Matrix Factorization

For further investigations it is useful to split up $\underline{\mathbf{C}}_{l}^{}$(z) into two parts, namely into a constant matrix $\underline{\mathbf{K}}_{l}^{}$ describing the orientation of the wave propagation and into a z-dependent diagonal matrix $\underline{\mathbf{D}}_{l}^{}$(z) representing the amplitude oscillation within the layer. Hence we write

 

$$\underline{\mathbf{C}}_l(z) = \underline{\mathbf{K}}_l\,\underline{\mathbf{D}}_l(z)$$ (C.9)

with

$$\underline{\mathbf{K}}_l = \begin{pmatrix}1 & 0 & 1 & 0 \\ 0 & 1... ...2+k_{l,z}^2}{\eta_0k_lk_{l,z}} & -\frac{k_xk_y}{\eta_0k_lk_{l,z}} \end{pmatrix}$$ (C.10)

$$\underline{\mathbf{D}}_l(z) =\begin{pmatrix}e^{+jk_{l,z}z} & 0 & ... ... 0 \\ 0 & 0 & e^{-jk_{l,z}z} & 0 \\ 0 & 0 & 0 & e^{-jk_{l,z}z} \end{pmatrix}.$$ (C.11)

Some interesting properties exist for the three matrices defined by (C.11) to (C.13),

 

$$\underline{\mathbf{D}}^{-1}_l(z)=\underline{\mathbf{D}}_l(-z),\qq... ...derline{\mathbf{I}},\qquad\underline{\mathbf{C}}_l(0)=\underline{\mathbf{K}}_l.$$ (C.12)

The validity of each of these relations is self-evident. However, as they will be used further on we explicitly summarized them in (C.14). Additionally, the orientation matrix $\underline{\mathbf{K}}_{l}^{}$ defined in (C.12) can be factorized as follows:

$$\underline{\mathbf{K}}_l = \begin{pmatrix}\underline{\mathbf{I}} ... ...atrix}k_xk_y & (k_y^2+k_{l,z}^2) \\ -(k_x^2+k_{l,z}^2) & -k_xk_y \end{pmatrix}.$$ (C.13)

Finally, the inverse matrix $\underline{\mathbf{C}}_{l}^{-1}$(z) of $\underline{\mathbf{C}}_{l}^{}$(z) is of interest. From (C.11) and (C.13) we obtain with (C.14)

 

$$\underline{\mathbf{C}}_l^{-1}(z) = \underline{\mathbf{D}}_l(-z)\,\underline{\mathbf{K}}_l^{-1}$$ (C.14)

whereby $\underline{\mathbf{K}}_{l}^{-1}$ is given by

$$\underline{\mathbf{K}}_l^{-1} = \frac{1}{2} \begin{pmatrix}\under... ...atrix}k_xk_y & (k_y^2+k_{l,z}^2) \\ -(k_x^2+k_{l,z}^2) & -k_xk_y \end{pmatrix}.$$ (C.15)

Note that $\underline{\mathbf{S}}_{l}^{-1}$ simply equals to $\underline{\mathbf{S}}_{l}^{-1}$ = - 1/$\eta_{0}^{2}$ $\underline{\mathbf{S}}_{l}^{}$.

Summarizing the results of the matrix notation and factorization (cf. (C.10), (C.11) and (C.16)) shows that the lateral field components u_l(z) are related to the electric amplitudes e_l by

 

$$\mathbf{u}_l(z) = \underline{\mathbf{K}}_l\,\underline{\mathbf{D}... ... \underline{\mathbf{D}}_l(-z)\,\underline{\mathbf{K}}^{-1}_l\, \mathbf{u}_l(z).$$ (C.16)