4. B . S . Abramovich and S. N. Gurbatov, Izv. Vyssh. Uchebn. Zaved. , Radiofiz. , 2_.33, No. 4, 442 (1980). 5. A . I . Saichev, Izv. Vyssh. Uchebn. Zaved. , Radiofiz. , 23 , No. 11, 1305 (1980).

EFFECTS OF INHOMOGENEITY DRIFT ALONG A BEAM

PROPAGATION LINE ON THE PERFORMANCE OF A

WAVE FRONT-REVERSAL SYSTEM

A. N. Malakhov and A. I. Saichev UDC 538.56:519.25

A study has been made of the effects of wind drift of turbulent inhomogeneities on the statistical

properties of waves reflected from a mirror that reverses the wave front. The mean field and

coherence function for the reflected wave have been calculated. It is found that the coherence

function for the reflected wave is anisotropic on account of the distinct drift direction of the in-

homogeneities. One way of statistical analysis of drift effects is examined for the case where the

largest relative displaeement of the inhomogeneities is less than the internal scale of the latter.

1. In recent times there has been increased interest in phase-conjugation systems that reverse the wave

fronts of incident waves [I, 2]. This has occurred because these systems efficiently suppress the amplitude

mud phase distortions in the beam arising, e.g., in propagation in turbulent medium. However, ideal compen-

sation of the turbulent distortions by a phase-conjugation system in unattainable in practice for many reasons.

Also, the size of the system is limited, and this goes with details of the design to cause many of the systems

to reverse only the wave front of the incident wave while not using information about the intensity profile. Also,

there is imperfection in the wave front reversal by real systems, while the inhomogeneities vary in time, and

as a result the incident and reflected beams, in general, propagate through diffelent inhomogeneities. There

are some discussions [3, 4] on the performance of such systems in relation to the details of the design and the

restricted dimensions. Here we examine the effect of wind drift on turbulent inhomogeneities. This effect be-

comes the more important the longer the beam propagation line. in all other respects we consider the system ideal.

2. Let the plane x = 0 contain an unbounded mirror that ideally reverses the wave front of an incident

wave (WFR mirror), while in the plane x = L a wave is emitted with the initial profile u(p). We are interested in a wave v (p) reflected from a WFR mirror in the plane x = L:

"r(x, s, ~ )= G~ O, p; x, ~ + G2 O, p; x, ~- - ~ dp.

(1)

Here G I and G 2 are, respectively, Green's functions for the reflected and incident waves. In the quasioptieal

approximation, and with the assumption that the turbulence is frozen, 7(x, s, p) satisfies the following equation on the basis of the reeiprosity theorem [5, 6]:

= - - - - e X , ~ "~, ax ~ -7-I \ 2 2

"r (o, s, e ) - - ~ ( s ) , (2)

where efx, p) a re the turbulent inhomogenei t ies in the d ie lec t r ic constant of the medium; c~ = 2V• c, veloci ty of light; and V• t r a n s v e r s e component of the dr i f t speed of the inhomogenei t ies . The beam usually va r i e s much m o r e slowly in longitudinal coordinate x than it does in the t r a n s v e r s e di rect ion, so the effects of dr if t along the beam propagat ion di rec t ion can be neglected if V N / V • << L0/aef f , where aeff a r e the effect ive d imen- sions of the beam and L 0 is the external sca le of the turbulent inhomogenei t ies , which we take as the sca le of

Gorki Univers i ty . Trans la ted f r o m Izves t iya Vysshikh Uchebnykh Zavedeni i , Radiofizika, Vol. 24, No. 11, ppo 1356-1361, November , 1981. Original a r t i c le submit ted Sep tember 23, 1980.

0033-8443/81/2411-0917507.50 �9 1982 Plenum Publishing Corpora t ion 917

beam var ia t ion in the longitudinal coordinate . Usual ly , the longitudinal sca le of the beam is g r e a t e r than L0, so the above inequality is sufficient , tn what follows we a s sume that it is obeyed.

