S-matrix completion by plane-wave CBED interpolation
PRISM
In PRISM1, Colin Ophus introduced the idea of simulating a sparse grid of plane wave CBED patterns, then combining those after simulation to form shifted probe CBEDs. We plan to implement this in Issue #8. Ophus mentions that the "interpolation factor", i.e. the sparsity factor in each dimension of the frequency plane, should be such that the simulated diffraction patterns will still overlap. This is necessary because otherwise there will be no contrast due to shifting the beam to a new location.
CBED Interpolation
The limit on interpolation factor is more pronounced for defocused probes since their CBED impulse responses are necessarily more localized in Fourier space (i.e. more dispersed spatially). In order to address the lack of position-dependent contrast, we propose to infill the S-matrix. That is, we will use simulated plane-wave CBEDs on a coarse frequency grid to interpolate CBED patterns for plane-waves at intermediate locations. These intermediate locations may then be given non-zero coefficients in the probe wavefunction's Fourier transform, and we will sum them as if we had simulated those off-grid plane waves directly.
We will do the interpolation by shifting each simulation to the origin (from the 2D input frequency position), performing a weighted sum using bilinear interpolation weights, then shifting the result to the desired (interpolated) frequency position.
Plan
Once #8 is implemented, we can implement interpolation as an optional step, by holding separate probe and simulation frequency grids, accepting a frequency grid at simulation time. Note that this enables us to use different offsets to simulate different plane-waves at every gradient iteration.