= Third-order correlation width of a Gaussian pulse
{title2=$\tau_3=\sqrt{3/2}\,\tau_I$}
The \b[third-order correlation width of a Gaussian pulse] converts a measured ideal <third-order intensity autocorrelation> width to the width of its assumed Gaussian intensity profile. If the pulse intensity has temporal <standard deviation> $\sigma_I$, its squared intensity gate has deviation $\sigma_I/\sqrt2$. Their <cross-correlation> has
$$
\sigma_3^2=\sigma_I^2+\frac{\sigma_I^2}{2}
=\frac32\sigma_I^2,
\qquad \tau_3=\sqrt{\frac32}\,\tau_I,
$$
where both $\tau$ values are intensity <full widths at half maximum>. The parent <third-order intensity autocorrelation> article derives the complete Gaussian trace by integration. The factor concerns that shape family and that correlation definition; it is not a universal pulse-width conversion.
A focused extension separates a known instrument response. Suppose an additional normalized Gaussian response of width $\tau_R$ convolves the ideal trace. Addition of Gaussian variances gives
$$
\tau_{\rm measured}^2=\frac32\tau_I^2+\tau_R^2,\qquad
\tau_I=\sqrt{\frac23\bigl(\tau_{\rm measured}^2-\tau_R^2\bigr)}.
$$
For an original chosen $\tau_{\rm measured}=120\,\mathrm{fs}$ and $\tau_R=60\,\mathrm{fs}$, the inferred pulse width is $84.9\,\mathrm{fs}$. Applying only the ideal conversion would give $98.0\,\mathrm{fs}$. This <Gaussian instrument broadening in third-order pulse correlation> assumes that the additional response really acts as a Gaussian <convolution>; it is not an empirical correction for every correlator.
The historical physical implementation used <second-harmonic generation> followed by <sum-frequency generation> and spatial delay encoding.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printed pp. 194–195, equation for A3 and Description/Figure 2]{ref} Optical gate width, <phase matching> and spatial overlap belong to its response model. A finer scan step alone does not demonstrate a smaller optical response width.
If the assumed response width equals or exceeds the measured width, the displayed subtraction does not provide a resolved positive pulse width. Non-Gaussian traces, satellites and chirped fields also require explicit modeling; an intensity-width conversion does not reconstruct the optical <wave phase>.
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