= Common attenuation in third-order pulse correlation
\b[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 signal
$$
A_3(\tau)=\int I^2(t)I(t+\tau)\,dt,
$$
replacing $I$ by $kI$ gives $A_{3,k}(\tau)=k^3A_3(\tau)$. 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>.<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 implementation]{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 $2/101\approx0.0198$. 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. Then
$$
\mathbb E[K^3]=1+3\mathbb E[\eta^2]+\mathbb E[\eta^3].
$$
For 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.
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