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A 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 $C$, considerThe 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 $\mathbb{E}{S}^2<\infty$, thenThis 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 $0.02/6\approx 0.003333$. 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 $\varphi$ the standard normal density and $\Phi$ the standard normal distribution function. The accepted central measurements have a truncated normal distribution, withTo derive the second identity, standardize $C$ and use ${\varphi }^{\prime }(x)=-x\varphi (x)$, so ${\int }_z^{\infty }x\varphi (x)dx=\varphi (z)$. 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 $c\mapsto 1/c$ is a convex function for $c>0$, Jensen's inequality and the cutoff implyThe 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.

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.^[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.

A shot-resolved pulse-contrast estimator normalizes each individual optical shot before combining its contrast with other shots. For a specified side-delay signal $S_j$ and positive central signal $C_j$, both background-corrected, defineThe estimand is the mean of the individual shot ratios under the stated sampling and acceptance rule. It differs from normalizing an accumulated trace, ${\hat{R}}_{\mathrm{sum}}={\sum }_j{S}_j/{\sum }_j{C}_j$, which has third-order correlation brightness weighting.

An original three-shot example uses central signals $(1,2,8)$ and side signals $(0.01,0.04,0.80)$. The individual ratios are $(0.01,0.02,0.10)$, with equal-shot mean about 0.0433. Normalizing the sums gives $0.85/11\approx 0.0773$. The latter gives the brightest shot most of the influence. Both are valid summaries of the chosen data, but answer different questions.

Single-shot acquisition matters when the profile fluctuates. The primary third-order correlator maps delay into a spatially resolved signal and records it on a CCD, using second-harmonic generation and sum-frequency generation to form the gate.^[ref] A scanned trace assembled from different shots does not by itself provide all the $r_j$ for a common ensemble.

Normalization introduces pulse-contrast ratio denominator noise. First-order uncertainty propagation for one ratio $r=S/C$ givesShared baseline estimates can correlate the numerator and denominator, so their covariance cannot automatically be set to zero.^[ref]

For stipulated $S=0.1$, $C=10$, independent $u(S)=0.02$ and $u(C)=1$, the local approximation gives $r=0.01$ and $u(r)\approx 0.00224$. Near a zero or poorly resolved denominator the linear approximation and even a finite mean ratio may fail. A central-signal acceptance threshold changes the selected ensemble and must be reported.

For independent shots with a well-behaved ratio distribution, sampling uncertainty in the equal-shot mean can be estimated from the observed ratio variation. Consecutive shots may instead share drift, calibration errors or correlated pulse fluctuations. Keep the acquisition timing, background treatment, saturation checks and rejected-shot count with the estimate; averaging cannot repair an undefined denominator or an unspecified selection rule.