Materials Science Research Lecture
Physical design meets convex optimization: Hidden structure in Maxwell's equations
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Link to join Webinar: https://caltech.zoom.us/j/85010413991
Webinar ID 850 1041 3991
In optimization theory, one clear dividing line between "easy" and "hard" problems is convexity. In convex optimization problems, all local optima are global optima, which can be found by efficient computational algorithms. By contrast, nonconvex problems can have highly oscillatory landscapes, and one must typically resort to local optimization techniques or black-box approaches. Nanophotonic design problems, like most physical design problems, 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. The key differential-equation constraints can be transformed to infinite sets of local conservation laws, which have a structure well-suited to quadratic and semidefinite programming. This approach offers a general framework for global bounds ("fundamental limits") for many design problems of interest. It also appears to offer a dramatically new approach to the design process itself.
Spectral degrees of freedom have further hidden structure. In fact, we can introduce a new viewpoint on electromagnetic scattering. We show that a special scattering matrix, the "T" matrix, can always be decomposed into a set of fictitious Drude–Lorentz oscillators with matrix-valued (spatially nonlocal) coefficients. For any application and any scatterer, the only designable degrees of freedom are these matrix coefficients. This natural encoding of causality and passivity offers new insights into what is possible in nanophotonics.
Throughout I will emphasize novel applications where we utilize these techniques, including: scaling laws for analog photonics, optimal quantum control, and a theory of the ultimate limits of near-field radiative heat transfer.
More about the Speaker:
Owen Miller is an Asst. Prof. of Applied Physics and the Energy Sciences Institute 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.
Contact: Jennifer Blankenship email@example.com