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6.3.3 VectorMatrix Notation
In Section 6.2.6 we introduced a vectormatrix notation
to achieve a compact form for the ODE system.
The notation is governed by the convention (6.30)
combined with the indexmapping of (6.31). Using this notation
the BCs (6.40) and (6.47)
at z = 0 and z = h, respectively, take the form
whereby all
matrices as well as the two
matrices
are diagonal with dimension
N_{ODE} x N_{ODE}.
For example,

(6.31) 
Note that the diagonal property of the matrices reflects the mutual
independence of the BCs of different harmonic frequencies (n, m).
Similarly, to the transition from (6.32) to (6.34)
we comprise the individual coefficient vectors to one common vector
u(z) (cf. (6.33)) and can thus further
condense (6.49) to

(6.32) 
The two boundary matrices
and
are both of dimension
N_{ODE}/2 x N_{ODE}, whereas the excitation vectors
a^{pq} and
0 are of dimension
N_{ODE}/2.
At the upper interface at z = 0 a different excitation vector
a^{pq}
occurs for each coherent source point (p, q). The simultaneous treatment
of multiple BCs will be described in
Section 6.4.3 which is one of the major advantages of the
proposed implementation of the differential method. Furthermore it is
worth mentioning that the excitation vector at the lower interface z = h
vanishes, i.e., we have the zerovector
0, which reflects that only
outgoing waves occur in the substrate due to the assumption
of an infinitely extended substrate (cf. Fig. 6.1).
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Up: 6.3 Formulation of the
Previous: 6.3.2 Resist/Substrate Interface
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417