Field and Wave Electromagnetics 2nd Edition by David Cheng

Exercise 43

When the slab thickness of a dielectric-slab waveguide is very small in terms of the operating wavelength, the field intensities decay very slowly away from the slab surface, and the propagation constant is nearly equal to that of the surrounding medium.
a) Show that if k y d 1, the following relations hold approximately for the dominant TE mode:

where

b) For a slab of thickness 5 (mm) and dielectric constant 3, estimate the distance from the slab surface at which the field intensities have decayed to 36.8% of their values at the surface for an operating frequency of 300 (MHz).