two-position, two-frequency mutual-coherence function in turbulence

1446 J. Opt. Soc. Am./Vol. 71, No. 12/December 1981

Two-position, two-frequency mutual-coherence function inturbulence

Ronald L. Fante

Avco Systems Division, Wilmington, Massachusetts 01887

Received May 24,1981

By using the extended Huygens-Fresnel principle, an expression has been obtained for the generalized two-posi-tion, two-frequency mutual-coherence function of an electromagnetic beam in a turbulent medium. That resulthas been examined for different states of source coherence and for different turbulence conditions.

1. INTRODUCTION

In this paper we derive the two-frequency, two-position mu-tual-coherence function (MCF) for an arbitrary source ra-diating into a turbulent medium. This quantity can be usedto compute the two-point correlation function for pulsepropagation, the transient intensity, and specific quantitiessuch as the coherence bandwidth and delay time.'

The earliest studies2-5 of transient propagation in randommedia were limited to plane-wave propagation in weaklyturbulent media. Later, Sreenivasiah et al.

6 and Dashen7

obtained results for plane-wave propagation in a stronglyturbulent medium by solving the Markov approximationequation for the special case of a quadratic wave-structurefunction. More recently, the two-frequency MCF for a planewave was obtained for turbulent media of arbitrary strengthand with arbitrary power-law-structure functions.810 Fur-thermore, Sreenvasiah and Ishimarull have now generalizedtheir previous results to an arbitrary beam and obtained thequadratic-structure function approximation to the two-fre-quency, two-position MCF in homogeneous turbulence.Because the assumption of a quadratic-structure function cansometimes lead to difficulties,'2 this paper removes this re-striction, along with the restriction of homogeneous turbu-lence, to obtain results for the two-point, two-frequency MCFin inhomogeneous turbulence with index-of-refraction fluc-tuations governed by a Kolmogorov spectrum. This situationis a realistic one for propagation through atmospheric tur-bulence.

2. ANALYTICAL FORMALISM

In the analysis that follows we consider a narrow-band,' 3

planar source radiating into a turbulent medium. Under thenarrow-band assumption, the temporal behavior of the electricfield U(z, r, t) at position (z, r), where r = (x, y), can bewritten as

U(z, r, t) = eiwOt f u(z, r, w)eiwtdw + c.c., (1)

where u(z, r, w) is the positive frequency portion of the tem-poral frequency spectrum and wo is the radian center fre-

quency. From Eq. (1) it is possible to compute measurablequantitiessuchas (U(z,r,t)U(z,r',t')),where ( ) denotesan ensemble average, provided'4 that we can calculate (u(z,r, w)u*(z, r', w')). Consequently, we proceed to calculate u(z,r, w) in a turbulent medium in terms of the source field u(o,r, w).

Suppose that we have a source field u(O, rl, k ) in the planez = 0, radiating at frequency w1 = ck,, where c is the speed oflight and r, = (xl, yj). By employing the extended Hu-ygens-Fresnel principle it can be shown that the field u(z, r,kj) in the plane z is given by 1 5 "16

u(z, r, k) =- JJ d2 rju(0, r, k,)21riz -T

Xexp [i2k (r-zrl)+ ](r, rik) + ]iklz(2)

where t(r, ri, ki) is the additional complex phase at wavenumber k = wi/c that is due to turbulence of a spherical wavepropagating from (o, rj) to (z, r).

