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Attenuation scaling of a third-order correlation ghost tests the nonlinear origin of a side lobe by changing a real delayed replica's strength while preserving the main pulse. The diagnostic prediction differs from attenuating the entire input. The quantity attenuated must be stated as an intensity or energy ratio, rather than confused with a field-amplitude ratio.

Use the nonoverlapping equal-shape model $I(t)=g(t)+\epsilon g(t-d)$, with $d>0$ and $0<\epsilon <1$. Let an independently controlled intensity transmission $a$ act on the delayed replica only. With the main pulse fixed, the third-order intensity autocorrelation has central, real-replica and third-order pulse correlation ghost lobe weights proportional toThe common factor is the same single-pulse correlation amplitude. Thus the real lobe has logarithmic slope 1 and its ghost slope 2 versus replica transmission, before normalization. Their exact ratio is $G/S=a\epsilon$.

An original example starts with $\epsilon =0.08$. Initially the weights are $(1.000512,0.08,0.0064)$. Halving the delayed replica's intensity gives $(1.000064,0.04,0.0016)$. The real side lobe halves while its ghost quarters. If each trace is divided by its own central peak, the predictions becomeThese normalized curves have the small additional central-peak dependence; treating them as exact pure powers would omit it.

In contrast, common attenuation in third-order pulse correlation sends $I\to kI$ for the complete two-pulse input. All three factors in $\int I^2(t)I(t+\tau )dt$ acquire the same scaling, so every unnormalized lobe scales as $k^3$. Halving the entire input therefore reduces every lobe eightfold while leaving the normalized shape unchanged. That experiment cannot demonstrate selective quadratic ghost scaling.

A suitable controlled test forms a delayed replica in a separate optical path, varies that path's transmission without changing its shape or delay, then recombines it with a monitored main pulse. Retain background-corrected, unsaturated readings and verify comparable detector response at both delays. The historical experiment's inserted-etalon postpulse supplied a physical delayed reference, not this particular attenuation sweep.^[ref]

Agreement with the scaling model supports the ghost interpretation under its assumptions. It does not exclude an additional real optical prepulse, unequal replica shapes, overlapping fields or an attenuation-dependent background. Those possibilities require further independent observations.

Cubic intensity-noise bias in pulse correlation is the difference between the average of a third-order signal and the signal predicted from the average intensity when a fluctuating common intensity multiplier is cubed. Assume every shot has the same pulse shape, $I_j(t)=K_j{I}_0(t)$, with nonnegative scalar $K_j$. The ideal third-order integral givesThe three intensity factors follow from the second-harmonic generation and sum-frequency generation gate.^[ref]

Let $K=\mu +\delta$, with $\mathbb{E}[\delta ]=0$, variance ${\sigma }^2$ and third central moment $m_3$. Expanding the cube givesFor $\mu >0$, write the coefficient of variation as $v=\sigma /\mu$ and, when $\sigma >0$, skewness as $\gamma =m_3/{\sigma }^3$. The relative multiplier is $1+3v^2+\gamma v^3$. For zero variance the multiplier is one without needing to define skewness.

Equally likely $K=0.9$ and $1.1$ have mean one and give mean cubic multiplier $1.03$. Rare bright shots can matter much more: stipulated probabilities 0.9 and 0.1 at $K=0.8$ and $2.8$ also give mean one, butThe extra signal need not mean that the mean input intensity increased. These are original probability examples, not measurements reported in the correlator paper.

For unchanged pulse shape, common fluctuations multiply every delay by the same expected factor. Normalizing the averaged trace by its peak cancels that factor. Shape changes correlated with brightness destroy this simplification. For example, two equally frequent shot classes with central signals 1 and 8 and side/center ratios 0.01 and 0.10 give ratio of averaged signals $(0.01+0.8)/(1+8)=0.09$, whereas averaging the two ratios gives 0.055. They answer different questions.

A controlled measurement records a synchronized input-intensity monitor and detector background alongside each trace. Compare intensity-binned signals in an unsaturated range and test whether normalized shapes agree. Common attenuation in third-order pulse correlation predicts the cubic scaling only while the nonlinear gate, pulse shape and readout retain the assumed response. A mean power reading alone does not determine the required third moment.