Research Initiation Award: Mathematical Modeling On the Geometric Optics Problem of Refraction

研究启动奖为传统黑人学院和大学的教师提供支持，他们正在建立一个研究项目。期望该奖项有助于进一步提高教师的研究能力和效率，改善其所在机构的研究和教学，并使本科生参与研究经验。该奖项授予霍华德大学在许多领域具有潜在的更广泛和社会影响。该项目旨在研究源自几何光学的数学模型。该研究跨越了数学、工程和物理领域，在光学系统的合成、设计和制造方面具有潜在的应用前景。本科生将获得研究经验，研究将被整合到一些数学课程中。折射问题是几何光学中的一个反问题，它涉及到具有不同光学性质的两种介质之间的界面表面的建模。这项工作的目标是获得和分析能够将光线从位于一种介质中的点源发射到一组方向或位于另一种介质中的特定目标的表面，以预先指定光线的输入和输出强度的方式。应用几何光学定律和能量守恒原理来建立数学模型，自然会得到蒙日-安培特尔型二阶非线性偏微分方程。曲面的确定将引发非线性偏微分方程解的存在性、唯一性和正则性问题以及质量输运问题。本课题的具体目标包括：开发各向异性介质中近场折射问题的计算算法，建立晶体等各向异性介质中几何光学问题的数学模型，寻找各向异性介质中近场和远场折射问题的数值解和解析解，研究各向异性介质中近场和远场折射问题解析解的规律性。各向异性介质的考虑将几何光学问题的数学理论推进到更一般的传播介质，使其更适用于光学器件、自由折射率光束整形器的设计和激光光学等光学应用。

Research Initiation Awards provide support for faculty at Historically Black Colleges and Universities who are building a research program. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at his home institution, and involves undergraduate students in research experiences. The award to Howard University has potential broader and societal impact in a number of areas. The project seeks to investigate mathematical models that originate from geometrical optics. The research crosses the fields of mathematics, engineering and physics and has potential applications in the synthesis, design and manufacturing of optical systems in an efficient and cost effective manner. Undergraduate students will gain research experiences and the research will be integrated in a number of mathematics courses.The refractor problem is an inverse problem in geometric optics that is related to modeling of interface surfaces between two media that have different optical properties. The goal of this work is to obtain and analyze surfaces which are capable of redirecting light rays emanating from a point source located in one of the media into a set of directions or onto a specific target located in the other media, in such a way that the input and output intensities of the rays are prespecified. The application of the laws of geometric optics and conservation of energy principles to formulate the mathematical model will naturally lead to second order nonlinear partial differential equations of Monge - Ampère type. The determination of the surfaces will incite problems of existence, uniqueness and regularity of solutions to nonlinear partial differential equations and mass transport problems. The specific objectives of this project include: developing a computational algorithm for the near field refractor problem in isotropic media, formulating mathematical models of geometrical optics problems in anisotropic media such as crystals, finding numerical and analytical solutions for near field and far field refractor problems in anisotropic media and investigating the regularity of analytical solutions for near field and far field refractor problems in anisotropic media. The consideration of anisotropic media will advance the mathematical theory of geometric optics problems to media of propagation that are more general and will make it more amenable to several optics applications including optical devices, design of freeform refractive beam shapers and laser optics.