Equation (2) is valid provided that the turbulent eddies arelarge in comparison with the wavelength and also requires thatkz>>1,Ir-r'l <<z,andkllr-r'1 3 z-4 <<1. Ifwenowas-sume that any source fluctuations are statistically indepen-dent of the fluctuations of the turbulent medium between thesource and the observation point, we can use Eq. (2) to showthat the two-point, two-frequency MCF is

r, 2(Z, r, r') = ,i** *J' d2 rjd2r 2F12 (0, ri, r2)

X G12(r, r'; ri, r2), (3)

where

rl 2 (z, r, r') - (u(z, r, kl)u*(z, r', k2)), (4)

G12(r, r', rk, r2)

2kjz2 exp |i(k, - k2)Z + ik' (r - rj)2 - ik2 (r' - r2 12(27rz)2 L~l1l2Jz 2\r ,I

X (exp [1'(r, ri, ki) + ,6*(rW, r2, k2 )J), (5)

where ( ) denotes an ensemble average. Equation (3) is the

0030-3941/81/121446-06$00.50 © 1981 Optical Society of America

Ronald L. Fante

Vol. 71, No. 12/December 1981/J. Opt. Soc. Am. 1447

Fresnel-zone generalization of earlier results 17 to include theeffects of turbulence and is valid for an arbitrary source-coherence state. If it is assumed that the turbulence is sta-tistically stationary and that the complex phase fluctuations4' induced by the turbulence are Gaussian random variables,it can be readily demonstrated that 18"19

(exp (' + 4*)) = exp [-'/2 D12(r - r', r, - r2)], (6)

where

D2(r -r', ri - r2) = (|4(r, ri, k) - (r', r2, k2)12).(7)

Ishimaru20 has used the method of smooth perturbationsto obtain a formal expression for D12. For isotropic turbu-lence that result is

D2(r - r', ri - r2)

= (27r)2 f kdk f 0 Z dz'(4n (k, z') {k 12 + k22

2kik 2J0 (k3) COS [Z'(Z Z') k2( (8)

where

0 - |(r-r') + (1 -Z (r -r2)|-( ( z'1 I (9)

4'n(k, z') is the wave-number spectrum of the index-of-re-fraction fluctuations of the turbulence, and Jo(x) is the Besselfunction of zero order. In deriving Eq. (8) it is implicitly as-sumed that the signal wavelength is smaller than the innerscale size of the turbulent eddies.

Because it has been assumed that 4' is a Gaussian randomvariable and because the method of smooth perturbations hasbeen used to evaluate D12, these results are strictly valid onlyin weak turbulence. However, in the limit when ki = k 2, ithas been found2l that Eqs. (6)-(8) are identical with the re-sults obtained using the Markov approximation, an approachthat is valid for turbulence of any strength. The range ofvalidity of Eqs. (6)-(8) for k 1 - k2 is uncertain, but it is ex-pected that if (k, - k 2)/(k, + k2) << 1 we can also safely usethese results in turbulence of arbitrary strength.

We evaluate Eq. (8) for turbulence having a von Kdrmdn 2 2

wave-number spectrum

0.033Cn 2(z') exp (k 4r2 )

(% (k, z') = (k2 + LO-2)"1 /6 (10)

where C, 2(z') is the index-of-refraction structure constantof the turbulence and Lo and lo are the outer and inner scalesizes of the turbulent eddies, respectively. If Eq. (10) is usedin Eq. (8) and the integration on k is performed the resultis

D2(r - r', ri - r2) ! 0.782Lo5/3(k, - k2)2Z1 1

X r dt C,2(t) + 8.69klk2z r dtC, 2(t)

X Re {km-5/3 [PFi (-6, 1,- 4 )-1il (11)

where t = z'/z, jFj is the hypergeometric function, Re denotesthe real part of, and

km2 =1

(12)(10)2 izt( -t) 1 - 1)\27r/ 2 Vk 1 k21

In deriving Eq. (11) it has been assumed that 1k, - k2Iz(2k k 2 Lo 2 )-1 << 1.

Equation (11) can be simplifed considerably under thefollowing assumptions:

(1) Either Ir - r'l or Ir - r2 l is much greater than theinner scale size, 10, of the turbulent eddies.

(2) The quantity A k, - k2 is sufficiently small that

A 2korm 2

ko Z(13)

where ko- (kl-k 2)/2 and rm is the larger of Irl-r 21 and Ir-rl1.

