Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 338 / 3

First include the mean-gain calculation preceding the numbered clauses. At each electron-multiplying CCD stage one electron contributes an expected $1+p$ electrons to the next stage. If $N_j$ is the charge after stage $j$, conditional expectation gives $\mathbb{E}[N_{j+1}\mid N_j]=(1+p)N_j$. Iteration from $n$ input electrons yieldsFor small $p$, $g\simeq e^{rp}$ if corrections of order $r{p}^2$ are negligible.

In the high-gain approximation the output from one input electron has an exponential distribution of scale $g$. Independent input electrons produce independent cascades, and their charges add. By convolution of independent random variables, assuming the formula for $n$ inputs,Together with the single-electron base case this provesIt is a gamma distribution with shape $n$ and scale $g$, so it is normalized and has expected value $ng$ and variance $n{g}^2$. The formula is a continuous approximation to discrete output charge, not an exact integer-valued branching distribution; for $n=0$ there is a point mass at zero instead.

Let $N$ be the input photoelectron count and $\mu =\eta Q$ its mean, with quantum efficiency $\eta$ and mean incident photon count $Q$. Independent arrivals give a Poisson distribution, so $\mathrm{Var}N=\mathbb{E}N=\mu$. Conditional on $N$, the preceding gamma distribution gives $\mathbb{E}[X\mid N]=gN$ and $\mathrm{Var}(X\mid N)=g^{2}N$. The law of total variance therefore yieldsOne contribution is the ordinary photon shot noise; the other is multiplication excess noise. Thus, neglecting read noise and backgrounds,This is an excess-noise factor $\sqrt{2}$ in analog operation: actual photon conversion efficiency has not been halved. At low occupancy, thresholding each pixel as zero or one event can avoid most multiplication noise, but high arrival rates produce coincident events that cannot be counted separately. That is why the high-rate result concerns charge measurement rather than ideal binary photon counting.

Treat $Q$ and $B$ as mean detected counts, or as photon counts with unit quantum efficiency. Let the independent patch measurements be $X\sim \mathrm{Poisson}(Q+B)$ and $Y\sim \mathrm{Poisson}(fB)$. For the specified unweighted background subtraction $\hat{Q}=X-Y$,The first error is a fixed bias of an estimator; the last expression is the sum of independent photon shot noise variances. Since $Q,B$ are already patch totals, no extra factor of the pixel count $n$ is needed, and read noise is neglected.

To obtain the systematic-error ceiling requested in the following clause, define the accuracy measure $Z$ using total root-mean-square error relative to the true source count. By the bias-variance decomposition of mean squared error,For $f\simeq 1$, the shot-noise term is approximately $Q+2B$, but the mismatch term must be retained.

There is a terminology qualification: the usual variance-based signal-to-noise ratio in photon counting is $Q/\sqrt{Q+(1+f)B}$ and does not include a fixed bias as noise. The printed next-part limit requires the root-mean-square accuracy convention above. A deterministic background mismatch contributes $b^2$ to mean squared error, not to the statistical variance.

Put $Q=R_{Q}T$ and $B=R_{B}T$ in the root-mean-square accuracy measure:For a fixed uncorrected $f\ne 1$ and $R_B>0$, the squared bias eventually dominates the shot-noise terms. ThereforeThe PDF omits the absolute value. Its expression is the positive accuracy ratio only if $f<1$; for $f>1$ the residual background changes sign, but an error magnitude and a signal-to-noise ratio in photon counting remain nonnegative. The printed signed formula can instead be read as source divided by signed bias.

For exact background matching $f=1$, there is no systematic-error signal-to-noise ceiling: $Z=R_Q\sqrt{T/(R_Q+2R_B)}$ grows without bound in this idealized model. With $R_B=0$ the same conclusion holds. Limits in which $f$ itself approaches one with exposure time are different from the fixed-mismatch limit used here.

Increasing exposure reduces fractional photon shot noise as $T^{-1/2}$, but a fixed background mismatch grows in proportion to the signal. The fractional systematic error is $|1-f|R_B/R_Q$, independent of exposure. The systematic-error signal-to-noise ceiling can therefore be poor for a faint source in a bright sky even when random fluctuations are tiny.

Better background subtraction requires matching sky location and time, correcting detector response, dithering or modelling spatial background variations. If $f$ were known exactly, one could instead use $X-Y/f$, which is unbiased and has variance $Q+B+B/f$. The ceiling arises from an uncorrected or unknown mismatch in the prescribed subtraction, not an unavoidable property of measuring two patches.

To reach an accuracy ratio $Z_{\ast }$ at all, a necessary condition in the fixed-mismatch model isEquality allows only an asymptotic approach; finite exposure adds positive random error. More exposure improves precision, but only better calibration removes the fixed background bias.

An H2RG detector is a hybrid infrared detector: a photosensitive array is bonded, pixel by pixel through indium contacts, to a separate silicon readout circuit. In infrared use the absorber is normally mercury cadmium telluride, whose composition sets the absorption cutoff. The H2RG readout architecture can also be used with other absorbing materials, so the readout name does not fix the detector’s wavelength response. The manufacturer’s H2RG specification gives a $2048\times 2048$ grid and $18\mu \mathrm{m}$ pixel pitch, together with reference-pixel and guide-window capabilities.

Each illuminated pixel contains a photodiode connected to its own readout node. Absorbed photons generate carriers; their accumulated charge changes the node voltage, with $|\Delta V|=|\Delta Q|/C$ for node capacitance $C$. A reset establishes the baseline; row and column addressing and multiplexed output amplifiers sample the pixel voltage. This is not the serial transfer of charge packets through adjacent pixels used by a charge-coupled device.

Nondestructive detector readout allows repeated measurements during an integration. Correlated double sampling removes the reset baseline, Fowler sampling differences groups of reads at the beginning and end, and up-the-ramp sampling estimates the signal from its slope and can flag cosmic-ray steps or saturation. The “R” denotes detector reference pixels, which monitor electronic offsets; “G” denotes a programmable guide window that can be read rapidly between full-array operations. Reference pixels do not measure sky background. Cooling suppresses dark current and thermal background. Read noise, nonlinearity, persistence and interpixel coupling still require calibration.