= Exponential-envelope tail-energy bound
{title2=$U_{\rm tail}\leq A\tau$}
An \b[exponential-envelope tail-energy bound] limits unobserved energy by integrating a justified pointwise envelope on the true optical power. If $0\leq P(t)\leq A e^{-(t-b)/\tau}$ for every $t\geq b$, with $A\geq0$ and $\tau>0$, then $U_{\rm tail}\leq A\tau$. The parent <bounding unobserved optical energy with an exponential tail model> derives this result and distinguishes it from a statistical uncertainty.
At a later acquisition endpoint $b+L$, the remaining allowance is
$$
U_{\rm after}\leq A\tau e^{-L/\tau}.
$$
For an original envelope with $A=0.12\,\mathrm W$ and $\tau=30\,\mathrm{ns}$, the initial allowance is $3.6\,\mathrm{nJ}$. Reducing the allowed missing energy to $0.036\,\mathrm{nJ}$ requires $L\geq30\log100\approx138.2\,\mathrm{ns}$. This <acquisition duration from an exponential tail-energy tolerance> reduces the bound by a factor100. It does not prove that99% of an arbitrary underlying pulse has been captured: the actual power may lie strictly below the envelope at early times.
Joint parameter constraints also matter. An original allowed set uses dimensionless $a=A/(0.05\,\mathrm W)$ and $q=\tau/(20\,\mathrm{ns})$ with $a\geq0$, $q>0$ and $a+q\leq6$. Since $aq\leq(a+q)^2/4$, every allowed envelope satisfies
$$
A\tau\leq9\,\mathrm{nJ},
$$
attained at $a=q=3$. Taking separate maxima $A\leq0.30\,\mathrm W$ and $\tau\leq120\,\mathrm{ns}$ would instead give the looser $36\,\mathrm{nJ}$ bound; those two maxima cannot occur together in the stated joint set.
The guarantee concerns optical power, not an uncorrected detector tail. <Optical tail and detector-response ambiguity> explains that distinction, while NIST's primary response-calibration procedure characterizes the electrical measurement chain separately.<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} A delayed extra pulse or an unbounded constant background requires its own model; neither is excluded by a local exponential fit.
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