The role of Berry´s geometric phase in the mode spectrum of a Fabry-Pérot resonator

Berry´s geometric phase is a universal phenomenon which appears naturally when dealing with three-dimensional rotations. Here we will demonstrate its unexpected role in the high-resolution transmission spectrum of a two-mirror Fabry-Pérot resonator. Using analytical three-dimensional vector solutions to Maxwell´s equations in spheroidal coordinates, we have calculated the first-order corrections to eigenmodes and eigenfrequencies in a Fabry-Pérot resonator in the short-wavelength limit. Analogous to atomic spectra, these analytical expressions show the “fine structure” of the spectrum of an optical resonator. Especially for high-order transverse modes with large orbital angular momentum as the one shown in the figure, the consequences are striking, leading to series of intricate mode patterns with Laguerre-Gaussian character split by their total angular momentum.