created by them in the quasistationary mode in geaeral outlines. Now we shall discuss the causes of the lattice shif t with respect to the interference pattern required for energy "transfer." In individual cases the ahift occurs as a r'eault of d natural aaieotropic medium in which rhe hulogram is created. The classical example of this ia described in the well-lcnown paper by Stabler and Amodei (298] devoted to the investigation of phase latticea ia a lithium niobate crystal. On illuminatioa of thia material ' 253 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R040500010010-4 charge carriera are relaeased in it, whi,ch is the cauae of local changes in the index of refraction. As a result of anisotropy of tFie given ataterial, tfie drift of the free carriers here is directed in an entirely defined direction which also causes a shift of the laCtice recorded in the preaence of two coherent beams. The authors [298] observed noticeable redistribution of the beam inteasities. It is true that they used the given phenomenon not f or correction of the wave front, but for discovery of the previously unknowa sign of the free charge carriers which can - be established by comparing the direction of the energy transfer process with the direction of conductivity of the given crystal. This phenomenon can hardly be exemplary of the dynamic correction - the setup times here are tens of seconds. ' In the general case of an isotropic medium a def ined lattice shif t can be obtained directly during interaction of the beams, obvioualy continuously shif ting the medium in the direction perpendicular to the lattice planes. The shift must appear as a result of unavoidable inertia of the processes of recording and erasing it. Later it will be seen,that {t is possible to obtain the desired r.esult in practice - using this procedure only when using such comparatively slow recording mechanisms as thermal recordings. According to (297] let us coasider this version. For the geometry of the interaction of the beams depicted in Figure 5.8, a, the - density of the thermal power dissipated as a result of absorption in the medium is Cjll + 1 2 + 2 I cos(27tx/11)I, where A- X/2n sin 8 is the period of the interfer- ence pattern, and the remaining notation was as before. The origin of the coordi- - nates is matched with one of the peaks of tiie interference pattern (it is possible - to neglect its small distortions along the Y and Z axes in the given investigation). A aimple analysis shows that in the case of stationary medium the dependence of the temperature on x in the greaence of such peak release sources has the form T' Tmean ' - 21I1i1, x x(~)~ cos (IM ) with setup time of the spatial modulation t= (TAri _ where Tmean is the mean temperature of the medium slowly increasing with tima, K is _ the coefficient of thermal conductivity, c is the heat capacity and p is the dpnsity of the medium. This leads to the occurrence of the phase lattice with modulation - amplitude Qn = 2~ lll~ K(~)a~l dT I� � If the medium is shifted continuously along the x-axis with a velocity V, the latti.ce setup time remains as before; its stationary positioa turns out to be shif ted by px s(A/2Tr)arctg wT, the depth of the modul tion qf the temperaCure and the index of refraction decreases in the ratio (1 + w~t2)-1/2, where w6 27rV/A. A comparison - of these data with (5) indicatea that the parameters of the "transfer" theory under the investigated conditions assume the values � t (lxx) n T +~0,go) d= erctg wz. As the velocitq of the medium increases, d increases, approaching the o timal value of 1T/2 from the point of view of the "transfer,01 and on the other hand ~aj decreases rapidly. From (7) it follows that the intensity of the energy transf ez process from beam to beam is def ined in the f iaal analysis by the value of I a sin 5 value reaches the maximum equal to 2nx(Z~)1I d7 I, for w0 . 1/T; thus, the optimal 254 FOR OEFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407102109: CIA-RDP82-00850R000500410010-4 _ apeed of the meium is Va ~ C ~x-p f+ln a. Tbe denaity oE the "~doaor" beam - for which the amplification of the second beam as a result of "transfer" compensates far its abaorption in the medium; for VO is equal to I~ = 4kst sin' 0/ C n I-~ 44 1). (8) All of the relations derived above are also valid for the case of "counter" beams (Figure 5.8, b) except here the index of refraction ie not modulated along the x-auis, but along the z-axis; movemeat of the ffiedium must be realized in the same direction. Relation of the Idea of Dynamic Holography to the Pheaomena of Stimalat-ed Scattering. Lasers Based on