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

本项目的目的是研究光在多维复宇称时间(PT)光子结构和无序晶格中传输的非线性动力学行为。本研究的数学策略是使用渐近法和摄动法来建立简化模型。利用新的计算方法，将用数值方法检查这些模型局部解的存在及其稳定性(如果可能，还将采用解析方法)。结果将直接与实验数据进行比较，从而为模型的准确性和对观测的定量洞察提供可行的信息。为了达到全面的数学理解，我们将考察模型和各种扰动机制的渐近有效性。具体地说，我们将研究光波在PT非线性薛定谔方程所描述的多维复杂PT周期结构中传播的局域化性质和稳定性分析。此外，我们还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，并研究被称为非线性Anderson局域化的现象。非线性和随机光子结构中的光波传播领域在全光信号处理、导航和交换等方面具有潜在的技术应用前景。光子学在制造高速、纳米级的开关和门方面有着巨大的希望。光学介质中空间或时间局域结构的控制是制造所有光学器件的关键问题。因此，许多资源被投入到利用新的光学材料建立创新的实验中，这些材料探索光学材料的动力学并控制光学材料中的非线性和随机性的影响。到目前为止，大部分研究都是线性的。最近的实验发现，如创造光子晶格的光学感应技术，已经开始允许研究人员解决许多与周期性介质中的波传播有关的重要问题，例如直到最近还被认为不可能的光波导阵列。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配套理论研究的需求很大。由于所涉及的尺度范围不同，用于研究光学现象的原始方程，麦克斯韦方程在计算上是难以求解的。因此，需要计算效率高的简化模型来定量地捕捉基本现象，提供清晰的物理和理论见解。这个项目的目的是通过探索、研究和模拟麦克斯韦方程的约化来解决光学光子系统的这些问题。重点是波在光子晶格和波导阵中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.