When the aforementioned conditions are satisfied it can beshown that Eq. (11) reduces to

D2(r- r', ri - r2) = 0.782A2Lo5/3z

x r dtC, 2(t) + 2.92klk2Z

X J dtCn2(t)t(r -r') + (1 - t)(r, - r2)15/3. (14)

Note that Eq. (14) reduces to the conventional wave-structurefunction' 9 when ki = k2, so that A = 0.

Let us now use Eqs. (5), (6), and (14) in Eq. (3), and thenmake the following changes of variables:

r + r'2

4 = r -r',

P r= -r2, =r + r22

k, -k2, k, ±i k2= k,-k 2 , ko = 2

We can then rewrite Eq. (3) as

rF2 (z, k, 4) = C(A) fJ -- J * d2pd2nrl 2 (o, p, n)

X exp (o (R-,q) - P) 2zz 2z

X [(R - 1)2 + 1/4(4 - p) 2 ] - y (ko, A, 4, p),

where

(k02-- I2

4k2 /\2C(A) = (2 )2 exp(iAz -a2),

y(ko, A, 4, p) = 1.46 (ko2 -Az

X 3' dtC,2(t)Itt + p(l - t)15/3,

1a = 0.39IL05/3Z r dtC, 2(t).

(15)

(16)

(17)

(18)

(19)

(20)

(21)

Ronald L. Fante


The result in Eq. (18) is the general expression for the two-frequency, two-position MCF in a random medium.

3. PARTIALLY COHERENT SOURCE

For a (temporally) stationary, partially coherent source, it isfound23 that the source MCF can be written as

ri(op, a) = f(ko, PI n)5(A), (22)

where b(A) is the Dirac delta function. Consequently, for thiscase Eq. (18) lecomes

rF 2 (Z, R, 4) = b(A)C(0) J ... fd2pd2 ,7f(ko, p, A)

X exp [i- k(R-71) -( -p)-y(ko, 0, 4, p) (23)

The temporal mutual-coherence function F(z, R, 4, t = t -t') follows directly2 4 from Eq. (23) through a Fourier trans-form.

If the source is also spatially incoherent, it is possible towrite

f(ko, p, ?1) = g(ko i)a( 2)(p), (24)

where 6(2 )(p) is the two-dimensional Dirac delta function. In

this case we can simplify Eq. (23) even further as

r 1 2 (i, R, 4) = b(A)C(O) exp[-y(ko, 0, iq, 0)]

X ffJ d2-qg(ko, n) exp i ° (R (25)

Finally, if g(ko, i7) = h(ko) exp (-_i2/w2), the integration in Eq.(25) can be performed to yield

rl 2 (z, R, 4) = 5(A)W2ko2 h (k 0)4-rz2

Xexp ( R-.I.R k 24 2 _ t 1 ' (26)z 4z 2 wP

where

ps 5/3 = 1.46ko2z 4' dtt5/ 3 C, 2 (t). (27)

Ronald L. Fante

r12(zl R. = C(A) SJ d2iq M(A, ho, i)

X exp{i ° (R - )-4+i2 [4 - q)2 +

- oy (o, A, 4, 0)}. (29)

4. COHERENT SOURCE

When the source is completely coherent it is evident that2 5

F1 2(0, ri, r 2) = u(0, r, kl)u*(0, r2 , k2). (30)

We shall now consider sources with a canonical spatial dis-tribution of the form

u0, rl, k) = A(hi) exp- r2 - ikr2 2 J (31)

where ao is the equivalent radius of the source, z = F is thepoint at which the source is focused when the wave numberkl= 1, and A(h1) isan amplitude function. By writing u(0,rl, k 1) in the form in Eq. (31) we have implicitly assumed thatthe spatial distribution of the source is invariant over thefrequency band of interest, although its amplitude, A(k 1), doesvary as the frequency is changed.

For a coherent source it is generally impossible to performall the integrations in Eq. (18) because of the nonanalyticityof -y in Eq. (20). Consequently, we consider a number oflimiting cases.

