Unstable resonators with Gosper-island boundary conditions : virtual-source computation of fractal eigenmodes

The Gosper island is a well-known fractal belonging to a family of self-similar “root 7” curves constructed from a simple iterative algorithm [1]. One begins with a regular hexagon (the initiator, corresponding to iteration n = 0) with sides of reference length l 0 , and then breaks each of these straight-edge elements into three equal segments of length l n = l 0 (1/7 1/2 ) n where n = 1, 2,3,... (the generator stages). If the total number of length elements after applying the generator n times is given by N n = 6 × 3 n , then the Hausdorff-Besicovich dimension of such a curve is calculated to be D = lim n→∞ -log(N n )/log(l n ) = 2log(3)/log(7) ≈ 1.1292.In this presentation, we report on our latest theoretical results predicting the modes of unstable resonators [2,3] when the small feedback mirror has a shape corresponding to increasing iterations of the Gosper island fractal. A fully two-dimensional generalization of Southwell's virtual source (2D-VS) method [4] (itself an approximation of Horwitz's asymptotic theory [5]) is deployed, whereby the resonator is unfolded into an equivalent sequence of apertures illuminated by a plane wave. Each aperture has a characteristic size (capturing a band of pattern spatial scalelengths), and it acts as a virtual source of diffracted waves that are computed using edge-wave decompositions within a circulation-integral method [6]. The empty-cavity eigenmodes are then constructed from a linear combination of the constituent single-aperture Fresnel patterns