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### Propagation Matrix

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     (C.19)

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