A. Thin Turbulent LayerFor a ground-based -or low-altitude source transmitting to asatellite along a nearly vertical path through the atmosphereit is appropriate to consider the turbulence as concentratedin a thin layer adjacent to the source. In this case, if Eq. (30)is used in Eq. (18), the result is

where

From Eq. (26) we see that the magnitude of the MCF of astationary, spatially incoherent source is dominated by tur-bulence effects if

2zPs <<-,

how(28)

where w is the spatial size of the source. Conversely, the fieldcoherence properties are unaffected by turbulence if p5 >>2z/how. Note from Eq. (27) that Ps X if all the turbulenceis concentrated on a thin layer adjacent to the source.Therefore, as expected physically, the turbulence adjacent toan incoherence source cannot affect the field-coherenceproperties, because the source is already spatially incoherent,so the turbulence cannot make it appear any more inco-herent.

Equation (18) can also be specialized to the case of a sourcethat is incoherent spatially but not necessarily temporallystationary. In this case P1 2(o, p, n) = M(A, k0, kk)6 2 (p).

Substitution of this into Eq. (18) gives

P1 2 (z, R, 4) = eilzA(kh)A*(k 2)P12 (z, R. 4),ko 2 p 2

P12 (z, R, 4) = 2 G(a, v)

X expjo- aA2- (hot + AR)2]4sz2

R0 +A _M4 =-4°.R +-I2R2± + J2)z 4z 2J

1 k A\S 2 '

ao2 F~z

=P( S + I2)

o= hp (,312Z2 + 201At4. R + , 2R 2)1/2,

p- = 1.46kh2 z 4' dtC 2 (t),

A .~ ik0

4ko 2sz

iA

2sz

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

G(a, v) = . dt exp (-at - t516 )Jo(utI/2 ), (41)


and Jo(x) is the Bessel function of zero order. When k, = k2so that A = 0, it is easily shown that Eq. (33) reduces to theconventional two-point coherence function for a thin turbu-lent layer.

Although G (a, v) cannot be evaluated in closed form, it ispossible to obtain an approximation to G by replacing t516 witht in Eq. (40). If this is done we obtain

J_ V2

exp 1+a (42)

By using Eq. (42) in Eq. (41) we can obtain an approximateexpression for P,2(z, R, t). For example, if Eq. (42) is usedin Eq. (41) and we then evaluate r12 in the plane-wave limit(ao -a, F - a) * we find that

That is, when calculating the single-point, two-frequencyMCF an extended turbulent medium can be simulated exactlyby a homogeneous thin screen located at an appropriate dis-tance zo from the source. The thin-screen approach was usedby Rino et al. 10 in their analysis of coherence bandwidth lossin transionospheric radio propagation. These authors hadoutstanding success in comparing their theoretical results withmeasured data. Our analysis shows that the reason for thissuccess is that the equivalent thin-screen model is exact, andnot an approximation, even for extended turbulence of arbi-trary variation. However, it is easily shown that an equivalentthin screen cannot be found for calculating the two-pointMCF.

Let us now examine Eq. (44) when the beam is collimated(F - c), the turbulence is homogeneous, and the observationpoint lies on the z axis (R = 0). Then if Eq. (42) is used toapproximate G(b, vo) we get

P12(Z, R, ~) ~-exp 1+iq ~o- l[A2zi (ho A) I

, (43)1 -iq

where q = k02p2/2Az. As A = ki-k 2 -Oit can shown that

Eq. (43) reduces to rii(z, R, t) exp (-_ 2/p 2), which is thequadratic approximation to the well-known result, ri, (z, R,t) = exp (- I (/p i 5/3) ,

B. Single-Point, Two-Frequency MCF in ExtendedTurbulenceWhen r = r' but kl 1d k2, the result in Eq. (18) becomes

r1 2(z, R, 0) = (kopo)24sz 2

X exp i 2z _ 4z-2 A) G (b, vo), (44)

where

b = P +2 s J'+

pokoiR

-1/ = .6k2z

po-51 3= 1.46 ho2z ,f dt (1 -t)5 I3 Cn 2 (t), (47)

and F12 is related to r12 through the relationship in Eq. (32).It is interesting to note from Eqs. (44)-(47) that the single-point, two-frequency MCF in extended turbulence is exactlyequal to the result that one would obtain if all the turbulencewere concentrated in a homogeneous thin layer with index-of-refraction structure constant C' 2 and normalized thicknesse such that

