# Dielectric interface in a waveguide

## Homework Statement

Consider a wave guide with a square cross section of dimensions a x a. Let the z axis be the axis of the wave guide. Suppose the region z < 0 is vacuum, and the region z >0 is a dielectric with permittivity $$\epsilon$$. Write a solution of the wave equations and boundary conditions such that there is an incident and reflected wave for z < 0 and a transmitted wave fro z > 0. All three waves are TE(1,0) waves. Determine the transmitted power as a fraction of the incident power. [Answer: $$S_{trans}/S_{inc} = 4kk^{'}/(k+k^{'})^2$$]

## The Attempt at a Solution

So I wrote the electric field as a superposition of the TE electric field for z < 0:
$$\vec{E} = \left[-\hat{y}\frac{\pi}{a}sin\left(\frac{\pi x}{a}\right)\right][\Psi_0e^{i(kz-\omega t)}+\Psi_0^{''}e^{-i(kz+\omega t)}]$$
Applying boundary condition that the parallel components of E must be continuous on the boundary at z=0, I got that $$\Psi_0^{'} = \Psi_0 + \Psi_0^{''}$$
Where the double primed one is the reflected wave, and the single prime is transmitted. This, however, is the only equation I can get by applying boundary conditions. Any thoughts? Also I get a really messy equation if I take the real part of E and B and cross them, nothing like the simple answer they give me.