FREQUENCY A

MOVING,

FLUCTUATIONS RANDOMLY

OF

A

P R O P A G A T I N G IN

WAVE

MEDIUM

INHOMOGENEOUS

V. I. S h i s h o v

UDC 621.371

The statistical c h a r a c t e r i s t i c s (mean s p e c t r u m and variance of the line centroid) of the f r e quency fluctuations of a monochromatic wave propagating through a moving, randomly inhomogeneous medium are determined. The calculations are c a r r i e d out f o r the weak f l i c k e r and saturated flicker r e g i m e s . The theoretical r e s u l t s a r e compared with the e x p e r i m e n t a l data on propagation of narrow-band radiation in the interplanetary plasma. It is concluded that the e x p e r i m e n t a l data a r e best d e s c r i b e d by the model of a power-law s p e c t r u m of plasma turbulence. i. S T A T E M E N T

OF

THE

PROBLEM

E x p e r i m e n t a l data have recently been published on the s p e c t r a l spreading and frequency fluctuations o f n a r r o w - b a n d radiation during t r a n s m i s s i o n through the plasma close to the sun [1-3]. This effect has been interpreted on the basis of a theory developed in the g e o m e t r i c a l approximation [4, 5]. R a d i o - a s t r o n o m i c a l studies of the f l i c k e r of radio s o u r c e s [6] indicate, however, that the intensity fluctuations of waves propagating through the n e a r - s o l a r plasma can be saturated (strong), in which case they cannot be described by the g e o m e t r i c - o p t i c s approximation; m o r e o v e r , even in the weak flicker r e g i m e the g e o m e t r i c - o p t i c s approximation is not always valid. Below we consider the frequency fluctuations of a plane wave propagating in amoving, randomly inhomogeneous medium, taking account of the latest r e s u l t s in the theory of wave propagation in a randomly inhomogeneous medium [7]. We adopt as the initial radiation a plane wave of unit intensity {the adjustment f o r sphericity of the wave will be discussed later): ~'[,=0 = exp (-- i 2 ~ o t ) .

(1)

A f t e r t r a n s m i s s i o n through a randomly inhomogeneous medium the plane wave e x p e r i e n c e s modulation, which we d e s c r i b e by the complex field amplitude E0(x, t). We c a r r y out the ensuing analysis on the basis of a sign a l - p r o c e s s i n g c i r c u i t (linear with r e s p e c t to intensity), which we d e s c r i b e by the relation

I(t, [) = i dr

-- =)

_ &

dq

_&

dt.,Eo(tOEo(t, ) exp [2r. i ( / - - [o)(t~ -- t._,)]F(: -- q)F* ('. -- t2),

(2)

-~o

in which cp(t) is the response of the low-pass filter, F(t) is the response of the high-pass filter, and I(t, f) is a random function of the time and frequency. 2. MEAN VALUE OF I We consider the case in which the frequency bandwidth Af0 of the receiver is much narrower than the radiation bandwidth after scattering in the medium, Averaging (2), we obtain the familiar expression 0

cc

=C

~ d~Be(~)expi2~i(f--fo)~],C= -co

0

S d~(~) S dt, F(t,)F*(t,), m~

~)

--Qo

in which BE (7) is the t e m p o r a l c o h e r e n c e function of the field. P. N. Lebedev P h y s i c s Institute, Academy of Sciences of the USSR. T r a n s l a t e d f r o m Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, VoL 19, No. 10, pp. 1507-1511, October, 1976. Original article submitted June 30, 1975. This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any mean~, electronic, mechani~l, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the puolisner lor ~ 1.no.

1056

If the medium moves with a uniform velocity V, then the t e m p o r a l coherence function emulates the spatial c o h e r e n c e function, which, in turn, b e a r s a v e r y simple relationship to the s t r u c t u r e function DS(&X) of the phase fluctuations [7]: B, (.X l) ---- Br(A x ~-- ,_Xt V) = exp [-- Ds (_Xx)]. (4) If DS(AX) can be r e p r e s e n t e d in quadratic f o r m , then I(f) has a Gaussian form. If, on the other hand. DS(AX) can be r e p r e s e n t e d by a power law (with exponent less than 2), then I(f) has a power-law tail. 3.

SHIFT

OF T H E

LINE

CENTROID

A variation of the radiation frequency can be manifested as broadening of the frequency band and as a shift of the line centroid. We c h a r a c t e r i z e the position of the centroid of a s p e c t r a l line by the quantity

?,t = 1 _ ~

(5)

d[l([, t ) ( , - - 'o).

The m e a n - s q u a r e value of 5f is < 6 t)" > = 1, S ~ d[, df.~ ( ; (L, t) ; (f2, t) > (It -- fo) (Y~ -- h).

