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In the bilocal-approximated Dyson's equation, both the observation point and the source point of the dyadic Green's function are within the random medium and can coincide with each other within the domain of integration. Physically, this corresponds to the fact that the scatterers act as sources of radiation. By taking into account the singularity of the dyadic Green's function, a strong permittivity fluctuation theory of electromagnetic propagation can be derived that is applicable for larger variances of the permittivity. In this section we apply the strong permittivity fluctuation theory (SPF) to study scattering from a layer of random medium with spherical correlation function. The first moment of the field is calculated by using the bilocal approximation, and the second moment of the field is calculated by using the distorted Born approximation. The correlation function is obtained by using simple physical arguments and is expressed in terms of the fractional volumes and the particle sizes of the constituents of the dielectric mixture. The lowfrequency assumption is made in the derivations. The low-frequency effective propagation constant h&') an imaginary part that is due to a combination of absorption and scattering. The scattering part is dependent on particle size and can be ignored in the very low frequency limit. It is also shown that the derived result of the effective permittivity in the very low frequency limit is identical to the Polder and van Santern mixing formula. It is also identical to the result of the effective medium theory. In this section we only study media with spherically symmetric correlation functions. Results for anisotropic correlation functions can be found in the references. 3.1 Random Medium with Spherically Symmetric Correlation Function

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Consider a medium with permittivity E(1') which is a random function of position. The vector wave equation is

(4.3.1)

the solution to the problem becomes direct. The predictor is now not necessary as for the solution only requires information of y at the previous steps, i.e. at Y , , ~ + ~ i = 1,2,..., k. Where the problem contains two variables, one nonintegrable, such that

where k o follows:

\7 x \7 x E -

k~ (E(7') - EU)

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The LineBreak element is added to the previous code to implement a line break between the two sections of text. Figure 3.13 shows each of the TextBlock examples in this section.

(4.3.2)

The permittivity Eg is chosen such that the effective permittivity calculated by the bilocal approximation differs from f g by a term that is required to be small. That small difference is dependent on particle size. The particle-size-

dependent term vanishes as k o ---. 0 so that Eg is the effective permittivity at the verylow frequency limit. Let Gg(r, r') be the dyadic Green's function that satisfies a vector wave equation with wavenumber kg with (4.3.3) Thus, (4.3.4) Note that both Eg and Gg are non-random quantities and are chosen to approximate well the effective background through conditions that are to be imposed later. Using (4.3.4) in (4.3.2), we have

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E(r) = Eo(r)

E(r')

(4.3.5)

If you want the user to be able to enter more than one line in a TextBox by pressing Enter, you ll want to add AcceptsReturn= True to your code listing.

where Eo(r) is the field that satisfies the homogeneous wave equation with wavenumber kg. We note that in (4.3.5) both r, the observation point, and

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