A ray analysis of optical resonators formed by two spherical mirrors

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.