(6)

Using (2) and p e r f o r m i n g partial integration, we obtain

"/. ( E , E*~E3 E*4 )?/(t, -- t2)?," (ta -- t4) F (~t -- t,) F* (~, - - t 2 ) F ( ~ -- ta) F* (~., -- tO,

(7)

where Ei = E (ti) and 5 r is the derivative of the delta function. F o r the ensuing calculations we must p a r t i c u l a r i z e the f o r m of the f o u r t h - o r d e r coherence function. We examine two c a s e s . a) Weak F l i c k e r Regime. In this r e g i m e we can neglect amplitude modulation of the wave in the zeroth approximation, keeping only phase modulation and thus writing E = e is,

(8)

S = S (x -- Vt),

w h e r e S is the p h a s e - l e a d fluctuations in the layer. Using (8) and assuming that S has a n o r m a l distribution function, we obtain t

t

O~ D s (~x ~

-o~" -~o"

~2)

O~(~' - - ~s)

'

where D S is the s t r u c t u r e function of the phase fluctuations. If the second derivative of D S exists and the time scale of DS is much g r e a t e r than the time scale of q~, then

(10)

02Ds(~) 1

( (~y)2 > = ------70"~=0" This e x p r e s s i o n is also valid f o r a spherical wave. b) Saturated F l i c k e r Regime. Now the f o u r t h - o r d e r coherence function is e x p r e s s e d in t e r m s of the s e c o n d - o r d e r coherence function [6]. On the basis of this fact we write 1

<(a/);>=~

t

t

J'

j"

--~

....

Y j" dt~ S dt~

d;,d-.~,(t--~,)?(t-~)

--oo

[ dBe(tt -- t2) [* ~/t~ I x(~, _ t,)l=l

F(~.,-- t~)l'

(11)

1057

We a s s u m e that the time scale of BE(r) is less than the time scale of F(r).

_.f_ ~

"r

In this case

"ii

((~I)')ffi-C.] . . a~,. S . d,l~(t--,,)~(t--,~) . .

S. dtttF("--t')l=lF(~--t')l'

(12)

dr,

0-a)

w h e re

,,

--~1

We introduce the time constant r0 of the low-pass filter: 0

0

'0=[ t ,: (') <"I' / .C ~, (t) at.

0-4)

If r0 >> (27tAro)-1, then 1 < (~,[)2 > . . . . .

.

(15)

F o r To << (2~Afo) -1

a fo !

< (~ I)" > =

(16)

2~t~

w h e r e At0 is given by the relation 0

0

E x p r e s s i o n s (12)-(17) are also valid for a spherical wave. 4.

COMPARISON

WITH

OBSERVED

DATA

Yakovlev and others [3] have published e x p e r i m e n t a l data on the shape of the mean energy distribution in the line and on the variance of the shifts of the line centroid. We f i r s t discuss the data on the shape of the mean e n e r g y distribution in the line. F i g u r e 1 gives the mean s p e c t r u m obtained in the cited w o r k [3] for an e l o n g a tion ,J = 2.7 ~ The right and Left sides of this s p e c t r u m have been averaged and plotted in logarithmic scale, The figure indicates that the s p e c t r u m has a definitely non-Gaussian f o r m {a Gaussian s p e c t r u m is r e p r e s e n t e d by the solid curves). The tail part of the s p e c t r u m is not inconsistent with a power-law behavior, the power exponent having a value a ~ 2.5. This r e s u l t a g r e e s with data on the temporal spectra of the flicker of radio s o u r c e s [8]. We point out that the power-Law variation of the mean s p e c t r u m c o r r e s p o n d s to a power-law behavior of the phase fluctuation s t r u c t u r e function:

Ds (~) .~,

,~,-1.

(18)

The same law a l s o applies to the spatial s t r u c t u r e function. Yakovlev and c o - w o r k e r s [3] also give values of the variance cr2 = ((At)2) of the fluctuations of the line c e n t e r frequency, r e m a r k i n g that a c o m p a r i s o n of a with the c h a r a c t e r i s t i c width A_.fof the mean s p e c t r u m on the assumption of a Gaussian distribution function for the frequency fluctuations (At = 2.6~) indicates roughly a twofold e x c e s s of the values of 2.5~ o v e r the values of A f in the weak f l i c k e r r e g i m e ($ 7~ This dispartiy is fully explainable if one b e a r s in mind the power-law behavior of DS(T); accordingly

a> Fig. 1. Mean s p e c t r u m (I(f)) of n a r r o w band radiation scattered in the i n t e r planetary medium for an elongation r = 2.7 ~. The dots r e p r e s e n t the e x p e r i mental r e s u l t s , the solid c u r v e s c o r r e spond to Gaussian functions, and the dashed line, to a p o w e r - l a w dependence. 0,0~

1058

I

0,4

~D

I

40 ft H.Z

the distribution function of the frequency fluctuations has s t r o n g e r tails than a G a u s s i a n function, and in c o r r e s p o n d e n c e with (9) the value of a is d e t e r m i n e d not only by the p a r a m e t e r s of the m e d i u m , but also by the time constant of the r e c e i v e r . LITERATURE 1.

