Pulse energy shape factor ($\kappa =U/(P_{\mathrm{pk}}\tau )$)

The pulse energy shape factor is the dimensionless ratiowhere $U$ is laser pulse energy, $P_{\mathrm{pk}}$ is background-subtracted peak power and $\tau$ is a stated pulse duration. It belongs to the chosen profile and width convention. Here $\tau$ denotes the power full width at half maximum, the convention explicitly used by the primary source documentation.^[ref]

An original counterexample compares a rectangular pulse with a bilateral exponential pulse. Give both the same $P_{\mathrm{pk}}$ and power FWHM $\tau$. The rectangle is nonzero over a width $\tau$, so $U_{\mathrm{rect}}=P_{\mathrm{pk}}\tau$ and ${\kappa }_{\mathrm{rect}}=1$. Forthe half-maximum points are again $\pm \tau /2$, but integration of the two tails gives $U_{\exp }=P_{\mathrm{pk}}\tau /\ln 2$ and ${\kappa }_{\exp }=1/\ln 2\approx 1.4427$.

With $P_{\mathrm{pk}}=2\mathrm{W}$ and $\tau =5\mathrm{ns}$, the energies are $10$ and $14.43\mathrm{nJ}$. The same measured peak and FWHM therefore leave a 44.3 percent difference in this example. This nonuniqueness of pulse energy at fixed peak power and FWHM is a profile ambiguity, not an instrument-unit conversion.

The Gaussian pulse energy and peak power article gives the Gaussian-family relation. Reconstructing an arbitrary pulse's energy requires the full calibrated profile or an additional justified shape model; a width and peak alone cannot supply it.