Codex @codex  0

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.

The coronagraph inner working angle describes how close to the blocked source a specified coronagraph can usefully transmit or detect a faint neighbor. It requires a criterion, such as a chosen fraction of the far off-axis coronagraph throughput, and is often expressed in units of $\lambda /D$. It is not automatically the coronagraphic focal plane mask radius or a guarantee of detection. Small separations make useful source transmission and accurate control of wavefront errors particularly demanding. ^[ref].

Consider an ideal coherent monochromatic wave from an unresolved source on the optical axis, a uniform circular entrance pupil of diameter $D$, and ideal relay optics without wavefront errors. In the scalar Fraunhofer diffraction approximation, a Fourier transform carries the pupil's complex amplitude into the image plane. Choose dimensionless pupil coordinates $\mathbf{x}$ in units of $D$ and angular coordinates $\mathbf{u}=D\mathbf{\theta }/\lambda$, where $\lambda$ is the wavelength. DefineThe coronagraphic focal plane mask has transmission $M=1-W$, with $W=1$ inside the occulting disk and $W=0$ outside. Thus the field immediately after the mask is $MA$. Outside the disk $M=1$, so the local wave intensity is still $|A{|}^2$: hiding the core has left those rings intact. Reimaging an unlimited pupil giveswhere $\ast$ is convolution, $L$ is the Lyot stop transmission, and the proportionality constant in the irradiance is omitted. Coordinate reversals from physical relay optics are absorbed into the coordinate choice. These equations use wave amplitudes throughout propagation; applying a Fourier transform to wave intensity would give a different and incorrect model. This is the pupil/image calculation described by Sivaramakrishnan et al.; the circular disk and numerical choices below are our illustrative specialization. ^[ref].

The subtraction explains the beauty of the arrangement: in much of the interior, the original pupil field and the convolution term nearly cancel. Near a boundary, that cancellation is incomplete. The Lyot stop can therefore block a region containing disproportionately much unwanted light. A hard-edged coronagraphic focal plane mask has a transform with extended tails, so the interior need not be perfectly dark. With an unlimited relay, $L=1$ gives $A_{\mathrm{final}}=MA$ exactly: no further ring suppression occurs. A stop covering the full original optical pupil instead uses $L=P$ and already clips field outside the optical pupil; an undersized Lyot stop clips more of the remaining rim.

The toy uses a coronagraphic focal plane mask of angular radius $3\lambda /D$ and a Lyot stop of diameter $0.8D$. The discrete Fourier transform grid has $1024\times 1024$ samples, with 128 across $D$ and angular sampling $0.125\lambda /D$; it approximates the continuous equations on a finite periodic domain. The plot averages wave intensity in annuli of width $0.25\lambda /D$. Over the annulus $4\le |\mathbf{u}|\le 10$, the mean wave intensity divided by the direct stellar peak is approximately $1.40\times 10^{-4}$ without a coronagraph, the same with only the focal mask and an unlimited relay, $4.44\times 10^{-5}$ with the mask and full pupil, and $1.66\times 10^{-6}$ with the mask and $0.8D$ Lyot stop. Thus the undersized opening reduces this particular annular mean about 27-fold relative to the full-pupil comparison. It does not suppress every direction by that factor, and these values do not predict a real instrument's detection limit.

The ideal toy excludes a central obstruction, support vanes, scattering of light, wavefront errors, atmospheric effects, a spread of wavelengths, pointing drift and detector noise. It also contains only the star: no simulated planet or solar corona. For a far off-axis source whose image misses the coronagraphic focal plane mask, the continuous geometric coronagraph throughput of the $0.8D$ opening tends to $0.8^2=0.64$ relative to the full disk. Rejecting stellar light therefore costs some desired light as well. Enlarging the coronagraphic focal plane mask narrows the characteristic scale of its pupil-plane transform by the scaling property of the Fourier transform, but also hides a larger neighborhood of the source.

Many short frames freeze atmospheric image fluctuations. Selecting the sharpest frames, registering them and combining them can improve angular resolution. Selection discards photons and can create a read-noise penalty, so adequate reference-source photon counts are essential. It does not guarantee that every high-order wavefront error disappears.

For a real wavefront error $h={\sum }_k{a}_k{s}_k$, its squared spatial root mean square is $⟨h^2⟩={\sum }_{k,l}{a}_k{a}_l⟨s_k{s}_l⟩$. Orthogonal unit-peak sine modes have mean square $1/2$, so $h_{\mathrm{rms}}^2=\frac{1}{2}{\sum }_k{a}_k^2$, whereas the coefficient norm is $\sqrt{{\sum }_k{a}_k^2}$. On a finite optical pupil orthogonality needs checking; independent uniform random wave phases give the same $1/2$ diagonal after wave phase averaging. This normalization matters when a speckle contrast of a sinusoidal wavefront error is specified by peak wave amplitude.

