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Common attenuation in third-order pulse correlation multiplies the entire input intensity profile by one constant factor before the correlation is formed. For the ideal signalreplacing $I$ by $kI$ gives $A_{3,k}(\tau )=k^3{A}_3(\tau )$. All unnormalized lobes scale together, so division by the central peak removes this common factor. The primary correlator paper defines the three-intensity-factor integral and implements it through second-harmonic generation and sum-frequency generation.^[ref]

The factor $k$ is an intensity transmission. An optical field-amplitude factor $a$ instead gives intensity factor $|a{|}^2$ and correlation factor $|a{|}^6$. Confusing these conventions changes the predicted power law. A spectral filter that reshapes or chirps the pulse is also not necessarily a common scalar attenuation.

An original example takes an ideal central signal 100 and a side lobe 1 in consistent detector-signal units. Halving the full input intensity gives central 12.5 and side 0.125; their ratio remains 0.01. In contrast, attenuation scaling of a third-order correlation ghost changes only the delayed replica and predicts different real-lobe and ghost powers. A common attenuator cannot perform that selective test.

A fixed background changes what normalization appears to show. Add one unremoved signal unit to both locations. Before attenuation the observed side/center ratio is $2/101\approx 0.0198$. After common intensity transmission $k=1/2$, it is $1.125/13.5=0.0833$. The apparent side feature has grown relative to the peak even though the optical profile did not change. Detector baseline subtraction and a verified linear readout are therefore part of the experiment.

Shot-to-shot intensity variation supplies another consequence. Let the common intensity multiplier be $K=1+\eta$, where $\eta$ has zero mean. ThenFor equally likely multipliers 0.9 and 1.1, the mean intensity is unchanged but the mean cubic signal is $1.03$ times the unit-intensity signal. This cubic intensity-noise bias in pulse correlation shows why cubing a mean input is not equivalent to averaging third-order signals. The calculation assumes the pulse shape is unchanged on each shot.

A controlled check records pulse intensity alongside the correlator signal, varies a broadband common attenuator, and tests cubic scaling in an unsaturated interval after background correction. Failure of that test can indicate detector nonlinearity, pulse reshaping or an uncontrolled background; it does not uniquely identify a particular fault. The ideal law applies only while the nonlinear gate and optical overlap retain their stipulated response.

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 areand the equal-shot mean $R_{\mathrm{shot}}(\tau )=N^{-1}{\sum }_j{r}_j(\tau )$. 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.^[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 givesThe 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: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.