6.2.1 First Maxwell Equation

The three vector components of the first Maxwell equation take the explicit form

 

$$\begin{aligned}\frac{\partial H_z(\mathbf{x})}{\partial y} - \fra... ...c{\partial H_x(\mathbf{x})}{\partial y}\right] &= E_z(\mathbf{x}).\end{aligned}$$

Note that for the vertical component we have used the reciprocal permittivity $\chi$(x) = $\varepsilon^{-1}_{}$(x). Insertion of the Fourier expansions (6.11) and (6.12) and comparison of the coefficients yields

 

$$\begin{aligned}\frac{d H_{y,nm}(z)}{d z} &= \phantom{-} j\omega\v... ...i_{(n-p)(m-q)}(z) \Big[k_{x,p}H_{y,pq}(z)-k_{y,q}H_{x,pq}(z)\Big],\end{aligned}$$

whereby the derivatives with respect to the lateral coordinates x and y were transformed into a multiplication of the Fourier coefficients with the respective harmonic frequency jk_{x, n} and jk_{y, m}, i.e.,

$$\frac{\partial}{\partial x}\Big[\;\cdot\;\Big] \;\,\longmapsto\,\... ...partial}{\partial y}\Big[\;\cdot\;\Big] \;\,\longmapsto\,\;jk_{y,m}[\;\cdot\;].$$ (6.13)

The first two equations of (6.15) constitute two coupled ODEs for the lateral magnetic field coefficients H_{x, nm}(z) and H_{y, nm}(z), whereas the third equation is an analytical expression for the vertical electric field coefficients E_{z, nm}(z). The convolution-like relations in the Fourier domain follow from the multiplication in the spatial domain (cp. (6.14) with (6.15)).