2. 3. 4.

5. 6.

7. 8.

CITED

J. V. HoIlweg and J. V. H a r r i n g t o n , J. Geophys. R e s . , 7.33, 7221 (1968). R. Goldstein, Science, 166, 598 (1969). O. I. Yakovlev, ]3. P. T r u s o v , V. A. Viktorov, A. I. Efimov, Yu. M. Kruglov, S. S. Matyugov, and V. M. R a z m a n o v , Kosmich. I s s l e d . , 12, 600 (1974}. V. G. G a v r i l e n k o and N. S. Stepanov, Radiote, kh. E l e k t r o n . , 1.88, 1105 ('1973). O. I. Yakoviev, Radio-Wave P r o p a g a t i o n in the Solar System [in Russian], Izd, Soy. Radio, Moscow(1974). M. H. Cohen and E. J. G u n d e r m a n n , A s t r o p h y s . J., 155, 645 (1969). A. M. P r o k h o r o v , V. F. ]3unkin, K. S. Gochediashvili, and V. I. Shishov, Usp. Fiz. Nauk, 114, 415 (1974). A. V. Pynzar,, V. I. Shishov, and T. D. Shishova, Astron. Zh., 5_~2, 1187 (1975).

QUASIOPTICA

L RADIO

l~. I . G e l ' f e r , Yu. S. E . F i n k e l , s h t e i n ,

VISION V.

IN T H E

Lebskii, a n d N. A.

PASSIVE

REGIME

UDC 535:621.378 Yakun'

The possibility of d i r e c t quasioptieal m i l l i m e t e r radio vision in the p a s s i v e r e g i m e is t h e o r e t i cally and e x p e r i m e n t a l l y investigated. E x p e r i m e n t s have been r e p o r t e d on studies of radio vision [1-3] f o r the active r e g i m e , in which an object e m i t s a c o h e r e n t wave f r o m the t r a n s m i t t e r . However. radio vision in the p a s s i v e r e g i m e , i.e., the c o n s t r u e tion of r a d i o images of objects derived f r o m t h e i r s e l f - r a d i a t i o n , is c l e a r l y the m o s t interesting case. In the c u r r e n t a r t i c l e the possibility of p a s s i v e o b s e r v a t i o n of distant objects with m i l l i m e t e r waves is evaluated as a function of the b o d y - m e d i u m t e m p e r a t u r e c o n t r a s t , the sensitivity of the r e c e i v e r , and observation conditions, and a l a b o r a t o r y e x p e r i m e n t on p a s s i v e radio vision is d e s c r i b e d and the e x p e r i m e n t a l r e s u l t s presented. The i m a g e s of the objects w e r e constructed using a standard m i l l i m e t e r antenna [3]. In c o n t r a s t to r a d a r s e t s , in r a d i o vision we d e t e r m i n e not the total e n e r g y received by the antenna, but r a t h e r the field intensity distribution in a plane conjugate to the plane of the object. In selecting the dimensions of the fixed element (for e x a m p l e , a horn), both e n e r g y relations, as well as the demands f o r m a x i m a l resolution of the s y s t e m , m u s t be taken into account. C l e a r l y , the optimal a r e a of the horn is d e t e r m i n e d by the size of the resolving e l e m e n t in the image plane. In this c a s e , the emitted power f r o m an object with s i m p l e configuration z units f r o m the antenna is given by [4] Peck_. kTa A f ,l e-~z ,

(1)

w h e r e k is B o l t z m a n n ' s constant, T a is the a p p a r e n t t e m p e r a t u r e of the body, Af is the r e c e i v e r frequency band, u = 0.4-0.7 is the antenna a r e a utilization f a c t o r [5], and ~ is the a b s o r p t i o n coefficient in f r e e space. The power r e c e i v e d by the antenna m u s t exceed the fluctuation sensitivity threshold of the r e c e i v e r ( r a d i o m e t e r ) APmin: Prec> n a Pm,n.

(2)

It is usually a s s u m e d that the w e a k e s t signal that can be detected by a r a d i o m e t e r is equal to the standard deviation a of the output fluctuations. It can be verified in the c o u r s e of an e x p e r i m e n t (ef. Fig. 6) that in radio G o r ' k i i State University. T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uchebnykh Zavedenii, Radiofizika. VoL 19, No. 10, pp. 1512-1517, October, 1976. Original a r t i c l e submitted July 16, 1975. This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechaiffcal, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $7.50.

1059