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Modal theory for the two-frequency mutual coherencefunction in random media: point sourceJasmin Oz aa Department of Electrical Engineering - Physical Electronics , Tel Aviv University , Tel Aviv,69978, IsraelPublished online: 19 Aug 2006.

To cite this article: Jasmin Oz (1997) Modal theory for the two-frequency mutual coherence function in random media: pointsource, Waves in Random Media, 7:1, 107-117, DOI: 10.1088/0959-7174/7/1/007

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Waves in Random Media7 (1997) 107–117. Printed in the UK PII: S0959-7171(97)73301-6

Modal theory for the two-frequency mutual coherencefunction in random media: point source

Jasmin OzDepartment of Electrical Engineering – Physical Electronics, Tel Aviv University, Tel Aviv69978, Israel

Received 2 April 1996, in final form 29 July 1996

Abstract. The recently introduced modal expansion representation for the two-frequencymutual coherence function is applied here to the solution of a point-source field in a randommedium. This approach reduces the solution for any structure function to an eigenvalue problemfor an ordinary differential equation. For the initial point source it is shown here that the modalexpansion yields a result similar to that for the initial plane wave, modified by a sphericalfree-space phase which contains a weighted coordinate that does not interact with the medium.Having established these general characteristics, special attention is paid to power-law mediaand, in particular, to a quadratic medium, for which a new exact solution is derived. Via acollective summation of this new modal solution, we rederive the alternative exact solutionwhich exists in the literature. We also discuss the new parameterization implied by the newmodal solution.

1. Introduction

Pulsed wave propagation in random media is mainly described by the two-frequencymutual coherence function0(ρ1, ρ2, z, k1, k2) where z is the propagation range,ρj arethe transversal coordinates of the observation points andkj = ωj/c are the wavenumbers.Here, we consider propagation in a statistically isotropic and homogeneous medium wherethe structure functionD is a function ofρd ≡ |ρ1 − ρ2| only. In the literature analyticalsolutions are available in the case of a quadratic structure function for an initial plane wave[1] and an initial beam [2], which contains the point source as a special case.

It has recently been shown in [3, 4] that by transforming to the transversal coordinates

ρd = ρ1 − ρ2 ρw = k1ρ1 − k2ρ2

k1 − k2(1)

the parabolic equation for0 becomes separable and the solution foranysource problem canbe represented as a superposition of modes. This reduces the partial differential equationfor 0 to an eigenvalue/eigenfunction problem for an ordinary cross-sectional differentialequation. The modal expansion theory has been formalized in section 2.1 of [3].

The case of plane-wave initial conditions has been addressed in [3, 4]. This initialcondition is the simplest to solve since the boundary conditions atz = 0 do not dependon the transversal vectorsρd and ρw. However, the modal solution has some spuriousconvergence properties nearz = 0 owing to the unbounded (non-physical) nature of theinitial conditions.

As will be shown in this paper, the solution for an initial point source has a rathersimple form since the initial conditions atz = 0 can be factorized into two functions

0959-7171/97/010107+11$19.50c© 1997 IOP Publishing Ltd 107

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108 J Oz

where one depends on the magnitudeρd of ρd and the other on the magnitudeρw of ρw.Consequently, the general solution for a general statistically isotropic structure function,considered in section 2 as a special case of the general theory [3], can be factorized intotwo multiplicative terms. One term, which depends only onρw, represents free-spacediffraction of a spherical wave and is thus independent of the medium properties. The otherterm, that senses the medium, depends only onρd and has essentially the same structureas the plane-wave solution in [3, 4]. Special attention will be paid to power-law mediaand, in particular, to the quadratic medium, for which a new exact solution is derived(section 3). This new solution is compared with the alternative exact solution which existsin the literature [2, 5]. The exact results for the quadratic medium will then be extended insection 4 to form an approximate solution for a general power-law medium for which noexact solution exists. Specifically, it will be shown that while the modal solution convergesslowly as kd → 0, one can nevertheless obtain a collective expression which is valideverywhere including, in particular, atkd → 0. This observation will be demonstrated bothanalytically and numerically. Finally, in section 5 we shall explore the parameter range forwhich the solution based on paraxial initial conditions can actually be considered to be apoint-source solution. The presentation ends with concluding remarks.