1eCh2 = f dtC 2(t), (48)

provided that the thin screen is located a distance zo from thesource given by

1I ZO5/3 S dt (1 - t)5 /3C, 2 (t)

1 d-C=) . (49)f a tC,,2(t

r12 (z, 0, 0)

[01QL5/6 '2

1 -i 662 4(11 + - - I- -- + I-Q2 o 0 4 -2 ~i 21 '

- (50)

where £L = koLo2/z is the Fresnel number of the largesteddies, 0o = koao2/z is the Fresnel number of the source, i =kopo2/z is the Fresnel number associated with the phase-coherence length po, and 6 = A/ho. Equation (50) is thegeneralization of the plane-wave results obtained by Rinoet al. 10 to the case of a finite source.

In general, as can be seen from Eq. (50), the relative co-herence bandwidth (i.e., the value of 6 at which I P121 is equalto e- 1 of its value at c = 0) of the finite beam is not signifi-cantly different from the plane-wave result (QO -O ) unlessthe turbulence is sufficiently strong that Q << 1 and. QQ oforder unity. These conditions are necessary but not alwayssufficient for the beam size to affect the coherence bandwidth.Whether they are sufficient depends on the value of QL/U.

In the plane-wave limit (Q0 - ) it can be seen from Eq.(50) that there are two distinct regimes of behavior of F 12depending on whether the numerator or denominator deter-mines the behavior. It can be shown that if the turbulencestrength is such that Q >> 4.4 QL 5/7, the numerator of Eq. (50)determines the wave-number coherence bandwidth 6c, whichis given by

1.19 (Q 5/12

BC1.9QL~ -

Equation (51) can be written in terms of A, Cn 2, etc. as

1.61

(C" 2LO5/3,)1/2

(51)

(52)

Similarly, the condition Q >> 4 .4QL -5/7 can be written in termsof the turbulence strength parameter ,12 = 1.23ko7 /6 Cn2 z 11

/6

as a12 << 0.65QL2 5/4 2.

In the opposite limit when Q << 4.4 QL-5/ 7 (or equivalently,c 1

2 >> 0.65 QL25/42) the denominator of Eq. (50) determinesthe wave-number coherence bandwidth as

Q 1.31 1.026C =2 -

c 2 (0f1)12/5 k o7/5 Cn 12/5Z 11/5

Results equivalent to Eqs. (50), (51), and (53) were obtained"by solving the parabolic wave equation in the Markov ap-proximation with a quadratic-structure function. Conse-

Ronald L. Fante

.,

(53)


quently, our results obtained using the Huygens-Fresnelprinciple provide yet another demonstration of the equiva-lence of solutions obtained using the Huygens-Fresnel prin-ciple and those derived with the Markov approximation.

C. Quadratic Approximation to Structure FunctionIn order to obtain an approximate result for F12 (z, R, t) inarbitrary extended turbulence it is possible to approximatethe exact function y in Eq. (20) by a quadratic approximation.This approach leads to results that predict trends correctlybut are not accurate to better than 30 or 40%. The techniquethat we use is to approximate Eq. (20) by

y - (1.46ko2Z)6/5 3I dt [C,2(t)]6/5 1 (t + p (1 - t) 12.

