= Pulse-energy bounds from tail duration and pedestal height
\b[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 $D_{\max}\geq\tau$. Suppose power outside that interval is at most $H<P/2$. Then
$$
\frac{P\tau}{2}\leq U\leq P\tau+H(D_{\max}-\tau).
$$
The 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 $20\leq U\leq50\,\mathrm{nJ}$. Suppose a calibrated measurement additionally determines core energy $U_c=32\,\mathrm{nJ}$. The <pulse-energy interval from incomplete tail measurements> tightens to
$$
32\leq U\leq32+0.40(30-5)=42\,\mathrm{nJ},
$$
with 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.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§§3.4–3.5, transfer and reference standards; §7.2, Leakage Effect]{ref} The numerical bounds above are conditional mathematical examples, not specifications of that apparatus.
Back to article page
