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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.

Pulse-energy bounds from tail duration and pedestal height convert independently justified constraints on low-level light into a bound on total laser pulse energy. Peak power and full width at half maximum alone do not constrain an arbitrarily long below-threshold tail, as unbounded pulse energy at fixed peak and FWHM demonstrates.

Consider a nonnegative isolated power profile with peak $P$, one contiguous half-maximum interval of duration $\tau$, and total support of duration at most $D_{\max }\ge \tau$. Suppose power outside that interval is at most $H<P/2$. ThenThe lower bound uses the half-maximum condition within the interval. The upper bound fills that interval to $P$ and every permitted remaining instant to $H$. These bounds assume the stated shape class; multiple half-maximum crossings require an explicit interval convention.

For an original $P=8\mathrm{W}$, $\tau =5\mathrm{ns}$, $H=0.40\mathrm{W}$ and $D_{\max }=30\mathrm{ns}$, they give $20\le U\le 50\mathrm{nJ}$. Suppose a calibrated measurement additionally determines core energy $U_c=32\mathrm{nJ}$. The pulse-energy interval from incomplete tail measurements tightens towith the displayed product using watts and nanoseconds. Reducing the independently established tail duration to $15\mathrm{ns}$ tightens the upper bound to $36\mathrm{nJ}$ without changing the measured peak or width.

A receiver with characterized low-level response and a sufficiently long acquisition window is needed to justify $H$ and $D_{\max }$. A trace that falls below a detector threshold establishes no later support cutoff by itself. NIST's pulse-energy calibration discussion separates temporal waveform measurement from background and leakage effects.^[ref] The numerical bounds above are conditional mathematical examples, not specifications of that apparatus.