= Gaussian instrument broadening in third-order pulse correlation
{c}
{title2=$\tau_M^2=\tfrac32\tau_I^2+\tau_R^2$}
\b[Gaussian instrument broadening in third-order pulse correlation] is the widening produced when an ideal <third-order intensity autocorrelation> is convolved with an additional Gaussian instrument response. It is a model of the measurement, not a universal correction for every correlator. The finite width of the optical nonlinear gate is already part of the ideal third-order trace and must not be counted again as an independent response.
Let a <Gaussian pulse> have intensity <standard deviation> $\sigma_I$. Its ideal trace is a Gaussian $C(\tau)=C_0\exp[-\tau^2/(2\sigma_3^2)]$ with $\sigma_3^2=3\sigma_I^2/2$. Take the additional normalized <impulse response> to be
$$
R(\tau)=\frac1{\sqrt{2\pi}\sigma_R}\exp\!\left(-\frac{\tau^2}{2\sigma_R^2}\right).
$$
The <convolution> $M=C*R$ is
$$
M(\tau)=C_0\frac{\sigma_3}{\sqrt{\sigma_3^2+\sigma_R^2}}
\exp\!\left[-\frac{\tau^2}{2(\sigma_3^2+\sigma_R^2)}\right].
$$
The response preserves integrated trace area while reducing peak height and increasing variance. Since every Gaussian <full width at half maximum> has the same conversion factor from standard deviation,
$$
\tau_M^2=\frac32\tau_I^2+\tau_R^2.
$$
For an original example, $\tau_I=80\,\mathrm{fs}$ and $\tau_R=60\,\mathrm{fs}$ give an ideal correlation width about $98.0\,\mathrm{fs}$ and a measured width about $114.9\,\mathrm{fs}$. The peak falls to approximately 0.853 of the ideal peak under this unit-area response. Dividing the measured width only by $\sqrt{3/2}$ would overestimate the pulse duration.
The correct Gaussian inversion is $\tau_I=\sqrt{(2/3)(\tau_M^2-\tau_R^2)}$ when the radicand is positive. Its conditioning worsens near the response floor: the derivatives with respect to $\tau_M$ and $\tau_R$ are $(2/3)\tau_M/\tau_I$ and $-(2/3)\tau_R/\tau_I$. Small width errors can therefore cause large errors in a narrowly resolved pulse. <Gaussian pulse-width deconvolution uncertainty> must include uncertainty of the characterized response, with covariance if relevant.
The primary correlator used <second-harmonic generation>, <sum-frequency generation>, crossed beams and spatial imaging to encode delay. It did not assert that every additional broadening mechanism was a Gaussian convolution.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[printed pp. 194–195, third-order integral, Description and Figure 2]{ref} To apply this model, characterize the response under the same optical alignment, spectral acceptance and detector conditions. Non-Gaussian wings, clipping or unresolved side lobes require a different model; a smaller sampling step cannot by itself remove optical broadening.
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