Probing Fundamental Performance Limits in Photonics Design

Abstract

The past few decades have seen revolutionary advances in the application of structural inverse design to photonics, where the design problem is formulated as an optimization. By leveraging advances in computing power and combining gradient-based optimization algorithms with efficiently calculated structural gradients via the adjoint method, photonics designers can now handle thousands to millions of structural parameters to engineer complex devices with high performance. As design sophistication continues to increase, one natural question arises: are there fundamental limits to photonic device performance, and if so, how close are current designs to achieving such limits? The answer to this question not only has intrinsic theoretical interest; it may also serve to guide future efforts in practical photonics design. Unfortunately, a naïve approach starting directly from the formulation of inverse design is unfeasible, as the resulting optimization problems are both high-dimensional and non-convex, making it impossible to determine the global optimum. To address this question, this thesis presents a general framework for evaluating fundamental performance limits of photonic structures. Given the material that a potential structure is made of and a pre-specified design domain, the framework produces a bound applicable to all possible structures that reside completely within the design domain. The key is converting the structure optimization to a field optimization: through targeted relaxations of Maxwell’s equations, we derive structure-agnostic quadratic constraints on possible polarization fields: In conjunction with the machinery of Lagrangian duality, these constraints can be used to bound any design objective that is a quadratic function of the polarization field. This framework is applied to a selection of important and representative problems in photonics design, including the maximization of scattering cross sections, enhancing the local density of states, planewave focusing, and wavefront shaping. The framework is not solely a numerical tool: we can also perform asymptotic analysis to extract fundamental scaling laws to device performance with regards to material and size. Topology optimization results are presented in conjunction with the bounds; in many cases these come within order of magnitude of each other, demonstrating the power of both inverse design and the limits framework.