Various Forme of Stim,latedScattering. Finally the time has come to explain what the logical development of the idea of holographic correctioa has led us to. For this purpose it is sufficient to compare two facts. Eirst, when ia- - vestigating the above-deacribed "transfer" proceas in the coordiaate aystem which is stationary with respect to.not the interference pattern but the medium, the in- teracting beams acquire a defiaed frequeacy difference as a reault of the Doppler , effect. Its magaitude, as ia easy to s2e, is w, which explains why ia this coor- - dinate system the interf ereace pattern is shifted at a velocity V. Secondly, as follows from (7), the differential amt~lificaiton coefficient of the radiation with respect to beam II does not at all de>>end on its inteasity and is completely deter- m3ned by the deasity of beam I. Therefore when the latter has 4uff icient power - that the "transfer," process takes place, in its preaence not only the specially _ formed radiation, is subj ect to application, but also the randomly scattered or "noise" radiation ef proper frequency and direction. The phenomenon consisting of the fact that on illumination cof the proper medium by a powerful coherent beam amplif icatioa of the radiation takes place with frequency usually somewhat shif ted with respect to the initial frequeacy ia, as is kaowa, ~ called induced-.scattering of light. A apecific amplif ication mechaaism was de- scribed above conaected with variation of the index of refractioa as a result of heating of the medium during abaorption of light. This type of iaduced acattering is actually kaown. It was discovered in 1967 and siace that time has been calledstim- ular.ed thermal scattering (STS) [299, 3001. In accordance with the above-pre- sented calculations, the maximum amplif ication of the scattered. radiation occurs on a frequency shift with respect to.the initial frequency in the antistokes di- rection (that is, larger) by w0 . The STS threahold on this frequency is def ined by the formula (8); for other frequencies the threshold increases proportionally to la sin B1maX/la sin 81 = 1/2(wT + 1/wT). The "transfer" observed in [298] in lithium niobate pertains to the phenomeua of induced scattering on conductien electrons in semiconductors [301]. On the other hand, al' types of induced scatteriag permit aaalogous 01holographic" interpretation and are distinguished only by the mechanisms which cause variation of the index of - reEraction and shift of the phase lattice with respect to the position of the in- terf erence pattern. The reaeon for the shif t almoat slways is movement of the in- terference pattern with respect to the medium as a result of the difference in frequencies of primary and scattered radiation (the source of the "seed-"-- photons with ahifted frequency for spontaneously occurring iaduced scattering which - occurs predominately in the forward and return directions, usually is the scattering 255 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047102109: CIA-RDP82-00850R400504010010-4 on the random dynamic fluctuations c� the index of refraction). The fact that for STS the frequency of the scattered radiation is shif ted with respect to the initial in the direction opposite to the usual shift direction fur aimulated Maadelstam- Brillouin scattering on hypersonic waves (SMBS) and stimul.ated Raman scattering (SRS) are explained simply by the fact that the non'6iaearitq parameter a has different signa in these cases. Thus, the dynamic holographic correction reduced to the well-known idea of solving the problem of divergence by constructing radiation converters based on induced acattering - no other is given. Our corrector on the thermal hologram in the moving meditmm is none other than the S.TS amplif ier with frequency shif t compensate3 as a result of the Doppler effect. Supplemeating this amplif ier by feedback - a resonator obviously we can construct the analogous laser. For STS, the frequency shift is so small that it actually is easy to compensate, shif ting the medium: according to the estimatea of [297] if ia the diagram in Figure 5.8, a the medium is a liquid of the type of an organic solvent, the required speed of its movement is a total of -10 cm/sec. If ttLe shif t is compensated, it is posaible to use the part of the initial beam transmitted through the shaper as the amplif ied. The f irst experiments with an amplif ier of this type are described in j3021; the source of primary radiatioa aplit inta two interacting beams was a single mode ruby laser operating in the "epike" mode of free lasing. With a total lasing pulse duration of ,.400 microseconds, the lattice relaxation time was ..100 microseconds, which, in turn, sigaif icaatly exceeded t`.hp time interval between - individual "spikea"; therefore for the exteat of the greater part of the pulse the lattice was in practice quasistationary. The highest energy eff iciency of "trans- fer" was achieved for the iaitial ratio of the beam iatensities of 10:1; af ter _ passage of a liquid moving at a speed of 8 cm/sec, this ratio became equal to 1:3, and the amplified beam power was about SOX of the total power of the i.nitial beams. High inertia of the thermal processes permits observation of the pheaomenon of aon- steady "transfer" predicted in [303] aad consistiag in the fact that for short- term interaction of the beama the energy transf er to the weaker oae takea place even if the medium is stationary. This has the following explanation: when the unshif ted lattice begins to be recorded and the scattered radiation appears, it, in accordance with (6), leads to additional phase incursiona of the initial beams. As has already been noted, for a beaan with lower intensity this incursion turns out to be larger; as a result, the interference pattern, while the processes of setting - up the thermal lattice are taking place, moves through the medium which causes "transfer." The direction of the displacement is such that the weak beam is sub- - jected to amplification. As applied to the problems of correction of the wave fronts this phenomenon was experimentally studied in references 1302, 304, 305,335]. Although the papers studying the "transfer" on the thermal lattices turned out to be highly useful for understanding the possibilities of dynamic holography, it is not possible now to count oa the fact that by uaing this process we will decrease the divergence by the radiation of real lasera. flere the mechanism of recording the lattic.e itself i~ unfavorable. The heat release required for ita formatfon has, in addition to everything else, a negative effect on the optical quality of the medium. As a result of uaavoidable nonuniformity of heatiag of the medium, varia- tions of the index of ro-.fraction averagzd over the lattice period appear; in the liquids, in addition, light scatteriag begins on the formed gas bubbles and so on. 256 FOCc OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R040500010010-4 All of this, in turn, leads to an increase ia tIie divergence of the beam itself into which the energy is "tranaferred."' Tfie ePfects of thia type begin to be manifes- ted in practice by comparatively levels of the beams [335]. ' tlow let us proceed to other types of induced scattering. For ,5TS the energy "transfer" to a weaker beam with the same frequency is achieved without special difficulty even in the quasistationary mode. In the preaence of SMBS and, in, for SRS everything looks somewhat different. The frequeacy shifts here are not so small, and the possibility of compensation for them using the Doppler eff ect is for a number of reasons for the most part speculative. It is already suff icient that in the former case the medium should be shif ted at the speed of sound in it, and in the latter case, many times f aster. Therefore by -using the given forms of induced scatteri:~, having lower thresholda in the majority 3f inedia than SPS, it ia possible to construct oaly lasers and aaaplifiers of radiation with shifted frequency. Nevertheless, the "transfer" with splitting of the initial beam into the donor and amplif iect beama ia realizable and in this case the required frequenL,y shif t of the later amplif ied beam can be abtained using the same induced scattering. When uaing this system [306] conversioa on S~MS was obtaiaed with transfer on the order of 80% of the doaor beam power to the amplif ied beari. As a resuJ.t of the conversion, the axial luminous intensitq increased by igore than 3 arders. As for the SMBS and S,RS lasers, their propertiea are quite well-knowa. ThesF� lasers - have, of course, many epecific.peculiarities distinguishing them fram standard in- verted medium lasers. First of all here although the primary radiation frequently is called pwaping radiation as before, theoretically different requirements are imposed on ita coherence the aources of the pumping of lasers based on induced scattering usually are other lasers with apectral radiation selection. Complex phenomena arise ss a result of the fact that in the case of induced scattering the theoretical role is played only by the magnitude of the frequency different Aw, but not the exact absolute values. Tharefore when the density of the converted ra- - diation itself begins to exceed the induced scattering threshold, lasing is excited on a fre4uency 2Ae,il from the iaitial frequeacy, and so on. Finallq, as already been mentioned in � 4.2, with respect to the nature of the interrelatton between the excitation and the generated radiatiog f ields the Converters based on _ iaduced scattering more resemble f low lasers than standard lasers with a stationary medium. In spite of all their peculiarities, the induced sc.attering lasers are, of course, _ the most genuine lasers. The principles of the eelection of the type and parame- ters of the reffionators remain the same as in ordinary lasers; the reeonator defor- mations exiat exactly the same, iucludiag those caused by heating of the medium. ]3eing the basic, theoretically unavoidable source of heatiaag, Che Stokes losses in the SRS and SP'!BS lasera are much leas than in the ordinary lasers which gives riae to the prospectiveness of this entire area. At the present time a number of experimentsl papers have already been published in which the conversion of the - radiation with the help of S1IBS and, ia particular, S.RS, has led to a decrease in the divergence (a broad bibliography exiets, in particular, in [218, 306]). It is not appropriate to enumerate all of them; let us only mention one interesting area of research, In 1973 when atudying the properties of the SRS converters I307] and $MBS conver- ters [308] with pumping by the radiation of multimode lasers, it was posaible to 257 FOR OFgICIAL i15E ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007102/49: CIA-RDP82-40850R040500014010-4 FOR OFFICIAL USE ONLY observe significant amplification of narrowly directional light beams without sig- nificant variation af their spatial structure. The nontrivialnese of the situation consists in the fact that the multimode pwmping field is nonuniform and actually % divided into a large number of randomly distributed spots with respect to volume, ! the characteristic dimensions of which are defined by the parameters of coherence and the geometry of the illumination. The theoretical analysis of the conditions ~ wnder which the nonuniformity of the pumping field does not imply a change in the spatial structure of the amglified beam (as a result of the statistical averaging ' of the effect of a large number of small nonuni fo rmi ties) was performed in ~ references [309, 3101. The firat purposeful experiment checking this model was described in [311]. 95.3. Method of Wave Front Reversal The Idea and Theoretical Possibilities of the Method. In 1971 another method of obtaining narrowly directional laaer emission was proposed [312]. The basis for , it was the argument zhat it is better to realize wave front correction not at the exit of a powerful laser amplifier, but at its entrance, giving the wave front a form such that after nassage through an optically nonuniform amplifying medium it _ becomes plane. Shifting the corrector fram the exlt to the entrance pezmtts the requirements on ttie beam stability of its elements to be lowered; in addition, even if the correction process has low energy Efficiency, the dissipated power will be small and will be felt little in the efficieney of the system as a whole. - In order to give the wave front of the beam coming to the amplifier exit the required shape, in [312] it was proposed that the procedure illustrated in Figure 5.10, a be used. The reference light wave 1 with plane (or other required) front shape is fed to the laser amplif;.er 2 containing optical nonuniformities ; from the direction of its exit and passes through it in the opposite direction, , being amplified and simultaneously acquiring the phase distortions. Then this wave is somehow transformed to a so-called reversed (or complex-con3ugate) wave 3 of the same freq.uency which has equiphasal surfaces comcnon to the reference wave at the input to the amplifier and similar amplitude distribution with respect to cross section but is directed in the opposite direction. It is easy to shcrw that on satisf action of certain quite reasonable requirements imposed on a laser ampli- ~ fier, coincidence of the equiphasal surfaces of these two waves with respect to ! one side of the amplifier guarantees their coin cidence also with respect to the other side. Thus, the reversed wave, passing through an amplifier containing nonuniformities, acquires the required shape of its front (in the given case , plane). Actually, let us consider the interrelation between the spatial structure of the reference wave and the reversed' wave. Although this interrelation is of the most universal nature, as usual we shall limit