OP: Mathematical Analysis of Nonlinear Optics in Periodic and Complex Media

光在周期介质和损益平衡的介质中的传输是光学和应用数学研究的前沿。周期性介质，例如光子晶体或具有周期性变化的折射率的介质，可以被操纵来控制光的传播，并实现例如光的捕获和路由。这一领域的研究进展在微观尺度的芯片数据处理方面具有很好的应用前景。通过在所谓的奇偶时间对称配置中明智地平衡增益和损耗，人们可以诱导光以新颖的方式工作，从而产生一种可能的“光学二极管”。这种仅能在一个方向上传输信号的光学二极管将是光学计算机的基本基础。在本项目中，我们从理论上研究了光在周期介质和增益损耗平衡介质中的新型非线性行为。通过发展新的数学方法，人们将对光在这种介质中的非线性传播有新的认识，从而可以评估它们在数据处理中的应用。从更广泛的角度来看，这个项目将促进研究生在这个重要的跨学科领域的培养。这个项目所涉及的问题是应用数学和光学的前沿。在数学方面，这个项目将发展新的数学方法来处理当代的非线性光学问题，例如一种复杂的指数渐近技术，用于研究二维周期介质中集总孤子的分叉和线性稳定性。在物理方面，这个项目的数学结果将使我们能够评估光孤子在各种物理应用中的潜力，例如在微观周期结构上的数据处理。此外，对实际的奇偶时间对称单模激光器的理论研究可能会直接影响这些激光器的非线性运转，并可能导致更先进的激光器件。

Light propagation in periodic media and media with balanced gain and loss is at the frontier of research in optics and applied mathematics. Periodic media, such as photonic crystals or media with periodically changing refractive index, can be manipulated to control light's propagation and achieve, for example, light trapping and routing. Progress in this area of research has promising applications for on-chip data processing at microscopic scales. By judiciously balancing the gain and loss in the so-called parity-time-symmetric configurations, one can induce the light to behave in novel ways, giving rise to a possibility of "optical diodes." Such an optical diode that is capable of transmitting a signal in only one direction would be an elemental base of optical computers. In this project, novel nonlinear behavior of light in periodic media and gain-loss-balanced media are theoretically investigated. By developing new mathematical methodologies, new insight will be gained on the nonlinear propagation of light in such media, so that their application for data processing can be assessed. From a broader perspective, this project will facilitate training of a graduate student in this important interdisciplinary area.The problems undertaken in this project are at the cutting edge of applied mathematics and optics. On the mathematical side, this project will develop novel mathematical methodologies for the treatment of contemporary nonlinear optics problems, such as a sophisticated exponential asymptotics technique for the bifurcation and linear stability of lump solitons in two-dimensional periodic media. On the physical side, the mathematical results from this project will allow us to assess the potential of optical solitons for various physical applications such as data processing on microscopic periodic structures. In addition, the theoretical investigation of actual parity-time-symmetric single-mode lasers could directly impact the nonlinear operation of those lasers and may lead to more advanced laser devices.