6.2.2 Second Maxwell Equation

In an exactly analogous way the components of the second Maxwell equation are first written explicitly:

 

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

and then transformed into the frequency domain. This yields two coupled ODEs for the lateral electric field coefficients E_{x, nm}(z) and E_{y, nm}(z) as well as one analytical expression for the vertical magnetic field coefficients H_{z, nm}(z):

 

$$\begin{aligned}\frac{d E_{y,nm}(z)}{d z} &= -j\omega\mu_0H_{x,nm}... ...{1}{\omega\mu_0}\Big[k_{x,n}E_{y,nm}(z) - k_{y,m}E_{x,nm}(z)\Big].\end{aligned}$$