ourselves to the case where the scalar approximation is applicable. Let us represent the field of the reference and reversed waves in the form E=A exp(-ikz) and E'=A' exn(ikz), respectively, where A and A' are comparatively slowly varying functions of the coordinates, the - direction of the z- axis is shown in the figure. Using the condition of similarity of the amplitude dtstribution with respect to Crosa section, it is easy to ; demonstrate that on any equiphasal surface coimnon to two waves, 'the values of 258 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2047/02/09: CIA-RDP82-00850R000540010010-4 FOR OFFICIAL USE ONLY A and A' are related by an expression of the type A'-yA*, where Y is a constant factor, the symbol * cie.notes complex conjugation (this is where the second name of the reversed wave finds its origin). 2 ~ ~ ~ - . 3 Z 1 a) . r---~ 4 1 .P4 ~n J._sJ b) ~ i -i I L _ ^J 2f I 4 c) . Figure 5.10. Correction of a wave front by the "reversal" - method: a) scheme for compensation of phaae distortions in a laser amplifier; 1-- reference beam, 2-- amplifier, 3-- conjugate wave; b) effect of decxeasing the conjugation preciaion in the presence of aperture irises; 1-- screen with aperture; 2-- reference wave, 3--�assembly in which "reversal" takes place, 4-- reversed wave; c) diagram.of a laser with self-induction of emisaion; 1-- laser amplifier, 2-- assembly in which "reversal" takes place, 3-- object, 4-- awdliary laser, 5-- front of the emiasion scattered by the object, 6-- front of the laser exit emission. _ Substituting the expressions for E and E' in the wave equation V2E+k2En0, we arrive at the eq uations 0^A-21k(8A/30-0, V2A1+2ik(2A'/8z)-0. Let us write out another equation complex conjugate ta the first: V2A*+2ik*(2A*/2z)m0. It is obvious that in the absence of absorption or amPlification, that is, when kis real, the values of A* and A' satisfy the same difiEerential equatian. Inasmuci, as on a common equiphasal surface these valuea coincide with accuracy to a canstant factor, the comparison must occur in the entire region of space where the reference and reve rsed wave do not go beyond the boundaries of the meditan containing only phase nonuniformities. Hence, in turn, it followe that in the entire indicated region these waves have common equiphasal aurfacea and similar distribution of the radia- tion intensity with respect to cross section. 259. FOR OFF[CIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY The theoretical possibility of using this remarkable property of conjugate waves for autocompensation of purely phase inhomogeneities ie�.umquestioned; in 1965, Kh. Kogel'nik demonstrated how, forming a reversed wave, it is possible to repro- duce an undistorted holographic image of the subjected observed through an optically inhomogeneous medium [313]. On transition to the problem of correction of phase distortions in lasers, definite difficulties arise. The main one of them naturally is connected with the fact thai the parameter k in the amplifying medium is complex and equations for A* and A' cease to coincide. It is primarily of interest that the presence of amplification uniformly distributed with respect to cross section involves only multiplicatiian of the complex amplitude of the wave passing through the anplifier by some additioaal constant; the form of its dis- trib ution remains the same as in the absence of amplification (see �1.3). Hence, it follows that the presence of anplification under the condition of uniformity of its distr,ibution w'th respect to cross section has no influence on the precision of coincidence of the equiphasal surfaces of the conjugate waves. The case is also important where although the amplification is nonuniform with respect to crosa section b ut varies little at distances on the order of the size of the Fresnel zone, the phase nonuniformities are not so large as to cause "mix- - ing" of the radiation with respect to cross sectian inside the amplifier. Then - the equiphasal surf aces of the reversed wave at the amplifier output continue approximately to coincide with the equiphasal surfaces of the reference beam; only the intensity distribution in them becomes different. Inasmuch as the intensity distribution has much less influence on the magnitude of the divergence than the radiation phase distribution (�1.1), the given correction met: od is unconditionally effective also in this situation. It can become meaninglesa only for the most , Lnfavorab le properties of the active medium. - The accuracy of reproductian of the initial structure of the reference wave at the amplifier output can also decrease as