MEMORANDUM

RM-5787-PR

~ FEBRUARY 19609

THE SECOND-ORDER

RYTOV APPROXIMATIONH. T. Yura

Thi, research ,t s.upportl-d by the United Slates Air Force under Project RAND-Con.tr;ct No. V" 1620.67.C.O 15 monitored by the Directorate of Operaticnal Requirementsand ) e.lopment Pians. Deputy Chief of Staff. Research and Develo-)ment, Hq USAF.View-, or roncluqions coniained in this study should not be interpreed as representingthe official opinion or policy of the Uniled States Air Force.

DISTRIBUTION STATEMENTThis document has wen approved for public release and sale; its distribution is unlimited.

I,. n' St* .WS o~(a

Thk i .titdv i- liteswed. a- a ftomitewt troeatnira ofr fh -tiijut-1. %%crwh% or piih.liratlmnl liii- iis CofrjpsIJIimig f Gct r O (Ii tel~ity of tit-, re'-ej iwi. v iilqiimIe-a;Irily *'ntor-i, liy h,1iiniiic mid i--mr,n- uif lw ite Ii

Published by The RAND Cotpirarn

Iii

PREFACE

This Memorandum, prepared as part of RAND's continuing study of

long-distance optical propagation through the atmosphere, gives a work-

able solution to the second-order Rytov approximation.

Recently there has been a controversy .i the ranges for which the

first-order Rytov approximation gives an adequate engineering estimate

of the effect of a turbulent medium on laser propagation. Therefore,

it is felt that an explicit derivation of the second-order Rytov approx-

imation should be of interest to those concerned with optical communi-

cation and laser radar. Furthermore, the second-order approximation

is necessary because it is the lowest order nontrivial approximation

which conservges average energy to the order of the approximation.

==

Vm

SUKMY

An explicit and useful formulation of the solution for the

second-order Rytov approximation is given. From this solution a con-

dition of validity for the Rytov solution is obtained. It is concluded

that, in general, both the Born and Ryrov approximations have ie same

do -mna validity.

CONTENTS

PREFACE..........................................................i

SUMMARY ........................................................... v

SectionI. INTRODUCTION ................................................

II. THlE SECOND-ORDER RYTOV APPROXIMATY)N ....................... 3

III. VALIDITY CONDITION .......................................... 8

REFERENCES ........................................................ 11

btlI. INTRODUCTION

A great deal of theoretical effort has been given to calculations

based on the so-called Rytov approximation. (-)However, the valid-

ity of the Rytov approximation has been questioned by a number of

authors. (2 ,4 ) In a recent paper Fried (5 ) continues to take issue with

(4)(2the arguments of Hufnagel and Stanley and Brown, who claiW Lnat

the Born and Rytov approximations have the same domain of validity.

In view of this continuing controversy, it is felt that an explicit

derivation should be given of the second-order Rytov approximation

(defined below). In this Memorandum the hierarcny of Rytov equations

is presented. The explicit solution (in quadrature) of the second-or-

der Rytov approximation is obtained. In Ref. 6 the Born approximation

was examined in detail, and it was assumed that the domain of validitv

of the Rytov solution was the same. Here we present an explicit demon-

stration that this Is indeed the case.

The solution obtained is correct through terms of second order in

nI , where n1 is the fluctuating part of the index of refraction. From

this solution a conditior of validity is obtained for the Rytov approx-

imation with the conclusion that both the Born and Rytov approximations

have the same domain of validity.

As in Ref. 6, the analysis is Lime independent and rustricted tc

weakly inhomogeneous media which are assumed to be statistically homo-

geneous and Isotropic, characterized by an index of refraction corre-

Jation function. Furthermore, the electrical conductivity and magnetic

permeability of the medium are taken as zero and one, respectively.

Scalar waves at optical wavelengths are considered, with the extensions

2

to vector fields being straightforward. For further details of the

tieory of optical propagatic-' through a turbulent medium, the reader

is referred to Ref. 6 and the refereices therein.

The scalar wave equation is

72U + k2 n2 (rU = 0 (1)

where U is a typical component cf the field, k is the optical wave

number, and

n(r) 1+ n (r) (2)I

where n1 is the fluctuating part oi the index of refraction assumed

to satisfy In 1 .< 1 and n1 = 0 (the bar over a quantity indicates the

ensemble average of the qtantity).

I

3

II. THE SECOND-ORDER RYTOV APPROXIMATION

The Rytov transformation consists of setting U exp[!t] in

Eq. (1) and thus obtaining

+ (V)2 + k2 [1 + n((3)) 0

(1-3)In the literature it is customary to seek a solution for ' as a

power series in nI. Let

l'0

= m (4)

mi0

where 1c is zero order in nl,i is first order in nl, and so forth.

