= Cubic intensity-noise bias in pulse correlation
\b[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, $I_j(t)=K_jI_0(t)$, with nonnegative scalar $K_j$. The ideal third-order integral gives
$$
A_{3,j}(\tau)=K_j^3 A_{3,0}(\tau),\qquad
\mathbb E[A_{3,j}(\tau)]=\mathbb E[K^3]A_{3,0}(\tau).
$$
The three intensity factors follow from the <second-harmonic generation> and <sum-frequency generation> gate.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printed p. 194, third-order correlation integral and apparatus]{ref}
Let $K=\mu+\delta$, with $\mathbb E[\delta]=0$, <variance> $\sigma^2$ and third central moment $m_3$. Expanding the cube gives
$$
\mathbb E[K^3]=\mu^3+3\mu\sigma^2+m_3.
$$
For $\mu>0$, 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, but
$$
\mathbb E[K^3]=0.9(0.8)^3+0.1(2.8)^3=2.656.
$$
The 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.
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