The Optical Ring Resonator

In the ring laser, a light beam is directed about a closed loop, typically by three or four mirrors, and regeneratively amplified at frequencies for which the circuit path equals an integral number of wavelengths.1"3 The only available analysis of a ring resonator with more than one spherical mirror appears to be that of Clark,4 who used a ray-optical approach to derive the stability conditions for a ring with mirrors of two different curvatures and spacings. However, the means for a complete analysis of any arbitrary ring resonator are contained implicitly in the optical-mode theory developed in recent years in connection with two-mirror resonators.0"8 The purpose of this note is to trace the connection between this theory and that of ring resonators in two different ways, and to derive the formulae defining the Gaussian (fundamental mode) beam in an arbitrary four-mirror resonator. A third method has been proposed recently by Collins in general form,9'10 but will not be employed here because of its greater complexity. In a two-mirror resonator, the wavefront curvatures of the light beam coincide with those of the mirrors, since the beams are reflected back on themselves. This is not the case in ring resonators, in which the beam is reflected obliquely. The boundary condition of the latter is merely that 907