 Gaussian instrument broadening in third-order pulse correlation

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 beThe convolution $M=C\ast R$ isThe 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,

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.^[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.