EVALUATION OF THE INTENSITY OF A WAVE DIFFRACTED FROM A DIELECTRIC SPHERE

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000200110012-5
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 10
• Document Creation Date: January 4, 2017
• Sequence Number: 12
• Publication Date: August 25, 1952
• Content Type: REPORT

Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
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OF A WAVE DTk'FRACTE~ FROM A ~~ELECTRTC ,SPHERE
EVALUATION OF THE TNT~?N~~T~ ,
autl~ar; L. Ae ZY;ekulin ~ ~
r f
r ~ ' Tekhnich~kikh N'auk~ No 9, , " ~ ,
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nl
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def.
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' rourco: T~vest~.ya Akademii Nauk SaS 9 ~
.4;
J.9~b~ pp 12~~-121~~
1
STAT
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`a Nonhorno~en~;i~ties in the a~Lmaspl�iax^e in ~ha form pf clau,cl f orrna~~ions (~labulss )
,
affect the propagation of ultrashort waves xn connection with ~tl~.i.s, we cansa.~er
~thc~ phenomenom of diffraction in the simplest case from a sphere with a dieleeM
~t
uric constant r which differs little from the dic~lec~tric
constant of the surrounding medium~(~~ar t�~~'~~~. ~ ,
Let a plane clectroma;netic wave, propa~atin~ on tree negative side of the
4
.r
z-axis ~ra.th v~.b~^atians of the electric fie~.d alonr~ t1zG x-axis (unit ampl.itl~d~ of
.1~
. vibrs,tions is assumed), be incident unan a cliclectric sphere of radius Zt v~rwth~
center Q at the of^~.~;~.n of the coordinate system. ~ y
of
In tl~c; spl~~;~^c:~al system (r, caorclinate3 (wa.~tl~ or~.~;in at U), the car�~ a
nts f tl~e elec~tz^ic ~'a.e].d ~of 'the diffracted wave are('' .
pane o ,.y
~ ~ ~ ~ ~2
~
~a)
1-Iere k M ~ is the wave number.. a~' t~�,te external ~.edium (with respect to
~
.
' ~ 1, ~tlze sphere) Inside ''the sphere
/
~ ~
~
The cyl:~nclrica~. f~unctic~nw
~
~
~ ~ l ~ ~
are nebye Harmed,, '
,1~
~ Vic limit oursclve~ to consicl~ra~tian of~the ,fe~.d , on the z~axis anly, fai^ which
~ ~ differs frr~m 1!r'~ only in tl~a~ cos i~ replaced by szn in ~xpr~ssiarx
' '
~ ~ ~In this work, we ara ~.n~eres~~ed :~.~a the ~l~ctroma~ne~t~c f~.al~ only at ~~a; cons:~der-
vi, ee ~ _ ~ ~ ~ ~ 1
~k ~ ~ ,
` ~ ~ ~ able dxst~~..ce~ from thu ~~t spl~~re ~(r . ~`It known, hat ~ ~ or
I t ~ ~ ,
' ;1 ,
r
cd~ 4#~ 1p~,_ _
.i w:,
ij
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a ;
, ,1,, r, ~ ~ ~ 1
~ ar ~
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~ ~ i ~ ~ ,ti. r l ~ ;
i'~, ~ ~ , ~ , .
,
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
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Formula takes on the ~'oxm
~ '
y~
/I"'~ I i,~j i
t i
a
~i '4
~,t
and moreover ~ M
1 ~ ~ ( )
r ~ ~ 4
+ ~ ,
l~
obtained from th~.s expression if the constant n ,
The coeffieient is ~
f the ri h~hand fide are shi~'ted from the subw o
in the numexatnr and denoma.nator o ~ h s
trahend to the minuend. (~.1 `
;
at for ~~*Ir~ ~
. sim lification is obtained if we assured th
' further P
usin the identity
.and by ~
a
~r,
e innin ~ with those of the second order (with ,
his re~ardin~ inf in~.tesmals, b ~ ~
. after all calculations that the difference
respect to we fa.nd
4
~ ~
~ fox brevity
where x� has been set equ ~ i ,
,series 1) and (2).converge very slowly if-the, wave a,
It ~,s will known that the
s'~derabl less t~ian the radius of the ,~phere N,.