a result of the presence af certain aperture irises. This effect is explained by Figure 5.10, b. Let only part of the ~ reference b eam cross section 2 pass th rough the aperture in the screen 1. Then if radiation losses are absent until the reference beam hits the node 3 which realizes "reversal," the. reversed wave 4 formed in this node passes entirely through the opening; conjugatton in the vicinity of the aperture is complete. Sowever, to the righ t of the screen the absence of radiation must be felt in the peripheral - sections of the cross section of the reversed wave near the screen the wave ! som.ehow diffracts at the aperture, and on going away from the aperture, the exactness of coincidence of the equiphasal surfaces gradually decreases. Obviously, I it is necessary to achieve the si tuatian where the greatest possible part of the reference beam entering the amplifier reaches the node realizing the "reversal" operation. This must be done especially carefully when using amplifiers with long path length of the emission travel through the active medium, in particular, multipass amplifiers of the type described in �4.3. The stipulations made cannot shake the general conclusion of the extraordinarily - prospectiveness of the "reversal" method. Let us note another advantage of this method. Preamplifi cation of the reference beam (before it hits the "reversal" node) makes it possible to get along with very small initial radiation intensities. This makes the idea of creating a laser with homing of emission on a given_ 26a FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-40850R040500010010-4 FOR OFFIC[AL USE ONLY object a remote target, a target in experiments in laser thermonuclear fusion and the like which is very popular at the -present time, highly realistic. As was pointed out in [312 in order to provide the hoidng ef fect, it is suf fi- = cient to expose the target to the coherent radiation of an awdliary laser and use the light scattered from the target in the range of the primary laser as a reference beam (see Figure 5.10, c; it possible to get along without the auxiliary laser, converting the basic emitter to the lasing mode for some period of time). . In this case, the optical distortions turn out to be compensated not only in the laser medium and the shaping system, but also in the atmospnere (if, of course, the inhomogeneities do not change while the light travela the distance to the " emitter and back). Obviously, this method can be used to solve the problems of automatic target tracking, and so on. It is necessary to note that the possibilities of creating emitters with a wave front which is "reversed" with respect to the wave front of the emission scattered from the target, began to be discussed in the 1960's (primarily as applied to the microwave band). However, we are talking only about the corresponding correction of the phase distribution directly at the output of a complex multielement system using phase control of the individual elementary emitters making up the system. The development of analogous ideas as applied to the optical band Uecame one of the.Principal areas of adaptive optics naentioned in the preceding seczion [334]. "Reversal" in Stimulated Backscattering. Let us proceed with the investigation _ of specific methods by means of which the "reversal" operation can be realized. Although historically the holographic methods of forming a complex-conjugate wave - began to be discussed first, which we shall discuss somewhat later, general attention has been attracted to the "reversing" method on a somewhat different level. In 1972 it came to be known that a wave reflected back in an uncontrolled induced Brillouin scattering process under defined conditions tvrns out to be conjugate with respect- to the initial wave [316]. Soon, by using the given effect, a ruby laser was built by the scheme presented in Figure 5.10, a[317].. Generaily speaking, at that time a number of observations had already been pub- lished indicating that with forced backscattering frequently something similar to the process of formation of a conjugate wave takes place. In the experiment, the results of which were revealed by Bespalov and Kubarev in 1966 [339], the back-reflected beam had, 3ust as the initial beam, diffraction divergence. Men- tions of the fact that the backscattered beam has approximately -thb same divergence as the initial beam (or the same cross section at a noticeable distance from the cell with the medium in which acattering took place) can also be found in the papers by Rank, et al. on stimulated Brillouin scattering and stimulated Thomson scattering [318, 310], and reference [319] on stimulatedRaman scattering, Although the presence of an entire series of such observations indicated that they are not accidental, a clearcut conclusion regarding conjugation of the scattered and i.nitial wave was formulated only in [316 in the same paper zhe possibility o� using this effect for autacompensation of phase distortions was clearly demon- strated. The experiment performed in [316] consisted in the follawing. Narrowly directional emission of a unimodal ruby laser was passed through an etched glass plate which naturally led to multiple increase in the divergence. Then the emission was focused on a?