= Third-order correlation brightness weighting
\b[Third-order correlation brightness weighting] is the way bright shots contribute disproportionately to an averaged nonlinear correlation trace. A normalized average trace is a <weighted mean> of shot-normalized shapes, with weights set by their central correlation signals rather than by how often the shots occur.
Let a nonnegative, background-corrected third-order trace on shot $j$ be $A_j(\tau)$. Define its central signal $B_j=A_j(0)>0$ and normalized shape $r_j(\tau)=A_j(\tau)/B_j$. Then the experimentally distinct averages are
$$
R_{\rm trace}(\tau)=\frac{\sum_j A_j(\tau)}{\sum_j A_j(0)}
=\sum_j w_jr_j(\tau),\qquad
w_j=\frac{B_j}{\sum_kB_k},
$$
and the equal-shot mean $R_{\rm shot}(\tau)=N^{-1}\sum_jr_j(\tau)$. The nonnegative $w_j$ sum to one, but are equal only when the $B_j$ are equal. Here “central” denotes the specified zero-delay signal, not whichever noisy sample happens to be the largest.
For identical pulse shapes multiplied by intensity factors $K_j$, the ideal integral $A_j(\tau)=\int I_j^2(t)I_j(t+\tau)\,dt$ scales as $K_j^3$. The primary correlator implements this three-intensity-factor gate using <second-harmonic generation> followed by <sum-frequency generation>.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printed p. 194, defining integral and nonlinear optical arrangement]{ref} If shapes change, $B_j$ can depend on duration and overlap as well as brightness, so pulse energy alone need not determine the weight.
An original two-class example has equally frequent classes with $(B,r)=(1,0.01)$ and $(8,0.10)$ at one chosen delay. Their equal-shot mean ratio is $0.055$, but the normalized summed trace gives
$$
R_{\rm trace}=\frac{1(0.01)+8(0.10)}{1+8}=0.09.
$$
The bright class supplies eight ninths of the central signal despite representing only half the shots. Neither result is an arithmetic mistake: they estimate different summaries of the stipulated ensemble.
For an ensemble with finite moments and $\mathbb E[B]>0$, the distinction can be expressed exactly through <covariance>:
$$
\frac{\mathbb E[Br]}{\mathbb E[B]}-\mathbb E[r]
=\frac{\operatorname{cov}(B,r)}{\mathbb E[B]}.
$$
Unchanged normalized shape makes the covariance zero, even with large brightness fluctuations. Shape–brightness dependence can make it nonzero; <cubic intensity-noise bias in pulse correlation> alone does not prove a contrast change.
A <shot-resolved pulse-contrast estimator> needs synchronized intensity information and a stated selection rule. Thresholding weak shots or dividing by a noisy central signal can introduce further selection and ratio errors. Report whether normalization occurred before or after averaging, retain <detector baseline subtraction>, and test readout linearity. A smooth averaged trace is not a representative single shot unless those assumptions are justified.
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