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Deriving Numerical Aperture = N.A.
The geometries of diffraction patterns must match the geometries of acceptance angle.
Condenser N.A. must at least match that of the objective to realize full resolution.
Complex frequency-based information can be decomposed into combinations of simple waveforms.
Object detail vs. acceptance angle
The finer the spatial detail in an object the wider the diffraction angle - easily seen with a laser pointer and a fine grating.