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.