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

卓越的研究：几何光学中的最佳传输和自由形状折射表面

• 批准号: 2200589
• 负责人: Henok Mawi
• 金额: $ 24.73万
• 依托单位: Howard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200589&HistoricalAwards=false
• 关键词: Excellence; Research; Optimal; Transport; Freeform

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.

该项目旨在研究数学模型，以设计自由（非对称）光学表面（透镜），以可控和有效的方式引导光强分布。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有着广泛的应用。该研究将有助于减少光学设备中所需光学元件的数量和尺寸，从而提高更轻，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，并为来自代表性不足的群体的学生提供培训机会，从事交叉数学、物理和工程的跨学科研究活动。该项目旨在研究几何光学问题，这些问题制定了通过反射或折射传输前后光强度之间的关系。本研究将介绍新的数值方法，并进一步讨论在各向同性和各向异性介质中的折射光学表面相关问题的解决方案的分析方面。数值技术将开发利用熵最优运输，其中包括增加一个惩罚项，使问题凸，也通过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进行。此外，从密切相关的非线性偏微分方程称为生成雅可比方程，其中之一是蒙格-安培方程的技术，将用于进一步的分析。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。