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Adaptive optics measures changing wavefront errors and corrects them using a deformable mirror. A wavefront sensor supplies measurements to a fast feedback controller; a guide star provides the reference. It improves astronomical seeing toward diffraction-limited resolution.

Samples spaced by pitch $p$ can distinguish spatial frequencies strictly below $1/(2p)$ cycles per unit length. Thus the shortest limiting wavelength is $2p$; exactly at the limit a sinusoid's phase can make all samples vanish. A practical deformable mirror has additional limits from actuator influence functions and finite pupil geometry.

The main optical path is telescope to deformable mirror to beam splitter to science camera. The splitter also directs reference light to a wavefront sensor; a controller reconstructs the wavefront error and feeds mirror commands back to the deformable mirror. The mirror is normally conjugate to a pupil so its actuators address the corresponding pupil phase. A separate steering mirror can handle overall image motion.

Adaptive optics corrects rapidly changing atmospheric wavefront errors to improve angular resolution and image concentration. A wavefront sensor estimates those errors and a feedback controller commands a deformable mirror to oppose them, aiming toward the diffraction limit of a telescope rather than the uncorrected astronomical seeing limit.

For a large aperture in the standard turbulence model, astronomical seeing is approximately $s=0.98\lambda /r_0$ radians, with Fried parameter $r_0$. Maintaining comparable correction requires a deformable mirror actuator pitch of order $r_0$, so the number of actuators across a diameter is proportional to $D/r_0$ and the total illuminated actuator count is proportional to $(D/r_0{)}^2$. ThereforeLonger wavelengths require fewer actuators and have slightly smaller uncorrected atmospheric angular blur, even though the telescope's diffraction-limited resolution scale $\lambda /D$ increases with wavelength.

The projected actuator pitch is $p=D/N$ in the question's sampling convention. A sinusoidal wavefront error needs at least two samples per spatial period, by the Nyquist–Shannon sampling theorem. Thus the shortest limiting correctable scale at the primary isThe Nyquist spatial frequency is $N/(2D)$ cycles per unit length. Exactly at that limiting frequency some phases are poorly sampled; actual deformable mirror performance also depends on actuator influence functions, so this is an ideal bandwidth limit.

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:A square actuator lattice has a square ideal frequency region: $|{\theta }_x|,|{\theta }_y|\le N\lambda /(2D)$. 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.

Atmospheric scintillation is the fluctuation of received stellar intensity produced by propagation through atmospheric turbulence. A phase-only deformable mirror does not generally remove these amplitude fluctuations.