Excellence in Research: Optimal Transport and Freeform Refractive Surfaces in Geometric Optics

该项目旨在研究数学模型，以设计自由形状（不对称）光学表面（透镜），以可控和有效的方式指导光强分布。该项目在天线、太阳能电池板、医疗设备、军事设备和许多其他相关领域的光学元件设计中具有广泛的应用。这项研究将有助于减少光学器件所需光学元件的数量和尺寸，从而改进更轻、更紧凑、更高效的光学系统的制造。该项目将加强霍华德大学的研究基础设施，并为来自弱势群体的学生提供培训机会，让他们参与数学、物理和工程交叉学科的研究活动。该项目旨在研究几何光学问题，即通过反射或折射传输前后光强度之间的关系。本研究将引入新的数值方法，并进一步讨论与各向同性和各向异性介质中折射光学表面有关的问题的解析解决方案。数值技术将通过利用熵最优传输来发展，其中包括添加一个惩罚项以使问题凸出，并使用闵可夫斯基型参数。分析研究将通过使用重定向表面的固有几何特性来进行。此外，来自密切相关的非线性偏微分方程的技术被称为生成雅可比方程，其中之一是蒙日-安培方程，将用于进一步分析。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project aims to study mathematical models to design freeform (asymmetric) optical surfaces (lenses) that direct light intensity distribution in a controlled and efficient way. This project has vast applications in the design of optical components in antennas, solar panels, medical devices, military equipment, and many other related areas. The research will help reduce the amount and size of optical components required in optical devices, thereby improving the manufacturing of lighter, more compact, and more efficient optical systems. The project will enhance the research infrastructure at Howard University and provide training opportunities for students from underrepresented groups in interdisciplinary research activities that intertwine mathematics, physics, and engineering. The project aims to study geometric optics problems that formulate relationships between the intensities of light before and after transport via reflection or refraction. This research will introduce new numerical approaches and further discuss analytical aspects of solutions to problems related to refractive optical surfaces in both isotropic and anisotropic media. The numerical techniques will be developed by exploiting entropic optimal transport, which involves adding a penalty term to make the problem convex and also by using Minkowski-type arguments. The analytical investigation will be conducted by using the intrinsic geometric properties of the redirecting surfaces. Additionally, techniques from the closely related nonlinear partial differential equations known as generated Jacobian equations, one of which is the Monge-Ampere equation, will be used for further analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.