Momentum-Space Scattering Extremizations

Studies into scatterings of photonic structures have been so far overwhelmingly focused on their dependencies on the spatial and spectral morphologies of the incident waves. In contrast, evolutions of scattering properties through another parameter space of incident directions (momentum space) has not attracted sufficient attention, with the underlying general governing rules still unrevealed. Here, from the perspective of quasi-normal modes (QNMs), the momentum-space scattering extremizations with respect to varying incident directions of plane waves are investigated. It is revealed that for effective single-QNM excitations, scatterings are maximized exactly along those directions where the QNM radiation reaches its maximum, with matched incident and radiation polarizations. For an arbitrary direction, when the incident polarization is tuned to be orthogonal to that of the mode radiation, the QNM cannot be excited and thus the scatterer becomes invisible with null scatterings. The principles revealed are protected by fundamental laws of reciprocity and energy conservation (optical theorem), which can be further expanded and applied for other branches of wave physics. Related to the question of how to orientate an arbitrary opaque object toward the sun to cast the largest or smallest geometric shadow, this study manages to solve an analogous problem in wave optics and answers the following fundamental question in Mie theory: under which incident direction and polarization will an object scatter the most or least light?image