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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 338 / 2 / b / iv / Solution

{title2=$\theta_{\max}=N\lambda/(2D)$}

With $N$ actuators across diameter $D$, the projected pitch is approximately $D/N$. The <Nyquist spatial frequency> is $N/(2D)$, so the largest correctable angular speckle displacement along an actuator row is $N\lambda/(2D)$. A square actuator array gives a square ideal spatial-frequency region. See the https://arxiv.org/abs/astro-ph/0510597[Bordé–Traub wavefront-control derivation].

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 Incoming links: Deformable-mirror control radius