3. We first find the mean reflected field. From (1) we have that

- c o

(3)

The equation for <7} is as follows in the diffusion approximat ion , as (2) impl ies :

Here the symbols are

0<7> i (v,vp)<~) + D ( s + = x ) < ~ , > = 0 , k T

<x(o, s, p)> =B(s).

D ( s ) = A ( 0 ) - - A ( s ) , A ( s ) = ~ (s(0, 0) a(x, s ) } d x .

The solution to the equation for <Y) is x

< T(x, s, l~)} = ~(s)exp { - - (k:/4) ~ D(az)dz}. 0

We subst i tu te this into (3) to get the following express ion for the mean ref lec ted field:

L

< v(p) } = u* (~) exp { --(k2/4).i" O (ax) dx }. 0

(4)

The maximum value of the argument of the structural function appearing here is -c~L. For approximate esti-

mates we take G ~ 10 -8 and the internal scale of the turbulence as l 0 ~ 1 cm; then (4) can be rewritten as fol- lows for lines L < 10 8 cm:

7

/. v(p) : . . . . u* (~)exp {- - ted ,t' ~sax-' dx }, (5) 0

where D = 0.41C2l~ 1/a [71. If the dr i f t speed is constant along the propagat ion line, (5) gives

< v (fa) > = u* (p) exp I - - (k2/3)DL a =21 (6)

and the mean ref lec ted field is v i r tua l ly the s ame as the complex-conjuga te of the init ial field at d is tances k2DLace 2 << I o r oeL << Pc(L), where 0c(L) = (k2DL) -I/2 is the coherence radius of the initial plane wave pa s s - ing along the turbulent line of length L. To e s t ima te the effects of d r i f t -ve loc i ty fluctuation on the r e c o n s t r u e - tion of the initial field u(0) in the ref lec ted wave, we a s s u m e that c~ is a random Gauss ian function of x with the covar ianee function fi(z), mean ( a >, and co r r e l a t i on length lop We cons ider the two l imit ing cases Zc~ >> L and lc~ << L. In the f i r s t case we ave rage (6) over the ensemble of the Gaussian r andom quantity c~ to get

{ (<a>L)2 } V 3 92e(L) <v(p)> = u * ( p ) exp - - 3,o{~-L-~+2~ ~ 3Pc~(L) + 2 ~ ' L e

(7)

where G 2 = fi(0) iS the dispersion of the fluctuations in the dimensionless drift velocity oz. For ~2 _ 0, (7) be-

comes (6) apart from the replacement of c~ 2 by <cr )2; for <a ) = 0 the mean reflected field falls as L increases

more slowly than in the case <G ) ~ 0, but the condition for the performance in the WFR mirror in restoring the initial field remains qualitatively as before: GL << Pc(L).

In the other limiting ease Ic~ << L, the integral in (6) can be considered a Gaussian random quantity by virtue of the central limit theorem, which gives

918

where

dz.

The last term can virtually always be neglected, so finally

<~> L2 / (8) < v (~) > = u* (p) exp 3 p~c(L) l

Th i s f o r m u l a a g r e e s w i t h (6) a p a r t f r o m r e p l a c e m e n t of oz 2 by (c~ 2 ).

4. The m e a n f i e ld d e s c r i b e s the d e g r e e of r e c o v e r y of the a b s o l u t e p h a s e c h a r a c t e r i s t i c s of the i n i t i a l f i e ld on r e f l e c t i o n f r o m the VvFR m i r r o r . The r e l a t i v e p h a s e c h a r a c t e r i s t i c s ( c o h e r e n t p r o p e r t i e s ) a r e e q u a l l y i m p o r t a n t , as a r e the s t a t i s t i c a l f e a t u r e s of the r e f l e c t e d - w a v e i n t e n s i t y . H e r e we need to c a l c u l a t e the h i g h e r m o m e n t s of the w a v e v(p) r e f l e c t e d f r o m the W F R m i r r o r . The f o l l o w i n g is one w a y of c a l c u l a t i n g the h i g h e r m o m e n t f u n c t i o n s of the r e f l e c t e d w a v e w i t h a l l o w a n c e fo r the w ind d r i f t in the i n h o m o g e n e i t i e s .

