= Solution
A <wavefront error> ripple of period $\Lambda$ generates a pair of speckles at angular displacement $\lambda/\Lambda$. Combining this with $\Lambda_{\min}=2D/N$ gives the <deformable-mirror control radius> along an actuator row or column:
$$
\boxed{\theta_{\max}=\frac{N\lambda}{2D}.}
$$
A square actuator lattice has a square ideal frequency region: $|\theta_x|,|\theta_y|\leq N\lambda/(2D)$. Its full width is $N\lambda/D$, approximately $N$ <diffraction-limited resolution> elements per side, or $N^2$ in area. The circular subset within the row-direction radius contains approximately $\pi N^2/4$ such elements. The primary aperture shapes each speckle's <point spread function>; it does not turn the square sampling limit into a circular one.
This full square describes phase-error control. Simultaneous amplitude and phase correction with a single pupil-plane <deformable mirror> generally requires restricting the dark region to a half-plane; it is a different constraint from the sampling bandwidth.
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