= Acquisition duration from an exponential tail-energy tolerance
{title2=$L_{\min}$}
An \b[acquisition duration from an exponential tail-energy tolerance] chooses how long to continue recording when a validated optical-power envelope bounds the remaining energy. Suppose after time $b$ the true power satisfies $P(t)\leq A e^{-(t-b)/\tau}$, with $A\geq0$ and $\tau>0$. The <exponential-envelope tail-energy bound> gives a remaining allowance $A\tau e^{-L/\tau}$ after extension $L\geq0$. For tolerance $\varepsilon>0$,
$$
L_{\min}=\begin{cases}
0,&A\tau\leq\varepsilon,\\
\tau\log(A\tau/\varepsilon),&A\tau>\varepsilon.
\end{cases}
$$
This is a conditional upper-bound calculation, not an inference from the last sample merely falling below a detector threshold.
An original uncertainty example has guaranteed parameter limits $A\leq0.10\,\mathrm W$ and $\tau\leq40\,\mathrm{ns}$. The remaining allowance increases with either parameter, so a $0.040\,\mathrm{nJ}$ tolerance requires $L\geq40\log100\approx184.2\,\mathrm{ns}$. Using only best-fit values $A=0.08\,\mathrm W$ and $\tau=30\,\mathrm{ns}$ would give about $122.8\,\mathrm{ns}$, which does not guarantee the same tolerance over the allowed set. Joint parameter constraints can tighten a bound, but their coverage must be justified.
In the positive-duration branch, the local sensitivities are
$$
\frac{\partial L}{\partial A}=\frac\tau A,\qquad
\frac{\partial L}{\partial\tau}=\log(A\tau/\varepsilon)+1,\qquad
\frac{\partial L}{\partial\varepsilon}=-\frac\tau\varepsilon.
$$
This <acquisition-time sensitivity to tail-model parameters> explains why an uncertain slow decay can matter more than a small amplitude correction. The derivatives change interpretation at the zero-duration boundary.
The envelope must apply to optical power throughout the unrecorded region. A later pulse or an unremoved persistent background can invalidate it. NIST's separate measurement of detector response illustrates why an electrical waveform's decay alone is insufficient.<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} State the chosen time origin, tolerance units, valid envelope parameters and any other excluded intervals when reporting the acquisition requirement.
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