Write down Maxwell's Equations for electric and magnetic fields $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ in the absence of charges and currents. Show that there are solutions of the form

$$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left\{\mathbf{E}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right\}, \quad \mathbf{B}(\mathbf{x}, t)=\operatorname{Re}\left\{\mathbf{B}_{0} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}\right\}$$

if $\mathbf{E}_{0}$ and $\mathbf{k}$ satisfy a constraint and if $\mathbf{B}_{0}$ and $\omega$ are then chosen appropriately.

Find the solution with $\mathbf{E}_{0}=E(1, i, 0)$, where $E$ is real, and $\mathbf{k}=k(0,0,1)$. Compute the Poynting vector and state its physical significance.