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The three vector components of the first Maxwell equation take the explicit form
Note that for the vertical component we have used the reciprocal permittivity
(x) = (x).
Insertion of the Fourier expansions (6.11) and
(6.12) and comparison of the coefficients yields
whereby the derivatives with respect to the lateral coordinates x and y
were transformed
into a multiplication of the Fourier coefficients with the respective
harmonic frequency jk_{x, n} and jk_{y, m}, i.e.,

(6.13) 
The first two equations of (6.15) constitute two coupled ODEs
for the lateral magnetic field coefficients
H_{x, nm}(z) and
H_{y, nm}(z), whereas
the third equation is an analytical expression for the vertical electric field
coefficients
E_{z, nm}(z). The convolutionlike relations in the Fourier domain
follow from the multiplication in the spatial domain
(cp. (6.14) with (6.15)).
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Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417