= Bounding unobserved optical energy with an exponential tail model
A finite optical record can bound later <radiant energy> if an independently justified <exponential decay> envelope applies at every later time. Let the recorded window end at $b$, and suppose the true nonnegative <radiant power> satisfies
$$
0\leq P(t)\leq A e^{-(t-b)/\tau}\quad(t\geq b),\qquad A\geq0,\quad\tau>0.
$$
Integration gives the <exponential-envelope tail-energy bound>
$$
\boxed{U_{\rm tail}=\int_b^\infty P(t)\,dt\leq A\tau}.
$$
Equality holds when the entire tail follows the envelope. This is a bound on an unobserved contribution, not automatically a <standard uncertainty> or a probabilistic interval.
For an original numerical example, suppose the energy inside the recorded window is known to lie in $[48,52]\,\mathrm{nJ}$, while a justified envelope has $A=0.20\,\mathrm W$ and $\tau=20\,\mathrm{ns}$. The missing contribution is at most $4\,\mathrm{nJ}$, so the total <laser pulse energy> lies in \b[$[48,56]\,\mathrm{nJ}$]. The <optical energy outside a time gate> fraction, considering only this late contribution, is at most $4/(48+4)\approx7.69\%$. Any unobserved early energy requires its own bound.
One way to justify the envelope mathematically is a <differential inequality for an exponential tail envelope>. Suppose $P$ is <absolutely continuous> on each finite interval after $b$, $P(b)\leq A$, and $P'(t)\leq-P(t)/\tau$ <almost everywhere>. Multiplying by the <integrating factor> $e^{(t-b)/\tau}$ gives
$$
\frac{d}{dt}\left[e^{(t-b)/\tau}P(t)\right]\leq0,
$$
so $e^{(t-b)/\tau}P(t)\leq P(b)\leq A$. The decay law must apply throughout the unobserved region; it is a physical assumption to validate, rather than a consequence of fitting the last few samples.
The distinction is testable by a counterexample. Append a later nonnegative pulse whose support lies entirely beyond the recorded interval. Its energy can be arbitrarily large while every recorded sample is unchanged. Such a pulse violates the global envelope but cannot be excluded by a good local exponential fit alone. Similarly, an envelope $Ae^{-(t-b)/\tau}+P_0$ with a positive constant background $P_0$ has an infinite integral over an infinite time interval. A finite observation horizon or an adequately defined and validated <background subtraction> is then essential.
Uncertain envelope parameters require a bound valid for their allowed combinations. If the allowed set is the rectangle $0\leq A\leq A_{\max}$ and $0<\tau\leq\tau_{\max}$, then $U_{\rm tail}\leq A_{\max}\tau_{\max}$. For $A_{\max}=0.25\,\mathrm W$ and $\tau_{\max}=25\,\mathrm{ns}$, this is $6.25\,\mathrm{nJ}$ and the preceding total interval becomes $[48,58.25]\,\mathrm{nJ}$. More restrictive joint parameter constraints can tighten this result; inserting separate best-fit values does not establish a guaranteed bound.
Finally, an electrical tail need not be an optical tail. In an original idealized <linear time-invariant system>, let a <photodetector> have unit-area <impulse response> $h(t)=\tau_d^{-1}e^{-t/\tau_d}$ for $t\geq0$, and zero otherwise. An impulsive optical input $U\delta(t)$, with $\delta$ the <Dirac delta function>, gives the calibrated output $s(t)=Uh(t)$ by <convolution>. Its electrical tail after $L$ has area $Ue^{-L/\tau_d}$ even though all incident optical energy arrived at time zero. This <optical tail and detector-response ambiguity> requires a separate response measurement. NIST's historical procedure characterized the <photodetector>, cable and <oscilloscope> together before convolving measured responses with trial waveforms. Appendix C describes typical optical excitation of about $120\,\mathrm{ps}$, a <photodetector> <rise time> near $5\,\mathrm{ns}$, and a $1\,\mathrm{GHz}$ <oscilloscope> with $0.4\,\mathrm{ns}$ <rise time>. The optical excitation was much shorter than the detector response, while the <oscilloscope> was fast enough that the detector dominated the measured waveform. Response data were collected at each gain setting and several signal levels.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37]{ref}
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