We no te f i r s t l y t ha t fo r c~ = 0 t h e r e is a s o l u t i o n y(x, s , p) = 5 (s) to (2), w h i c h e n a b l e s us to a s s u m e t h a t 7(x, s , ~) fo r c~ s u f f i c i e n t l y sma i1 is a f a i r l y n a r r o w f u n c t i o n of S l o c a l i z e d in s = 0. It s e e m s tha t the s c a l e of 7(x, s , p) in s is the m a x i m u m d r i f t of the i n h o m o g e n e i t i e s a l ong a l ine of l e n g t h x - ozx; if th i s is so , then f o r ozL < l 0 we c a n expand the c o e f f i c i e n t to 7 on the r i g h t s i d e of (2) as a T a y l o r s e r i e s in s and a and r e s t r i c t o u r s e l v e s to the f i r s t n o n v a n i s h i n g t e r m s :

O-f i (Vs V:,) ~; = ik _~ ik

= s ( x , ~), -~(0, s, p ) = ~ ( s ) .

(9)

We t r a n s f e r f r o m this e q u a t i o n to the e q u a t i o n for Y(x, n, p ) = ~ ~(x, s, p)e-~k(~s>ds : - - o o

OJ 1 ik 0-7 -~- (nv~) g -k - f (V• e Vn) J ~ -~- x (~r V• ~) J,

J(o , n, p ) = 1. (lO)

In what follows for simplicity we assume that oz = const. We solve (I0) by the characteristics method to get

Y(x , n, p) = exp { ikx(~n) - - ik(=p) + ik(g~o(x, n, p))}, (11)

where P0(x, n, P) are the transverse coordinates in the plane x = 0 for the geometrical-optics ray that has co-

ordinates p in the x plane and a vector angle of inclination n to the x axis. In what follows we are interested

only in the statistical features of J and y, and for convenience we replace the true functions P0 by the statisti-

cally equivalent ones p0(x, n, p), which satisfy the boundary conditions at x = 0:

dpo/dx = n,,, du~/dx = (1 /2 )7• Pc),

p,,(o) = p, no(0) = - n. (12)

It is readily shown that the solution to (ii) leads to (6) for the mean reflected field, which confirms the as- sumptions made above in deriving (II). Also, the J dependence of (II) in terms only of the coordinates of the

geometrical-optics rays does not mean in genera[ that use of the J of (Ii) in calculating the statistical proper-

ties of the reflected wave is equivalent to the geometrical-optics approximation.

5. The solution to (II) enables one to express the moment function M of the reflected field v(p) in terms

of the reciprocal characteristic function of the transverse coordinates M for the geometrical-optics rays. As

an example we give here only an expression following from (I) and (ii) for the coherence function of a planar

reflected wave:

F (s) ~ ( v (p) v* (? --i- s) > = ] tt ]:~ ( exp {lie (~r ~L)) - - ik (~s) } ) , (13)

where ~4(L) = P0(L, 0, s) -O0(L, 0, 0); therefore, the determination of the coherence function for the reflected wave amounts to finding the characteristic function for the relative diffusion of two rays. The statistical prop-

erties of the relative diffusion of two rays have been previously discussed [8]. For example, if s > L 0 is the

external scale of a turbulence, then the rays propagate statistically independently and (13) becomes

919

w h e r e

is equal to (6) fo r u*(p) = u*.

r = I < "' > 1:',

< v > = ~ t * < e x p l i k ( = p o ( L , 0 , 0 ) ) } >

If s < L0, w e have to c o n s i d e r the s t a t i s t i c a l d e p e n d e n c e be tween the r a y s in c a l c u l a t i n g the c o h e r e n c e funct ion . If < 22 ) << 1 is the m e a n s q u a r e of the a m p l i t u d e of the i n i t i a l l y p l ane wave c a l c u l a t e d in the g e o m e t r i - e a l - o p t i e s a p p r o x i m a t i o n a f t e r p a s s i n g t h rough a t u r b u l e n t l ine of l ength L, then in c a l c u l a t i n g the jo in t d i f fu - s ion of the r a y s we use the f i r s t a p p r o x i m a t i o n in g e o m e t r i c a l op t i c s [9], wh ich g ives

i ' ( s ) = 1< v ) 13 exp {(k2/12)L;(~ Vs)2D(s)}.

