 Exponential-envelope tail-energy bound

An exponential-envelope tail-energy bound limits unobserved energy by integrating a justified pointwise envelope on the true optical power. If $0\le P(t)\le A{e}^{-(t-b)/\tau }$ for every $t\ge b$, with $A\ge 0$ and $\tau >0$, then $U_{\mathrm{tail}}\le A\tau$. 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 isFor 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 $L\ge 30\log 100\approx 138.2\mathrm{ns}$. 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 $a\ge 0$, $q>0$ and $a+q\le 6$. Since $aq\le (a+q{)}^2/4$, every allowed envelope satisfiesattained at $a=q=3$. Taking separate maxima $A\le 0.30\mathrm{W}$ and $\tau \le 120\mathrm{ns}$ 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.^[ref] A delayed extra pulse or an unbounded constant background requires its own model; neither is excluded by a local exponential fit.