JPRS ID: 10212 TRANSLATION OPTICAL AND THE PROBLEM OF DIVERGENCE OF LASER EMISSION BY YURIY ALEKSEYEVICH ANAN'YEV
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23 December 1981
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OPTICAL CAVITIES AND THE PROBLEM
OF DIVERGENCE OF LASER EMISSION
By
 Yuriy Alekseyevich Anan'yev
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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 2328
[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 HuygensFresnel 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).
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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 Diffaction 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 SolidState Lasers (88).
Divergence of Radiation of SolidState 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 aTaoDimensional Resonator that
Have a Caustic (160). Z~aoDimensional Resonator with Central Aperture
(163). ThreeTiimension Resonator with Coupling Aperture. Discuasion
of Results (166). ,
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 � 3.5. Some Problems of MultipleMirror 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 FreeRunning 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 FlowThrough Laser (215).
� 4.3. Unsteady Resonators in Laswera with Controlled Spectral7'emporal
Emission Characteristics 218
Simplest Types of Lasers with Control Elements (218). Lasers with
Thiee�Mirror Optical Cavity (221). External SignalControlled
Lasers (224). Multipass Amplifiers (226).
CHAPTER S. OPTICAL INHOMOGFNEITY aF ACTIVE MEDIA AND METHODS OF
CORRECTING WAVE FRONTS 230
� 5.1. Thermal Deformations of SolidState 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
OpticalMechanical 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
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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 2328
[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 socalled
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.
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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 longdistance 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�105 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 19611966, 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 overestimatRd.
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.
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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 socalled unstable resonators are then discussed. The buJlding of these
 lasers is the result of an area of research, representatives of which ha.ve 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 Qsimilar 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
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 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 inustreta.in 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$ve found some application. However, the first reaonator in optics
was a ccriventional FabryPerdt interferometer made up of two flat mirrors (Figure
1, c). One mirror is partly transparent, and the stimulated emiasion is coupled
out thraugh it. The principal difference o� this resonator from the ring cavity
is that the feedback circuit passes through the same acti.ve medium, and the emission
is'repeatedly amplified.
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Us eful
a) Amplifier hsi 0%11.
Feedback
.ircul3c_J
b)
c)
 Figure 1. Diagrams of oscillators: a) oscillator
in electronics; b) oscillator in optics; c) laser
with flat optical cavity.
The FabryPerot interferometer has been such a successful resonator system that
even today it is the most widely u;sed type of laser cavity. Such popularity is
 due not only to extreme simplicity, but also to the capability of attaining high
energy characteristics of output emiasion (the reasona for this will be taken up
in section 1.4). Hawever, the situation with regard to divergence of emiasion
etimulated in flatmirror cavities was trivial. The origia of comparatively
 large divergence is explained to a$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 f3.at 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,ffaxis 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,
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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
, steadystate 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 socalled 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 FabryPerot
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 socalled stable resonators,
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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
singleLuode 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 heliumneon 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 steadystate 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
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~ 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' +j1""
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
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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 talc.es 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 singlestage 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 steadystate 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 socalled 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
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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 the.final 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 nonring 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 socalled fastflow 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 medixm 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
selfbalanced 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 itbecame
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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 preeminent heliumneon 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 presentday 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 solidstate 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
iaagnitude greater than that of the familiar heliumneon 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 solidstate lasers. The developers of hi.ghpower
gas lasers are now coming up against juat about the same difficulties aa werE
 previously encountered by the derelopers of the solidstate 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 solidstate lasers.
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CHAPTER l. GENERAL INFORMATION
g' 1.1. Laws of Propagation of Light Beams and Angular Divergence of Radiation
HuygensI'resnel Principle. We shall use the scalar theory of diffraction to study
the propagation laws of lasergenerated 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 HupgensFresnel 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
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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 opti.cal resonatora it is necessary to consider socalled
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 zaxis so that the plane xy is its equiphasal surface:
u(xl, yl) = C= const. Then (2) acquires the form
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Fi,gure 1.1, Huygena.4resnel principle.