length of the ~,nc~,dent :wave is con ~ Y
- ~
' ffrdction of ul~ras~ort waves .discussed,. 1`,
Thies is what obtains ~,n the prob~.em of d~. n
;~4
. , ,-:R,

r, here. r:
erefr~re for c~rl~,nd~a.ca1 function i.n the eguala~ty ~ ~ ,
B assumption kR ~ l ar~d tl~
S{4
y
With re5 ect t0 the
eb e s s asymptotic formulas, Tk~e summ~.ng p
F
we can use D y '
C~)9 a.
c.: a
in ~x ressi;on is divided into three parts
~ ~.ndex,~ p
~
~ ,
.4
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4 , ' r 4~ ~
' MINtl ~ M'
~ i ~r,
' ~r t . s i, i. r ti
, ,
r ' ~ ~ ~ .r ~~~,1
r
+r
~ i i ~ I
, ~ ~
~ ,
r
i ~ i
L . M ~ 1 1
y
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a, a ~ ~Y~
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~
' n` , _ ~
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;f' ~ iii ~ ~ ~I 1 ~ ~ ~ r i i,,
` ~ r ~ ~C~ r i i'. + ~ ~ ' fir.
j1 i ',4 I ~ tiq! i Aa ~ ~ - .
i ! I i ~ ~ ~ y.
i
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
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-
fi. q,
r
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t ~ '4 vt r
. ' ~ ~ i~iM fv
~ i1
~ ;~1
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,~i~~
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the Deb e formu~.ae for the three re~iar~~, t
beoauee of the d~.ffe~^~an~ faun of ,,4;
/
~ ~ ~
0 of two cone ecut~.ve member a of the
the absolute value of the rats
Compara.ng
,
fa.:~t region
asymptoaa�c sera.es in the ~ ~ ~
M ~ '
~ a ~
a~
� simi.lar ratio for the second region
withG~a ~ ,
~
+JJ~1

~ " r~' ~~y r
w~M C 1 ~ n.~,,,...,.nn~w~.,...~w,,,n~MA.��~ ~ ~ ~
V " ~ r~ ~~//~yy~yyy V t
1N ~
~ ~ :r
~ ~
the boundary of these regions ~ ~.j
and seating appro~,~.mately' at o
~
, ,
J ~
~ / ~ ~ fi
we fa.nd . ~~f a~ ~ } ~ '1
~ ~ ~
~ r
bird ~ ~ ~ } and s ecand .,I~
he same procedure for ahe a ~
~r repeata.ng a
wha.ch determine the boundaries
e desired inda.ces and ~
regions, we find th
;'N
~
f th e three ~e gi ons , ~
o ~ ~ ! ~ ~ ,
. ~ ~~a
~ ,,q~ 'consists of ,w
xe above the compa~ent E ~ fee equaaa.on ~ }
In acccirdance with t~ ~ ~
� first sum, :.using ~h~ ~o~,lawing ,
w~:~.1 ca~.cula~e, the f
t~ree~ sums; S� ~ l~ 3}A GJe ~ , "r""r
~ ~ ~
~
Debye ~f'ormulas ~ _
~ ~ _ ,
T r '
at~.on we roxirnat3.on
haws as a first app
formu~.a ~.nta consider ,
Then, taking r
~ h ~ a
r
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r
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r; ~
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i
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wit, ~ ' ';~~r ~ ~ ~ i
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
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' :b the ma ~n~,tude off' the sun ~.n the side of th~.s oquat~.an~ ,
~,pt ~xs eva~,ua ~ &
changes by a sm~~~~.va~.uv
~~.nc;e by assumpt~,an x a,s much gxeater than
a
;~s
~ Th~refaxe by gxaup~.ng the companemts of th~,s sum y~,
fox a un~,t incxease a ,
oven I) ~ ~
41
in paa.xs, we have (fax an 1)
,
r~ a~~ ~ ~ ~
~ ~ ~ ~ .,,V
5) 4 M t,',
+r ~ ~ i
~I
y
we have taken rota cons~.dexat~.an 'that
Here, j~'
1
s odd the summing in the 1eft~hand side of the equal~.ty should be
r
;a~
de u to and then the component correspond~.ng to the index' ~
shou~,d be added� xn the two sums of the right hand. side of .the latter~~~:equia~.a.ty,
~x
the index nskips~" to the values 1, 3, ~ ? ~ We can again return to 'the
v ues 1 2 raved relationshi
~ ~ ~ 3, , if we use the easily p p ~ R ~
a ~ J
~ ~ A ! '1+
and assumes the values ,;,r
where F " 7 is a slow~.y vaxying funct~.on of ,
o
~ and ~ �
~ � ' 91.n ~ We sha~.~.' ~ ~ ~
In the case under consideration, F~~ ~ ~ ~ a ~ ~ ~ ,h
' 'yM
u us of finite d~.~ferences (3) in order,to calculate ,
use the. methods of the calc 1
.
the sum, in tho right hand tide of 'the last ~ela~~l.onsh~.pa ,
- I�~
We .introduce the funetian
~ i~ ~
w.
_ which has th~~ e~~t~~"than~`'a~. ~ ~ ~~~1~ ~ represents approxa.ma~el~r the ~ ; ,
~ ~ Y ~
ctuall ca~,culat~.ons show, that for a s~.awly vary~,ng flu~ctican .
. ~ , t~.on 'to be summed,... A y,
~ Mrr~ ' ~ brk ~t~"~ ~ w~~ ~ 5 , ; li
~ t, ,
a,
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
~~h" '
;r,
A
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ry~~ ~ ~
Y p~ ~
ai
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'r"r' p ~ ~ ,i,~
wA f,~ 4 ~ ~ �n '
~ ~t;~
~ n
,i;
~ Burn
his used 1;a c~.au~. ate the ' Cis"
Tb~s s;pproac (~.l ~ ~
xs essent~~a].J.y determ3.ned
~ as ~th~.s i'ormu~.a shows,
The order o~ the Burn ~ ~ e when ~ ~ we re~,ain
~ Far the ~ransa.tion~. cas ~
b~ the quan~a~~y ~ ~ ~ ' .
y ~ ~ r
e foz~nux.as ; ~h,
r, ~ ~ ~r, i
~.rs~ three members in the pebY
, ~ ~
w a .w~
Mr ~ e z
~ , ~ ,awl
~ .~.e.., ~ L
f ~ Y M S ~ , l r~, 11rrMr
~ . ~ ~
YM I,
~ ~
' ~
j ere
~
V 4
k
1
for cyl~.ndrical functions ,13
The ~e~.l~~a'~ formu~.~ I'~~
i'~4;~
I.
P I
. e/ Y`
fMN"'. ~
~
,Y
U y ~ ~ ail {
~ andin~
~ ire ~a.nd 'the corre~~
~e ~ur~c~t3.ons in ~ormule ~ ~ ,~~'~P"
~ubsta.tu~~~ the / ~ ~ ~
,
the sum ~ ~ ~
e~cpressxo~? for ~ 2
1 [I ~ ~ Jp1//.~ ~ ~ r
e session in the .brackets. wi . ,
~ ~ the order o~ th p -
" ....,For . - , 9 orde the fire
~ a~ d the r ~o~ en sum
ta.a~.~' ~ upon the quart y o -
I dcpei~d es s en ~ ~ y~
red by the product r
1 ~ wi~.~. be de'terr~.
,I
� I ~a tt~.I; ~ ' ~ ~
i
i'
ar~ ,
i` ~ o-1 ~I ~ ~ ~
i
r~~ ~ ' gygl I
h
I ~ ~
f ~
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I i I� r I.,
~ ~ i ~ ~
4 ~ '
r~. ~
t
'i~ r
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11 Ili ~ ~ ~
1
1 1.
