= Pulse-contrast denominator acceptance rule
A \b[pulse-contrast denominator acceptance rule] decides which shots may be normalized by their measured central signal. For background-corrected side signal $S$ and central signal $C$, consider
$$
A=\{C\geq c_{\min}\},\qquad c_{\min}>0,\qquad R=S/C.
$$
The rule prevents normalization by an arbitrarily small accepted denominator. It also changes the estimand: averaging accepted ratios estimates $\mathbb E[R\mid A]$, a <conditional expectation>. An unconditional <expected value> is a different target and need not exist. The <pulse-contrast ratio denominator noise> explains why even the unfiltered ratio mean can differ from the ratio of mean signals.
The rule provides a precise stability guarantee when its moment assumptions hold. Interpret the indicator-weighted integrand below as zero outside $A$. If $p_A=\mathbb P(A)>0$ and $\mathbb E S^2<\infty$, then
$$
\mathbb E[R^2\mid A]
=\frac{\mathbb E[\mathbf1_A S^2/C^2]}{p_A}
\leq\frac{\mathbb E S^2}{p_Ac_{\min}^2}<\infty.
$$
This proves <finite ratio moments under a positive denominator cutoff>. The accepted ratio has finite <variance> and finite first absolute moment, the latter by the <Cauchy-Schwarz inequality>. No independence of $S$ and $C$ was used. The bound concerns mathematical stability, not whether accepted shots represent the full pulse population.
For an original brightness-selection example, suppose half the shots have $(C,S)=(2,0.02)$ and half have $(10,0.02)$. The two ratios are $0.01$ and $0.002$, so their <expected value> over all shots is $0.006$. A cutoff $c_{\min}=5$ retains only the bright group, giving accepted <conditional expectation> $0.002$. Meanwhile the ratio of overall mean signals is $0.02/6\approx0.003333$. These three quantities answer different questions. Physical dependence between brightness and temporal shape can make the choice consequential even without readout noise.
<Selection bias from a noisy denominator threshold> arises even at fixed true brightness. Consider another original model with constant $S=s>0$ and measured $C\sim N(\mu,\sigma^2)$, where $\mu>0$. Write $z=(c_{\min}-\mu)/\sigma$, with $\phi$ the <standard normal density> and $\Phi$ the <standard normal distribution function>. The accepted central measurements have a <truncated normal distribution>, with
$$
p_A=1-\Phi(z),\qquad
\mathbb E[C\mid A]=\mu+\sigma\frac{\phi(z)}{1-\Phi(z)}.
$$
To derive the second identity, standardize $C$ and use $\phi'(x)=-x\phi(x)$, so $\int_z^\infty x\phi(x)\,dx=\phi(z)$. The density-to-tail quotient is an <inverse Mills ratio>.
Choose $\mu=5$, $\sigma=1$, $c_{\min}=6$ and $s=0.05$ in arbitrary signal units. The true-signal ratio is $s/\mu=0.01$. Only $15.87\%$ of measured shots pass, and the <conditional expectation> of their measured denominator is $6.5251$. Since $c\mapsto1/c$ is a <convex function> for $c>0$, <Jensen's inequality> and the cutoff imply
$$
\frac{s}{\mathbb E[C\mid A]}\leq\mathbb E[s/C\mid A]\leq\frac{s}{c_{\min}},
\qquad
\boxed{0.0076627\ \lesssim\ \mathbb E[R\mid A]\ \leq\ 1/120<0.01.}
$$
The accepted ratios are necessarily biased downward relative to this model's true-signal ratio. The cutoff selects upward central-readout errors; increasing the accepted-shot count does not remove that conditioning.
\Image[/pulse-denominator-threshold-selection.png]
{title=A central-signal cutoff selects upward readout fluctuations and lowers the accepted normalized ratio}
{description=Original Gaussian readout model with true central signal 5, standard deviation 1, cutoff 6 and constant side signal 0.05. Shading marks the accepted tail. The ratio panel shows pointwise accepted ratios and bounds on their conditional mean; it is not measured correlator data.}
In the historical single-shot correlator, nonlinear mixing encoded delays spatially and an imaging lens recorded the third-harmonic signal on a filtered <CCD> detector.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[Printed pp. 194–195, Description and Figure 2]{ref} Selecting a trace by its central pixels conditions subsequent normalization on the same recorded data. Preserve the threshold, background treatment, number attempted and number accepted alongside the <shot-resolved pulse-contrast estimator>. For independent shots sharing the same acceptance probability, the accepted count has a <binomial distribution>, carrying information that disappears if only normalized accepted traces are retained.
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