JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Decay of Mutual Coherence in Turbulent Media*

LEONARD S. TAYLOR

Case Instiitte of Teclnology, Cleveland, Ohio 44106(Received 29 August 1966)

An expression is derived for the mutual coherence factor for a plane electromagnetic wave propagating ina region of dielectric turbulence. It is assumed that the medium is locally homogeneous and isotropic and thatthe scale of turbulence I is much larger than the wavelength X. The formula is valid within ranges L deter-mined by assuming small scattering angles (L<<14/X') and [(2ir/X)'((An)')lL]1<<1 where ((An)') is themean-square fluctuation of the refractive index. It is shown that this same expression can be derived withinthe limits of geometric optics for small separation distances.INDEX HEADINGS: Coherence; Atmospheric optics; Geometrical optics; Scattering; Refractive index.

W HEN an electromagnetic wave enters a regionV of dielectric turbulence, the coherence betweenspatially-separated wave vectors decays owing to ran-dom scattering by the fluctuations of refractive index.In an image-transmission system, the over-all transferfunction is the product of the lens transfer functionand the mutual-coherence factor' for the wave. Thus,for example, the decay of mutual coherence duringpropagation through atmospheric turbulence is re-quired for calculations of image degradation in opticaltransmission systems.

In this paper we consider the propagation of electro-magnetic waves with wavelength X through a dielectricwith scale of turbulence l>>A and mean-square variationof index of refraction ((An1)2). In the geometrical-opticsapproximation such problems are studied by assumingthat only a random variation of phase incrementAnL/X over the linear path from zero to L in theturbulent dielectric need be considered. It can be shown2

that this approach is valid when »>>(NL)4 . In the"next" method3 a formal integral solution of Maxwell'sequations is obtained and the Born (single-scattering)approximation is applied. This approximation dependsupon the smallness of a mean-square phase term7r2((A) 2)LI/M. Using the Born approximation4 we findthat the wave intensity attenuates, because of scatter-ing, as exp(-aL), where ae¢-((z)S)l/X2 for l>AX. Thusas the distance increases, the field becomes increasinglystochastic.

These considerations are embodied in the resultsobtained in a notable but rather inaccessible paper byKeller5 who showed that the field of a plane wave canbe represented as the sum of coherent (mean) plusincoherent (stochastic) components. The coherent com-ponent propagates with wave number kk 0o=2wr/X; k

* Work supported by the National Aeronautics and SpaceAdministration.

It R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52(1964).

2 V. I. Tatarski, Wave Propagation in a Tutrbident Medium(McGraw-Hill Book Company, New York, 1961).

3 A. W. Wheelon, J. Res. Nati. Bur. Std. 63D, 205 (1959).4 L. A. Chernov, Wave Propagation in a Random Medium

(McGraw-Hill Book Company, New York, 1960).6J. B. Keller, in Proc. Symposia in Applied Matltematics, XVI.

(American Mathematical Society, Providence, Rhode Island,1964), p. 145.

is complex with an imaginary part ki so that thecoherent component decays rapidly and is replaced bya stochastic wave. It is shown that k 4 =a/2. Thus inthe "multiple-scatter" zone the field is essentiallystochastic as expected. We apply the methods suggestedby these considerations and obtain a result which isidentical with the leading terms of the power-seriesexpansion of a result which has been obtained by anumber of other investigators' "8 7 in a variety of ways.The difference between this and previous calculationsis that we account strictly for the terms of higherorder, which are dropped from the previously publishedequations. As a result we find that the formula can bejustified only in the single-scatter zone defined by theleading terms of the power-series expansion and de-termined by the smallness of the mean-square phaseterm described above.

The first derivation of the formula was by, Hufnageland Stanley' who claimed validity for all ranges withinthe sagittal approximation, L<<14/N3. Chase8 showed,however, that this demonstration was incorrect. Hiscalculation may be taken to be strong (though notconclusive) evidence that the formula is not valid forall ranges.

A more recent attempt was made by Beran7 who alsoconcluded that the formula was valid for long ranges.However, in this calculation it was assumed that thewave propagates between iteration intervals with thesame wave number as in the uniform medium. Thisassumption is unjustifiable; the wave is attenuatedand phase delayed by the turbulence.5 9 These effectscannot be neglected, no matter how small the iterationinterval. They are taken explicitly into account in thepresent formulation; our final expressions differ fromthose of Beran by inclusion of these factors. Becausethese terms are available only to first order in theindex-of-refraction-perturbation parameter we find thatthe formula is valid only in the single-scatter region.

