 Nonuniqueness of pulse energy at fixed peak power and FWHM

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 $P_{\mathrm{pk}}>0$, power FWHM $\tau >0$, and $0<\alpha <1/2$. Choose any $D\ge \tau$. Let the power equal $P_{\mathrm{pk}}$ for $|t|\le \tau /2$, equal $\alpha P_{\mathrm{pk}}$ for $\tau /2<|t|\le D/2$, 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, butWith $\alpha =0.10$ and $D=101\tau$, the energy is $11P_{\mathrm{pk}}\tau$, compared with $P_{\mathrm{pk}}\tau$ 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.^[ref] A measured full profile, a justified shape family or additional tail-duration bounds supply information that peak and FWHM omit.