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Electric Field

We assume that a homogeneous plane wave coming from a certain direction k0 = (kx  ky  k0, z)T strikes onto a planar homogeneous layer l. Within the layer the electric field consists of two plane waves traveling in opposite directions kl+ and kl-. This situation is schematically illustrated in Figure C.1. The electric phasor El(x) is thus written as

 
$\displaystyle \mathbf{E}_l(\mathbf{x}) = \left(\mathbf{E}_l^+ e^{+jk_{l,z}z} + \mathbf{E}_l^- e^{-jk_{l,z}z}\right) e^{j(k_xx+k_yy)},$ (C.1)

whereby El+ and El- are the wave amplitudes traveling downwards and upwards the layer. The two wavevectors are given by

$\displaystyle \mathbf{k}_l^+ = (k_x \;\, k_y \, +k_{l,z})^{\mathrm{T}}{}\qquad{\text{and}}\qquad{}\mathbf{k}_l^- = (k_x \;\, k_y \, -k_{l,z})^{\mathrm{T}},$ (C.2)

and the vertical wavevector component

$\displaystyle k_{l,z} = \sqrt{k_l^2 - k_x^2 - k_y^2}$ (C.3)

depends on the wavenumber kl = k0nl of the layer material with refractive index nl.


  
Figure C.1: In a homogeneous planar layer with refractive index nl the electric field consists of two plane waves El+ and El- traveling downwards and upwards, respectively.
\resizebox{9cm}{!}{
\psfrag{z}{\large $z$ }
\psfrag{k0}{\large $\mathbf{k}_0$ }
...
...}{\large $\mathbf{E}_l^+$ , $\mathbf{k}_l^+$ }
\includegraphics{STonelayer.eps}}

Due to the transverseness of the plane waves the vertical amplitude components can be expressed by the lateral ones, i.e.,

 
$\displaystyle \left.\begin{aligned}k_xE^+_{l,x} + k_yE^+_{l,y} + k_{l,z}E^+_{l,...
...yle+\frac{k_x}{k_{l,z}} E^-_{l,x} + \frac{k_y}{k_{l,z}} E^-_{l,y}.\end{aligned}$ (C.4)

Hence it suffices to study only the lateral field components of (C.1) given by

 
$\displaystyle \begin{aligned}E_{l,x}(z) &= E^+_{l,x} e^{+jk_{l,z}z} + E^-_{l,x}...
...E_{l,y}(z) &= E^+_{l,y} e^{+jk_{l,z}z} + E^-_{l,y} e^{-jk_{l,z}z}.\end{aligned}$    

In these two equations El, x(z) and El, y(z) refer to the z-dependent part of the electric field phasor. As can be seen from (C.1) the complete phasor is obtained by multiplication with the exponential factor exp(j(kxx + kyy)), which describes the lateral dependence.


next up previous contents
Next: Magnetic Field Up: C.1 One Homogeneous Planar Previous: C.1 One Homogeneous Planar
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
1998-04-17