Fundamental limits on the performance of electromagnetic devices

Formulating the design of electromagnetic devices as an optimization problem has opened up the possibility of achieving performance metrics that have eluded common design techniques. However, since these optimization problems can only be approximately solved, it is not clear what performance metrics (e.g. transmission and cross-talk) are achievable for an optical device, subject to constraints on device footprint, refractive index contrast, and feature size. In this work, we show how to efficiently calculate provable bounds on performance metrics that are expressible as a quadratic form in the electric field for a general class of linear-optical devices using Lagrangian duality and convex relaxations. We illustrate these techniques by calculating bounds on some representative 1D and 2D electromagnetic design problems: the reflectivity of a 1D mirror and the focusing efficiency of a 2D lens.