# Electromagnetic field confined inside a closed optical resonator

I am currently studying Maxwell's equations. Out of interest, I was reading the introduction to the textbook The Quantum Theory of Light, third edition, by Louden. When discussing the photon, the author says the following:

The idea of the photon is most easily expressed for an electromagnetic field confined inside a closed optical resonator, or perfectly-reflecting cavity. The field excitations are then limited to an infinite discrete set of spatial modes determined by the boundary conditions at the cavity walls.

So does this mean that there is some form of Maxwell's equations that show that "the field excitations are limited to an infinite discrete set of spatial modes determined by the boundary conditions"? If so, then would someone please elaborate on this further (showing and explaining the mathematics) for the sake of clarity? Otherwise, what is the relevant equation that shows what the author is alluding to?

I would greatly appreciate it if people would please take the time to explain this.

Possibly related: Modeling the field inside a cavity resonator

• any optics textbook has the relevant derivations. See, e.g., the book by Saleh. You are looking for fabry-perot cavities or just simply a resonator. As light behaves as a wave, within a resonator you are only allowed to have standing waves which are phase multiples of the optical phase length of the cavity. The higher the finesse of the cavity (the closer to perfectly reflecting) the closer these modes become dirac delta like (as the photon lifetime inside the cavity tends to an infinite time and the interference becomes stronger) Commented Jul 3, 2020 at 14:42
• @JoséAndrade I've got the 3rd edition of the Saleh textbook (currently studying). Which page is it? Also, feel free to post an answer if you want, so that I may award the bounty. Commented Jul 3, 2020 at 14:47