WHISPERING-GALLERY LASER RESONATORS .2. ANALYSIS OF MIRRORS WITH NONUNIFORM CURVATURE

Although the whispering-gallery mirror shows great promise for use at short wavelengths, undesired beam divergence at the mirror exit can be a significant issue in the design of laser resonators. Fortunately, this problem can be greatly alleviated by appropriate choice of the mirror geometry. In particular, elongated mirror shapes can give reduced beam divergence without requiring too large an overall size. For circular geometries, beam propagation along the mirror surface is conventionally described in terms of normal modes. Elliptical mirrors are seen to admit a similar analysis, thereby providing a convenient prototype for the elongated geometries of interest. They are found to offer a substantial, though ultimately limited, reduction of beam divergence. Further improvement forces consideration of more general mirror shapes; this in turn requires a different approach to the analysis, as a normal-mode decomposition is no longer available. An equation much like the paraxial wave equation is developed to describe beam propagation in whispering galleries of arbitrary shape.