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For further investigations it is useful to split up
(z) into
two parts, namely into a constant matrix
describing the
orientation of the wave propagation and into a zdependent diagonal matrix
(z) representing the amplitude oscillation within the layer.
Hence we write

(C.9) 
with
Some interesting properties exist for the three matrices defined
by (C.11) to (C.13),

(C.12) 
The validity of each of these relations is selfevident. However, as they
will be used further on we explicitly summarized them in (C.14).
Additionally, the orientation matrix
defined in
(C.12) can be factorized as follows:

(C.13) 
Finally, the inverse matrix
(z) of
(z)
is of interest. From (C.11) and (C.13) we obtain
with (C.14)

(C.14) 
whereby
is given by

(C.15) 
Note that
simply equals to
=  1/ .
Summarizing the results of the matrix notation and factorization
(cf. (C.10), (C.11) and (C.16))
shows that the lateral field components
u_{l}(z) are related to
the electric amplitudes
e_{l} by

(C.16) 
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Up: C.1 One Homogeneous Planar
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Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417