The 374th forum: Why Simulations Shouldn't Always Model Experiments
Professor of Applied Mathematics and Physics
Massachusetts Institute of Technology
Steven G. Johnson received his Ph.D. in physics from MIT in 2001, where he was also an undergraduate student (receiving B.S. degrees in physics, mathematics, and EECS in 1995). He joined the MIT faculty in applied mathematics in 2004, and was awarded tenure in 2011. He works on the influence of complex geometries (particularly in the nanoscale) on the solutions of partial differential equations, especially for wave phenomena and electromagnetism. This includes analytical theory, numerics, and design of devices and phenomena. He is also known for his work in high-performance computing, such as his development of the FFTW fast Fourier transform library (for which he received the 1999 J. H. Wilkinson Prize for Numerical Software). In 2009, he received the Edmund F. Kelly Research Award from the MIT Mathematics Department.
The core of photonics and metamaterials computational design and modeling is a Maxwell solver: given sources (currents) as a function of time or frequency, calculate the resulting electromagnetic fields in a given geometry. A straightforward way to employ such a solver is to perform numerical experiments that directly mirror laboratory experiments, e.g. transmission or reflection calculations. Even extraction of effective-medium "metamaterial" parameters is commonly done by the direct analog of ellipsometry measurements. Although this approach is very useful, there are also way to exploit Maxwell solvers that have no direct experimental analogue, taking advantage of the extreme flexibility in sources and "measurements" provided by the computer. In this talk, we discuss some old and new analytical transformations that allow Maxwell solvers to be used in ways very different from experiments. "Standard" techniques that are not as well known as they should be include "adjoint" solves for sensitivity analysis and optimization. More recent ideas include ways to use "unphysical" complex-frequency solvers to extract results over a whole bandwidth simultaneously and laser modeling that skips time evolution of the gain medium to jump directly to the steady state. Often, the very thing that makes a problem seemingly intractable, e.g. the huge difference in timescales between the gain media and the optical frequencies, are what makes efficient computational solvers possible.
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