Spontaneous spatial fractal patterns: Towards nonparaxial nonlinear ring cavities

Spontaneous pattern formation in optical ring cavities containing a nonlinear (e.g., Kerr-type) material [see Fig. 1(a)] has been studied extensively for the past three decades. A notable trend in the literature over recent years has been a shift away from the (bulk) cavity + boundary condition models of McLaughlin et al. [1] toward the (longitudinally-averaged) mean field descriptions of Lugiato and Lefever [2]. While this latter approach is analytically more tractable, it does not yield Turing instability spectra with the multiple-minimum characteristic proposed as necessary for spontaneous fractal (i.e., multi-scale) pattern formation [3,4] [see Fig. 1(b)]. Here, we revisit the approach taken by McLaughlin et al. [1] but instead allow for the full generality of Helmholtz (broadband) as opposed to paraxial (narrowband) diffraction. Such a restoration of spatial symmetry (whereby diffraction occurs in both transverse and longitudinal dimensions) allows a much more reliable description of small-scale spatial structure in the circulating cavity field. Our analysis also goes some way toward addressing the issue of fractal pattern formation in systems with finite light-medium interaction lengths [3,4]. Linear analysis has predicted the threshold condition for spatial pattern emergence, and simulations have begun to investigate these new instabilities. References [1] D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985). [2] L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987). [3] J. G. Huang and G. S. McDonald, Phys. Rev. Lett. 95, 174101 (2005). [4] J. G. Huang, J. M. Christian, and G. S. McDonald, J. Nonlin. Opt. Mat. Phys. 21, 1250018 (2012).