Three-dimensional self-similar fractal light in canonical resonators

Unstable canonical resonators can possess eigenmodes with a fractal intensity structure [Karman et al., Nature 402, 138 (1999)]. In one particular transverse plane, the intensity is not merely statistically fractal, but self-similar [Courtial and Padgett, PRL 85, 5320 (2000)]. This can be explained using a combination of diffraction and imaging with magnification greater than one: each round trip through the resonator adds approximately the same diffraction pattern to the resonator, which gets then magnified through imaging, resulting after many round trips in the diffraction pattern being present on a cascade of length scales, a hallmark of fractals. Here we show that the same mechanism also shapes the intensity cross-section in the longitudinal direction. Combined with the mechanism for shaping the transverse intensity distribution, this results in three-dimensional, self-similar, fractal intensity structure in the lowest-loss eigenmode. Because the transverse and longitudinal magnifications are different, the scaling properties in the transverse and longitudinal directions are different. We demonstrate this structure using computer simulations.