-o11ow glass light gui3e with me;:hane (Figure 5.11). 261 FOR OFFIC[AL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 FOR OFFICIAL USE ONLY' The beara "reflected" back as a result of etimulated Br111auin scattering through the same lens and scattering plate, acquired the initial structure with very small divergenee. -s Subsequent studies demonstrated that conjugation of the incident and reflected wave also occurs for other types of stimulated scattering [320, 32I]. The mechanism of this phenomenon became Lmderstandable. This mechanism is explained as follows. When a beam, the average density of which greatly exceeds the induced scattering threshold, is incident on a nonlinear medium, a set of scatCered Iight waves appears in the medium. On propagation through a region exposed to primary emissian, these waves are amplified. The better the field distribution of any of them is "inscribed" in the outlines of this region, the more this wave is amplified. Obviously the waves, all of the field distribution peaks of which fall exactly at the initial radiation intenaity peaks, are subject to the greatest amplification. Inasmuch as the propagation of all of the waves is described by like equations, only the waves with the same shape of front can have intensity distribution exactly coinciding inside a defined volume. Therefore a wave canjugate to the initial wave is isolated among the backsca`tered waves. Figure 5.11. Diagram of the experiment in [316]: 1-- iris 6x6 mm, 2-- light dividing plate, 3-- etched plate, 4-- lens, 5-- cell with light guide, 6-- systems for measur- ing the parameters of the primar-y and reflected radiation. From this primitive explanaticn, it becomes tmderetandable that the degree of isolation of a conjugate wave against Che general background of scattered emission must depend on the configuration of the exposed zone, the average density of the primary emission and the nature of its distribution inside this zone. The role of a lens and etched plate in the above-described experiment urill also become under- standab le the lens and the etched plate provide the required average density and nature of distribution. us note that without the plate, the con3ugation effect was not observed in [316] the scattered emission had much greater divergence than the prlmary emission. - Cfi the other hand, if the initial beam has suitable charcacteristics immediately (or acq uires them, on being p ropagated in a nonuniform scattering meditmm), the lens and speci al scattering medium can be unnecessary then the device realizing ' the "revarsing" operation reduces to a cell with a nonlinear mediun. As an ' example we have reference [321] no separate scattering medium was used there. 262 FOR OFFICIAL USE O1NLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02109: CIA-RDP82-00850R000500010010-4 FOR OFFICIAI. USE ONLY Those desiring to familiarize themselves in more detail with the theory of the process of formation of a conjugate wave during stimulated backscattering are referred to the articles 1322, 323, 3401, and so on; in this book we shall proceed with a brief discussion of the general advantages.,and disadvantages characteriting the given "reversing" procedure. Its basic advantage is, of course, simplicity: as we have seen, in additian to the cell with nonlinear medium ane or twa primitive optical elements may be needed. A theoretically important item is the absence of any requirements on the optical quality of the nonlinear mediwn neither the initial nor the induced inhomogeneitiea can decrease the conjugation precision. Now let us proceed to the shadow aspects of using the "reversal" effect during stimulated backscattering. One of them ahould already be understandable to the reader: a conjugate wave is isolated from "noise" only with suitable nature of distribution of the enri.ssion in the interaction zone, which, in turn, is indiffer- ent to the spatial structure of the initial beam. The fact that the differences