Consider a plane polarized electromagnetic wave described by the vector and scalar potentials, \[\mathbf{A}(\mathbf{r},t) = \mathbf{A}_0\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} \qquad\text{and}\qquad \phi(\mathbf{r},t) = \phi_0\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)},\tag{1.1}\] with constant $\mathbf{A}_0$ and $\phi_0$. Use Maxwell's equations to find a relationship between $\mathbf{A}_0$ and $\phi_0$.
Find a gauge transformation such that the new vector potential is “transversely polarised”, i.e.\ $\mathbf{A}_0\cdot\mathbf{k}=0$ (Coulomb gauge, $\nabla\cdot\mathbf{A}=0$). What is the scalar potential $\phi$ in this gauge?