2. Modal expansion for initial point-source conditions

We shall consider the fieldU(ρ, z, k) generated by a point source located at(ρ = 0, z = z0)

with z0 < 0. Within the parabolic approximation, the field near thez-axis atz < 0 is givenby

U = 1

z − z0exp

{ik[(z − z0) + ρ2/2(z − z0)]

}. (2)

It is assumed that forz > 0 the medium is random and characterized by a structure functionD(ρd).

We define the two-frequency mutual coherence function with a reference atz0 as

0(ρ1, ρ2, z, k1, k2) ≡ 〈U(ρ1, z, k1)U∗(ρ2, z, k2)〉 exp[−ikd(z − z0)] (3)

wherekd ≡ k1 − k2. Henceforth, the solution will be described in terms of the normalizedfunction 01 = 0 exp( 1

8k2dA(0)z). The initial conditions for0 are

0|z=0 = 01|z=0 =(

1

z0

)2

exp(−ik1ρ

21/2z0

)exp

(ik2ρ

22/2z0

)exp(−ikdz0) . (4)

In order to apply the procedure outlined in [3], the boundary conditions in (4) areexpressed in the(ρd, ρw) coordinates of equation (1), giving

01|z=0 =(

1

z0

)2

exp(−ikdρ

2w/2z0

)exp

(−ikwρ2d/2z0

)exp(−ikdz0) (5)

wherek−1w = k−1

1 − k−12 .

Note that the initial conditions in (5) depend on the magnitudes ofρd andρw only. Itfollows from equation (16) of [3] that01 is given by

01 = kd

2

∑n

exp

(i

2Knz

)fn(ρd)

∫ ∞

0An(E)J0(

√Ekdρw) exp

(− i

2Ez

)dE. (6)

Here, Kn are the eigenvalues andfn are the normalized eigenfunctions of theeigenfunction/eigenvalue problem given in equation (12) of [3], with indexm set to zero

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Two-frequency mutual coherence function: point source 109

as a result of the azimuthal symmetry of the initial condition. From equation (15) of [3] itfollows that theAn(E) are given by

An(E) =∫ ∞

0

∫ ∞

0dρd dρwρdρwf ∗

n (ρd)J0(√

Ekdρw) 01|z=0 . (7)

Substituting (4) into (7) and using the identity∫ ∞

0e−axJ0(b

√x) dx = 1

ae−b2/4a

we obtain

An(E) = Cn

i

kdexp

(i

2Ez0

)exp(−ikdz0) (8)

where

Cn =(

fn,−1

z0exp(−ikwρ2

d/2z0)

)=

∫ ∞

0dρd ρdf

∗n (ρd)

(−1

z0

)exp(−ikwρ2

d/2z0). (9)

Finally, substituting (8) into (7) we arrive at the final expression for01 for any isotropicmedium

01 = i

2

∑n

Cn exp

(i

2Knz

)fn(ρd)

∫ ∞

0exp

(− i

2E(z − z0)

)J0(

√Ekdρw) dE (10)

giving

01 = exp(−ikdz0)exp[ikdρ

2w/(z − z0)]

z − z0

∑n

Cn exp

(i

2Knz

)fn(ρd). (11)

Equation (10) is our main result in this section. It represents01 for the point-sourcecase as a product of two terms: a free-space spherical term

exp(−ikdz0)exp[ikdρ

2w/(z − z0)]

z − z0

(discussed below) and a superposition of transverse mode functionsfn(ρd) with modalpropagation coefficientsKn. As has been noted in [3], ImKn > 0, and it increases as themode indexn increases. Thus, the modal fields decay as a function ofz and for largez

the field is dominated by only the lowest-order moden = 0. We shall not discuss the fieldparameterization implied by (10) here since this solution has essentially the same featuresas the plane-wave case explored extensively in [3, 4]. In fact, the modal series in (10)resembles the modal series for the plane-wave case (see equation (21) of [3]) except for theexpansion coefficients. In the plane-wave case theAn are given by the inner product of 1andfn(ρd) (see equation (19) of [3]).

