Generalized confocal resonator theory

The theory of the confocal resonator is extended to include the effect of unequal aperture size and unequal radii of curvature of the two reflectors. The latter is equivalent to a periodic sequence of lenses with unequal focal lengths. This treatment is in Cartesian coordinates as previously used. In an appendix the modes and resonant formulas are written in cylindrical coordinates. The effect of unequal aperture size of the two reflectors is shown to produce mode patterns of unequal size on the two reflectors of a confocal resonator. The previous computations for diffraction losses are found to be applicable. Generalization of the theory to the case of reflectors of unequal curvature shows the existence of low-loss regions and high-loss regions as the reflector spacing is varied. One of the high diffraction loss regions occurs when the reflector spacing is between the two unequal radii of curvature. Such a region is interpretable in terms of instabilities in a periodic sequence of lenses of unequal focal length. An estimate of diffraction losses is obtained for the low-loss regions. The presence of a high diffraction loss region or unstable region should be of importance in the design of resonators or of a periodic sequence of lenses.