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

本计画的目的是研究光在多维复杂宇称时间光子结构及无序晶格中传播的非线性动力学行为。本研究的数学策略是使用渐近和摄动方法来建立简化模型。使用新的计算方法，这些模型将被检查数值局部化解决方案的存在及其稳定性（在可能的情况下，也将进行分析方法）。结果将直接与实验数据进行比较，从而提供有关模型准确性的可行信息和对观测的定量见解。将检查模型和各种微扰状态的渐进有效性，以获得全面的数学理解。具体而言，我们将研究光波在由PT非线性薛定谔方程建模的非线性多维复杂PT周期结构中传播的局域化特性和稳定性分析。此外，我们将研究非线性三维随机光子晶格中的波局域化和动力学稳定性，并研究称为非线性安德森局域化的现象。光波在非线性和随机光子结构中的传播领域在全光信号处理、导航和交换等技术应用方面具有潜力。光子学在构建高速、纳米级开关和门方面前景广阔。在光学介质中控制空间或时间局部化结构是制造所有光学器件的关键问题。 因此，许多资源已投入到利用新光学材料建立创新实验，探索光学材料中的非线性和随机性的动态和控制效果。迄今为止，大部分研究都是在线性范围内进行的。 最近的实验发现，如光感应技术，以创建光子晶格，已经开始允许研究人员解决许多重要的问题，有关波在周期性介质，如光波导阵列，直到最近被认为是不可能的传播。 然而，即使有了目前详细的实验，以足够高的空间和时间分辨率探测光学现象，以获得更好的理论理解仍然是一个挑战。因此，对能够准确描述实验的同伴理论研究有很大的需求。 由于所涉及的尺度范围不同，研究光学现象的原始方程，麦克斯韦方程在计算上是无法求解的。 因此，需要计算效率高的简化模型，定量地捕捉基本现象，提供清晰的物理和理论见解。本计画的目的是透过探索、研究及模拟麦克斯韦方程式的简化，来解决光学光子系统的这些问题。重点是在光子晶格和波导阵列中的波传播的建模和计算方面。

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.