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6.2.6 VectorMatrix Notation
Any further considerations are much simplified by switching over to a
vectormatrix notation. For that we introduce the following four
N_{ODE}/4dimensional vectors comprising the
(2N_{x} + 1) x (2N_{y} + 1)
Fourier coefficients of the four relevant field components:
Within each of these vectors the coefficients are arranged accordingly to

(6.22) 
The mapping is unique and invertible, i.e., any index pair (n, m) corresponds
to one unequivocal vectorentry i, and vice versa.
The same methodology can be applied to the coefficients of the ODE system
(6.22). Each of the eight sets
r_{nm, pq}^{}(z) and
g_{nm, pq}^{}(z) with
()
{(xx),(xy),(yx),(yy)}
is arranged in one
N_{ODE}/4 x N_{ODE}/4matrix.
The ODE system can thus be written in the compact form

(6.23) 
The overall shape of the system matrix reflects the fundamental structure of the
Maxwell equations since the derivative of the electric field depends only
on the derivative of the magnetic field and vice versa.
Comprising the four field coefficient vectors of (6.30)
to one
N_{ODE}dimensional vector

(6.24) 
we obtain the most compact and convenient form^{c}

(6.25) 
Here the
N_{ODE} x N_{ODE}dimensional system matrix
(z) follows from (6.34). Its
entries h_{ij}(z) are determined by the Fourier coefficients
(z) and
(z) of the permittivity and its reciprocal, respectively.
With the mapping (6.31) we obtain, e.g,

(6.26) 
whereby
1i, j < N_{ODE}/4. The given example takes the
explicit form (c.f. (6.22))

(6.27) 
Footnotes
 ... form^{c}
 A prime as superscript
denotes derivation with respect to the independent variable that is the vertical
coordinate z in our case.
Next: 6.3 Formulation of the
Up: 6.2 Lateral Discretization of
Previous: 6.2.5 Truncation of the
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417