1 ~
1
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`~'i f ~ J~ ri.' ~
A
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t~
r ~
i d ~ ~ ilii,
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
~y~~,
,,e,t,t M
is
Y s;
r
i ''~~i
k
I
r
j 1 ^
1 ,
I
~ ~ ( ~ ~s1;
~ Iny`{
! 1 1
1 1 ~
P~'
. r ~ ~ ~ ~
4'
;,3ti
~y
/J ~ 1NMMA ~ ~ the asymptat~a ~Y~~
x+"~,nally, ~ the thixd xa~~.an fax ~{.,',1'~~,_.'
formulas far cy~.~.ndx~,ca~. fune~tions ,y~,e~,d
t1' ~ ~ ~
V ~ ~
c~ ~ and ~ ~ - cosh ~b
wo set '~'G ~p ,than cosh ~ ~ ~ o
r,,
and '
~ ~ f~' Fax a lame x~ ~x ~ 1), the functions ~
~ ~o ~ ~ a a
~ decrease ra id~.y with ail incr~~se a~' 4
4 x } wa.l~. p ~
o
~.~e sum- 5~ assumes the Farm I
M
~ ~
~ ~
~:f
iy.
r 1'
i~
' airs we re' resent (as before) the latter sum for
Grouping th~ componsnta ~.n p ~ p
even ~ in the Form ~ ~ ~ ~
~ ~ ~
~
's odd the e~cpression written wall have a negative sign .As was
~f ~ ~ ~
,r
revious r the problem reduces to the ,calculation of the sum
shown p ~ ~ ~
V'im' ~~,N~l1~A ~ .~4'
1~`~~ ~P\r1, 1 i0. ~rkA~ M~ !M'.`W ~ ~ ' . j,~ '.,1~
`~+r~ ~ / AAA ,
~ ~ �a
where ~ , ~
M /
rhtMw ~ ~ {t~
~
. ~
~'.in~ly, the formals. for ~ wi1~. hav� the farm
3
~ ~ Fj
~ ~ w
~ ~
~
r~
s taken fox 'the index ~ ~ ~
..:where i
the factor ,which ~a '
r ~~.`.r ~ a Here, as ire the e~pressa~~'ons for S~ and 5~~ ~we~ seo r ~
-
.esi
~.i~,~ f
- ~ ' nes the ardor of these sums ~
rms.
de e o
~,~I ~ ~
I ~ ,k
,i~ e., 1`i
~ ~ ~ j.
, ,
~ ~y,
~ ,,pro
f~
t ~ ~ ~
~ ~ ~ ~

M� ~
,
k , , i >>I: ~wi
t.~ l ~ ~~~~1 r ~ ~ ~ i
i
i
i,
i ~
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i". -
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
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~ 1,
i ~
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-I
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4t 1~
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f
1 I. ~ ~
1 ~
. ~A.
11
' I~
assa.ble to calcu~ to the sums t
a (b), (a)~ (9), and (1~) make p
Fa rmu~, a ~
nte~sity o~ the d'~ffracted wave
d to determ~~ne the fie~.d i P+,
~ ` ~ and ~ ~ an
.~w, 1~
q
I ~(w. ~ ,
, t~ VV
it
1~
~
b anoth
the solution of this problem Y
J ~
he author plus to return to

f ~ ~ Later, t
error of the resuJ. t5 obtained.
method and to evaluate the R found far th�
1~ l~,t~de ~ ~
' sion we note that 'the order of amP r,
In cnnclu , ~ ;
' rn elementary aonsideratinns~ According
' field ~,ntensity maY be obtan.ned fra ,
e:~ectr~.c
cidence of the wave upon the
~ ula the reflectance for normal in
to Fresnel s form ,
two media i~ ~ `
ur, f ace of s eparata.on of ~
~ ~
,,,,,,,w..IM~.}.w.r _ ~ l
~ ensities of the incident and ref-
e) (r) es a cta.vely axe the n,nt . ~
Here x and 1 p e refractive inde~cQS
~ ~ is the difference a.n th ~ ~
'
cted wages and ~ ~
le
. ~R~
of tha ~twa media.
