= Shot-resolved pulse-contrast estimator
A \b[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, define
$$
r_j=\frac{S_j}{C_j},\qquad
\widehat R_{\rm shot}=\frac1N\sum_{j=1}^{N}r_j.
$$
The estimand is the mean of the individual shot ratios under the stated sampling and acceptance rule. It differs from normalizing an accumulated trace, $\widehat R_{\rm sum}=\sum_jS_j/\sum_jC_j$, 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 $0.85/11\approx0.0773$. 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.<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 Fig. 2]{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 $r=S/C$ gives
$$
u^2(r)\simeq\frac{u^2(S)}{C^2}
+\frac{S^2u^2(C)}{C^4}
-\frac{2S\operatorname{cov}(S,C)}{C^3}.
$$
Shared baseline estimates can correlate the numerator and denominator, so their <covariance> cannot automatically be set to zero.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, sensitivity coefficients and correlated inputs]{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 $u(r)\approx0.00224$. 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.
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