 Pulse-energy bounds from tail duration and pedestal height

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.