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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.

The Rayleigh criterion gives a separation in the sensor plane $\delta =f\Delta \theta =2.44\mu \mathrm{m}$, using the $5\mathrm{mm}$ image distance. Counting one square pixel per such spatial resolution element givesThis is an estimate of the number of optically distinguishable elements. To sample all the spatial information, specify a signal sampling criterion as well. For incoherent imaging through an ideal circular aperture, the optical transfer function has cutoff spatial frequency ${\nu }_c=D/(\lambda f)$. The Nyquist–Shannon sampling theorem requires a pixel pitchA common intermediate rule, two pixels per Rayleigh separation, gives about $8.1$ million pixels; it is a heuristic and does not quite sample the full ideal optical transfer function bandwidth. Thus about $2$ million is the elementary spatial resolution element count, while $12$ million is the ideal monochrome Nyquist–Shannon sampling theorem answer to “full use.” Color-filter layout and finite pixel area would impose further practical conditions.

A spatial resolution element is one independently resolved angular or spatial scale under a specified resolution criterion. For an ideal telescope of diameter $D$, its angular scale is of order $\lambda /D$ and its angular area of order $(\lambda /D{)}^2$. A photodetector detector pixel or spaxel can be smaller than a resolution element without adding independent spatial information.