| Summary: | The seminal work published by Alan Turing in 1952 provides the framework for our understanding of spontaneous pattern emergence in Nature. He proposed that when a reaction-diffusion system’s uniform states become sufficiently stressed, arbitrarily-small disturbances combined with inherent feedback processes can lead to spontaneous self-organization into finite-amplitude patterns with a single dominant scalelength. In this thesis, Turing’s universal mechanism is applied to a range of distinct nonlinear optical cavities. Of principal interest is a Fabry-Pérot (FP) resonator, which comprises a thin slice of diffusive Kerr-type material placed between two partially reflecting mirrors and pumped by an external plane wave. This model is a non-trivial generalization of the single-feedback mirror system, with the inclusion of the second mirror facilitating a disproportionate increase in complexity: here, the interplay between diffraction, counterpropagation and diffusive nonlinearity must be supplemented by more involved boundary conditions accommodating periodic pumping, mirror losses, interferomic mistuning, and time delays. The presented research analyzes the thin-slice FP cavity mathematically and computationally, subject to plane wave pumping. Linear stability techniques are deployed to obtain the threshold spatial instability spectrum predicting the emergence of static patterns, and which exhibits a discrete-island type of structure. Simulations then consider the full dynamics of the system, testing theoretical analyses in the cases of instantaneous and diffusive medium responses. The emergence of both simple (single-scale) and fractal (multi-scale) spatial patterns is demonstrated, and specialist software used to assist with quantifying their dimension characteristics in terms of system parameters. The first steps are also taken towards understanding the role played by nonparaxiality when considering spatial fractal pattern formation in dispersive systems with a finite light-medium interaction length. The classic Schr¨odinger-type governing equations for bulk ring cavities are reformulated as Helmholtz-type problems, capturing a family of higher-order nonlinear effects that describe wavelength-scale spatial structure in the circulating field.
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