Can electromagnetic waves be interpreted as a collection of infinite coupled oscillators? I recently learned, for the first time, that the behavior of waves can be derived by treating them as an infinite set of coupled oscillators. This makes sense for waves whose medium is matter; it's easy to see how water waves, for instance, can be seen as the individual water molecules tugging on their neighbors as they oscillate.
At the same time, however, I've only ever seen electromagnetic waves treated continuously, and again this makes the most sense, since electromagnetic waves are just oscillations in a continuous field. But I wonder, is there an interpretation of EM waves as a set of discrete coupled oscillators? Would such a treatment be useful in any way?
 A: Yes.
Consider electromagnetic fields in a square box of volume $V = L^3$.  One can then impose periodic boundary conditions on the vector potential, and one arrives at the following expansion in terms of Fourier modes:
\begin{align}
  \mathbf A(t,\mathbf x) = \sum_\mathbf k\sum_r \left(\frac{\hbar c^2}{2V\omega_\mathbf k}\right)^{1/2} \mathbf \epsilon_r(\mathbf k)(a_r(\mathbf k)e^{i(\mathbf k\cdot \mathbf x-\omega_\mathbf k t)} + a_r^\dagger(\mathbf k)e^{-i(\mathbf k\cdot \mathbf x-\omega_\mathbf k t)})
\end{align}
where $\omega_\mathbf k = c|\mathbf k|$ and the $\mathbf \epsilon$'s are polarization vectors.  Because we are quantizing in a box with periodic boundary conditions, the sum over $\mathbf k$ is over the countable (discrete) set of wavevectors given by
\begin{align}
  \mathbf k = \frac{2\pi}{L}(n_1, n_2, n_3), \qquad n_i = 0,\pm 1, \pm 2, \dots
\end{align}
We then impose the creation-annihiliation commutation relations on the Fourier modes $a_r^\dagger(\mathbf k)$ and $a_r(\mathbf k)$, so we can treat the field as being composed of the excitations of a discrete number of oscillators.
