= Nonuniqueness of pulse energy at fixed peak power and FWHM
\b[Nonuniqueness of pulse energy at fixed peak power and FWHM] means that those two measurements do not determine the integral of an otherwise unspecified power profile. The <pulse energy shape factor> gives a rectangular/exponential counterexample. A stronger original construction shows there is no finite universal upper bound without restrictions on lower-level tails.
Fix $P_{\rm pk}>0$, power FWHM $\tau>0$, and $0<\alpha<1/2$. Choose any $D\geq\tau$. Let the power equal $P_{\rm pk}$ for $|t|\leq\tau/2$, equal $\alpha P_{\rm pk}$ for $\tau/2<|t|\leq D/2$, and zero afterward. The <optical pulse pedestal> is part of the stipulated signal, not an unrelated background. Its level stays below half maximum, so the peak and FWHM remain unchanged, but
$$
U=P_{\rm pk}[\tau+\alpha(D-\tau)].
$$
With $\alpha=0.10$ and $D=101\tau$, the energy is $11P_{\rm pk}\tau$, compared with $P_{\rm pk}\tau$ for the pedestal-free rectangle. Increasing finite $D$ gives <unbounded pulse energy at fixed peak and FWHM> across this family of finite-duration ideal profiles.
This is an identifiability result, not a claim that a particular laser has arbitrarily long tails. The primary source definition of power-FWHM fixes a threshold width, not the integrated energy below that threshold.<Plane wave and beam source - Simulation object in Ansys Optics documentation>[PULSELENGTH definition]{ref} A measured full profile, a justified shape family or additional tail-duration bounds supply information that peak and FWHM omit.
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