Diffraction of surface waves by a semi-infinite dielectric slab

The discontinuity at the end of the slab is regarded as the junction of an open dielectric-filled waveguide and a free-space waveguide. A variational expression is set up for the terminal impedance representing the effect of the discontinuity on the surface wave. Close upper and lower bounds for the impedance are obtained. A variational expression is also set up for the transfer impedance between the surface wave and one of the modes of the continuous mode spectrum associated with the free-space waveguide. The transfer impedances yield the modal amplitudes of the fields. The synthesis of all the modal components gives the electromagnetic field scattered by the end of the slab. The synthesis is carried out for the far field by the method of steepest descents. Terminal impedances and forward radiation patterns are plotted as functions of the thickness for several permittivities
\epsilon_{r}=2.49; 10; 100
. The approximation obtained by the variational expression is excellent for small values of
\epsilon_{r}
and begins to be bad for
\epsilon_{r} > 10
. For very large values of
\epsilon_{r}
it is possible to obtain a rough estimate of the impedance, immediately, from the values for
\epsilon_{r} = 10
.