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

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.

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.

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.

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.

Astronomical seeing is the atmospheric angular blurring of a point-source image, usually reported in arcseconds. Operationally it is commonly characterized by the full width at half maximum of a long-exposure stellar point spread function. The full observed width can also contain telescope and instrumental contributions; the atmospheric seeing is the turbulence contribution.

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.

For the unnormalized Zernike polynomial with $n=4,m=0$, the three radial terms areSince $\cos 0\varphi =1$,Along a central chord with signed coordinate $s\in [-1,1]$, replace $\rho$ by $|s|$. The profile is even, equals one at the center and both edges, and has minima $-1/2$ at $s=\pm 1/\sqrt{2}$. This Zernike spherical mode corresponds to primary spherical aberration, balanced by defocus and piston terms rather than simply the raw quartic Seidel term.

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.

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 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.

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.