Exercise 7.6.4: Measuring a sheet's permittivity with its surface wave

Minima are ${\lambda }_g/2$ apart: ${\lambda }_g=9.61\mathrm{mm}$. With ${\lambda }_0=12.5\mathrm{mm}$ and $k_0=502.7rad/m$,

$w=\alpha d/2=0.6272$. Solve $u\tan u=0.6272$ by bisection on $(0,\pi /2)$: $u=0.7180$, $h=2u/d=478.7rad/m$. Then

$\mathrm{NA}=\sqrt{1.60}=1.265$: TE${}_1$ and TM${}_1$ start at $c/(2d\mathrm{NA})=39.5\mathrm{GHz}$. At $24\mathrm{GHz}$ only TE${}_0$ and TM${}_0$ propagate, and the measurement sees a single mode if the launcher excites TE. The decay length is $1/\alpha =2.4\mathrm{mm}$.

Repeating (b) with ${\lambda }_g$ $1\%$ smaller and larger, $9.514\mathrm{mm}$ and $9.706\mathrm{mm}$, gives ${\epsilon }_r=2.651$ and $2.547$: a $1\%$ error in ${\lambda }_g$ costs about $2\%$ in ${\epsilon }_r$.

$V=\frac{k_{0}d}{2}\mathrm{NA}=0.7540\times 1.264=0.953$ and $\sqrt{u^2+w^2}=\sqrt{0.5155+0.3934}=0.953$. The value fits PMMA, about $2.6$ at microwave frequencies.