FUNDAMENTALS OF WAVE PROPAGATION IN RANDOM MEDIA in .NET

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FUNDAMENTALS OF WAVE PROPAGATION IN RANDOM MEDIA
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x > 0 is considered; re ections are assumed to be negligible. Integration of (3.55) gives (
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C x Y x exp ik0 x exp ik0 em y dy
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3:57
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where Y x being Heaveside s step function [12]. To calculate now the mean value of the (3.57), the only random term is the x second potential. For xed x, 0 em y dy being a linear functional of the centered Gaussian random function m x , is a centered Gaussian random variable j. If so, heik0 j i is the characteristic function of this random variable. As j is Gaussian, that is, heik0 j i e2k0 hj we can evaluate now *  y x 2 + x  2  e2 dy G y y0 dy0 hj i e  m y dy 
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The mean wave function is thus expressed in terms of the covariance function of the refractive index. Higher order moments such as hC x C x0 i are easily obtained, using the characteristic function of a multivariant Gaussian distribution [16]. We now introduce the covariance function ' &  x x0    G x x exp  
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8 9 y x < 1 = 2 hC x i Y x exp ik0 x exp k0 e2 dy G y y0 dy0 : : 2 ;
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where is the range of index correlation. The mean wave function can now be calculated as: n x o x 2 hC x i Y x exp ik0 x exp e2 k0 2 e 1 : 3:62
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AN EXACT SOLUTION OF 1D-EQUATION
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The dimensionless parameter which determines the behavior of the solution is ek0 . There are two interesting limiting approximations: a) ek0 ( 1. It is a long wavelength approximation (l ) ) and corresponds to weak interactions in quantum eld theory. A uniform approximation for hC x i is then
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2 hC x i Y x exp ik0 x expf e2 k0 xg:
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3:63
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As follows from (3.63), the initial excitation is damped with an extinction length
2 xex e2 k0 1 :
3:64
Let us compare xex and the wavelength l $ k0 1 xex 1 1 1 ) 1: $ 2 l e k0 e ek0 3:65
The decaying is thus very slow; it is due to phase mixing and is not related to any dissipative mechanism. The mean wave function hC x i can also be written as
2 hC x i Y x expfi k0 ie2 k0 xg:
3:66
The effect of randomness on the mean wave function, as follows from (3.66), is simply a renormalization of the wave number. The renormalized wave 2 number is now equal to k k0 ie2 k0 , which has a small imaginary part (because ek0 ( 1). In the next section, we shall obtain this wave approximation as a sum of an in nite series extracted from the perturbation expansion of the mean propagator. b) ek0 ) 1. It is a short wavelength approximation (l ( corresponding to strong interactions in quantum eld theory. A uniform approximation for hC x i is then ' 12 2 2 hC x i Y x exp ik0 x exp e k0 x : 2 & 3:67
The initial excitation is damped again, with an extinction length xex ek0 1 $ l=e; the damping decay is more rapid than in the preceding case. This approximation is equivalent to a renormalization of the wave number because x2 appears in the second exponent in (3.67).
FUNDAMENTALS OF WAVE PROPAGATION IN RANDOM MEDIA
Next we need the FT of the exact mean function (3.62)
hC k i
exp ik0 x hC x idx
1 1 0
x 2 2 exp i k0 k exp e2 k0 2 x exp e2 k2 x exp e2 k0 2 e dx:
3:68
Expanding the last exponential term in a uniformly convergent series and integrating (3.68), yields: