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This thesis concerns the study of two kinds of phase transitions, parity-time(PT ) symmetric phase transitions and topological phase transitions, using photonic systems. In the first part of the thesis, we perform a theoretical study of the nonlinear dynamics of a nonlinear optical isolator device based on coupled microcavities with gain and loss. This reveals a correspondence between the boundary of asymptotic stability in the nonlinear regime, where gain saturation is present, and the PT breaking transition in the underlying linear system. We establish a correspondence between the onset of optical isolation and the PT phase transition of the linear system. In the second part of the thesis, we study a new class of nonlinear optical isolator based on self-induced topological phase transitions. We show that topological phase transitions can be used to help design nonlinear photonic structures exhibiting power thresholds and discontinuities in their transmittance. This provides a novel route to devising nonlinear optical isolators. We study three representative designs: (i) a waveguide array implementing a nonlinear 1D Su-Schrieffer-Heeger model, (ii) a waveguide array implementing a nonlinear 2D Haldane model, and (iii) a 2D lattice of coupled-ring waveguides. In the first two cases, we find a correspondence between the topological transition of the underlying linear lattice and the power threshold of the transmittance, and show that the transmission behavior is attributable to the emergence of a self-induced topological soliton. In the third case, we show that the topological transition produces a discontinuity in the transmittance curve, which can be exploited to achieve sharp jumps in the power-dependent isolation ratio. In the third part, we study the realization of an unpaired Dirac cone at the center of the first Brillouin zone, using a gyromagnetic photonic crystal with broken square sub-lattice symmetry and broken time reversal symmetry. The behavior of the Dirac modes can be described by a gyromagnetic effective medium model with near-zero refractive index. When two domains are subjected to opposite magnetic biases, there exist unidirectional edge states along the domain wall. This establishes a novel link between topological edge states and the surface waves of homogenous magneto-optical media.

Two kinds of phase transition in photonic systems with application to optical isolation