F o r s < 10, the i n t e r n a l s c a l e of the t u r b u l e n c e , we put D(s) = 4Ds 2 - Bs 4 and ge t

F (s) = I u I-' exp { - (k~/3) BL a [~=s ~ + 2 (~s)~-]}. (14)

As would be e x p e c t e d , the c o h e r e n c e funct ion f o r the r e f l e c t e d w a v e is a n i s o t r o p i c , b e c a u s e t h e r e i s the d i s - t inc t d i r e c t i o n a , a long wh ich the c o h e r e n c e p r o p e r t i e s of the r e f l e c t e d wave a r e w o r s e . The mean s q u a r e of the f luc tua t ions in the l o g a r i t h m of the a m p l i t u d e of the i n c i d e n t wave is {){2 ) = 4BLa/3 in the f i r s t a p p r o x i m a - t ion in g e o m e t r i c a l op t i c s [9], so the l a t t e r equa t ion can be r e w r i t t e n as

P (s) = ] u 12 exp {-- (k~/4) < 7.. ~ > [a~s 2 + 2 (~s)q}.

We canno t u s e the f i r s t a p p r o x i m a t i o n of g e o m e t r i c a l op t i c s in the r e g i o n of s t r o n g f l uc tua t i ons in the i n t e n s i t y ((~2) >> 1), and the r e l a t i v e d i s t a n c e be tween the r a y s ~ (L) does not have a G a u s s i a n s t a t i s t i c . N e v e r - t h e i e s s , i t is known that the c u m u l a n d and, in p a r t i c u l a r , the G a u s s i a n a p p r o x i m a t i o n s g ive q u a l i t a t i v e l y and even q u a n t i t a t i v e l y c o r r e c t r e s u l t s when u sed to c a l c u l a t e the s t a t i s t i c a l c h a r a c t e r i s t i c s of r a n d o m p r o c e s s e s [10]. T h e r e f o r e , in the G a u s s i a n a p p r o x i m a t i o n [101 we r e w r i t e (13) as

r (s) = I u 12 exp {-- (k2/2) [ ( ( ~ (L)12,_. (~s)21 ,. (1 5)

H e r e we have u sed the obvious equa l i t y ( ~ ( L ) ) = s ; as p r e v i o u s l y , we e x a m i n e the b e h a v i o r of F(S) fo r s < l 0. We a s s u m e a t so tha t >c < l o to ge t f r o m (12) a s y s t e m of equa t ions fo r the c o m p o n e n t s of the v e c t o r ~ :

dx/dx = ~, d~/dx = ( l /2)(~v• P0),

z (O) = s , ~ (0) = 0.

Then we get the Einstein-Focker-Planck equation for the probability density W(x, ~4, #) in the diffusion ap- proximation [81:

dW/c)x ~ (xv~)W = Bzu&~ W + 2B(zv~I~W,

W ( O , . , g) = a (x - s ) ; (~ . ) .

A c c o r d i n g to (16), the mean ((oct<ix)) 2 ) a p p e a r i n g in (15) s a t i s f i e s the c l o s e d s y s t e m of equa t ions

d a < ( ~ ) 2 ) = 3 B < ( ~ ) ~ ) + 4 B~ ~ < x ~" >, dx s

d 8 ( ~' )

dx:'

d @ d k I x~(0) ) = s2' ( : ( ~ ) ~ ) = d " ~ ( x2 ) ~=o = o, k = t, 2.

This is r e a d i l y so lved . F o r m u l a (14) fo l lows f r o m th is and (15) fo r BL a s m a l l . In the r e g i o n of s t r o n g f l u c t u a - t ions in the intensity we have BL a >> 1 and (<X 2) >>1), and t h e r e <(c~(L)) 2) i n c r e a s e s e x p o n e n t i a l l y wi th the

3 exponent ~4"-gg~; c o r r e s p o n d i n g l y , the c o h e r e n c e r a d i u s of the r e f l e c t e d wave d e c r e a s e s exponen t i a i Iy as L ink c r e a s e s i f koel o > 1.

