Given an action of the form
\begin{equation}S=-\frac{1}{4}\int d^4x\eta^{\mu\nu}\eta^{\lambda\rho}F_{\mu\lambda}F_{\nu\rho}\end{equation}
where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, $\eta_{\mu\nu}=g_{\mu\nu}/a^2(\eta)$, where $g_{\mu\nu}$ is given by the line element:
\begin{equation}ds^2=a^2(\eta)[d\eta^2-(dx^i)^2]\end{equation}
I would like to solve for $A_{\mu}$, and standard solution is
\begin{equation}A_{\mu}^{(\alpha)}=e_{\mu}^{(\alpha)}e^{ik_\nu x^\nu}.\end{equation}
I am interested in knowing how to derive this result.
My approach is first write the Lagrangian from action and use EL eq
\begin{equation}\frac{\partial \mathcal{L}}{\partial A_{\mu}}-\frac{d}{d x^{\nu}}\frac{\partial \mathcal{L}}{\partial(\partial_{\nu}A_{\mu})}=0\end{equation}
My main problem is mathematical difficulty in evaluating the EL eq. Can anyone please help me on this?