There are many useful ways we can think about optics and waves. “Modes,” like resonator modes or propagating modes in fibers, can give particularly powerful and simple descriptions. We then only need amplitudes of a few functions rather than field values at each of a large number of points. But, for scattering from complex nanophotonic structures or use of multiple beams or waves in space to send more information, these resonator or propagating modes are not enough, and can lead to confusion and even major error (for example, in correctly counting the number of “channels” we have to communicate). We need to think about scattering or communicating waves from some source space to some receiving space. That means we need pairs of functions, one in each space. Fortunately, there are elegant and efficient ways of establishing these pairs, both mathematically and physically. Once we do so, we find many simple heuristic behaviors; we can correctly count numbers of usable channels, establishing clear limits; and we can find the “right” fundamental basis pairs for describing optical systems. That basis is fundamental because it leads, for example, to new “Kirchhoff” radiation laws and a simpler “Einstein A&B” coefficient argument, both of which work mode by mode, but only for these new basis function pairs. Along the way, we establish a new “M-gauge” for electromagnetism and an elegant quantization of the electromagnetic field without arbitrary “boxes.” The talk is summarizing a recent paper [1] that introduces all of the necessary math, physics, and results.

Prof. Olav Solgaard, Organizer Fall 2019

#### AMO Seminar Sub-Series first Monday of each month)

Monica Schleier-Smith, Organizer Fall 2019