# Modal theory for the two-frequency mutual coherence function in random media: point source

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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) 107117. 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(1, 2, z, k1, k2) where z is the propagation range,j arethe transversal coordinates of the observation points andkj = j/c are the wavenumbers.Here, we consider propagation in a statistically isotropic and homogeneous medium wherethe structure functionD is a function ofd |1 2| 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

d = 1 2 w =k11 k22

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 vectorsd and w. 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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where one depends on the magnituded of d and the other on the magnitudew of w.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 onw, represents free-spacediffraction of a spherical wave and is thus independent of the medium properties. The otherterm, that senses the medium, depends only ond 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(, z, k) generated by a point source located at( = 0, z = z0)with z0 < 0. Within the parabolic approximation, the field near thez-axis atz < 0 is givenby

U = 1z z0 exp

{ik[(z z0) + 2/2(z z0)]

}. (2)

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

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

0(1, 2, z, k1, k2) U(1, z, k1)U (2, z, k2) exp[ikd(z z0)] (3)wherekd k1 k2. Henceforth, the solution will be described in terms of the normalizedfunction 01 = 0 exp( 18k2dA(0)z). The initial conditions for0 are

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

1

z0

)2exp

(ik121/2z0)exp(ik222/2z0)exp(ikdz0) . (4)In order to apply the procedure outlined in [3], the boundary conditions in (4) are

expressed in the(d, w) coordinates of equation (1), giving

01|z=0 =(

1

z0

)2exp

(ikd2w/2z0)exp(ikw2d/2z0)exp(ikdz0) (5)wherek1w = k11 k12 .

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

01 = kd2

n

exp

(i

2Knz

)fn(d)

0

An(E)J0(

Ekdw) 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

0

dd dwdwfn (d)J0(

Ekdw) 01|z=0 . (7)

Substituting (4) into (7) and using the identity 0

eaxJ0(b

x) dx = 1a

eb2/4a

we obtain

An(E) = Cn ikd

exp

(i

2Ez0

)exp(ikdz0) (8)

where

Cn =(

fn,1z0

exp(ikw2d/2z0))

=

0dd df

n (d)

(1z0

)exp(ikw2d/2z0). (9)

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

01 = i2

n

Cn exp

(i

2Knz

)fn(d)

0

exp

( i

2E(z z0)

)J0(

Ekdw) dE (10)

giving

01 = exp(ikdz0) exp[ikd2w/(z z0)]

z z0

n

Cn exp

(i

2Knz

)fn(d). (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[ikd2w/(z z0)]

z z0(discussed below) and a superposition of transverse mode functionsfn(d) 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 largezthe 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(d) (see equation (19) of [3]).

The free-space spherical phase factor

exp(ikdz0) exp[ikd2w/(z z0)]

z z0does not appear for an initial plane wave and contains no parameters of the random medium.As has been pointed out in [3],w is a centre-of-mass coordinate which propagates throughthe medium without any scattering and thus it is not affected byD(d). As a result, the partof 01 at z = 0 which containsw 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(d, w) coordinates, given by

0(z, 1, 2) = 01(z, 1, 2) = exp(ikdz0)exp[ikd2w/(z z0)]

z z0exp[ikw2d/(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 ford = 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(d) = C2d, whereC is a dimensional constant, theeigenfunction/eigenvalue problem can be solved explicitly. The eigenfunctionsfn and theeigenvaluesKn are given by (see equations (42)(45) of [3]):

fn = Ln(2d) exp(

22d

)Kn = 2

i

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

(13)

whereLn are Laguerre polynomials and we have used the notation

2 = i2k2wkdC (14)

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

Cn = 1z0

0

exp(y/2) exp(ikwy/2z0)Ln(y) dy (15)

where we denotey 2d. Equation (14) involves the standard integral 0

ebxLn(x) dx = (b 1)nb(n+1)

so that

Cn = ( )n (2/z0)1 + ikw/z0 = ( )

n (2/z0)1 2i/K0z0 (16)

where we denote, for brevity,

(

1 ikw/z01 + ikw/z0

)=

(1 + 2i/K0z01 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[ikd2w/(z z0)]

z z02/z0

1 2i/K0z0 exp(

i

2K0z

22d

)F(d, z)

F (d, z) = 11 + exp(iK0z) exp

(2d exp(iK0z)

1 + exp(iK0z))

.

(19)

The result in (18) can be interpreted as the first mode multiplied by a functionF(d, 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[ikd2w/(z z0)]z z0 cos0 sec

(1

2K0z + 0

) exp

(i

22d tan

(1

2K0z + 0

))0 = i

2arg = 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 bysettingR = 0.

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

01|kd0 =1

(z z0)2 exp[ikc d/(z z0)] exp[ 1

12k2C2d(z z0)

](21)

which coincides with the expression given in equation (20-81b) of [5]. The singular termexp[ik22d/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(d) = Cd (whereC is a dimensional constantand 1< 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 cased = 0. We shall present approximateexpressions for both the modal series and the collective solution. From (10), the expressionfor 01 at d = 0 (note thatw becomes in this case) is

01 = exp(ikdz0)exp...

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