Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box

Appendix A

Modes of a perfectly conducting closed cavity: A quick review

There are many different modes, each of which will have a different resonant frequency corresponding to some particular complicated arrangement of the electric and magnetic fields. Each of these arrangements is called a resonant mode. The resonance frequency of each mode can be calculated by solving Maxwell's equations for the electric and magnetic fields in the cavity.

—Richard Phillips Feynman [206]

In Section 2.2 we showed that Maxwell equations involve only fields at the same point in Fourier space. Electrodynamics then becomes extremely simple. But that was all done in free space. For a cavity, the fields have to satisfy the constraints imposed by the boundary conditions on the walls. Here we show how the Fourier-space procedure that simplified electrodynamics in free space can be generalized for a perfectly conducting closed cavity (a perfect cavity for short). We will see that the plane waves appearing in the Fourier transforms in free space can just be replaced by a different sort of wave, whose shape is determined by the boundary conditions of the cavity. These waves are the cavity modes. They are extremely useful, especially for quantizing the electromagnetic field, because Maxwell equations for the electromagnetic field in the cavity involve only fields at the same point in mode space. From this point of view, the plane waves that appeared in the Fourier transforms in Section 2.2 are just modes ...