Then it can be shown that the *m's satisfym

+ (V*O)2 + k2 = 0 (5)

+ 21 * 1 + 2k2nl(r) - 0 (6)

V21 + 27K0 • 7 2 +k2 n(r) + (V* 0)2 (7)

andM-1

2, +* 0, m = 3,4,5... (8)m 0 m + in-

p=O

Equation (5) gives the solution to Eq. (1) for the case nI 0. Thusif U0

= exp 10, then Eq. (5) is equivalent to v2U0 + k2U0 = 0.

The function is what is usually referred to in the literature

as the Rytov approximation. In this Memorandum, is referred to as

4

the first-order Rvtov approximation, while is refe-red to as the

2

second-order Rytov approximation.

It cin be shown (i ) that within an arbitrary constant the general

solution of Eq. (6) is given by

2 k dr G(r - P) n(r') Ur) (9)U0(r)

where the integration in Eq. (9) extends over The region of space

where n1 (r) is different from zero and

ikl'-r"

G(r - r') = 4 - 1 (10)

is the Green's futction in the absence of turbulence, which satisfies

the outgoing radiation condition at infinity.

Others have failed to obtain a useful workable solution to

Eq. (7) (e.g., Schmeltzer (3)). It is noted that Eq. (7) can be solved

by the use of a transformatioa suggested by a comparison of the Born

and Rytov expansions. Upon equating second-order terms in the expansions

we have2

-2 '2 21 (11)

wht.. e is given in Eq. (9). Upon substituting Eq. (11) into Eq.

(7, and using Eq. (6), we find

- (.2 2 2k2 n2 V) " 2 (k nI + ,1) (12)

I. -

I

' e "' ,

I5Let

U 0 (r)

• ,nd substitute this into Eq. (12). We find that W satiifies

2W + k2W = 2 (kn 2 + 2kn1 U0

Hence,

-21 2. r d;' 2(14)W(r) k G(r r') L 2 1 (r) (r') Uo(r) dr (14)

Thus, the sucond-order Rytov approximation is given by Eq. (11),

where _2 and W are given by Eqs. (13) and (14), respectively.

The transformation employed to solve Eq. (7) suggests that the solu-

tion to Eq. (14) should be equal to the second-order Born approximation.

The Born expansion for U(r) 4s given by

U(r ) Ui(r) (15)i=O

where

2+ k U0 0 (16)

2 r2rUl(r) - 2k2 dr G(r -r nljr") Uo(r') (17)

and for i I

U (r) - 2 dr' GI' - r' n ' U

2 2 ~,--k dr' G(r - r' n (r )U .(r) (18)

For i I Eq. (18) gives

= - 2k dr' G(r r?) nlI(r') UI(r')

- k2 [ dr G(r - r') n (r ) U0 (r') (19)

Comparing Eq. (14) with Eq. (19), it is seen that to within an arbitrary

constant W(r) = U2 (r). Hence, to within an arbitrary constant the

second-order Rytov approximation is given by

U 2 (r) $ j(r) U 2 r) rU 1 (r) 2 (0

2 0 r- 2 (20) U0 r)

u 0(r) 00

The average field is determined, to terms of second order in nl,

as

=exp + 2)OU0 ~1 + 21

exp + ~(21)

where "2(r) 2 (r)/0 ".(r) For example, when the field in the

absence of turbulence is a plane wave

_B B(0) k2\ eikR ekR

i r + dR(R) eik-R (22)

2 p..

7

where Bn (R) = n1(7) n1(r + R) is the correlation function of the

index of refra-Lion Iluctuations.

It is shown in Ref. 6 that one must include the second-order

contribution to the field (;.) in order to obtain an adequate engi-

neering approximation to the scatfered field in the presence of

turbulence. Furthermore, a second-order expansion for the field

is the lowest order nontrivial approximation which conserves average

energy to the order of the ,, )roximation. That is, the field

U(r) = cxp L. O(r) + '.I(r) + ,+2(r) (23)

which contains all terms through second order in nl, gives the correct

average field, correct phase and intensity statistics, conserves aver-

age energy, and satisfies the optical theorem to this order. In the

present notation the optical theorem is, through terms of second

order in n,, given by

2Rec 2 + Iti12 = 0 (24)*

In the usual notation the optical theorem is: c - Imf(O),where c is the total scattering cross section per unit olume andf(O) is the forward scattering amplitude per unit volume.

