 Third-order correlation width of a Gaussian pulse

The 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 ${\sigma }_I/\sqrt{2}$. Their cross-correlation haswhere 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 givesFor an original chosen ${\tau }_{\mathrm{measured}}=120\mathrm{fs}$ 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.^[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.