Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 338 / 3

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.

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 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.

An optical pupil ripple of spatial period $\Lambda$ diffracts starlight into two sidebands displaced by $\theta \simeq \lambda /\Lambda$ along the ripple's spatial frequency vector. Substituting the axial Nyquist period gives the deformable-mirror control radiusThe square actuator lattice has component cutoffs $|u_x|,|u_y|\le N/2$, hence an ideal square corrected region with this half-width in each photodetector direction. Its diagonal reaches $\sqrt{2}$ times the axial radius; an isotropic conservative restriction is the inscribed disk. Using $\lambda /D$ as the scale of a spatial resolution element, the square is about $N$ elements on a side and $N^2$ in area; the disk has about $\pi N^2/4$ elements. These are angular area counts, not the quadrant coefficient count in part (ii).

A circular illuminated optical pupil has fewer active actuator degrees of freedom than the complete square array, and influence functions alter the usable boundary. Moreover one real wave phase deformable mirror produces conjugately related corrections at opposite speckles: arbitrary complex-field correction of both independent wave amplitude and wave phase errors generally needs additional control, or a restricted half-plane. The square therefore describes ideal spatial-frequency access, rather than a guarantee that every intensity element inside it can be independently set to zero. The Fourier-domain square control region is derived in Bordé and Traub's speckle-nulling analysis.

Use the ideal detection convention of one collected electron per detected photon; otherwise replace photon totals by detected electron totals using the quantum efficiency. Let $A$ be the summed measurement on the star patch and $B^{\prime }$ the independent background-only patch. After additive offsets are removed, their means are $Q+B$ and $B$. Independent Poisson distribution counts have variance equal to their mean, and a single read of $n$ detector pixels contributes $n{r}^2$ to each patch's variance. ThereforeThe two factors of two arise from the independently measured background and the second patch's read noise, not from doubling the stellar signal. The formula assumes equal background means, independent detector pixels and patch measurements, one read per patch, and negligible dark current or other noise. Several exposure reads or correlated detector pixels require their own variance and covariance terms.

With constant detected rates, $Q=R_{Q}T$, $B=R_{B}T$. Squaring the previous signal-to-noise ratio in photon counting givesFor $R_Q>0$, nonnegative background and target $Z>0$, choose the positive root:The other root is nonpositive. With no read noise, this reduces to $T=Z^2(R_Q+2R_B)/R_Q^2$; in a read-dominated short exposure it approaches $Z\sqrt{2n}r/R_Q$. If $R_Q=0$, no positive target stellar signal-to-noise is attainable. A zero target can be assigned zero exposure, with its limiting interpretation when all noise terms also vanish. Real quantum efficiency or multiple reads modifies the rates or read-variance term before solving the quadratic.

Let $k=m/2$. After offset subtraction, each summed image has mean $N$ analogue-to-digital units per detector pixel. For a constant detector conversion gain $g$ and independent Poisson distribution detected counts, $\mathrm{Var}(A)=\mathrm{Var}(B)=N/g$: summing $k$ independent frames adds both their means and variances. Here $N$ is the mean of the sum, not one frame.

Write $A=N+a$, $B=N+b$. At high counts, the delta method gives $A/B\simeq 1+a/N-b/N$. Independence therefore givesThus the formula is a high-signal gain estimation from flat-field ratios, not an exact identity for a ratio of random variables. A Poisson distribution denominator can even be zero at low counts. The exposures must be linear and unsaturated, offsets removed, and read noise negligible. Being close to saturation increases the signal but does not establish linearity; clipping or charge-induced correlations biases a variance estimate. A large ensemble of suitable detector pixels supplies the measured standard deviation. In normal photodetector terminology $g$ is collected electrons per ADU, consistent with detected photons here only under the stated unit-yield convention; see the instrument gain definition.

