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Dielectric interface in a waveguide

1. 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 [tex] \epsilon [/tex]. 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: [tex] S_{trans}/S_{inc} = 4kk^{'}/(k+k^{'})^2[/tex]]


2. Homework Equations



3. The Attempt at a Solution
So I wrote the electric field as a superposition of the TE electric field for z < 0:
[tex]\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)}][/tex]
Applying boundary condition that the parallel components of E must be continuous on the boundary at z=0, I got that [tex]\Psi_0^{'} = \Psi_0 + \Psi_0^{''} [/tex]
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.
 

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