= Deformable-mirror Fourier-mode counting
For actuator pitch $D/N$, full-cycle <spatial frequency> components are limited to $N/2$ by sampling. Integer lattice counting in a circular <frequency> disk gives approximately $\pi N^2/4$ signed points; conjugate pairing halves the number of pairs, and two real <wave phase> quadratures per pair restore approximately $\pi N^2/4$ real coefficients. Counting only positive-quadrant <frequency> representatives gives $\pi N^2/16$, but it omits other orientations or quadratures unless a restricted convention is specified. Area approximations have boundary corrections and do not give exact integer ranks. A square sampling cutoff and a circular illuminated actuator footprint are distinct geometric restrictions.
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