Quantum theory for cavity trapping light 
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*On the level of relativistic quantum mechenics i.e. till Dirac equation, are there any quantum theories to understand the working principle of Fabry-Pérot cavities, approximately?

*Is it fair to think about it as potential barrier problem? Although it's a very rough approximation.

*This cavity is also used to trap light in case of lasers, to get monochromatic light sources.

*I have tried searching for it, I have only found explanation by QED, which I haven't studied yet.

I will be thankful for any suggestions.
 A: For most of the physics of Fabry-Pérot cavities, classical optics is sufficient. You can calculate transmission and reflection coefficients (using Parratt's formalism or Abele's transfer matrix method). You can get the fields etc. from similar calculations.
You only start to need quantum theory when you introduce quantized light (i.e. photons) or interaction with matter. This is then the subject of cavity QED and I'm afraid you can't get around the QED part in that case.
The OP's question seems to be asking for an in-between formalism analogous to Schrödinger's equation in quantum mechanics. However, Maxwell's equations are a wave equation analgous to Schrödinger's equation already. So the "classical optics" case in the electromagnetic Fabry-Perot cavity corresponds to the "quantum mechanics" treatment of a Schrödinger Fabry-Perot potential barrier.

Is it fair to think about it as potential barrier problem?

Yes. On the level of wave equations, you only have to replace Schrödinger's equation by Maxwell's equation with a refractive index replacing the potential.
On the level of cavity QED, this is an advanced topic. Here are two resources which may be helpful with respect to this concrete question: doi, doi.
