A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors

Abstract

A paraxial resonance equation is derived. This gives the mirror separation as a function of the radii of curvature of the mirrors and an integer N which is the number of return transits necessary to form a closed path of rays. Differentiating the paraxial resonance equation gives a formula for the relative mode density as a function of mirror separation. It is shown that the output power from a laser incorporating solid mirrors is inversely proportional to the mode density. In the case of hole coupling, the output power follows the same general profile but dips in power occur at the mirror separations corresponding to the resonance configurations of modes characterized by low values of N. Further confirmation of the paraxial resonance equation is obtained from passive resonators in which conic interference fringes and sudden increases in transmitted intensity are found to occur at the predicted mirror separations for low values of N corresponding to mode-degenerate configurations. The positions of the vertices of the ray traces are found to correspond to the patterns of discrete spots which are obtained in the output of a CO(2) laser incorporating Brewster angle windows and a solid germanium mirror. The laser configurations which give maximum output power are plotted as a cliff of constant height above the g(1)g(2) plane of the stability diagram, where g(1) and g(2) are the configuration coordinates. The relative merits of all possible cavity configurations having one mirror in common are shown as a set of equipower contours, and the hyperbolic curves of constant N are also superimposed on the stability diagram. The advantages of simplicity and directness in using the ray model are made clear.

Dynamics of mode formation in an open resonator

“…The parameters of resonators where beats are lacking in Fig. 7a conform with the condition of paraxial resonance derived in the context of resonator described by the methods of geometric optics [9]:…”

“…The evolution of the relaxation oscillation amplitude can be aperiodic or occur as beating. The wave-based resonator simulation reveals "paraxial resonances" predicted in [9] by the methods of ray optics. It has been shown that the conditions of paraxial resonance are matched by aperiodic processes of evolution of relaxation oscillation amplitude in the course of mode formation.…”

“…Two principal experiments that proved the existence of MP modes and paraxial resonances in the stationary state were performed in [9] by Ramsay and Degnan. In the first experiment, they measured the transparency of the passive resonator by changing its length.…”

“…This places emphasis upon the relation between the wave and the geometric optics in describing stable resonators, upon single-pass (SP) wave LG modes and upon multipass (MP) geometric paraxial resonances. The authors of a fundamental paper on geometric optics of stable resonators [9] only mention this relation in the most general of expressions. This point also has not been touched upon in the above-cited monographs, which show a clear preference for the wave approach.…”

Off-axis re-entrant cavity ring-down spectroscopy with a mid-infrared continuous-wave optical parametric oscillator

“…Many other techniques have also been proposed to increase the resolution and measurement speed of the conventional on-axis CRDS technique, such as: laser-locked CRDS system [14], fast frequency scanning of the laser [15], rapidly swept cavity design [16], or a combination of the cavity modulation and fast laser scanning approaches [17]. This paper reports off-axis light injection with re-entrant configuration [18,19] of the ring-down cavity has been used. The off-axis technique provides enhancement of the spectral resolution without cavity length modulation [10,20,21].…”