Published: Feb. 21, 2020

Owen Miller, Department of Applied Physics, Yale University

Upper bounds to electromagnetic response via convexity, causality, and duality

Nanophotonics, the study of light interacting with materials patterned at the scale of the wavelength, is developing at a rapid pace, with ever more materials, form factors, and structural degrees of freedom now available. These large design spaces offer transformative technological possibility, but finding optimal material distributions can be difficult or impossible. And incorporation of extraneous constraints such as fabrication tolerances further hinder various approaches. An alternative approach is to attempt to find global upper bounds, which now appears tractable and efficient in many scenarios.

I will discuss the application of ideas from convex optimization, complex analysis, and quadratic programming to discover upper bounds for a broad array of nanophotonic applications. The key is to find “hidden” constraints in Maxwell’s equations, relaxing the problem to one whose global optimum can be computed efficiently (and often analytically). Three applications are discussed: maximal scattering at a single frequency, maximal response over any frequency bandwidth, and the minimal “size” of an electromagnetic mode. For the last case, I will show how Lagrangian duality enables incorporation of minimum features sizes and multi-frequency considerations, constraints that cannot be accounted for in any other approach that we know of.