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� 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 righthand side, does
not converge in general. Actually, we shall break down the integration surface into
socalled 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
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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 iI 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 socalled
"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
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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 maxitaumrange (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
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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,
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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
farfield 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
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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 riao ?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�
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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).
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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)
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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 cofactors, 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 signvariable value. Let us limit
ourselves to the simplest case of unifornt distribution of the complex amplitude of
 the type
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cos(mf2a)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(uxml i la) ka(aa{M~ 1 2 )
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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, 4r 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
signvariable 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 wi.th
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) lus (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
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All of the crass ternts oscillate arS,th dEfference fxeq,uenci.es; 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 highorder 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
aignvariable 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.
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 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 35 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 socalled 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.
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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 highQ (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 Qfactor aimiltaneous stimulation of
too large a number of them can lead to almost incoherent emission.
 Opticalband 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 highQ 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 highQ 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 naturaloscillations.
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 highestQ)
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 opti.cal.band cavities than the Q4actor concept,
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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 twomirror 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. ma.de 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 opticalband 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 highorder modea and for largemirrer resonators,
that is, they even encompassed cases where machine methods were useless in view of
the extraordinary awkwardness of the calculations.
, All of the abovementioned 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
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 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
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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 zjF!/ 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 11 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 deai.red 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
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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 case.in 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 ABCDmatrix 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.
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~ 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; with.an inErease in the radial index p,
the reoion of noticeable intensity shi,fts in the direction of larger, r,and so on.
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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 dashdot 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.
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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 IluygensFresn=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 dashdot 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 righthand
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
leftliand 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.
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Finally, from tlte same HuygensFresnel 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 li 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 twomirror 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
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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
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from the point of view of beam divergence1 fox laxge apertwces lasing takes place
in them on highorder 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
twomirror 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 oncurt: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 aboveinvesti
gated symmetric resonators, but any other twomirror 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!'
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. %
.
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 = 1Z/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 twomirror 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.
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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
recordsmall 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
"se1fimaging" (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
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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 Qfactor 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 Qfactor 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.
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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 twodimensional case
 of interest to us;
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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 radiusvector 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 omi.tt3.ng 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
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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 steadystate  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.
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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(xal)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 abovepresented 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 cylindri.cal 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
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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, finescale 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 abovedescribert 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
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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 aboveindicated
condition, is the socalled wavegui3e 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
waveguide 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 waveguide 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
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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~mnA1e =
A v?,,,N312; 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.
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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].
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.
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, cf, 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
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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 solidstate lasers performed in the
1960's.
Early Observations of Stimula*ed Emission of SolidState Lasers. The first experi
ments demonstr ated that the atL;;ular divergence of the emission of solidstate lusers
was significantly greater (us.ally 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 [3740]. 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
105 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 highspeed pltotography, recording
the f ield distribution for individual spikes. Even with highspeed 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
lowestorder 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.
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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 secondorder 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 higherorder 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 abovedescribed 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 [5153]. 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
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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 solidstate lasers for many
years.
Divergence of the Radiation of SolidState 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 lowangular light scattering is 0.010.1 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 520'. The core is surrounded by a system of com
paratively intense rings [5760, 39, 46] with angular radii equal to the radii of
the rings in the FabryPerot 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 [5759] 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 farf 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
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the amplifiers [63] and lasers [6368] 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 highparallel beam from an
extprnal source af ter single passage of it through the same sample [6769].
In references [6668], 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 L1/ ~ 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 (socalled 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 speci.men.
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 twofold 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
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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 lar.ge misalignments, it breaks down into a number of individual spots [77
80]. Resonators with unifor.m acti.ve 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 Qnodes, 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 [3943, 5759, 62, 7780], 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 steadystate 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
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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 soiidstate 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
Sf J Z> 0 1 2 d 4 5
nm~:  ~5
d.5  x
LJL
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 xaxis, 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,
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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 singlefrequency 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 differentfrequencies). 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.
