A plane electromagnetic wave of frequency $\omega$ and wavevector $\mathbf{k}$ has an electromagnetic potential given by

$$A^{a}=A \epsilon^{a} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$

where $A$ is the amplitude of the wave and $\epsilon^{a}$ is the polarization vector. Explain carefully why there are two independent polarization states for such a wave, and why $|\mathbf{k}|^{2}=\omega^{2}$.

A wave travels in the positive $z$-direction with polarization vector $\epsilon^{a}=(0,1, i, 0)$. It is incident at $z=0$ on a plane surface which conducts perfectly in the $x$-direction, but not at all in the $y$-direction. Find an expression for the electromagnetic potential of the radiation that is reflected from this surface.