Spontaneous Spatial Fractals: Universal Contexts and Applications

Reaction-diffusion systems can exhibit a Turing instability in which homogeneous states develop large-amplitude emergent patterns. These patterns are typically characterized by a single dominant length scale that corresponds to a single minimum in a modulational instability threshold curve. However, several nonlinear systems possess a hierarchy of local Turing minima. It was proposed [1] that this hierarchy could signature the spontaneous emergence of spatial fractal patterns (with detail spanning many decades of scale). The proposal was tested, and confirmed true [1], in an optical context involving a Kerr slice and a feedback mirror [2]. Firstly, we present the first evidence of spontaneous spatial fractals in ring cavities, and for a range of nonlinear materials [3]. This classic system is modelled in the thin-slice limit, with a pump detuning parameter determining the level of dispersion. Periodic pumping and losses are introduced via the conventional ring-cavity boundary condition, and a spatial filter allows the control of complex pattern formation. Characteristics of the resulting fractal patterns depend upon system parameters (such as diffusion length, pump intensity, and mirror reflectivity). One key result is that spatial fractals can emerge even in purely-absorptive regimes – see Fig. 1(a). Fig. 1 (a) Spontaneous transformation of classic Turing patterns (top – stripes; bottom – hexagons) toward fractals in nonlinear ring cavities. (b) Lowest-loss eigenmode patterns for kaleidoscope lasers with Neq = 30 and M = 1.5. Secondly, we summarize examination of further optical fractal generating contexts. Generalization