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~ . 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
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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 [8789], 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 steadystate 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 fourlevel 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~
~ vp11n~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 1cos2qiTrz/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
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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 (otherwi.se
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
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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 loweroXder 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 higherorder 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 highorder 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 wi.th
simultaneous excitatian of both modes.
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_ 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 rectangular mirrors, with
in the framework of the abovediscussed 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
twodimensional 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 fourlevel 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].
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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 observed 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 farfield pattern.
lIn real lasers such effects can be manifested only in complete absence of any reson
ator deformati.ons, including thermal.
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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 steadystate 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
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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 abovepresented 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, itwas 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 equivalentto.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
highorder 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.
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 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 wi.de?y 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 HuygensFresnel 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 wi.th 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)
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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 mirrors,
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
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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 tfirough.it,
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 twodimensional 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 abovediscussed 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.
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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, 1041061. In the last of the aboveenumerated 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 increa.se 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 secondorder
~ 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.
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, 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,
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tt"
10j
10''r
>0if
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  PI 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
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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 selfconsistent 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� anomalousdtsQerston, 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.
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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 smallscale 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 mutu.al 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
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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 opticalgeometric 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 vari.at~,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.
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~ 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 largescale optical inhomogeneity, the resultant distortions of the wave front
of the steadystate 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
_ succcs. 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 withaspherical 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 oth.er 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
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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, bresona
tors with dihedxal reflectors [120, 121]; cresonator with cen
tral "indentatioa" [122]; dresoaator 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 lefthand
reflectorthe "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 righthaad 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 highordar 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 heliumneon 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 itsreflectors, 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 sharplynothing 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 Lastmentioned version, the possibility of using the resona
tors depicted in Figure 2.22, ac, is unquestioned. 'However, they are all char
acterizee by the same deficiency as the ordinary atable resonator itself: under
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FoR oFrIc;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  Lz(1 
lln R) a socalled 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 heliumneon laser; the ratio L/R is plotted along the xaxis,
the line L/R  1 is the "stability" limit.* Oa approaching it fromthe 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
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 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 higherorder 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 correcticns.
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
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_ VVK vtr 1%.U%a. vou, vV....
waveleagth, but also the directioa of propagation 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 reflection., 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 [133140] 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 valuesof 0 's t(m'+ 1)9diff/2
(see Sections 2.2, 1.1; Adiff `X/Za), the magaitude of the.loases introducedby
(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 themagnitude 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�
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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: 1active sample;
2plane mirror; 3epherical mirror;'4 iris wfth aperturef,~ `
lens; 6Fabry and Perot etalon; 7planeparallel plate.
 Figure 2.25. Shape of the passbaad of different anSle selectors; Gaussian 
selector (dashdot 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; firstorder 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, secoiidorder wave aberrations) in the same variation range
of Tt turns out to be greater than in the case of a singlepass amplifier by only
21.5 times; higherorder 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): aamplitude dis tribu
 tion; bphase 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.
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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 onesided 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 INequiv S S; curve IINeQutv ~ 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 Vtype 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 Qfactor alternately exchange places
(similarly to how this occurs in Figure 2.21). This exchange takes place near
1"SQ
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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 twodimensional resonator [160J and a threedimensioaal 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 threedimensional case these laws are also maiatained for large Nequiv at
the same ti.me as in the twodimeasional case, beginning with a defiaed value of
Nequiv, the curves cease to intersectthe 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
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~
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: alosses in the
twodimensional resonator, M= 1.86 [157]; blosses 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);
cthe eigenvalues in the threedimensional 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
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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 abovedescribed 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.
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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 aboveenumerated 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 deviations, 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.
