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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
ul(z +
z) =
(z +
z)el and
el =
(- z)
ul(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 two-dimensionsa
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 complex-conjugate 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(
jkl, z
z), which correctly describes
the damping
exp(
Im[kl, z]
z) and the oscillation
exp(
jRe[kl, z]
z) of the plane wave traveling upwards and
downwards the layer.
Footnotes
- ... two-dimensionsa
- A two-dimensional
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 three-dimensional 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
1998-04-17