= Third-order intensity autocorrelation
{title2=$A_3(\tau)=\int I^2(t)I(t+\tau)\,dt$}
A \b[third-order intensity autocorrelation] measures the delay-dependent intensity product
$$
A_3(\tau)=\int_{\mathbb R}I^2(t)I(t+\tau)\,dt.
$$
The name counts three intensity factors. A physical implementation can use two second-order nonlinear processes: <second-harmonic generation> creates a gate proportional to $I^2$, and <sum-frequency generation> mixes it with a delayed fundamental replica. It therefore need not use a third-order material susceptibility. The historical device spatially encoded delay in crossing beams and imaged the sum-frequency signal onto a <charge-coupled device>.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printedpp194–195, equation forA3, Description and Figure2]{ref}
An original Gaussian calculation illustrates response width. Set $I(t)=I_0e^{-t^2/(2\sigma^2)}$. Completing the square in the integral gives
$$
A_3(\tau)=I_0^3\sigma\sqrt{\frac{2\pi}{3}}
\exp\!\left(-\frac{\tau^2}{3\sigma^2}\right).
$$
Hence the <third-order correlation width of a Gaussian pulse> obeys $\tau_3=\sqrt{3/2}\,\tau_I$ for intensity <full widths at half maximum>. A $90\,\mathrm{fs}$ correlation width would imply $\tau_I\approx73.5\,\mathrm{fs}$ within this ideal Gaussian model. The square-root-of-two conversion appropriate to second-order Gaussian <intensity autocorrelation> would be different.
For a general pulse, $A_3(\tau)$ need not equal $A_3(-\tau)$; this asymmetry can distinguish a delayed feature from an earlier one. It is still a <cross-correlation> with a finite-width nonlinear gate, rather than a direct reading of $I(\tau)$. The existing <optical pulse temporal contrast> derivation shows how a delayed weak replica also produces an opposite-delay <third-order pulse correlation ghost>.
Characterize <phase matching>, beam overlap, spatial uniformity and the temporal response before interpreting a narrow feature. Scaling the input intensity by $k$ scales this ideal signal by $k^3$, so input fluctuations also matter. The measured trace does not uniquely reconstruct an arbitrary complex optical <electric field>.
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