= Charge-transfer efficiency
{title2=$\eta_{\rm tr}$}
The <charge-transfer efficiency> is the fraction of a packet delivered into its intended next site or time slot in one transfer, under stated conditions. It is distinct from <quantum efficiency>, which concerns conversion of incident <photons> into collected <charge carriers>. Define the complementary inefficiency as $\epsilon=1-\eta_{\rm tr}$.
In an original model with constant $0\leq\eta_{\rm tr}\leq1$, let $N_j$ denote the mean number of original packet <Electrons> still delivered on time after $j$ transfers. Assuming no reinjection into that packet,
$$
N_{j+1}=\eta_{\rm tr}N_j,\qquad N_{\rm out}=N_{\rm in}\eta_{\rm tr}^{n}.
$$
Repeated substitution proves the result; for different efficiencies the fraction is $\prod_{j=1}^{n}\eta_j$. For $\epsilon\ll1$, a <Taylor expansion> of $\ln(1-\epsilon)$ gives $\eta_{\rm tr}^n\simeq e^{-n\epsilon}$, while the simpler $1-n\epsilon$ requires $n\epsilon\ll1$.
Choose $N_{\rm in}=10000$ and $n=2000$. For $\eta_{\rm tr}=0.9999$, the expected on-time output is $8187.23$ <Electrons>; for $\eta_{\rm tr}=0.99999$, it is $9801.99$. These are model means, so fractional <Electron> counts do not describe an individual packet. Even a small per-transfer imperfection can imprint distance-dependent image errors.
If every original <Electron> independently stays on time with <probability> $p=\eta_{\rm tr}^n$, a fixed initial count gives a <binomial distribution> with <variance> $N_{\rm in}p(1-p)$. An initial <Poisson distribution> with mean $\mu$ instead gives an on-time <Poisson distribution> with mean and <variance> $\mu p$, by <Poisson thinning>. A deterministic attenuation formula alone does not specify noise. Real trapping can correlate packets and depends on timing, illumination and <temperature>; the independent constant-efficiency model then fails.
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