A medium with index of refraction n₁ occupies the half-space y 0 is filled with a medium with index of refraction n₂. A propagating electric field bound to the y = 0 plane can have the form

(a) Find the field H(r, t) that accompanies E(r, t).
(b) Show that two types of solutions exist, one where E has only a z-component and one where E has no z-component. For the latter, show that the polarization of the wave is elliptical.
(c) Show that the time-averaged Poynting vector points exclusively along the x-axis for both solutions.

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E(r, t) = E₂ exp(-By) exp[i (kx - wt)] E₁ exp(ay) expli(kx - wt)] y > 0 y < 0.