Gertler Shai, Kuang Zeyu, Christie Colin, Li Hao, Miller Owen D
Department of Applied Physics and Energy Sciences Institute, Yale University, New Haven, CT 06511, USA.
Sci Adv. 2025 Jan 3;11(1):eadl3237. doi: 10.1126/sciadv.adl3237. Epub 2025 Jan 1.
Photonic design is a process of mathematical optimization of a desired objective (beam formation, mode conversion, etc.) subject to the constraint of Maxwell's equations. Finding the optimal design is challenging: Generically, these problems are highly nonconvex and finding global optima is NP hard. Here, we show that the associated optimization problem can be transformed to a sparse-matrix, quadratically constrained quadratic program (QCQP). Sparse QCQPs can be tackled with convex optimization techniques (such as semidefinite programming) that have thrived for identifying global bounds and high-performance designs in many areas of science and engineering but seemed inapplicable to the design problems of wave physics. We apply our formulation to prototypical photonic design problems, showing the possibility to compute fundamental limits for large-area metasurfaces, as well as the identification of designs approaching global optimality. Our approach appears directly extensible to any design problem whose governing dynamics are bilinear differential equations, as arise in structural optimization, fluid dynamics, and quantum control.
光子设计是在麦克斯韦方程组的约束下，对期望目标（波束形成、模式转换等）进行数学优化的过程。寻找最优设计具有挑战性：一般来说，这些问题是高度非凸的，寻找全局最优解是NP难问题。在这里，我们表明相关的优化问题可以转化为一个稀疏矩阵、二次约束二次规划（QCQP）问题。稀疏QCQP问题可以用凸优化技术（如半定规划）来解决，这些技术在许多科学和工程领域中已经成功地用于确定全局边界和高性能设计，但似乎不适用于波动物理学的设计问题。我们将我们的公式应用于典型的光子设计问题，展示了计算大面积超表面基本极限的可能性，以及识别接近全局最优的设计的可能性。我们的方法似乎可以直接扩展到任何其控制动力学为双线性微分方程的设计问题，如结构优化、流体动力学和量子控制中出现的问题。
