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is a centered Gaussian random value and hexp ik0 j i is its characteristic function ( where
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Turning back to the initial Equation (3.132), we get e hC r i
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that solves the problem. This functional integral can also be approximated for numerical purposes, for example, by multiple integrals:
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1   3 i 4pi t 2n e dy exp ik0 y hC r in k0 k0 0 & ' ik0 2 2 2 r r2 r1 r rn 1 . . . exp 4 t 1 ( ) n 1 X 1 2 2 Gi;j d3 r1 d3 r2 d3 rn 1 exp k0 t 2 i;j 1
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where Gi; j G ri ; rj and t y=n. The extension to higher order moments is straightforward, using characteristic functions of multivariate Gaussian distributions. Approximate Evaluations of the Functional Integral (3.137) a) Short Wavelength Approximation. If the range of the covariance function is much longer than the wavelength, we use a functional saddle point method to approximate the function 1 2 exp4 k0 2 2
y t 0 0
G r t ; r t0 dt dt0 5
by a quadratic function of r t r0 t , where r0 t is the function that makes the exponent stationary. It is then possible to calculate exactly this approximate functional integral. b) Long Wavelength Expansion. The multiple integral (3.138) reminds us of the formula for the partition function of a gas in thermodynamic equilibrium. We can write ( ) n 1 n 1 X X XX Y 1 2 2 Gi; j 1 Fij 1 Fij Fij Fikl . . . exp k0 t 2 i; j 1 i; j 1 with 1 2 Fij exp k0 t 2 Gi; j 2 & ' 1: 3:141
The resulting integrals are then represented by the Mayer s diagrams [3,4]. This method can also be related to the perturbation method of Section 3.2.
3.7. THE ELECTROMAGNETIC WAVE EQUATION In this section we consider the full electromagnetic wave equation with a random refractive index
2 E r r r E r k0 1 em r E r j r :
where j r is related to the actual current density j r by j r iom0 j r , o is the angular frequency, o 2pf , and m0 4p 10 7 is the permeability of free space. This equation is not equivalent to the reduced scalar wave equation because of the term r r E r , which is important when the refractive index changes much over a wavelength. We shall therefore only consider the case of long wavelengths such that jk0 j ( 1, and use the Bourret s approximation. This problem has already been treated by Tatarskii [16 18] but the results presented here do not agree. Taking the FT of (3.142) we get 2 2 k0 k2 dij ki kj Ej k ek0 m k k0 Ei k0 dk ji k : The unperturbed propagator Gij k satis es the following equation
2 k0 k2 dij ki kj Gjl k dil : 0 0
This equation is easily solved as Gjl k
  1 ki kj dij 2 : 2 k0 k0 k2
The Bourret s equation for the mean perturbed propagator hGjl k i is
or hG k i G k G
0 0 4 k e2 k0
G k k G k d k hG k i
where G k is the FT of the covariance function. After a few transformations, Equation (3.146b) becomes [31]:
2 4 k0 k2 dij ki kj e2 k0
  ! ki0 kj0 G k k0 dij 2 d3 k0 hGjl k i dil : 2 k0 k02 k0
Let us now denote the tensor Tij k as:   ki0 kj0 3 0 G k k0 Tij k dij 2 d k 2 k0 k02 k0 and assume that the covariance function is isotropic. Then the tensor Tij k is the convolution product of an isotropic tensor and an isotropic function; it is thus an isotropic tensor and can be written as: Tij k w k dij m k ki kj : 2 k0 3:148
The Bourret s equation for the mean propagator becomes now
2 4 2 k0 k2 e2 k0 w k dij 1 e2 k0 m k ki kj hGjl k i dil :