small electromagnetic beam by a
us consider naw the scattering of a
. Let ~ ara~.lel beam ~,s ~
tional area of ,the a.ncident p
rica~. surfaced if the cross sec
sphe
at a 'distance r from th.e center
urea r
er ref7.ectiar~ its crossasectiona~. ,
~ t
nL1x , a,ft
N ' ~ and Corxespanding~:y the i~,ten~i y' ~ , ,
'here wi~.l in,crea~e to a va~.ue nQx s
of .the sp ~ ~ ~ ere l(r) as
(r) az9ry ~ I~ ..Qx , wh ~4
,a~
d beam will be d~.c.reased: 1 ~ ,
of the refl~cte _ , ,
t ~a~ao~rs .from simp~.e geo- , ,
e wave ref~:ected: by :the gphere� ~ j~,
the intonsi~y o~ ~h
s` b~~;inning with rhos e of
~ (neg~,~ect3~ng `infiri~.~,es~m ~ r~
metry that to the
ected~ wave E~
the patio of the ampla.tudes ,of t e re 1,~ , ,
the second ardex)~ ~
1
� ent E(~ assumes the form. ~ ~ ~ (13)
ar~c~.d ~ ~ ~ ~
~
"i' o~ s ,il
h ~ ~ ~ ~ ~
J i i ~ ~ 'v i
~ 1 { ~ ,N
i
y 1 ~
~
a.
~ - 1 ~ a ~ _ ~
r i i 'r' w r i t i i i
i i~ p 1 ~ r i v r
~ 1 ~ ~ ti~h~~ ~ { t i ) fl ~
7 Ali ,~FYd i~ i i Ar 4~ ~ -
~ ~ P i 'N ~ ti ~ it ~ ~ 11 ~d 11'S i ~ '
~ I , 1 s I~ r
i. ~ ' ~
5 i~t~~~ i~ ~~`,IF~~~r , IMP
i ~ 7 ~ " is
i
r r
~ , , ,
;
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
r
<
i ~
,r~~ 1
~
.
~
1 1 ~ MudYM ~ ~ 1R M~~I ~1I~ w~~~
Y ~'vx r Rt the ardex o~ th� rat~.a of the amp/ x
the result obta~a.hed prev~.ausly.
u m~ ilea be a~ata~ned from sample considerations. Let s~'~
Maxe general res l ~Y ,
bounded b an axb~,trary canve~ surface having princ~.pa1
us consider a d~,elactr~.c Y
' and R which are large ~.n comparison with the wavelength
rad~.i of curvature
' ~ ~ ~ ,
~t
of t11e incident wave
a of a circular eleatramagnetac beam upon the surface of
Far ~vrmal inc~denc ~ f
t ~
the da.electric, we havQ as before
~1
�E, ~ ~ l~ '
a ~ l,S. ~ ~
s f curvature ~~f the normal cross-section ai' the surface and .
where R ~.s the radio a ,,Y
r t;,
distance (a~,.ang the normal) from the surface to the point of r
d r R a.s the
observation� Using the f+'uler .theorem
r Cod' ~ S ~ rt
,u
I~~
, .
tin with respect to the crassws~ectional area of the '
we calculata (a.ntegra g
' beam reflected by the surfaces
1 '
Qmitt~.ng the simple ca~.cul anions; we find r
~ ~
CrJ ~ / ~ I ~ r~
and the ratio of the ampl~.tudee des~,red assumes the form
,.q ~
Received 1~ June 1946
Secta~on fdr Sc~.entific Development of
Electric Corr~nuna.cat~ns ~~ablems,
.
Academy of Sciences ,USSR
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Declassified in Part - Sanit ,
�ed Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012 ~
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 Declassified in Part -Sanitized Copy Approved for Release 2012/05/04 :CIA-RDP82-000398000200110012-5
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