8

MiI. VALIDITY CONDITION

To obtain the condition of validity for the Rytov solution

U - exp "'O + 1 + V 2 + .."', it is sufficient to require that each

successive term in the expansion of '+ be smaller than the preceding

(2) .1term. In view of this, and since the various terms -in the expansion

of V; are random functions which can be described only in statistical

terms, it is assumed that the condition of validity on the Rytov solu-

tion is related to the relative magnitude of the square value of suc-

cessive terms. Therefore, the condition of validity for the Rytov

solutions is taken to be

.2 2

22

I I i

1 r :- 1 (27)

2 2It is noted that when ' l1 the successive terms in the asynptotic

2 1

expansion for (i.e., + 'I1 + 2 + .') will all be of the same

order of magnitude., The solution given by Eq. (23) is therefore valid

only when Eq. (27) is satisfied.

The condition of validity of the Born approximation is given by

exactly the same expression (Eq. (57) of Ref. 6).

We conclude that, in general, both the Born and Rytov approxi-

mations have the same domain of validity. A critical range R isC

determined by solving Eq. (27) with an equality sign. For R < Rc,

the Born and Rytov solutions are valid, while for R > R bothc

solutions fail to yieid a valid solution. Treatments which apply

the Rytov approximation beyond R , and therefore predict large dlis-cpersion of intensity, ( 7 are incorrect.

REFERENCES

1. Tatarski, V. I., Wave Propagation in a Turbulent Medium, McGraw-

Hill Book Company, Inc., New York, 1961.

2. Brown, W. P., Jr., "Comments on the Validity of the Rytov Approx-

imation," J. Opt. Soc. Amer., Vol. 57, No. 12, December 1967,pp. 1539-1543.

3. Schmeltzer, R. A., "Means, Variances, and Covariances for Laser

Beam Propagation through a Random Medium," Quart. Appl. Math.,

Vol. 24, No. 4, January 1967, pp. 339-354.

4. Hufnagel, R. E., and N. R. Stanley, "Modulation Transfer Function

Associated with Image Transmission through Turbulent Media,"

J. Opt. Soc. Amer., Vol. 54, No. 1, January 1964, pp. 52-61.

5. Fried, D. L., "A Diffusion Analysis for the Propagation of Mutual

Coherence," J. Opt. Soc. Amer., Vol. 58, No. 7, July 1968,

pp. 961-969.

6. Yura, H. T., E&ectromagnetic Field and Intensity Ftuctuations in

a Weakly Inhomogeneous Medium, The RAND Corporation, RM-5697,

July 1968.

7. Fried, D. L., "Aperture Averaging of Scintillation," J. Opt. Soc.Amer., Vol. 57, No. 2, February 1967, pp. 81-89.

A//

i4

/

DOCUMENT CONTROL OATAI ORIGINATING ACTIVITY 2a. REPORT SECURITY CLASSIFICATION

I UNCLASSIFIEDTHE RAND CORPORATION 2b. GROUP

3. REPORT TITLE

THE SECOND-ORDER RYTOV APPROXIMATION

4. AUTHOR(S) (Last name, first name, Initial)

Yura, H. T.

5. I RT DATE 16. TOTAL No. OF PAGES 6o. Na. OF REFS.February 1969 18 7

7. CONTRACT OR GRANT No. 8. ORIGINATOR*S REPORT No.

F44620-67-C-0045 RH-5787-PR

9a AVAILABILITY/ LIMITATION NOTICES 9b. SPONSORING AGENCY

United States Air Force

DDC-1 Project RAND

I0. ABSTRACT II. KEY WORDS

An explicit and useful formulation of the Turbulencesolution of the second-order Rytov ap- Optics

proximation for estimating the effect of Wave pr"pagationa turbulent medium on laser propagation. AtmosphereThe first-order Rytov pproximation ha,- Physics

been much used in tteor'.'etical analyses

biit recently there has been a controversyon the ranges for -hich it gives an ade-quate estimate. RM-5697-PR showed thatthe calculations of the EM field mustinclude all che second-order terms ofthe fluctuating part of the index of re-fraction, proved that the Born approxima-.tion is valid only over a limited range(a few kilometers at optical wavelengths),and assumed that the same validity con-dition applied to the Rytov approximation;this study explicitly demonstrates thatthis is the case. The second-order Rytovapproximation is obtained by equating thesecond-order terms in the Rytov and Bornexpansions and is explicitly solved inquad'-ature. The field equation that re-sults gives the correct average field,phase and intensity statistics, conservesaverage energy, and satisfies the opticaltheorem--within the do.ain of validity,which is the same for the Rytov and theborn approxiuations. Analyses that applythe Rytov approximation over longer dis-tances, and therefore predict large dis-persian, are incorrect.