Adaptive optics

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.

A deformable mirror changes its optical surface using an array of actuators. For reflection near normal incidence, surface displacement changes optical path length by twice that displacement.

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.

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 Bordé–Traub wavefront-control derivation.

A wavefront sensor measures information from which a wavefront error can be reconstructed. For example, a Shack–Hartmann wavefront sensor measures local wavefront slopes using lenslets.

A guide star provides photons for a wavefront sensor. Its brightness and angular separation from the target limit adaptive optics sky coverage.

A laser guide star is an artificial atmospheric beacon for adaptive optics. It improves sky coverage but samples a finite-distance cone rather than the full incoming stellar beam. Conventional systems still require a natural reference for absolute image motion.

Adaptive-optics sky coverage is the fraction of targets for which a sufficiently bright nearby guide star allows useful correction. Anisoplanatism limits the offset, and photon statistics limit the sensing speed. A laser guide star improves coverage but does not remove all low-order and finite-distance limitations.