The free-space spherical phase factor

exp(−ikdz0)exp[ikdρ

2w/(z − z0)]

z − z0

does not appear for an initial plane wave and contains no parameters of the random medium.As has been pointed out in [3],ρw is a ‘centre-of-mass’ coordinate which propagates throughthe medium without any scattering and thus it is not affected byD(ρd). As a result, the partof 01 at z = 0 which containsρw simply undergoes mere free-space diffraction describedby the spherical phase factor in (10). Indeed, the same spherical factor appears in thefree-space mutual coherence function when expressed in the(ρd, ρw) coordinates, given by

0(z, ρ1, ρ2) = 01(z, ρ1, ρ2) = exp(−ikdz0)exp[ikdρ

2w/(z − z0)]

z − z0

exp[ikwρ2d/(z − z0)]

z − z0. (12)

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The free-space spherical phase becomes singular in the narrow-band limit where we havekw = −k1k2/kd ' k2/kd. This singularity must be cancelled out by the series. In the nextsection we find01 for a medium characterized by a quadratic structure function. It willbe demonstrated in that case that the singularity of the phase in the monochromatic limitcancels out as a result of the series terms. For non-quadratic media,01 will be found forρd = 0 and kd 6= 0. Under these conditions the phase is regular. The solution is thenextended to includekd = 0 and, since it assumes the correct value at this point, it is thusvalid for all kd.

3. Quadratic structure function

3.1. Modal expansion solution

If the structure function is given byD(ρd) = Cρ2d, whereC is a dimensional constant, the

eigenfunction/eigenvalue problem can be solved explicitly. The eigenfunctionsfn and theeigenvaluesKn are given by (see equations (42)–(45) of [3]):

fn = Ln(λρ2d) exp

(−λ

2ρ2

d

)Kn = 2

√i

2Ckd(2n + 1) = K0(2n + 1) n = 0, 1, 2 . . .

(13)

whereLn are Laguerre polynomials and we have used the notation

λ2 = i

2k2

wkdC (14)

defined in [3, 4].In view of (9), the coefficientsCn are given by

Cn = − 1

z0

∫ ∞

0exp(−y/2) exp(−ikwy/2λz0)Ln(y) dy (15)

where we denotey ≡ λρ2d. Equation (14) involves the standard integral∫ ∞

0e−bxLn(x) dx = (b − 1)nb−(n+1)

so that

Cn = (−γ )n(−2/z0)

1 + ikw/λz0= (−γ )n

(−2/z0)

1 − 2i/K0z0(16)

where we denote, for brevity,

γ ≡(

1 − ikw/λz0

1 + ikw/λz0

)=

(1 + 2i/K0z0

1 − 2i/K0z0

). (17)

The modal solution for01 is obtained from (10) upon the insertion offn, Kn andCn.This solution provides a new parameterization of the two-frequency coherence function.Since for the point-source problem considered here01 is similar to the plane-wave solutionpresented in [3, 4], we shall mention these parameters only briefly. In the far zone, beyondzA of equation (22) of [3], the solution is dominated by the first term of the modal series.This implies that the longitudinal phase accumulation and the amplitude decay are readilydescribed by the simple exponential exp((i/2)K0z). A second feature is the coherencebandwidth, i.e. the frequency separation for which01 decays to a specified value. Thisquantity will be referred to in section 4.3.

