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

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.