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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.

Write spatial frequency in cycles across the optical pupil as $u=D/\Lambda$. The previous result gives a maximum axial spatial frequency $N/2$. If one counts one representative per positive-quadrant integer lattice point inside an isotropic disk $u_x^2+u_y^2\le (N/2{)}^2$, the continuum area estimate isThis is the geometric counting convention producing the printed formula. Lattice boundaries, the zero-frequency piston and axis points give lower-order corrections, so the formula is not an exact integer count.

The mode definition in the PDF is $D=j\Lambda /2$, hence $u=j/2$ and $j_{\max }=N$. It does not itself specify a two-dimensional lattice, its orientation, its independent wave phase components or a circular rather than square spatial frequency cutoff. Full-cycle integer frequencies correspond only to the even values of $j$ in that definition. A general real wavefront error also needs both sine and cosine quadratures, and distinct oblique orientations cannot all be identified with a single positive quadrant: the Fourier transform of a real wavefront has conjugate coefficients at $\mathbf{u}$ and $-\mathbf{u}$, rather than identifying every independent sign change of a component.

For comparison, a circular spatial frequency disk has approximately $\pi N^2/4$ signed full-cycle lattice points, or $\pi N^2/8$ conjugate pairs; two real wave phase components per pair restore approximately $\pi N^2/4$ real degrees of freedom. A square spatial frequency support gives a different count. The circular physical optical pupil similarly has only about $\pi N^2/4$ illuminated actuators. The printed $M$ is a quadrant area estimate under an extra counting convention; it is not a uniquely determined count of all independently correctable wavefront modes from the stated one-dimensional mode condition.

An optical pupil ripple of spatial period $\Lambda$ diffracts starlight into two sidebands displaced by $\theta \simeq \lambda /\Lambda$ along the ripple's spatial frequency vector. Substituting the axial Nyquist period gives the deformable-mirror control radiusThe square actuator lattice has component cutoffs $|u_x|,|u_y|\le N/2$, hence an ideal square corrected region with this half-width in each photodetector direction. Its diagonal reaches $\sqrt{2}$ times the axial radius; an isotropic conservative restriction is the inscribed disk. Using $\lambda /D$ as the scale of a spatial resolution element, the square is about $N$ elements on a side and $N^2$ in area; the disk has about $\pi N^2/4$ elements. These are angular area counts, not the quadrant coefficient count in part (ii).

A circular illuminated optical pupil has fewer active actuator degrees of freedom than the complete square array, and influence functions alter the usable boundary. Moreover one real wave phase deformable mirror produces conjugately related corrections at opposite speckles: arbitrary complex-field correction of both independent wave amplitude and wave phase errors generally needs additional control, or a restricted half-plane. The square therefore describes ideal spatial-frequency access, rather than a guarantee that every intensity element inside it can be independently set to zero. The Fourier-domain square control region is derived in Bordé and Traub's speckle-nulling analysis.

Spatial frequency is the number of oscillation cycles per unit length. A ripple with spatial period $\Lambda$ has frequency $\nu =1/\Lambda$; its angular wavenumber is $k=2\pi \nu$. In two dimensions the spatial-frequency vector specifies the orientation and period of a plane-wave pattern.