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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.

Anisoplanatism is the loss of correlation between wavefront errors along different viewing directions. Correcting a guide star therefore leaves a larger residual for a sufficiently separated target.

An ordinary passive primary must keep its optical figure while the direction of gravity relative to the mirror changes as the telescope tracks. A thin unsupported disk bends enough to introduce wavefront errors, degrading the point spread function. Increasing thickness strongly raises its resistance to bending.

For an isotropic elastic plate of thickness $t$, Young's modulus $E_Y$ and Poisson's ratio $\nu$, the flexural rigidity is $\mathcal{B}=E_Y{t}^3/[12(1-{\nu }^2)]$. Under its own weight the load per area scales as $\rho g_{0}t$. With support spans of order the diameter $D$, plate bending therefore givesNumerical coefficients depend on the support and boundary conditions. A larger $t$ greatly reduces gravitational sag, even though it adds weight. Near normal incidence a mirror displacement $h$ changes optical path length by approximately $2h$, so surface errors must be much smaller than the observing wavelength. Thickness provides passive stiffness needed to preserve the mirror figure.

Both methods reduce structured stellar residuals, rather than removing all noise. In angular differential imaging, changing atmospheric turbulence, imperfect adaptive optics, flexure and thermal drift change the point spread function between frames. The reference then fails to represent the instantaneous stellar field. Small sky rotation makes close companions contaminate their own reference, causing differential-imaging self-subtraction. Extended disks are especially susceptible; subtraction can alter shape as well as total flux. More images help independent photon shot noise, but do not necessarily average away correlated residuals.

Simultaneous spectral differential imaging avoids the time delay, but different channels have non-common-path wavefront errors. Chromatic optical aberrations, wavelength-dependent amplitude errors and out-of-pupil propagation prevent a perfect radial rescaling of speckle patterns. Filter throughput, detector calibration, image registration and atmospheric dispersion also leave subtraction residuals. Nearby bands align the stellar field better but give less positional diversity; wider separation gives more displacement but larger chromatic mismatch. A smooth-spectrum companion, or one too close for appreciable rescaled displacement, can undergo severe differential-imaging self-subtraction.

Artificial-companion injection through the complete processing pipeline and forward modelling can calibrate lost throughput and photometric or astrometric biases. They do not guarantee that every correlated residual is a real source. Independent epochs or spectral evidence remain valuable.

The Strehl ratio isusing the same total flux, pupil, and wavelength for both images. A value near one means that little light has been redistributed from the ideal central peak by wavefront errors. For small residual phase variance ${\sigma }_{\varphi }^2$, the Maréchal approximation is $S\simeq e^{-{\sigma }_{\varphi }^2}$, with ${\sigma }_{\varphi }=2\pi h_{\mathrm{rms}}/\lambda$ for the residual optical-path root mean square error.

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.

Adaptive-optics sky coverage is limited by the need for a guide star bright enough to measure the wavefront error on the atmospheric evolution time. A faint guide gives noisy measurements; a guide too far from the target samples a different turbulent column. That angular mismatch is anisoplanatism, with useful separation characterized by the isoplanatic angle. Thus a conventional single-guide system cannot provide equally good correction at every target position.

A laser guide star supplies an artificial bright reference and improves coverage. However, its finite-distance beam does not sample the full stellar turbulence column, and conventional laser systems need a natural reference for absolute image motion. The required reference brightness, angular separation, and desired correction quality therefore still constrain observations.

The supplied speckle contrast of a sinusoidal wavefront error gives $h_0=\lambda \sqrt{C}/\pi$. Each sinusoid has root mean square error $h_0/\sqrt{2}$. For orthogonal modes, or for uncorrelated phases so that cross terms average to zero, the total mean square adds:Therefore the assumed equal-contrast modal model givesIf the individual contrasts are not equal, the corresponding formula is $h_{\mathrm{rms}}^2={\lambda }^2{\sum }_i{C}_i/(2{\pi }^2)$. The error here is optical-path wavefront error; for a near-normal reflecting mirror, physical surface error is half the optical-path error.

Count independent sinusoidal ripples by their spatial Fourier series indices. A mode with $n$ cycles across $D$ has $\Lambda =D/n$, so the given half-wave index is $j=2n$. The Nyquist spatial frequency permits $n\le N/2$. In two dimensions, a circular cutoff therefore contains approximately $\pi (N/2{)}^2$ integer wavevectors.

For a real wavefront error, the wavevectors $\mathbf{n}$ and $-\mathbf{n}$ describe the same ripple with conjugate coefficients, so count each pair once. The continuum mode-counting approximation givesThis counts one ripple with amplitude and phase per conjugate pair, not one real coefficient. Exact finite-grid counts are integers and have boundary corrections; the formula is the circular area estimate implicit in the question, with piston excluded.

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.

A speckle pattern is a granular intensity pattern from coherent wave interference. In stellar imaging, residual wavefront errors produce speckles around the central point spread function.

The Strehl ratio compares the peak of an observed point spread function with the ideal diffraction-limited system peak for the same pupil and total flux. It measures how strongly wavefront errors redistribute light away from the central peak.

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.

The real Zernike polynomials form an orthogonal polynomial basis on the unit Euclidean disk, with radial factor $R_n^m(\rho )$ and angular factor $\cos m\varphi$ or $\sin m\varphi$. Here $0\le m\le n$ and $n-m$ is even. They decompose wavefront errors into modes useful for optical correction. Normalizations differ, so coefficients should specify the chosen convention.