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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 twodimensional resonators made of cylindrical mirrors with
comparatively small Nequiv. The calculations were performed by the aumerical
method of FoxL 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 higherorder 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
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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
f4
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 resoaator 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; bphase 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 twodimensional
unstable resonators with small Nequiv slight smoothiag of the edge is sufficient
(let us remember that far large Nequiv the degeaeraCion in the twodimensional
resonators is absent even without any smoothing). This conclusion can be
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directly generalized to the case of a threedimeasional 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 threedimensional 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)  Z1/ ~ ~�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 solidstate 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 0.11 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.
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. . _
Everything that has been stated together with certain other results of the
aboveenumerated 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 steadystate 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 cease.to be unimodal only in the presence of such large dis
turbances that the fo naulas (14), (15) become inapplicable even for tbR 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 siaglefrequencq 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 halfinteger. In this case the difference of the losses of the two highest
1,58
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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 twodimen
sional resonator for Nequiv '30, M= 2.38 [166].
In conclusion let us aote that the abovediscussed 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 selffocusing 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 socalled aaodizing of the aperture). 9
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�3.4. Unstable Resonators with Central Coupling Aperture
Initial Premises. Oscillations of a ZfaoDimensional 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 smallfield 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 highestQ 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
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eigenvalues of the integral equation of a resonator made of convex cylindrical
mtrrors of infinite dimensions have the form
_ Fv ~~l) = D, ( t etn/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
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CHAPTER 4. AFPLICATIONS OF UNSTABLE RESONATORS
� 4.1. Unstable Resonators in Pulsed Freerunning 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 singlemode 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 IItype. 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.
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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 (socalled 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 oneway 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 largescale 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
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shall ].i.mit ouraelves to the analysis af the simplest case where th,e pumpiag is
uniformly distributed withrespect 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('1F 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:
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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 integrationleads 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
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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 fourlevel 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 $hows the efficiency of the resonator g calculated using the last
formula as a function of the value of M for a number of values of Q0L and kampL (let
us remember that in the case of a fourlevel medium the ratio k~p/cto is the amount
the lasing threshold is exceeded in the abseace of losses to radiatior:, that is,
for M 9 1). The data for a telescopic resonator are compared with the data pertain
ing to a planar resonator with the same total losaes and lasing tbreshold (the
= reflection coef�icient of the output mirror R' is 1/M2).
= Eron Figure 4.3 it follows that the eff iciency of the energy conversion in a tele
scopic resonator in the given case is somewhat lesa, but it is very close to the
eff iciency in the corresponding planar resona.tor. Thus, for a laser with a tele
scopic resonator, the formulas of � 1.4 can be used under the condition of replace
ment of R' in them by 1/M2. In particular, for QQL that is aot too large, the value
of X can be determined using the expression
Xln M i_ ln M+ asL
n 34L ( kyaL j' (3)
 the optimal magnif ication of the resonator, by theformula
ln M ra) z os L/cy�,o7vp i~r (4)
.
Key: a. opt
finally, thc maximum value of X, just as in the case of a plane resonator is
approximately equal to
Xma: ~ (1  YQolkj~~~. (5)
The abovediscussed analysis was performed ia ref ereace [187], being the f irst
example of calculation of the energy characteristics of lasers with unstable reson
 ators. Later, more complex cases were iavestigated wbich require very awkward
machine calcuLations (we shall touch on the methods of performing these calculations
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in g 4.2). Thus, in I1881 calculati,ong wese made of the efficiency for nonuniform
distribution of the pumpiag with re.spect ta cross sectioa and large excesses
over the threshold; it turned out that the formalae (3)(5) rEmain in force under
the crnndition of aubstitution of the value of kamp averaged over the cross sectioa
in them. Analogou$ 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
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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 singlermode 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 ofunifoxm 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 abovediscussed 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, 192197], 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 twomirror 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 sorved as a prototype
 for the sPries manufactured GOS1001 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 1520"; 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.
