Wave propagation in linear and nonlinear optical parity-time (PT) periodic media

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The purpose of this project is to study the nonlinear dynamical behavior of light propagating in multi-dimensional complex parity-time (PT) photonic structures and disordered lattices. The mathematical strategy in this research is to use asymptotic and perturbation methods to develop reduced models. Using novel computational approaches these models will be examined numerically for the existence of localized solutions and their stability properties (where possible an analytic approach will also be undertaken). The results will be directly compared with experimental data, thus providing viable information on the accuracy of the model and quantitative insight to observations. The asymptotic validity of the model and various perturbative regimes will be examined in order to achieve a comprehensive mathematical understanding. Specifically, we will study localization properties and stability analysis of optical waves propagating in nonlinear multi-dimensional complex PT periodic structures that is modeled by the PT nonlinear Schroedinger equation. Moreover, we will investigate wave localization and dynamic stability in nonlinear three-dimensional random photonic lattices and study the phenomenon known as nonlinear Anderson localization. The field of optical wave propagation in nonlinear and random photonic structures has the potential for technological applications such as all-optical signal processing, navigation and switching. Photonics hold great promise for building high-speed, nanoscale switches and gates. Control of spatially or temporally localized structures within optical media is a critical issue in fabricating all optical devices. As such many resources have been put into building innovative experiments utilizing new optical materials that explore the dynamics and control the effects of nonlinearity and randomness in optical materials. To date, much of the research has been in the linear regime. Recent experimental discoveries such as the optical induction technique to create photonic lattices, have begun to allow researchers addressing many important issues related to wave propagation in periodic media such as optical waveguide arrays which until recently were thought to be impossible. However, even with the current detailed experimentation, probing optical phenomena at a sufficiently high spatial and temporal resolution to gain an improved theoretical understanding remains a challenge. There is thus great demand for companion theoretical investigations capable of accurately describing experiments. Due to the disparate range of scales involved, the primitive equations for investigating optical phenomena, Maxwell's equations are computationally prohibitive to solve. Thus there is a need for computationally efficient reduced models that quantitatively capture the essential phenomena, providing clear physical and theoretical insight. The purpose of this project is to address these issues for optical photonic systems by exploring, investigating, and simulating reductions of Maxwell's equations. The emphasis is on the modeling and computational aspects of wave propagation in photonic lattices and waveguide arrays.