= Pulse energy shape factor
{title2=$\kappa=U/(P_{\rm pk}\tau)$}
The \b[pulse energy shape factor] is the dimensionless ratio
$$
\kappa=\frac{U}{P_{\rm pk}\tau},
$$
where $U$ is <laser pulse energy>, $P_{\rm pk}$ 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.<Plane wave and beam source - Simulation object in Ansys Optics documentation>[PULSELENGTH definition]{ref}
An original counterexample compares a rectangular pulse with a bilateral exponential pulse. Give both the same $P_{\rm pk}$ and power FWHM $\tau$. The rectangle is nonzero over a width $\tau$, so $U_{\rm rect}=P_{\rm pk}\tau$ and $\kappa_{\rm rect}=1$. For
$$
P_{\rm exp}(t)=P_{\rm pk}2^{-2|t|/\tau},
$$
the half-maximum points are again $\pm\tau/2$, but integration of the two tails gives $U_{\rm exp}=P_{\rm pk}\tau/\ln2$ and $\kappa_{\rm exp}=1/\ln2\approx1.4427$.
With $P_{\rm pk}=2\,\mathrm W$ 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.
\Image[/pulse-shape-energy-comparison.png]
{title=Equal peak power and FWHM do not determine pulse energy: exponential tails change the shape factor.}
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.
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