Another derivation of the same formula was givenby Fried and Cloud6 who relied on Rytov's method.

OD. L. Fried and J. D. Cloud, in Proc. Conf. on AtmosplhericLitmitations to Optical Propagation, Boulder, Col. (1965).

7 .M J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).8 D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).' L. S. Taylor, J. Math. Phys. 4, 824 (1963).

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Mklarchl967 DECAY OF MUTUAL COHERENCE IN TURBULENT MEDIA

However, Brown'0 showed that Rytov's method isnot valid in the optical limit beyond the single-scatterregion and surmised that this limitation is valid forall frequencies. In a related work" we have shown thatthis surmise is correct and furthermore that Rytov'smethod does not in general correctly yield even allfirst-order terms in the perturbation parameter. Thus,no proof relying upon the Rytov method can be invokedto validate the formula in question beyond the single-scatter zone.

In the last section of this paper we examine thegeometric-optics derivation of the formula with morecare than hitherto employed. We find that, even withinthe restrictions of this approach, the formula is limitedto the single-scatter zone if no restrictions are placedupon the lateral separation in the mutual coherence.Only for vanishing separation does the geometric-opticsformulation yield the formula in question without addi-tional restrictions upon the range.

The failure of any wave solution in the multiple-scatter region must be emphasized. It is a strikingfeature of present development that no rigorous solu-tions of Max-well's equations are available beyond thesingle-scatter region in turbulent media. Thus validsolutions of optical-propagation problems do not extendbeyond a few dozen meters in the lower atmosphere.

BASIC SOLUTION

When the magnetic field is eliminated from Max-well's equations, the electric vector is found to satisfy

therefore

M(rlr2 ) = (u (r,t)u* (r2,))= -v (rl)vc* (r2)+(vi (rl)v* (r2)). (4)

We consider a plane wave which is initially uniformin the z= 0 plane and which propagates in the z direc-tion through a region of homogeneous isotropic dielec-tric turbulence between z=0 and z= L. As describedin the previous section, the coherent component propa-gates as exp(-jcot+jk-r) where l, the effective propa-gation factor for the turbulent medium is complex

k=kr+jki. (5)

We use the notation (x,y,z) = (p,z) and assume thatthe turbulence is stationary with respect to p transla-tions. Thus, using k= kj , we find

vc (p,,L)vc* (p2,L) = v0(pl,0)vc*(P2,0) expj(k* r,-k" * r2)=Vc(P1,0)V0 *(P2,0) exp(-2kiL)= exp(-2kjL), (6)

where we have assumed amplitude at z= 0. As indicatedpreviously, the factor 2kj is the attention coefficient afor scattering. Using Eq. (6) and the assumption oflaterally stationary turbulence, we write the mutual-coherence factor in the plane at z= L

M (pa-P2, L)= exp(-2kiL) +(vi(pL)v* (p2,L)). (7)

The incoherent component vj(p,L) is obtained by useof the derivation due to Keller,5 which we brieflyreview for completeness. Apply the definitions of Eq.(3) to Eq. (1). We find that

V2E-- - -V -E *eVI.c2 0t2 i -E

(1)

We assume as usual that the gradients of the dielectricconstant are small so that we may set the right sideof Eq. (1) equal to zero and write n'2== Then thescalar components of E may be represented as the(complex) disturbance u(r,t) which satisfies the waveequation

V2 u- (n2/c2) (&u/ al2) = 0. (2)

For the time-harmonic wave u(r,t) = e-jtv(r) propagat-ing in a medium with random index of refractionn(r), v(r) and i(r) are random functions whose ex-pectations we denote by (n(r)), (v(r)). The coherent(mean) and incoherent fields va, Sv are defined by

v.( r) = (v (r))

Vj(r)= v(r)-V,(r). (3)

The mutual-coherence factor for X is by definition'2

10 W. P. Brown, Jr., J. Opt. Soc. Am. 56, 1045 (1966).11 L. S. Taylor, J. Res. Natl. Bur. Std. Radio Sci./USNC (April,

1967).1" M. Born and E. Wolf, Principles of Optics (Pergamon Press,

Inc., New York, 1959).

