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Matrix Notation

We introduce now a compact matrix notation for the two relations (C.5) and (C.7) by combining them like

$\displaystyle \begin{pmatrix}E_{l,x}(z)\\ E_{l,y}(z)\\ H_{l,x}(z)\\ H_{l,y}(z) ...
...x}\!\begin{pmatrix}E^+_{l,x}\\ E^+_{l,y}\\ E^-_{l,x}\\ E^-_{l,y} \end{pmatrix}.$ (C.6)

Summarizing the z-dependent part of the lateral components of the electric and magnetic field in ul(z) and, similarly, the electric amplitudes propagating upwards and downwards the layer in el, i.e.,

$\displaystyle \mathbf{u}_l(z) = \Big( E_{l,x}(z)\;\,E_{l,y}(z)\;\,H_{l,x}(z)\;\...
..._l = \Big( E^+_{l,x}\;\,E^+_{l,y}\;\,E^-_{l,x}\;\,E^-_{l,y} \Big)^{\mathrm{T}},$ (C.7)

we simply obtain

$\displaystyle \mathbf{u}_l(z) = \underline{\mathbf{C}}_l(z)\,\mathbf{e}_l.$ (C.8)

The matrix $ \underline{\mathbf{C}}_{l}^{}$(z) definitely describes the field propagation in the layer l, i.e., the electric and magnetic field can be calculated at any vertical position z within the layer from the above equation (C.10).

Heinrich Kirchauer, Institute for Microelectronics, TU Vienna