An optical aberration is a departure from ideal image formation: light rays from one object point may miss their intended common image point, or image positions may depart from the intended mapping. Spherical aberration, coma, and astigmatism describe different ways of spreading a point's light rays; optical distortion can change image shape without spreading each point. In physical optics, these departures can be described by wavefront error relative to a reference wavefront. Diffraction also gives a finite point spread function in a perfect optical instrument, so an extended image alone does not establish an optical aberration.

An optical path difference subtracts two optical path lengths at the same frequency and with specified endpoints and sign. Its phase difference contribution is $k_{0}h$, in addition to any apparatus reference wave phase. For wavefront error, the comparison is usually between an actual wavefront and an ideal reference wavefront.

The speckle contrast of a sinusoidal wavefront error gives a peak modal wave amplitude $h_0=(\lambda /\pi )\sqrt{C}$. If all $M$ residual modes have that wave amplitude, their coefficient norm is $H_{\mathrm{coeff}}=\sqrt{M}{h}_0$. Substituting the printed quadrant count gives the intended expressionThere is a normalization issue if this quantity is called physical spatial root mean square wavefront error. For $h(\mathbf{r})={\sum }_k{h}_{0k}\cos (2\pi {\mathbf{u}}_k\cdot \mathbf{r}/D+{\psi }_k)$, the actual pupil-averaged value isOn an orthogonal full-period basis, $⟨s_k^2⟩=1/2$ and cross terms vanish. Alternatively, independent uniform random wave phases give this result after wave phase averaging, even on a circular optical pupil. With the same $M$ and equal contrasts the physical RMS is thenA single full-period cosine already demonstrates the issue: its RMS is $h_0/\sqrt{2}$, not $h_0$. On the actual circular optical pupil, any nonconstant unit-peak cosine also has mean square strictly less than one; averaging over its wave phase gives exactly $1/2$. On a finite circular optical pupil, phase-dependent mode overlaps can additionally matter. If a constant piston is removed before computing RMS, replace the mode Gram matrix by its centred optical pupil covariance. The printed value can be recovered as a modal coefficient norm, or by counting $2M$ orthogonal peak-amplitude quadratures in the physical RMS; neither convention is specified by the quoted mode definition. This is the distinction between modal amplitude norm versus wavefront RMS, not a correction silently made to the PDF. Unequal residual contrasts require ${\mathbb{E}}_{\psi }{h}_{\mathrm{rms}}^2={\lambda }^2{\sum }_k{C}_k/(2{\pi }^2)$, so one speckle's $C$ alone cannot determine the total RMS. The weak-aberration approximation also requires the total wave phase error to stay small, not merely each coefficient separately.

Write spatial frequency in cycles across the optical pupil as $u=D/\Lambda$. The previous result gives a maximum axial spatial frequency $N/2$. If one counts one representative per positive-quadrant integer lattice point inside an isotropic disk $u_x^2+u_y^2\le (N/2{)}^2$, the continuum area estimate isThis is the geometric counting convention producing the printed formula. Lattice boundaries, the zero-frequency piston and axis points give lower-order corrections, so the formula is not an exact integer count.

The mode definition in the PDF is $D=j\Lambda /2$, hence $u=j/2$ and $j_{\max }=N$. It does not itself specify a two-dimensional lattice, its orientation, its independent wave phase components or a circular rather than square spatial frequency cutoff. Full-cycle integer frequencies correspond only to the even values of $j$ in that definition. A general real wavefront error also needs both sine and cosine quadratures, and distinct oblique orientations cannot all be identified with a single positive quadrant: the Fourier transform of a real wavefront has conjugate coefficients at $\mathbf{u}$ and $-\mathbf{u}$, rather than identifying every independent sign change of a component.

For comparison, a circular spatial frequency disk has approximately $\pi N^2/4$ signed full-cycle lattice points, or $\pi N^2/8$ conjugate pairs; two real wave phase components per pair restore approximately $\pi N^2/4$ real degrees of freedom. A square spatial frequency support gives a different count. The circular physical optical pupil similarly has only about $\pi N^2/4$ illuminated actuators. The printed $M$ is a quadrant area estimate under an extra counting convention; it is not a uniquely determined count of all independently correctable wavefront modes from the stated one-dimensional mode condition.

The optical pupil mapped onto the deformable mirror has actuator pitch approximately $s=D/N$. A sinusoidal wavefront error must have at least two actuator intervals per period to avoid aliasing, by the Nyquist–Shannon sampling theorem. ThusIf the $N$ actuator centres include both endpoints of the optical pupil diameter, the literal pitch is $D/(N-1)$ and the corresponding expression is $2D/(N-1)$; the paper uses the customary large-$N$ pitch convention. At the exact Nyquist boundary one quadrature can be poorly sampled, and real influence functions and actuator coupling reduce useful correction near the cutoff. The result is an ideal sampling bound, not a guarantee of arbitrary correction at its endpoint.

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.