(V 2+kju 2)Vu = ko2 (n2 vj)+k-2k[jI1 2) - n2]V (8)

where ko=w/c. Thus writing n(r)=1+eq(r) where eis a small parameter, we obtain

(V2+k02)vi= 2eko¶QAvi)-gvi+ (&H)-t)vc1+0(e2 ). (9)

Thus vi is 0(f) so we may simplify to

(V2+ko2)sv= 2ek02 ((g)-g)v,+0(E). (10)

(Throughout this paper we assume that the turbulenceis homogeneous over the entire region 0< z < L. Thuswe write (,u)=(g(r)). The generalization to the caseof only local homogeneity is straightforward.)

The solution of Eq. (10) is

v1(r)=2Eko2 G(r,r')Lg)-pg(r')]ve(r')dr+0(E2), ( 1)

where G(r,r') is the free-space Green's function. Usingthe above results from Keller we find

(v (rn)v* (r!!))

- 4E&ko4WJ G(rs,r")G* (r2,r') [(g (r')y (r"))- Q,)2]

XvC(r')vC*(r"')dr'dr"+0(E3 ). (12)

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6LEONARD S. TAYLOR

Without loss of generality we may take ()==O, using a trivial adjustment of the constants ko, e. Using thenormalized correlation function

C (r'-r") = (u (r') u (r"))/VQ2 ), (13)we obtain, correct to 0(e2),

M(pi-P2, L) =exp (-2kiL)+4ko4eN2(f2 fG(rt~rD)G* (r2r")C(r'-r")e xpj kz'-k*z')dr'd'. (14)

Introducing the Green's functions explicitly and noting Mi(pt- p, L) (pL)for p= P-p we find

oe2(A2 ff expj(koj rl-r'l -koj r 2 -r"I +kzP-k*z)M(p,L) = exp(- 2kiL)+ j G (r'-r`)dr'dr". (15)

For koly>> we may follow the usual procedure, which assumes small scattering angles, and limit the region ofintegration to cones with vertices at Pl, P2 and with apertures (kod)-' where I is the scale of turbulence. This sagittalapproximation2 imposes only the weak condition LK<14K/. For convenience we take rl= (p,O,L), r 2= (0,OL). Then

r'- == L-z'+ 2[(Xt p)2 y'2]/ (L- z')=r2 -r"> -L-z"l+l (XI1"2+y" 2)/(L--z ). (16)

Because of the assumption of locally isotropic turbulence C(r'-r")=C(r), = fr- r" Thus

kcN e2(L,2) ffdr'dr'tC(r)

M(p,L) = exp (- 2kiL)±+ f f C-I2 J(L -z') (L-z")

Xexpj ko(z"-z')kz' -k*z"+1ko [(x-p)2 +y" V (17)L-z-iAs usual, we introduce relative and "center-of-mass" coordinates

XX -X f, yy'-y" 1, 5=2t -Z

IV= (x'+x Y= (Yy'+Y), Z=2(iZ-z"), (18)whence

ko4 J<L2) I .L p e-_p[j(k-ko)z-2k,Z]C(r)M (p,L) =exp- 2kiL)± J JoI . J L-- )-Z+-) dzdZdxdy

4712 - J J-J¢: L-Z-12z)(LZ+z)

X -0 f(Y + p2-p -+ (Y+yl2)2 (K- x/2)2t (Y- y/2)21XJ J expl ik{o - dXdY. (19)

- 2 L-Z---z L-Z+Iz J

The integration in X and Y is readily evaluated using an identity proven by Chernov4 in the course of a similarcalculation. We find

l(pL) = exp(- 2kjL)+J J f r1 exp[j(k7-ko)z- 2kiZ]dzdZ(x-p)2+y2

JX r [ ikfC(rDddy. (20)

We apply the method of stationary phase'2 to the x, y integration in Eq. (20), justifying the asymptotic ap-proximation by noting that we are assuming kz>N and that the principal contribution to Mf(pL) is given atsmall z. The integration over Z follows immediately and we obtain

M(pL) =exp(- 2kiL)+ (ko2/ki)Qt,:')[1E-exp(- 2kiL)]f C(p,O,z) exp-j (kr-ko)zldz. (21)

Equation (21) predicts that M(p,L) will remain terms of higher order in e. Physically it is clear thatconstant after a few distances of order k'-1, an obvious the coherent wave cannot be the source of the field atimpossibility. The difficulty is that we have included ranges such that exp(- kL)<<1 and w-e must restrict

3CJ6 Val, 57

March1967 DECAY OF MUTUAL COHERENCE IN TURBULENT MEDIA

ourselves to the leading terms in keL. kA is 0(e2), so thatwe consistently assume (kjL)2 <<1 and write correctto 0(E,2 )

