= Pulse-energy interval from incomplete tail measurements
A \b[pulse-energy interval from incomplete tail measurements] combines measured energy with bounds on optical power outside the acquisition window. It describes a set of compatible total energies, not automatically a <standard uncertainty> or confidence interval. The <pulse-energy bounds from tail duration and pedestal height> article treats a particular peak-and-width model; a window-based construction need not assume that pulse shape.
Suppose true <radiant energy> inside a measured window $G$ lies in a justified interval $[L,H]$, with $0\leq L\leq H$. Partition the unobserved times into regions of durations $\Delta_j$ with independently established nonnegative <radiant power> caps $M_j$. Then
$$
0\leq U_{\rm missing}\leq\sum_jM_j\Delta_j,\qquad
\boxed{L\leq U\leq H+\sum_jM_j\Delta_j}.
$$
For an original measurement, let the observed-window energy lie between $48$ and $52\,\mathrm{nJ}$. An unobserved early region lasting $2\,\mathrm{ns}$ at most $0.50\,\mathrm W$ contributes at most $1\,\mathrm{nJ}$; a late region lasting $20\,\mathrm{ns}$ at most $0.20\,\mathrm W$ contributes at most $4\,\mathrm{nJ}$. The total interval is therefore \b[$48$ to $57\,\mathrm{nJ}$]. For these assumptions the fraction of <optical energy outside a time gate> equal to $G$ is at most $5/(48+5)\approx9.43\%$.
A finite support cutoff is sufficient but not necessary for a finite bound. <Bounding unobserved optical energy with an exponential tail model> uses a validated cap $P(t)\leq P_b e^{-(t-b)/\tau}$ after the window endpoint $b$, which integrates to $P_b\tau$. Without a justified duration or integrable envelope, one below-threshold reading cannot bound all later energy.
Establish the caps with characterized detector sensitivity, baseline and temporal coverage; do not identify detector ringing directly with optical tails. NIST's response analysis explicitly models the electrical trace as a convolution of the optical waveform with detector response.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[AppendixC.1, printedp36, impulse response and convolution; AppendixC.2, response-measurement procedure]{ref} The interval above follows from chosen optical bounds, not from an uncorrected finite-bandwidth detector trace.
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