1.

2. 3.

LITERATURE CITED

J . W. H a r d y , TI IER, 666, NOo 6, 31 (1978). W a v e f r o n t R e v e r s a l fo r Op t i ca l R a d i a t i o n in N o n l i n e a r M e d i a [in R u s s i a n ] , S b o r n i k I P F A N , Gork i (1979). R. L. F a n t e , & Opt. Soe. A m . , 6___6_6 , 730 (1976)o

920

4. A.V. PolovinkinandA.I. Saichev, Izv. Vyssh. Uchebn. Zaved., Radiofiz. (in print). 5. V.I. Gel'fgat, Akust. Zh., 22, No. i, 123 (1976). 6. A.I. Saichev, izv. Vyssh. Uchebn. Zaved., Radiofiz., 2_~I, No. 9, 1290 (1978).

7. S.M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Introduction to Statistical Radio Physics, Part 2: Random Fields [in Russian], Nauka, Moscow (1978).

8. V.I. Klyatskin~ Stochastic Equations and Waves in Randomly Inhomogeneous Media [in Russian], Nauka, Moscow (1980).

9. V. !. Tatarskii, Wave Propagation in a Turbulent Atmosphere [in Russian], Nauka, Moscow (1967). i0o A.N. Malakhov, Cumulant Analysis of Random Non-Gaussian Processes and Their Transformations [in

Russian], Soy. Radio, Moscow (1978).

REFLECTION OF A NONSTATIONARY WAVE BEAM

FROM A RANDOMLY !NHOMOGENEOUS PLANE-LAYERED MEDIUM

S. N. Gurbatov UDC 538.56:519.25

The discussion concerns the reflection of a signal bounded in space and time from a randomly inhomogeneous plane-layered medium with small-scale inhomogeneities. The signal at the exit from this medium is represented as an expansion in terms of planar monochromatic waves in the incident signal on the basis of the reflection coefficients. The intensity of the reflected wave is expressed in terms of the correlation of the reflection coefficient for two planar monochro- matic waves with different frequencies and wave vectors. We consider a scattering condition where beam diffraction is not important. The transient response in the buildup of the steady- state scattering mode is discussed.

It has been shown [1-6] that virtually all the incident wave is reflected if a plane-layered medium with random small-scale inhomogeneities is of sufficient optical thickness, i.e., the layer acts as a mirror. The dynamic characteristics of such a mirror have been discussed [7, 8], viz., by- analysis of the reflection of planar nonstationary signals [7] and monochromatic beams [8]. Here we examine the reflection of nonstationary wave beams from such a medium, i.e., signals bounded in space and in time. The solution to the problem is of independent interest and also provides an answer to questions related to the buildup of steady-state scatter- ing modes.

A layer z ~ [0, L] of randomly inhomogeneous medium receives from the right a two-dimensional non- stationary wave beam whose field in the plane of incidence is E0(t , x); a natural representation of the wave field for a plane-layered medium is an expansion in terms of planar monochromatic waves:

(1)

If the electric-field vector is parallel to the plane of incidence, and also for acoustic waves, we have as follows [9] for the complex amplitudes A:

d 2 A dz 2 + ( , ~ + k 2 A ~ ( z ) ) A = O ; k=o~/c , k ~ : ] / k 2 _ • (2)

Here As(z) a r e the f luc tua t ions in the d i e l e c t r i c c o n s t a n t s(z) ( ( s ) = 1, ( A s ) = 0); we a s s u m e that s = 1 ou ts ide the l a y e r , i . e . , the med i a a re matched in the plane of inc idence . We expand the inc iden t f ield as p l ana r waves

and introduce the reflection coefficient R(w, ~) for a planar monochromatic wave, where for the reflected field we have

Gorki University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 24, No. 11, ppo 1362-1367, November, 1981. Original article submitted September 23, 1980.

0033-8443 /81 /2411-0921507 .50 �9 1982 P l e n u m Pub l i sh ing C o r p o r a t i o n 921