Propagation Matrix

Now it is easy to find a relation like

 

$$\mathbf{u}(z+\Delta z) = \underline{\mathbf{L}}_l(\Delta z) \mathbf{u}(z),$$ (C.17)

which connects the lateral field components at two different vertical points that are at a distance $\Delta$z from each other. The matrix $\underline{\mathbf{L}}_{l}^{}$($\Delta$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 + $\Delta$z) = $\underline{\mathbf{K}}_{l}^{}$ $\underline{\mathbf{D}}_{l}^{}$(z + $\Delta$z)e_l and e_l = $\underline{\mathbf{D}}_{l}^{}$(- z) $\underline{\mathbf{K}}^{-1}_{l}$u_l(z), which combines with the second relation of (C.14) to

$$\mathbf{u}_l(z+\Delta z) = \underline{\mathbf{K}}_l\,\underline{\mathbf{D}}_l(\Delta z)\,\underline{\mathbf{K}}_l^{-1}\mathbf{u}_l(z).$$ (C.18)

Hence an explicit expression for the propagation matrix $\underline{\mathbf{L}}_{l}^{}$($\Delta$z) can be derived to

$$\underline{\mathbf{L}}_l(\Delta z) = \frac{1}{2} \begin{pmatrix}\underline{\mathbf{I}}\:& \phantom{-... ...}}^{-1} \\ \underline{\mathbf{I}} & -\underline{\mathbf{S}}^{-1} \end{pmatrix}$$

$$= \frac{1}{2} \begin{pmatrix}e^{+jk_{l,z}\Delta z}\underline{\mat... ...^{-1} \\ \underline{\mathbf{I}} & -\underline{\mathbf{S}}_l^{-1} \end{pmatrix}$$

$$= \frac{1}{2} \begin{pmatrix}(e^{+jk_{l,z}\Delta z} + e^{-jk_{l,z... ...z} + e^{-jk_{l,z}\Delta z}) \underline{\mathbf{I}}\phantom{^{-1}} \end{pmatrix}$$

$$= \begin{pmatrix}\cos(k_{l,z}\Delta z)\underline{\mathbf{I}} & j\... ...ine{\mathbf{S}}_l & \cos(k_{l,z}\Delta z) \underline{\mathbf{I}} \end{pmatrix}.$$ (C.19)

The propagation matrix has some interesting properties. For example, in case of a vanishing distance $\Delta$z = 0 we obtain $\underline{\mathbf{L}}_{l}^{}$(0) = $\underline{\mathbf{I}}$. It can also be shown that the reciprocity relation $\underline{\mathbf{L}}_{l}^{-1}$($\Delta$z) = $\underline{\mathbf{L}}_{l}^{}$(- $\Delta$z) exists. Additionally, in two-dimensions^a the matrices $\underline{\mathbf{I}}$ and $\underline{\mathbf{S}}_{l}^{}$ are replaced by its determinants, i.e., $\underline{\mathbf{I}}$ $\mapsto$ det$\underline{\mathbf{I}}$ = 1 and $\underline{\mathbf{S}}_{l}^{}$ $\mapsto$ det$\underline{\mathbf{S}}_{l}^{}$ = 1/$\eta_{0}^{2}$. The resulting expression for the propagation matrix (C.21) can be found in many textbooks, e.g., in [11, p. 58]. Finally, $\underline{\mathbf{L}}_{l}^{}$($\Delta$z) is a normal matrix as $\underline{\mathbf{L}}_{l}^{}$($\Delta$z)$\underline{\mathbf{L}}_{l}^{\mathrm{H}}$($\Delta$z) = $\underline{\mathbf{L}}_{l}^{\mathrm{H}}$($\Delta$z)$\underline{\mathbf{L}}_{l}^{}$($\Delta$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 $\underline{\mathbf{L}}_{l}^{}$($\Delta$z) consist of two pairs $\lambda_{1}^{}$ and $\lambda_{2}^{}$ that are both of manifold two. They are inverse as the determinant and thus their product equals unity, i.e., det$\underline{\mathbf{L}}_{l}^{}$($\Delta$z) = $\lambda_{1}^{2}$$\lambda_{2}^{2}$ = 1. The two values $\lambda_{1}^{}$ and $\lambda_{2}^{}$ can easily be calculated and equal to $\lambda_{1,2}^{}$ = exp( $\pm$ jk_{l, z}$\Delta$z), which correctly describes the damping exp( $\mp$ Im[k_{l, z}]$\Delta$z) and the oscillation exp( $\pm$ jRe[k_{l, z}]$\Delta$z) of the plane wave traveling upwards and downwards the layer.

Footnotes

... two-dimensions^a

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.