Higher-order modes of the kaleidoscope laser

Summary form only given. The recent discovery that the modes of hard-aperture unstable cavity lasers have fractal character and the subsequent extension of the work into two transverse dimensions highlight the need for the accurate calculation of unstable resonator modes. Although we have developed techniques for computing higher order modes in one transverse dimension, our 2D work has so far been based on the Fox-Li method, which only yields the modes with the lowest loss. Nevertheless, this limited set of modal patterns has exhibited such complexity and beauty that we have christened the device the "kaleidoscope laser." We present a new approach that enables the 2D higher-order modes of the kaleidoscope laser to be generated. We generalize W.H. Southwell\´s (1986) "Virtual Source" (VS) method from one to two transverse dimensions. In the VS method, mode patterns are built up from the successive diffraction of a plane wave through a sequence of apertures representing the unfolded cavity. Each aperture creates diffracted edge waves, and the family of eigenmodes is formed from different weighted sums of the edge waves, together with a plane-wave component. Although many recent experiments have used unstable cavities with relatively low Fresnel numbers, it is normally assumed in the VS method that the Fresnel number is large. We have found that the method gives surprisingly good predictions for Fresnel numbers as low as unity. We have also developed a generalization of the VS method to handle apertures of arbitrary shape. Results are presented for a variety of transverse geometries in a graphical format that provides a physical picture of the underlying mathematics and that justifies the assumptions that have been invoked.