Generalized Ewald-Oseen Extinction Theorem

After computing the singular solution K, the elements of the s vector can be expressed in terms of an arbitrary constant, and this constant will be determined from the Ewald-Oseen extinction theorem. Certainly, this strategy tacitly assumes that the system of equations (2.185) reduces to a single scalar equation and to prove this assertion we introduce the notations 

Using the fact that for a vector plane wave polarized along the x-axis, the incident field coefficients are given by 

Taking into account the expression of s / given by (2.181), we deduce that the above set of inhomogeneous equations reduces to a single scalar equation 

the procedure is to first find the singular solution of the Lorentz-Lorenz law together with the dispersion relation for the effective wave number Kg, and then to determine the arbitrary constant from the scalar equations (2.190) or (2.191). After the conditional average of the scattered field coefficients has been evaluated, the coherent reflected field can be computed by taking the configurational average of (2.171), i.e., 

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Substituting (2.177) and (2.182) into (2.176) gives two types of terms. One type of terms has a exp(jA s 2oi) dependence and corresponds to waves traveling with the wave number of the incident wave, while the other type of terms has a exp(jifg o/) dependence and corresponds to waves traveling with the wave number of the effective medium. The terms with wave number should balance each other giving the generalized Ewald-Oseen extinction theorem,... 

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