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Two-frequency mutual coherence function: point source 111

3.2. Collective solution

The modal series can be summed into a closed form using the identity

exp[−xt/(1 − t)]

1 − t=

∞∑n=0

Ln(x)tn. (18)

Inserting (17) into (10) we obtain

01 = exp(−ikdz0)exp[ikdρ

2w/(z − z0)]

z − z0

−2/z0

1 − 2i/K0z0exp

(i

2K0z − λ

2ρ2

d

)F(ρd, z)

F (ρd, z) = 1

1 + exp(iK0z)γexp

(λρ2

d exp(iK0z)γ

1 + exp(iK0z)γ

).

(19)

The result in (18) can be interpreted as the first mode multiplied by a functionF(ρd, z)

which accounts for the collective effect of the rest of the modes. Alternatively, (18) can bepresented in a form compatible with the expression in [2]:

01 = −exp(−ikdz0)

z0

exp[ikdρ2w/(z − z0)]

z − z0cosθ0 sec

(1

2K0z + θ0

)× exp

(iλ

2ρ2

d tan

(1

2K0z + θ0

))θ0 = i

2argγ = arctan(2/K0z0) .

(20)

As can be verified, (19) coincides with the expression for01 for a finite beam given inequation (16) of [2] if this expression is reduced to the case for an initial point source bysettingαR = 0.

The monochromatic limit is obtained by approximatingk1k2 ' k2 andkd/k � 1. Theresult is

01|kd→0 = 1

(z − z0)2exp[ikρc · ρd/(z − z0)] exp

[− 1

12k2Cρ2

d(z − z0)

](21)

which coincides with the expression given in equation (20-81b) of [5]. The singular termexp[ik2ρ2

d/2kd(z − z0)] which appears originally in the spherical phase was cancelled outby the first-order term in the tangent in (19) in the limitkd → 0 (see appendix).

4. General power-law structure function

4.1. Modal expansion solution

When the structure function is given asD(ρd) = Cρνd (whereC is a dimensional constant

and 1< ν 6 2) no analytical solution for01 exists. Analytic expressions for the solutionand the relevant parameters can nevertheless be derived by means of the approximationtechniques outlined in [4].

In this section we shall be concerned with the caseρd = 0. We shall present approximateexpressions for both the modal series and the collective solution. From (10), the expressionfor 01 at ρd = 0 (note thatρw becomesρ in this case) is

01 = exp(−ikdz0)exp[ikdρ

2/(z − z0)]

z − z0

∞∑n=0

Cn exp

(i

2Knz

)f̄n(0) (22)

wherefn(0) are in general unknown constants.

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112 J Oz

Based on dimensional reasoning, it has been shown in [3] that the propagationcoefficientsKn are given by (see equation (29) of [3])

Kn = εn

(i

2kdC

)2/(2+ν)

|kw|(2−ν)/(2+ν) (23)

whereεn are real positive numbers. In particular, using the variational approach forε0 andthe WKB method for the highern, we derived in [4] the following approximations:

ε0 '(

2 + ν

2

)2ν

(2 + ν)[ 1

4(2ν)2/(2+ν) + (2ν)−ν/(2+ν)]

0(4/(2 + ν))εn = ε0(2n + 1)2ν/(2+ν). (24)

Finally, the expansion coefficientsCn are found by projecting the initial conditions ontothe eigenfunction via (9). As has been pointed out in section 3.1 of [4], the eigenfunctionsfn for a power-law structure function are some functionsf̄n of the dimensionless variabley = λρ

(2+ν)/2d (compare also with equation (12) in the previous section). They are generally

unknown with the exception of the caseν = 2. However, based on the dimensionalarguments used in section 3.1 of [4], the functionsfn(ρd) = f̄n(y) may be taken asLn(y) exp(− 1

2y) to a first approximation whenever evaluating expansion coefficients ofthe modal series.

Thus, by settingy = λρ(2+ν)/2d and substituting into (9) we obtain (compare with

equation (9) of [4])

Cn ' − 1

z0

∫ ∞

0exp

[− i

2kwy4/(2+ν)/λ4/(2+ν)z0

]Ln(y) exp(−y/2)y(2−ν)/(2+ν) dy. (25)

This integral is some function of the constantkw/λ4/(2+ν)z0. Since the form of (24) iscompatible with (14) we may, to a first approximation, estimate the integral by assuming thatthe functional dependence onkw/λ4/(2+ν)z0 is the same as in the quadratic case. Extendingγ as already given in (16) to anyν and using the relationKnkw = −εnλ

4/(2+ν) (see equation(29) of [3]) and (22) we arrive at

γ ≡(

1 − ikw/λ4/(2+ν)z0

1 + ikw/λ4/(2+ν)z0

)=

(1 + iε0/K0z0

1 − iε0/K0z0

)(26)

and thus (see (15))

Cn ' (−γ )n(−2/z0)

1 + ikw/λ4/(2+ν)z0= (−γ )n

(−2/z0)

1 − iε0/K0z0. (27)

Furthermore, using the arguments mentioned above, in (21) we may usef0 = f̄ (0) 'L(y) exp(−y/2)|y=0 = 1.

4.2. Collective solution

We apply the Poisson summation as in section 4.2 of [3] by simply replacingA(n) thereinby C(n) and evaluate the resulting typical integrals by their end-point contributions. As in(18), the result is the product of the first mode and a factorF(0, z) which accounts for thecollective effect of the rest of the modes:

01 = exp(−ikdz0)exp[ikdρ

2/(z − z0)]

z − z0

(−2/z0)

1 − iε0/K0z0exp

(i

2K0z

)F(0, z)

F (0, z) = 1

1 + exp[i2ν/(2 + ν)K0z]γ.

(28)

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Two-frequency mutual coherence function: point source 113

The functionF(ρd, z) appeared already in (18) and was extended in (27) to a generalν.Equation (27) may be recast in a form compatible with (19):

01(ρd = 0) ' −exp(−ikdz0)

z0

exp[ikdρ2/(z − z0)]

z − z0cosθ0 sec

(ν

2 + νK0z + θ0

)× exp

[i

2K0z

(2 − ν

2 + ν

)]θ0 = i

2argγ = arctan

(ε0

K0z0

).

(29)

In the above equations, the variational expression forε0 in (23) may be used.In the limit |ε0/K0z0| � 1, (27) reduces to

01 = exp(ikdρ2/z)

z2i

K0

ε0exp

(i

2K0z

)(1

1 − exp[i2ν/(2 + ν)K0z]

). (30)

We remark here that, since the modes decay exponentially withK0z, the limit z � |z0| alsoleads in practice to the limit|ε0/K0z0| � 1 (but the converse is not true). In view of this,(29) describes01 asz � |z0|.

As already mentioned at the end of section 2, we assume to be in the region wherekd 6= 0 andρd = 0 to avoid the problem of the singularity. However, the solution extendedto include all values ofkd is regular everywhere and assumes the correct value atkd = 0.We thus conclude that the solution in (28) is valid for allkd.

4.3. Numerical example

Figures 1 and 2 depict the magnitude and phase of01 for ν = 2 andν = 5/3, respectively,based on (29). The graphs were plotted as functions ofη ≡ (kd/kcoh

d1 )ν/(2+ν) wherekcohd1 is

the plane-wave coherence bandwidth [5] given by

kcohd1 = 2(4+ν)/νC−2/νk−2(2−ν)/νz−(2+ν)/ν . (31)

The results are plotted forρd = 0 andρ = 0. For ρ 6= 0, |01| remains the same but thephase increases bykdρ

2/z. As mentioned in [3] and at the end of section 2, the modal seriesrequires more and more terms for smallη. The number of modes needed is determined fromequation (39) of [3]. In particular, forη > 1 only the first mode is needed to describe01

completely, as is demonstrated in the graphs. The figures compare the collective solutionwith the modal solution forν = 2 andν = 5/3 and it is shown that the addition of the secondmode to the first mode improves the range of validity of the modal series significantly.

