In optimization theory, one clear dividing line between "easy" and "hard" problems is convexity. Convex optimization problems do not have local optima that are not global optima, and multiple decades of development have led to efficient computational machinery for solving convex problems. By contrast, nonconvex problems can have highly oscillatory landscapes, and one must typically use local optimization techniques or black-box algorithms. Nanophotonic design problems, and many design problems across physics, reside squarely in the latter category of nonconvex optimization problems.
Or do they? I will show that there is a surprising amount of mathematical structure hidden in the typical differential equations of physics, and that this structure enables new connections to modern techniques in convex optimization. I will describe how the key constraints in these design problems can be transformed from the typical differential-equation descriptions to infinite sets of local conservation laws, and that the latter have a structure amenable to quadratic and semidefinite programming. I describe how this approach can lead to global bounds ("fundamental limits") for many design problems of interest, and potentially to dramatically new approaches to identifying designs themselves. Next, specific to electromagnetic scattering, I will describe a unique construction of scattering matrices that leads to new methods for identifying fundamental limits across any bandwidth of interest.
Throughout I will emphasize novel applications where we have applied these techniques, including: minimal-thickness perfect absorbers, scaling laws for analog photonics, speed limits in quantum optimal control, and a new theory of the ultimate limits of near-field radiative heat transfer.
Owen Miller is an Asst. Prof. of Applied Physics and Physics at Yale. His research interests center around developing large-scale computational and analytical design techniques for discovering novel structures and new phenomena in nanophotonics. He is the recipient of AFOSR and DARPA young investigator awards, as well as the Yale Graduate Mentor award.