## 1. Introduction

[2] In the paper the technique to solve Markov parabolic equation to the spaced position and frequency second-order coherence function [*Rytov et al*., 1978, *Ishimaru*, 1978] is developed, which is based on the quasi-classic approximation in terms of complex trajectories. Rather than separation of variables employed by *Oz and Heyman* [1996, 1997a, 1997b, 1997c], this technique is valid including the case of a nonhomogeneous background medium and does not demand the statistical homogeneity of fluctuations. It has no constraints relevant to the initial conditions in the form of an incident plane wave and produces in automatic fashion known solutions [*Sreenivasiah et al*., 1976; *Knepp*, 1983; *Bronstein and Mazar*, 2002; *Bitjukov et al*., 2002].

[3] The present paper further extends the method of the investigation of the two-position, two-frequency coherence function, previously developed by the authors [*Bitjukov et al*., 2002] for a particular case of plane wave, to the case of an incident field of a general type. In that paper there was considered a simple case of an incident plane wave propagating along the axis of the parabolic equation such that the appropriate second-order Markov moment equation did not contain differential operations in the central transversal variable and the solution did not depend on this variable. In this case the technique of complex trajectories (complex geometrical optics) has the most transparent form. The technique allowed constructing the solution to the cases of realistic behavior of the structure function of fluctuations tending to a positive constant rather than to the infinity as the difference argument increases. The extension of the technique to the general case (where the dependence on the central variable also occurs) that will be considered here employs more complicated complex trajectories in five-dimensional complex space.

## 2. Main Equations and Relationships

[4] Classic two-frequency second-order Markov's parabolic equation [*Ishimaru*, 1978] is considered for the homogeneous background medium, where possible statistical nonhomogeneity of fluctuations in the longitudinal direction is allowed as follows:

Here ∇_{s} and ∇_{d} are the operators of gradient with respect to the sum and difference transversal co-ordinates **r**_{+} and **r**_{−}, , *k*_{1,2} are the vacuum wave numbers for frequencies ω_{1,2}. It was accepted that the background medium had the unity dielectric permittivity Quantity *A*(*z*, **r**_{−}) is the effective transversal correlation function of the fluctuations δ-correlated in z-direction.

[5] If complex amplitude *U*(*z*, **ρ**, ω) of the field *E*(*z*, **ρ**, ω) is introduced according to

the coherence function of the amplitudes *U*(*z*, **ρ**, ω)

is expressed through the solution Γ_{1} of equation (1) as follows:

To further treat equation (1) for function Γ_{1}, the dimensionless variables are introduced according to: and Here *l*_{ε} is the effective spatial scale of fluctuations. Employing these dimensionless variables ς, **r**, **R** equation (1) can be rewritten as

In equation (5) parameter is the dimensionless parameter, which is assumed to be the large parameter of the problem; is the following dimensionless mistuning and the dimensionless central wave number is given by Dimensionless correlation function of fluctuations is expressed through the effective transversal correlation function of δ-correlated fluctuations *A*(*z*, **r**_{−}).

[6] To finally rewrite equation (5) in the form enabling asymptotic solution at large *K*, additional rescaling of the central transversal dimensionless variable **R** should be performed. If taking account of the relationship between quantities *K*, and :

rescaled central transversal variable **ρ** is introduced according to

Employing (7), equation (5) is finally rewritten as

Here

is the effective transversal structure function of fluctuations δ-correlated in z-direction.

[7] Equation (8) allows asymptotic solution at Physically, the large *K* means that random inhomogeneities of a medium are of large spatial scale in terms of vacuum wavelengths for both frequencies ω_{1} and ω_{2}. At the same time this is one of the limitations of Markov's diffusive approximation. When constructing the asymptotic solution at large *K*, it should be accepted that for any finite , but quantity in equation (8) can be expanded into a series as follows:

Analysis of possible forms of the coherence function of the high frequency incident field indicates the following most general form of the solution to equation (8):

Standard asymptotic procedure of substituting (11) into (8) and taking account of (10) results in the “eikonal” equations for ”phase” functions ψ and Ψ

and the main transport equation for the amplitude *U*_{0}

As far as higher-order transport equations for amplitudes *U*_{n}, in the series (11) are concerned, they can be also written taking account of the full expansion (10), but they are fairly space consuming and are not exposed here in order to save the space.

