Longitudinal modes of a cavity ring resonator

An analysis is made of the polarizations, spatial field distribution, and frequency spectrum of longitudinal (axial) modes of a cavity ring resonator with an arbitrary configuration of the axial contour. Use is made of the integral equation method based on the Huygens-Fresnel principle. It is shown that in the longitudinal mode case it is possible to decouple the equations governing the polarization and spatial field distribution. The two-dimensional integral equation for the field distribution has inseparable variables. Its solution is found on a reference plane which intersects normally the axial contour at an arbitrarily selected point. It is found that the equal-amplitude curves on the reference plane are ellipses and–in contrast to the planar resonator case–the directions of the principal axes of these ellipses need not coincide with the directions of the principal curvatures of the wave fronts. The important case of a four-mirror with a symmetric axial contour is discussed in detail.