4.4. Coherence bandwidth

The coherence bandwidthkcohd1 is obtained by requiring the magnitude of (29) atρd = 0

to have decayed to a specified value, such as e−1 of its monochromatic value which, from(20), equals(z − z0)

−2.An explicit result may be obtained in the limit|ε0/K0z0| � 1. Applying this to (27)

and noting thatγ → −1, by multiplying both sides of (28) by(z − z0)2 we obtain the

following expression which defineskcohd1 implicitly:

2

∣∣∣∣K0

ε0(z − z0) exp

(i

2K0z

)(1

exp[i2ν/(2 + ν)K0z] − 1

)∣∣∣∣kcoh

d1

= e−1. (32)

From the form of (31) we infer thatK0z|kcohd1

is a function of the ratioz/z0. Comparingthis with the condition onK0z|kcoh

d1for an initial plane wave given in equation (32) of [3]

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114 J Oz

(a)

(b)

Figure 1. The magnitude (top) and phase (bottom) of01/01|η=0 for a quadratic medium(ν = 2). Parameters:λ = 0.6943 µm, −z0 = 1 m, z = 10 km, C = 10−14 m−1. Brokencurve: first mode only. Chain curve: superposition of the first two modes. Full curve: collectivesolution in (29).

it follows that the coherence bandwidth of an initial point source equals the plane-wavecoherence bandwidth times some function ofz/z0. If, in addition, we havez � |z0|, (31)becomes a function of the variableK0z alone. As discussed in [3], we assume that thedecay of01 is mainly due to the decay of the first mode. Thus, neglecting the exponent inthe denominator of (31), equation (32) simplifies to

2

∣∣∣∣K0

ε0z exp

(i

2K0z

)∣∣∣∣kcoh

d1

= e−1. (33)

Solving (33) results in a coherence bandwidth identical to that in plane-wave case, multipliedby a factor close to 2.2, for both quadratic and Kolmogorov media. In figures 1(a) and 2(a)it is seen that the value ofη at which |01| decays to e−1 is indeed approximately equal to2.2. As one can also see from the graph by comparing the first mode with the collective

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Two-frequency mutual coherence function: point source 115

(a)

(b)

Figure 2. The magnitude (top) and phase (bottom) of01 for a Kolmogorov medium(ν = 5/3).Parameters:λ = 0.6943µm, −z0 = 1 m, z = 10 km, C = 10−17 m−2/3. Broken curve: firstmode only. Chain curve: superposition of the first two modes. Full curve: collective solutionin (29).

solution, the assumption that the decay of01 is dominated by the first mode is true. Theusefulness of the modal series for parameterization is thus demonstrated.

5. Validity of the paraxial boundary conditions for the point-source problem

Up to now we have solved01 for the case of the quadratic phase initial conditions in (4).These conditions, however, provide a good model for a point-source field only ifρ � |z0|in the z = 0 plane. Here, we state the conditions under which the main contribution to thesolution originates from the paraxial regionρ � |z0| in order for01 to be indeed consideredas the solution to a point-source problem.

The boundary condition is used when evaluating the expansion coefficientsCn of

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116 J Oz

the series. If the point-source initial conditions can be approximated paraxially over thecharacteristic spread offn with ρd, then theCn obtained are also valid for a true point-sourceproblem.