[8] Equations (12–14) can be solved by the method of characteristics, which, according to *Kravtsov and Orlov* [1980], is also applied to the case of complex characteristics. Within this method, the Hamilton-Jacobi equation [see *Kravtsov and Orlov*, 1980, equations (2.1') and (2.1”)] in our case has the form given by (12). Then, according to equations (2.3)–(2.5) from *Kravtsov and Orlov* [1980], presenting the appropriate Hamilton equations for characteristics, complex characteristics corresponding to the equation (12), are given by the following set of the first-order equations:

These equations define complex trajectories in 5-dimensional space (ς, **r**, **ρ**). According to equation (15) it can be accepted that the points along a trajectory are parameterised by the real parameter The trajectories start at initial complex points (**r**_{0}, **ρ**_{0}) at and arrive at real points of observation (ς, **r**, **ρ**), so that (**r**_{0}, **ρ**_{0}) are the initial conditions to equations (15–17). In fact, (**r**_{0}, **ρ**_{0}) are being determined as the function of the real point of observation (ς, **r**, **ρ**) when the “inverse” homing problem is considered.

[9] Initial conditions to the moments (**ρ**_{r}, **p**_{ρ}) from the set of equations (18–20) are defined by a given initial distribution of the eikonal function of the coherence function of the incident field at (see equation (11)) as follows:

Finally, the initial condition to *p*_{ς} is obtained from equation (12) employing also relationships (21, 22) at which yield

When dealing with the complex trajectories the analytical continuation of a structure function to the complex domain of its argument is an important point, and it should be specially discussed in each particular case. However, this continuation is straightforward when the structure function of fluctuations is given explicitly by the analytic function. Then the appropriate Loran series can be written for different domains of convergence in the complex domain.

[10] Once characteristic equations (15–20) with the initial conditions outlined above have been solved eikonal ψ from equation (12) is then obtained by integrating along the appropriate trajectory as follows:

[11] When considering the second eikonal Ψ governed by equation (13), it turns out that along the trajectories satisfying equations (15–20) with the appropriate initial conditions, so that along any trajectory

and Ψ_{0}(**r**_{0}, **ρ**_{0}, ) is defined by a given initial distribution of the second eikonal of the coherence function of the incident field at (see equation (11)). In particular, if this eikonal is not presented in the incident field then it is also not present in the solution of the problem given by representation (11). As will be seen, this holds in the case of an incident field of a spherical wave that will be considered below. However, Ψ may be present in other cases, for instance, when the incident field is a plane wave propagating in the direction different than z-direction.

[12] Finally, the solution to the main transport equation (14) is also obtained by integrating along the constructed trajectories as follows:

In equation (26), *U*_{00}(**r**_{0}, **ρ**_{0}, ) is the distribution of the “amplitude” of the coherence function of the incident field, and the quantity under the sign of the square root is the determinant of the appropriate matrix of the derivatives of the points along a trajectory by the initial conditions to this trajectory.

## 3. Spaced Position and Frequency Coherence Function to the Spherical Wave

[13] To demonstrate how the outlined technique works, the case where the incident field is a spherical wave written in the small-angle approximation respectively z-direction is considered here. Its coherence function is given by

with

Here ς_{s} is the dimensionless distance from the source of the spherical wave to the plane According to (27) in the coherence function of the incident spherical wave, then, according to (25), eikonal Ψ at identically equals zero

so that Ψ is not present in the solution to the spherical wave (27–29).

[14] With equations (28, 29) characterizing the incident spherical wave the initial conditions (21–23) for ray equations (15–20) become as follows:

If also the statistical homogeneity of fluctuations along ς-direction as well as the statistical isotropy of fluctuations in the transversal planes is accepted such that ray equations (15–20) with the initial conditions (31–33) can be solved in the closed form. With it all, it turns out that vector **r**(ς) is always in the plane of the initial vector **r**_{0} (**r**(ς) is collinear to **r**_{0}), so that in cylindrical variables for **r** its absolute value is given by the relationship

whereas its polar angle φ is the same as the angle φ_{0} for the initial vector **r**_{0} that is As far as the central variable is concerned, it is expressed through **r**(ς) as follows:

To further proceed in consideration of the two-frequency, two-position coherence function to the spherical wave, the model of the structure function of fluctuations (*r*) should be specified. In the next subsection a simple power law model of fluctuations will be considered which allows further analytic assessment.

### 3.1. Inverse Power Law Model of Fluctuations

[15] The following model of the effective transversal structure function of fluctuations is considered below:

It has a positive spatial spectrum, and also has a realistic behavior at *r* → ∞, tending to a positive constant 2σ^{2}.

