Exercise 7.2.3: A metal-plate lens

${\lambda }_0=3\mathrm{cm}$. $n=0.5$ requires $({\lambda }_0/2b{)}^2=0.75$:

With $E$ parallel to the plates only TE modes, which have $E_x$, are excited, and of these only TE${}_1$ propagates (and the field of a broad incoming wave, even across the channel, couples only to odd $n$ anyway).

$Z_{TE}={\eta }_0/0.5=2{\eta }_0$, so $\Gamma \approx \frac{2{\eta }_0-{\eta }_0}{2{\eta }_0+{\eta }_0}=\frac{1}{3}$ and $|\Gamma {|}^2=11\%$ of the power is reflected at the face (the estimate ignores the mismatch between the uniform incoming field and the $\sin (\pi y/b)$ mode).

A length $d$ of the stack advances the phase by $k_{0}d(1-n)$ relative to air. A full cycle needs $d={\lambda }_0/(1-n)=3/0.5=6\mathrm{cm}$.

At another frequency the same $6\mathrm{cm}$ step gives $k_{0}d(1-n(f))$:

$n={\tau }_1/k_0=s$ at $10\mathrm{GHz}$: $\sqrt{1-(8.66/10{)}^2}=0.500$, and $v_p=c/n=6\times 10^{8}m/s$.