In principle, all modes are excited by the initial condition atz = 0. However, the modeswith the higher mode indices decay rapidly withz and do not contribute significantly inthe expansion. Thus, at givenz, there exist a number ofrelevantmodes which have beendiscussed in detail in section 4 of [4]. The effective transverse width of the superpositionof the relevant modes as a function ofz has been found (equation (33) of [4]) to be

ρ0(z) =(

4

Ck1k2z

)1/ν

. (34)

The condition under which the solution via paraxial boundary conditions describes atrue point-source problem is

ρ0(z) � |z0| (35)

which provides the following lower bound onz:

z � 4

Ck1k2|z0|ν . (36)

On the other hand, the propagation range on the right-hand side of (35), which we denoteby zz0, should be weighted against the propagation rangezLm

defined in equation (34) of [4]and given by (35) with the substitution|z0| → Lm. zLm

is the range beyond which effectsdue to the finite outer scaleLm, such as continuous mode indices, have to be taken intoaccount. Thus, provided|z0| > Lm, zz0 is not greater thanzLm

, andz0 does not impose anyother restriction (such as outer scale effects) on the modal solution. Under this condition,01 is indeed a point solution over the rangez > zLm

.

6. Summary

In this paper we have presented the solution of the two-frequency coherence function foran initial point source based on the modal approach presented in [3, 4]. In section 2 wepresented some general features which hold for any statistically isotropic and homogeneousmedium. In particular, the general structure of the modal expansion has the form given in(10), wherefn andKn are the modal eigenfunction and the modal propagation coefficient,respectively. The expansion for the point source has essentially the same structure as that fora plane wave in [3, 4], multiplied by a free-space spherical phase term. Particular attentionhas been paid to general power-law media. Two types of solution have been considered,namely a modal series and a collective solution. The modal series converges slowly inthe quasi-monochromatic regionkd → 0. On the other hand, the collective solution basedon the series is regular in that limit, but requires some additional analysis via the Poissonsummation formula.

The theory has been demonstrated for the case of a quadratic structure function. Anew exact modal solution has been derived (section 3.1) which coincides, upon collectivesummation, with the solution in the literature (section 3.2). The results have been extended toa general power-law medium (section 4). We have also considered the coherence bandwidth(section 4), while in section 5 the restrictions for the solution based on paraxial boundaryconditions to describe a point-source problem correctly were stated.

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Two-frequency mutual coherence function: point source 117

Acknowledgments

The author would like to thank Ehud Heyman for his helpful comments on the manuscript.

Appendix

At kd → 0, we haveθ0 ' π/2 − 12K0z0 with K0 ' O(k

1/2d ). Inserting this approximation

into (19) and retaining terms up to the first order in the secant and up to the second orderin the tangent yields

sec

(1

2K0z + θ0

)' sec

(π

2+ 1

2K0(z − z0)

)' − 1

12K0(z − z0)

tan

(1

2K0z + θ0

)' tan

(π

2+ 1

2K0(z − z0)

)' − 1

12K0(z − z0)

+ 1

6K0(z − z0).

Noting from (1) thatρw ' (k/kd)ρd + ρ and substituting into (19), we obtain01 in thequasi-monochromatic limit

01 → 1

(z − z0)2exp[ikρc · ρd/(z − z0)] exp[ik2ρ2

d/2kd(z − z0)]

× exp

[i

2k2ρ2

d

√iC/2kd

(−1/

√iCkd/2(z − z0) + 1

3

√iCkd/2(z − z0)

)].

The singular terms containingk2ρ2d/kd in the exponent cancel out, leading to the final result

given in (20).

References

[1] Sreenivasiah I, Ishimaru A and Hong S T 1976 Two-frequency mutual coherence function and pulsepropagation in a random medium: an analytic solution to the plane wave caseRadio Sci. 11 775–8

[2] Sreenivasiah I and Ishimaru A 1979 Beam wave two-frequency mutual coherence function and pulsepropagation in a random medium: an analytic solutionAppl. Opt. 18 1613–8

[3] Oz J and Heyman 1997 Modal theory for the two-frequency mutual coherence function in random media:general theory and plane wave solution: IWaves Random Media7 79–93

[4] Oz J and Heyman E 1997 Modal theory for the two-frequency mutual coherence function in random media:general theory and plane wave solution: IIWaves Random Media7 95–106

[5] Ishimaru A 1978Wave Propagation and Scattering in Random Mediavols 1 and 2 (New York: Academic)

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