  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, the evolution of scattering
properties through another parameter space of incident directions (momentum
space) has attracted comparably little attention, though of profound importance
for various scattering-related applications. Here we investigate, from the
perspective of quasi-normal modes (QNMs), the momentum-space scattering
extremizations with respect to varying incident directions of plane waves. 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 we have 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.
