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6.2 Lateral Discretization of the Maxwell Equations
In this section we discuss the lateral discretization of
the Maxwell equations. For a summary confer also the first item
of the enumeration listed on page .
Due to the assumption of a periodic simulation domain the inhomogeneous
permittivity
(x) as well as its reciprocal
(x) = (x) can be represented by Fourier series

(6.10) 
whereby the periods a and b in x and ydirection, respectively,
are the lateral extensions of the simulation domain. Furthermore the incident
field is periodic due to the periodic property
of the photomask. Because of the periodicity of both the excitation and the
permittivity also the disturbance inside the simulation domain must be
periodic (cf. (6.10)). Hence, we can expand the electric
as well as the magnetic phasor,
E(x) and
H(x), respectively, into Fourier
series, i.e.,

(6.11) 
with discrete lateral wavevector components

(6.12) 
These expansions are similar to those introduced for the vectorvalued extension
of Fourier optics in Section 4.1.5 and the analysis of the
stratified medium in Appendix C. The fundamental difference to
(4.55) and (C.1) lies in the fact
that we cannot consider planar waves traveling through a homogeneous medium.
In the simple homogeneous case the vertical dependence is known and is described
by the exponential function with real or complex argument depending on whether
the medium is ideally dielectric or absorptive, respectively. In case of
inhomogeneous materials the eigenfunctions of the PDE
are not known and can, in general, only be determined numerically.
Insertion of the material expansions (6.11) and the field
expansions (6.12) into the first and second Maxwell
equations (6.5) transforms the PDE valid
in the spatial domain to a system of ODEs valid inside the Fourier domain.
Next: 6.2.1 First Maxwell Equation
Up: 6. Differential Method
Previous: 6.1.2 Operation Principle
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417