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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 solidstate
laser with telescopic resonator1152]: 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 yazis.
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 1.52 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 23� 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 with.active 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].
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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 twomirror 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 ha.ve 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
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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 propagated 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 twodimeasional 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.
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~
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 singlemode 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 lasre 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 "hiAhorder
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
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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 rt 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 largescale 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.
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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 19711972 there were alreadp lasers with
' brightness of ~1017 watts/cm2steradian) [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.
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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 weYlknown, 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 [207209]. 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 th.is 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,....
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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,
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_ 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 [207209] 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 abovepresented relations it followa that the time of formatioa of a diffrac.
tiondirectional beam with fixed dimensions of the resoaator decrea.ses 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 useunstable 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 limit.by 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 19691973, their subjecta were lowpresaure 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.
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 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
5'103 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).
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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. Th.is 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 fastflow 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 fastflow 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.
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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 through.an 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].
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 These arguments are formulated in 11541, and tfiey, wpse confirmed by tbe experimental
data of the abovequoted 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 selfbalanced
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
_ singlecomponent 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 singlecomponent 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 eolidstate 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 Qf 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 selfconsisteat 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
largescale 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.
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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 selfconsistent 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 finding 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� ea.ch of these procedures pr"ant very convinciag arguments
in its favor, it is now diff icult to determine which of them is actually more
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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
[222227, 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 twomirror 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 aJ.ong 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 oMirror 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
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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
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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 twodimensional. 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 threedimenaional 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 twodimensional 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 tb,and side has been completely con
stxucted.
For optimization of the reaonator, its magnification M, the widtha of the left and
righthand 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
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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.mLr 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 lefthand 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 lefthand 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 thand 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;thand 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 leadsto 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'
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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 tiiand paxt of the resonator subj ect to variation wh.ea
= investigating the righthand 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 righthand side.
 The construction of the solution in the righthand 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 thand 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 lefthand 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 oacillatcry.energy 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 righthand side, losiag very little in
efficiericy, and on the other hand gaining sigaificantly in size of the mirrors.
For the aboveiadicated 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 righthand
 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�thand 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 tside 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 thand side, but also the righthand 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
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X alread}r rea.chea 60%. I.t makes no speci.al 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
lefthand 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 aboveinveszigated 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 0.60.7 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 righthand side.
Problem of Forining Uniform Field Distribution over the Cavity of a
FlowThrough Laser. The aboveinvestigated 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 fastflow 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 righthand side k2 = 15 cm:
a) dependence of the magnitude of X on the length of the lefthand
side for M a 1.33; b) the value of R as a function of parameter M
f or Ql s 3 cm.
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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 twodimensional 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 abovedescribed 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
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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 sa.me 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 90degree 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 flowtype
C02laser 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 twodimen
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 flowtype 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
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aperture it is sufficient to resort to aeymatetric output of the radiation. In the
caae of a thzeedimeneional 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 twodimensional 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 lefthand 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 lefthand side almost does not change. As for the righthaad
 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 righthand side of the ordinary resonators. However, as a result of much
greater radiation density the eff iciency of the righthand 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 prospectiveness of the application of unstable reson
ators inveatigated in 9 3.6 with field rotation in continuousaction 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
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also given timespectral 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 giantpulse _ laser with unstable resonator
 belongs to the year of 1969 [189]. This was a solid state neodymiumdoped 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�105 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 fastacting shutters
usually have small aperture, the prospectiveness of using such systems to construct
spikeless, monopulsed and other controlledQ lasers becomes obvious. Thus, at the
beginning of the 1970's highly improved a giantpwlse neodymium_., glass lasers were
 built with radiation brightness of _1017 watts/(cm2steradian) [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 4switched 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).
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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 hardly lead to good results. ,
Thua; ia giantpulseJ.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. �
1SelfQ 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:
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Lasera with ThreeMirror 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 threemirror 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 twodimensional 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 twodimensional 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
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 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 selfexcitation 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 threedimensionp.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 twodimeasional
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
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of thia line with iatense singlefrequeacy 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 selfexcitation 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 neodymiumdoped 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�105 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 selfexcitation 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 abovementioned division
into the "master oscillator" and "amplif ier" is purely arbitxarp inasmuch as the
radiation control of the "amplif ier" is realized whea the selfexcitation 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 threemirror 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.
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Ending with threemirror 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 smallcross 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 SignalControlled 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 selfexcitation 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].1lmowa method of
obtaining the given spectraltime characteristics. This method was used success
fully in plaaar resonators (see, f or example [240243]), 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 spectraltime 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 selfexcitation 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
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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 selfexcitation 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 theirlimita 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 neodymiumdoped 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 solidstate 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 C02lasers : 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 remark.able 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
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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
aelfeaccitation in the abaence of aa external signal it would 'oe a"pure" three or
fivepaes 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
aboveindicated 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 smallscale 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
selfexcitation 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 singlepass amplifisrs with telescopes hetaeen them (see 92.6,
 Figure 2.28) [239].
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a)
. . .
.
C)
Figure 4.19. Telescopic amplifiera: a) multipass, b) threepass,
c) twopass 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 singlepass 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 tb.ing here is not so much efficiency as it
is the weakaignal gain, which preferablq should be maximized.
The amplif ication of the weak signal in the npaas 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 wi.th 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 selfexcitation 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.
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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 threepass 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 selfexcitatioa 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 xcomponent can be present equally with the ycomponenc 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 Qewitched operation. The
largecross section planar active elements are not too euitable
for monopulsed lasers tr.is 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 ang~:l.ar divergence of the powerful liquid lasers
almost unattainable. Z'herefore the liquid media are widelv used only in
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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.
OpticalMechanical 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 selfadjustment 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 contiauousaction 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 lowpower 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 heliumneon 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, this radiation then "spreads" over the entire cross section. If the
resonator was misaligned, the intenaity distribu;lion with respect to cross sec
tion turns out to be asymmetric, which ia uaed for generating the error signal
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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 socalled 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 largescale 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
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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 lowinertia
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
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~ 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 threedimenQional 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 through.an 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 thialayer 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 19691972 [289
291]. �
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_ Among the other results contained in the mentioaed papera, the realization of
phase correction of the radiation of the heliumneon 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 thinlayer 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 threedimensional 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 threedimensional
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.
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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 latt:Lce.is 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.larLiceit acquires_the structure of the second beam entirelq similarly to
how this occura in the case of a thin hologram (aee Figure 5.7, ac). 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 abovediscussed 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.
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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
obviouslyis 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
righthand 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 threedimeneional 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 righthand
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
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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 rightsides 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 ormally,
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 righthand 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.
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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) onesided 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 welllcnown paper by Stabler and Amodei (298] devoted to the investigation of
phase latticea ia a lithium niobate crystal. On illuminatioa of thia material
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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 xaxis 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 TY.is
value reaches the maximum equal to 2nx(Z~)1I d7 I, for w0 . 1/T; thus, the optimal
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_ apeed of the meium is Va ~ C ~xp 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
xauis, but along the zaxis; movemeat of the ffiedium must be realized in the same
direction.
Relation of the Idea of Dynamic Holography to the Pheaomena of Stimalated 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 abovedeacribed "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 abovepre
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
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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 wellknown 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.
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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 small.energy 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
particul.ar, 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 wellknowa. 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
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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 socalled reversed (or complexcon3ugate) 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
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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
selfinduction 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^A21k(8A/300, 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.
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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_
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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 complexconjugate 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
backreflected 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).
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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 primary 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 abovedescribed experiment urill also become under
standab le the lens and the etched plate provide the required average density
and nature of distribution. I.et 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.
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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 b.ad 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.
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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
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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