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Now it is easy to find a relation like

(C.17) 
which connects the lateral field components at two different vertical points that
are at a distance z from each other.
The matrix
(z) is called the propagation matrix
of the layer l and is obtained in the following way:
Evaluating (C.18) at the two different points gives
u_{l}(z + z) = (z + z)e_{l} and
e_{l} = ( z) u_{l}(z),
which combines with the second relation of (C.14) to

(C.18) 
Hence an explicit expression for the propagation matrix
(z) can be derived to
The propagation matrix has some interesting properties. For example,
in case of a vanishing distance
z = 0 we obtain
(0) = .
It can also be shown that the reciprocity relation
(z) = ( z) exists.
Additionally, in twodimensions^{a}
the matrices
and
are replaced by its
determinants, i.e.,
det
= 1 and
det
= 1/. The resulting
expression for the propagation matrix (C.21) can
be found in many textbooks, e.g., in [11, p. 58].
Finally,
(z) is a normal matrix as
(z)(z) = (z)(z) holds,
whereby the superscript
H denotes Hermitian or complexconjugate transposition. This property
proves the existence of an eigenvalue decomposition. The four eigenvalues
of
(z) consist of two pairs
and
that are both of manifold two. They are inverse
as the determinant and thus their product equals unity, i.e.,
det(z) =
= 1.
The two values
and
can easily be calculated and
equal to
= exp(
jk_{l, z}z), which correctly describes
the damping
exp(
Im[k_{l, z}]z) and the oscillation
exp(
jRe[k_{l, z}]z) of the plane wave traveling upwards and
downwards the layer.
Footnotes
 ... twodimensions^{a}
 A twodimensional
analysis always suffices because the field lies
in one common plane and the coordinates can be chosen in such a way
that one lateral field vector component vanishes.
However, we present a threedimensional analysis as the
results can directly be used as boundary conditions for the
differential method.
Next: C.2 Stack of Homogeneous
Up: C.1 One Homogeneous Planar
Previous: Matrix Factorization
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417