(54)

If Eq. (54) is used in Eq. (18) we obtain

F12 (zR, R) = 4sDoz2

X exp li 2 - aA 2 -Q 2 (hot + AR)2

[ (B# + y lX exp 4Do ]' 5

where a, R, s, , were defined previously and

B = A-im + i , (56)4z 2sz2

Do = s+ n +(57)4 4sz 2

I = (1.46 ko2Z)6/5 J t2 [Cn 2 (t)]6 /5 dt, (58)

m = 2(1.46 ko2z) 615 f t (1- t) [Cn2(t)] 6/5 dt, (59)

n= (1.46 ko2z) 6/5 3 (1 - t) 2 [C. 2 (t)]6/5 dt. (60)

The result in Eq. (55) is equivalent to the results of Sreeni-vasiah and Ishimarull, who used a quadratic approximationto the structure function and then solved the parabolic waveequation in the Markov approximation. We can check thevalidity of Eq. (55) by considering the plane-wave limit (ao, F - ) and setting A = 0. We then find that

r12 (z, R, t) = exp [-(1 + m + n)02], (61)

where from Eqs. (58)-(60) we see that

I + m + n = J dt[1.46 ko2zCn2 (t)]6 /5. (62)

Equation (61) is the correct quadratic approximation to theplane-wave MCF and predicts an e-1 coherence length thatagrees with the exact result in the limit when Cn2(t) is inde-pendent of position. However, we caution again that Eq. (55)

is accurate for predicting trends only, and results obtainedwith this formula should not be considered as numericallyaccurate.

Ronald L. Fante

5. SUMMARY AND DISCUSSION

In this paper we have used the extended Huygens--Fresnelprinciple to obtain an expression for the two-position, two-frequency MCF in a random medium; this quantity gives acomplete description of the second moment of the field. Wehave noted, in the special limit when the wave-structurefunction is a quadratic function of It(r - r') + (1 - t)(r, -r 2) J, that our solution agrees with an existing 1l solution ob-tained by solving the quadratic approximation to the parabolicwave equation in the Markov approximation, even in strongturbulence. This is interesting because one might expect thatour solution would be valid only in relatively weak turbulencebecause we have assumed that the complex phase t in Eqs.(5)-(7) is a Gaussian random variable and then used a deri-vation of the wave-structure function D12 based on the methodof smooth perturbations, a method known to lead to certaininconsistencies in strong turbulence. The answer to thequestion of why our solution agrees so well with the more ac-curate solution obtained by the Markov approximation waspartially answered in several recent papers. 26' 27 First, it hasbeen demonstrated that if 4 X + iS, the phase fluctuationsS dominate the behavior in strong turbulence; furthermore,the phase behavior is computed rather accurately by themethod of smooth perturbations, whereas the log amplitudeX is not. The second reason for the success of our approxi-mations is that it can be shown that the extended Huygens-Fresnel principle implicitly contains the central limit theorem,provided that the source is sufficiently large that certainconstraints, given in Ref. 27, are satisfied. That is, the inte-gration in Eq. (2) essentially sums a number of randomfunctions, exp[it(r, ri, kj)], so that in strong turbulence, thesum obeys Gaussian statistics; this is effectively independentof the statistics assumed for 4.

We have also noted in Section 4.B that the coherencebandwidth can exhibit different behavior, depending on therelative sizes of Q and QL. The results in Eqs. (50)-(53) canbe stated in another fashion: there exists a path length zcsuch that for paths of length z < z, the coherence bandwidthis proportional to (C, 2 z)-1/2 whereas for z > zc the coherencebandwidth is proportional to (C, 2zl/ 6)-6 /5. The value zc atwhich this transition occurs is z, = 0.77 ko- 0 2 35 (Cn2)- 0 41 2.For a horizontal path near the ground it is found at visiblewavelengths that z, is of the order of 4 or 5 km, whereas atmillimeter wavelengths z, is of the order of 3 km. The valueof zc is lower at millimeter wavelengths than at visible wave-lengths because even though ko is much smaller at millimeterwavelengths the value of C,

2 is larger (than at visible wave-lengths) because humidity effects. The rapid decrease in thecoherence bandwidth beyond about 5 km places a limitationon the path length over which wideband signals can betransmitted without severe distortion.28

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Ronald L. Fante

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two-position, two-frequency mutual-coherence function in turbulence

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