[16] Analytical continuation of (36) into the complex domain of its argument is straightforward. In particular, its Loran series in the circle (here this is the same as its Tailor series) is

and for the circle it is given by

The rigorous form (36) shows how Loran series (37) can be analytically continued into the domain or, how the series (38) can be continued into

[17] Employing main terms of representations (37, 38) explicit calculation of the integral in (34) can be carried out so that the final explicit representations for the coherence function can be obtained for the cases of small and large values of the difference variable *r*.

[18] By means of representation (37), complex trajectories are constructed which are launched and landed at Using the main term in (37) (that is equivalent of considering the model of the quadratic structure function) results in the following explicit form of the equation (34):

Equations (35, 39) describe in the explicit form the complex trajectories In other words, they also show how the initial complex values **r**_{0}, **ρ**_{0} at should be chosen for the trajectory to come at the real point of observation (ς, **ρ**, **r**). The limiting case of an incident plane wave corresponds to ς_{s} → ∞. In this limiting case, equation (39) becomes exactly equation (34) from *Bitjukov et al.* [2002], which describes complex trajectories in the difference variable in the case of the incident field of a plane wave.

[19] Having (39) and taking account of (31–33, 35) allows explicit calculation of the integral for the eikonal function given by (24), as well as the amplitude expressed through (26). Then, when putting all together including (30), this yields the main term of the asymptotic representation of the two-frequency, two-position coherence function for small |*r*|,|*r*_{0}| in the model (36) as follows:

where

Being the solution to the spherical wave in the medium with fluctuations having the structure function (36) at small |*r*|,|*r*_{0}| (the main term in (37)), representation of the coherence function (40–43) at the same time gives the rigorous solution to the case of a spherical wave in the medium with quadratic structure function of fluctuations considered by *Knepp* [1983]. Moreover, in the limiting case of a plane incident wave representation (40–43), properly renormalized, gives equation (38) from *Bitjukov et al*. [2002] that is the solution to the coherence function of a plane wave propagating in the medium with the quadratic structure function of fluctuations initially obtained by *Sreenivasiah et al*., [1976]. As can be shown, for the quadratic structure function and the incident field of a plane wave propagating along z-axis, the higher-order amplitudes in (11) identically equal zero, so that in this case our asymptotic technique produces the rigorous solution in automatic fashion.

[20] However, the most interesting is the case with trajectories located at when the structure function of fluctuations is given by representation (38). This is the case of realistic behavior of the structure function of fluctuations at large values of its spaced position *r*. Considering trajectory equations (34, 35) with the model of the structure function of fluctuations given by the two terms in the series (38) yields the following explicit form for trajectories instead of (34):

Equation (44) together with (35) explicitly describe complex trajectories which now start and finish at large |*r*|, |*r*_{0}|. In the limiting case of the incident field of a plane wave (ς_{0} → ∞) equation (44) is reduced to equation (44) from *Bitjukov et al.* [2002].

[21] Having (44) and again taking account of (31–33, 35) allows explicit calculation of the integral for the eikonal function ψ given by (24), as well as the amplitude expressed through (26), now for the case of large |*r*|, |*r*_{0}|. Then again, when putting all together including (30), this yields the main term of the asymptotic representation of the two-frequency, two-position coherence function now for large |*r*|, |*r*_{0}| in the model (36) as follows:

where

In equation (47) for amplitude *U*_{0} trajectory is defined implicitly by equation (44). In the limiting case of plane incident field and with proper renormalization equations (45–47) are reduced to the coherence function presented by equation (45) from *Bitjukov et al*. [2002].

### 3.2. Single-Frequency Spaced Position Coherence Function to the Spherical Wave

[22] To conclude the consideration, it is discussed here how the general technique outlined in Sections 2 and 3 works in the limiting single-frequency case, when This is the case of the pure spatial coherency comprehensively studied by Tatarskii [see *Rytov et al*., 1978, chap. VII]. Equation (45.18) from this book represents the general solution of the single frequency problem for any given incident field in the form of appropriate Fourier integral in the central transversal variable. When applied to the problem of a spherical wave in the small angle approximation whose coherence function is given by (27–29) with Fourier transform of the coherence function of the incident spherical wave is proportional to the appropriate δ-function, so that integration in spectral parameter in (45.18) from *Rytov et al. *[1978, chap. VII] can be easily performed. This finally yields the following solution to the single-frequency coherence function (in our notations):

On the other hand, when considering the same problem by means of the technique discussed in this paper, in the single frequency case ray equations (15–20) give the following trajectories:

Then, performing necessary integrations for eikonal function ψ given by (24) along trajectories (49) and calculating the appropriate determinant in the amplitude factor *U*_{0} given by (26) for trajectories (49, 50) results exactly in representation (48) for the coherence function, if (4) and (11) (with have been also taken into account.