A stable multiplicative sensitivity factor $s_i$ at detector pixel $i$ changes both means to $N_i=s_i{N}_0$. The noise-free ratio is still one, so flat-field correction is not required merely to remove that fixed pattern from the mean ratio. This is the principal advantage of using two matched flats rather than the variance of a single image.

However, the photon shot noise is not cancelled. At high counts the conditional ratio variance is $2/(g{N}_i)$. Pooling detector pixels gives $E^2\simeq (2/g)⟨1/N_i⟩$, where brackets denote a spatial average. If the formula uses $\overline{N}=⟨N_i⟩$, it returnsThe inequality follows from the Jensen inequality for $1/x$ on positive signals. For small fractional response variation, the relative bias is approximately minus its squared coefficient of variation. A locally nearly uniform region or a fit using detector pixel-dependent variances avoids this bias from mixing the harmonic mean with the arithmetic mean. Stable response cancellation also presumes the conversion gain itself is uniform; gain variations, offset errors or changing sensitivity do not obey that simple cancellation. The fixed response pattern cancels in the ratio mean; unequal shot-noise levels can still bias a globally pooled gain estimate.

For $k=m/2$ independent reads in each sum, read noise contributes $k{r}^2/g^2$ in squared ADU units. HenceUsing the shot-noise-only formula therefore givesThe condition for negligible contamination is $gN\gg k{r}^2$, equivalently a large number of collected electrons per individual frame compared with $r^2$. Summing more identical-level frames does not change that ratio because both $N$ and the read variance scale with $k$. Measure read noise from paired bias images, or fit mean versus noise variance across several illumination levels: the shot-noise slope determines detector gain, while the read term gives an intercept. If $r$ is known, the corrected positive solution satisfies $E^2{N}^2{g}^2-2Ng-2k{r}^2=0$. Offset subtraction removes the mean bias but not the read variance. Correlated read errors need a different covariance calculation.

If spatial illumination is stable between the two sums, it acts just like the sensitivity factor: the mean ratio remains one but each detector pixel has a different shot-noise variance. A whole-image standard deviation is disproportionately influenced by faint detector pixels, so substituting a global arithmetic mean signal generally underestimates detector conversion gain. If the illumination pattern differs between the sums, their noise-free ratio is not constant; its structure adds apparent variance and causes a further systematic error.

Use locally uniform signal bins or subregions, remove offsets, mask defective or saturated detector pixels, and propagate the local shot and read variances. In the stable-pattern case a useful alternative to pooling raw ratios is the normalized differenceAt high signal and negligible read noise, treating the denominator at its mean gives $\mathrm{Var}(U_i)\simeq (2N_i/g)/(2N_i)=1/g$, independent of the illumination. Thus a variance fit or $g\simeq 1/\mathrm{Var}(U)$ can use a spatially varying flat without the arithmetic/harmonic mismatch. Measured noisy denominators cause higher-order corrections, so independent or smoothed signal estimates can be preferable. Fit and subtract a genuine changing illumination pattern before noise analysis, using sufficient smoothing not to fit away the random noise. Arbitrary flat-field normalization without propagating its noise does not by itself recover the original gain.

If temporal variation is spatially uniform, summing all frames in each group still gives independent Poisson distribution totals, but their means may now be $N_A,N_B$ with ratio $c=N_A/N_B\ne 1$. At high counts and negligible read noise, direct propagation givesAfter dividing the ratio image by its measured mean $c$, the appropriate estimator isThe original formula is recovered only for $N_A=N_B=N$. Uniform brightness drift does not automatically create a spatial pattern or extra ratio scatter beyond these changed noise levels; its leading problem is unequal group normalization. Interleave the groups in time or pair exposures of comparable flux, measure their separate means, and propagate shot and read noise when rescaling. If the spatial pattern or photodetector sensitivity also evolves, the ratio develops systematic structure that must be modeled or those frames rejected. A normalization inferred from finitely many detector pixels introduces a small shared correlation, which vanishes in the large-pixel limit and can be included in a precise uncertainty analysis.