$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$
We consider only linear homogeneous isotropic media without charges or currents; then Maxwell's equations \begin{align} &\varepsilon \mu\, \boldsymbol{E}_t = \nabla \times \boldsymbol{B}, \label{eq-5.B.1}\\ &\boldsymbol{B}_t = -\nabla \times \boldsymbol{E}, \label{eq-5.B.2}\\ &\nabla \cdot \boldsymbol{E}=0, \label{eq-5.B.3}\\ &\nabla \cdot \boldsymbol{B}=0. \label{eq-5.B.4} \end{align} Then using magnetic vector potential $\boldsymbol{A}$ defined from \begin{gather} \boldsymbol{A}_t = -\boldsymbol{E},\label{eq-5.B.5}\\ \nabla \times \boldsymbol{A}_t = \boldsymbol{B} \label{eq-5.B.6} \end{gather} we arrive to \begin{align} &\boldsymbol{A}_{tt}= c^2 \Delta \boldsymbol{A}, \label{eq-5.B.7}\\ &\nabla \cdot \boldsymbol{A}=0 \label{eq-5.B.8} \end{align} with $c=\frac{1}{\sqrt{\varepsilon \mu}}$; $\varepsilon$ and $\mu$ are electric permittivity and magnetic permittivity respectively.
Then we can apply what we learned in Sections Section 5.1--Section 5.3: \begin{gather} \boldsymbol{A} \sim e^{ik \phi} \sum_{n\ge 0} \boldsymbol{A}_{n} k^{-n}. \label{eq-5.B.9} \end{gather}
Remark 1. Maxwell system is overdetermined (there are two extra equations (\ref{eq-5.B.3}) and (\ref{eq-5.B.4}) but they are compatible with (\ref{eq-5.B.1})-(\ref{eq-5.B.2}). Also (\ref{eq-5.B.5})--(\ref{eq-5.B.6}) and (\ref{eq-5.B.7})--(\ref{eq-5.B.8}) are overdetermined but again compatible.
Different boundary conditions are formulated using $\boldsymbol{E}$ and $\boldsymbol{B}$; there should be two scalar boundary conditions (normally for (\ref{eq-5.B.7}) there should be one vector (that is three scalar), however (\ref{eq-5.B.8}) implies one scalar condition.
Again we can apply what we learned in Section Section 5.4.