Consider an inhomogeneous dielectric medium, i.e., one for which the
dielectric constant is a function of position, \(\kappa_{e}=\kappa_{e}(x, y,
z)\). Show that the fields obey the wave equations
$$
\begin{aligned}
&\nabla^{2} \mathbf{E}-\frac{\kappa_{e}}{c^{2}} \frac{\partial^{2}
\mathbf{E}}{\partial t^{2}}=-\nabla\left(\frac{\nabla
\kappa_{e}}{\kappa_{\theta}} \cdot \mathbf{E}\right) \\
&\nabla^{2} \mathbf{B}-\frac{\kappa_{e}}{c^{2}} \frac{\partial^{2}
\mathbf{B}}{\partial t^{2}}=-\frac{\nabla \kappa_{e}}{\kappa_{e}}
\times(\nabla \times \mathbf{B})
\end{aligned}
$$
where, in general, the terms on the right-hand sides couple the cartesian
components of the fields. Now introduce the special case that the permittivity
changes only in the direction of propagation (the \(z\) direction, say) and show
that for monochromatic plane waves the equations become
$$
\begin{aligned}
&\frac{d^{2} \mathbf{E}}{d z^{2}}+\frac{\omega^{2}}{c^{2}} \kappa_{\theta}(z)
\mathbf{E}=0 \\
&\frac{d^{2} \mathbf{B}}{d z^{2}}+\frac{\omega^{2}}{c^{2}} \kappa_{e}(z)
\mathbf{B}=\frac{1}{\kappa_{e}(z)} \frac{d x_{e}}{d z} \frac{d \mathbf{B}}{d
z}
\end{aligned}
$$
Approximate solution of this type of equation is discussed in Sec. \(9.1 .\)