M (p,L)= 1-2kjL+2ko 2 2(2 )L

L

X| C(p,0,z) expE-j(k,-ko)]dz. (22)

The complex propagation factor Y is the solution' ofthe equation

k2=ko,(1 +e'2(,g))± (4e0(,2)k04 k)

xf exp(jkor) sinkrC(r)dr. (23)

When the correlation length is much larger than thewavelength, kr-ko is 0(t).5 We can write Eq. (23) inthe form

X {exp (2jkor) exp-[ki-j (k< ko)]r

-exp[kj-j(k 7 -k,)j]rjC(r)dr. (24)

If we impose the condition koe(u')jl<K1, we may[remembering kei-C¼'2)] set the second and third ex-ponentials equal to unity and find

ki= ka2eQ2)f2 C (r)dr. (25)

It is readily seen that the condition kae((Q2))11<<1 issatisfied if (kjL)2<<1, L»>1 so we have not imposed anyfurther restriction. As expected ki=a/2, where a is theattenuation coefficient for scattering4 derived, usually,from single-scatter theory. With the same approxima-tion we may also set the exponential in the integrandin Eq. (22) equal to unity and obtain from Eqs. (22)and (25)

JI(p,L) = 1- 2k6'e2'(g2)L

Xf [C(0,0,Z)-C(p,0,z)]dz. (26)

In writing Eq. (26), we have replaced the upper limit inEq. (22) by z, since L>>t and we have used the assump-tion of isotropic turbulence to write C(r) =C(0,0,z).

The present analysis may be considered to be anextension of the Born method: viz; we have replacedthe unperturbed fields which appear at the right in Eq.(12) by the coherent fields which include the effects ofattenuation and phase delay due to previous traversalof the stochastic medium. The condition (ktL)'<<1 isclosely related to the condition required for the single-scatter approximation. Using Eq. (25), we find thatthis condition can be written as [ko'e'(Qz)'L]K<<1. Theexpression in the brackets is recognizable as thephase term whose smallness is required in the Bornapproximation. 3 Of course, Eq. (26) cannot be deriveddirectly from the single-scatter approximation, in whichki=k,-ko=0-

RELATION TO THE GEOMETRICOPTICS SOLUTION

Hufnagel and Stanley' provided an interesting deriva-tion for the mutual-coherence factor based upon geo-metric optics. We review this derivation briefly in thepresent notation in order to clarify the range of validityof the resulting formula:

Beginning with Eq. (2), we substitute u(rt)=A(r)Xexpj(*oz-ad). Then, to 0(e),

V2A +2jhko3(A/az)+2kua'eA = . (27)

The effects of diffraction and scintillation are ignored.1 2

by dropping the term V2A .As a result, for unit amplitude

A (x,y,L) = exp[ikoef a(xyiz')dz'] (28)

With our previous choice of coordinates and assumnp-tions of local homogeneity and isotropy we now find

M(p,L) (exp {ikeej [(p,0,z)-g (0,0,z)]dz}). (29)

By the central limit theorem, the integral has gaussiandistribution with mean zero. As a result,

M(pL)-exp {-2 koi e2f dzij dz2, ((,0,z,) -Ap (0,0,Z)][g(P,0,z2) -y (0,0,z2)]) |

exp {ko2e2(2)f dzif dz2EC(0,0,zi-Z2)-C(p,0,z,-z2)1 . (30)

Introducing the definitions of Eq. (18), we obtain for L>>1

M(pL)-=ex pj -2ko2eNj2)Lf [C(0,0,z)-C(po0z)]dz}. (31)

307

308 LEONARD S. TAYLOR Vol 57

Equation (31) is the previously obtained result expressed in our notation, for which validity has been claimedwithin the limits imposed by the condition under which diffraction can be ignored, P2/X>>L. In order to maintaincorrespondence with the equations of geometrical optics, we have eliminated terms of O(E2). Thus, because Eq.(27) is correct only to 0(e), the result given in Eq. (31) is useful only to 0(e2). Therefore we must expand Eq. (31),eliminating higher-order terms in A. We then find that neglecting the effects of scintillation and diffraction leadsback to Eq. (26), with the requirement

[2ko2E2([42)Lf) [C(0o0,z)-C(P)z) ]dz ]<<1. (32)

Equation (32) is satisfied for all p if (kL)2 <<1, where ki is given by Eq. (25), in agreement with our previousresult. (We observe, however, that the formula will also be valid to longer ranges for small separation p<l.)