Many photonic design problems are sparse QCQPs.

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.