between the th resholds of the atimulated scattering and destruction of the medium ;(caused by optical breakdown or something sin~i.lar) are usually not too large imposes its restrictiona also. As a result, the dynamic range of adnd.ssible radiation densities of the reference wave turns out to be small. This is especially inconvenient when solving the "homing" problem where the reference beam is formed from scattered light, the intensity of which, depending on the properties and degree of remoteness of the target, can vary within the broadest limits. Finally, an important deficiency of stimulated scattering ftom the point of view of the possibilities of its use in lasers with wave front "reversal," which theoretically cannot be eliminated, is the presence of a frequency shift. The situation is especially with stimulated Raman scattering. Let us begin with - the fact that all of our preceding arguments were based on at least example equality of the frequencies of the initial and the scatteted light. For stimulated Raman scattering the frequenciea are so different th at even in the case of exact con3uga- tion of the waves at the inFut of an amplifier containing signifi cant phaae in- homogeneities., coincidence of the fronts ai'. its output can turn out to b2 far from - ideal. In addition, the exactnesa of cottjugation in the proceas of stimulated scattering itself decreases: coincidence of a large nwnber of intensity peaks of two waves in a thick layer of nonlinear medium in general becomes impossible. Added to this fact is the fact that the frequency shift during stimulated Rama:z scattering greatly exceeds the band width of all of the active media used in power- _ f ul lasers; therefore in the diagram in Figure 5.10, a, c the li ght can be amplified only on one of two pasaes through the medium. On the basis of all of the mentioned facts, the work in the given area offers little promise. - The situation is much more favorable with atimulated Brillouin and Thomson scatter- ing the frequency shifts are too small for them to have a real influence on the exactness of conjugation. They are also less than the amplification band widChs of many active media; therefore amplification can be realized on both passes through the medium. However, other unfortunate situations arise in the final analysis the small frequency shift is the consequence of high inertia of the processes leading to stimulated scattering. Therefore for very rapidly vary- ing wave fronts neither stimulated Brillouin nor Thomson scatCering can be used. 263 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-00850R040500010010-4 FOR OFFICIAL U5E ONLY a) ; b) J a) 2 r. ; d) 6 Figure 5.12. Methods of "reversal": a) "reversal" system for small deviations from plane shape of the wave front; b) four- wave interaction system (1 hologram, 2-- beam subject to "reversal," 3, 4-- reference beams with plane wave fronta, 5-- desired conjugate beam); c) "reversal" system using addi- tional diffraction procedure in the case of a thin hologram , (1 thin flat hologram, 2-- beam eubject to "reversal," - 3-- reference beam with plane wave front, 4-- beam scattered in one of the additional diffraction procedures, 5-- mirror); i d) conjugation with parametric amplification of light , (1 nonlinear crystal, 2-- light dividing mirror, 3-- double frequency emission, 4-- beam subject to "reversal," 5, 6-- signal wave and idler wave). ' "Reversal" by Metliods of. Classical Optics and Holography. Now let us proceed to othPr methods of "r.eversal," which, in contrast to the preceding one, can be talled sCimulated as a result of the uncontrolled stimulated scattering. Initially we ' shall mention one possibility of using the methods of classical optics, of interest -I but not having special practical significance [312]. The simplest device capable ~ of performing the "reversal" operation with small deviations from a plane wave front is depicted In FiRure 5.12, a; it cansists of a lens, in the focal nlane of which a mirror� has Ueen installed. On the system axis the mirror has a projection (or depression) of height a/4 covering only the central peak of the diffraction pattern which appears on exposure of the lens to a parallel light beam. This 264 - FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000500010010-4 APPROVED FOR RELEASE: 2407/02109: CIA-RDP82-00850R000500410010-4 FOR OFFICIAL USE ONLY corrector operates as follaws. Let an almost parallel beam incident on the lens from the right have amglitude distribution A exp(i~), where A=const, 0 is the phase distortion which is variable with respect to the beam cross section. For 1�1