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An 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 $P(t)\le A{e}^{-(t-b)/\tau }$, with $A\ge 0$ and $\tau >0$. The exponential-envelope tail-energy bound gives a remaining allowance $A\tau e^{-L/\tau }$ after extension $L\ge 0$. For tolerance $\epsilon >0$,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 $A\le 0.10\mathrm{W}$ and $\tau \le 40\mathrm{ns}$. The remaining allowance increases with either parameter, so a $0.040\mathrm{nJ}$ tolerance requires $L\ge 40\log 100\approx 184.2\mathrm{ns}$. 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 areThis 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.^[ref] State the chosen time origin, tolerance units, valid envelope parameters and any other excluded intervals when reporting the acquisition requirement.

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 satisfiesIntegration gives the exponential-envelope tail-energy boundEquality 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 $[48,56]\mathrm{nJ}$. The optical energy outside a time gate fraction, considering only this late contribution, is at most $4/(48+4)\approx 7.69\%$. 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 $b$, $P(b)\le A$, and $P^{\prime }(t)\le -P(t)/\tau$ almost everywhere. Multiplying by the integrating factor $e^{(t-b)/\tau }$ givesso $e^{(t-b)/\tau }P(t)\le P(b)\le A$. 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 $A{e}^{-(t-b)/\tau }+P_0$ 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 $0\le A\le A_{\max }$ and $0<\tau \le {\tau }_{\max }$, then $U_{\mathrm{tail}}\le A_{\max }{\tau }_{\max }$. 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 $t\ge 0$, 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 $U{e}^{-L/{\tau }_d}$ 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.^[ref]

A differential inequality for an exponential tail envelope supplies a sufficient decay condition for a global power bound. Let nonnegative $P$ be absolutely continuous on every finite interval after $b$, with $P(b)\le A$ andalmost everywhere, for constant $\tau >0$. The integrating factor argument in bounding unobserved optical energy with an exponential tail model gives $P(t)\le A{e}^{-(t-b)/\tau }$. Corners in the waveform are allowed under absolute continuity; an unmodeled upward jump is not.

The inequality is stronger than merely fitting a decay constant at a few sampled times. For example, $P(t)=A{e}^{-2(t-b)/\tau }$ obeys it and carries only $A\tau /2$ after $b$. The exponential-envelope tail-energy bound $A\tau$ remains valid but need not be tight. Conversely, a later pulse can violate the differential condition while leaving all earlier samples unchanged.

A useful extension is a time-dependent decay-rate envelope. For locally integrable $k(t)\ge 0$, the condition $P^{\prime }\le -kP$ givesThis follows from the same integrating factor and is a scalar comparison bound. If $k(t)\ge k_{\min }>0$ throughout the tail, its integrated energy is at most $A/k_{\min }$.

For an original piecewise rate, choose $A=0.20\mathrm{W}$, $k=1/(10\mathrm{ns})$ for the first $10\mathrm{ns}$ and $k=1/(5\mathrm{ns})$ thereafter. Integrating the resulting continuous envelope yieldsUsing only the minimum rate would give $2.0\mathrm{nJ}$. Faster guaranteed late decay tightens the result.

This is a conditional mathematical proof. A measured electrical waveform obeying a similar slope rule does not establish the optical condition without accounting for the detector impulse response. The primary NIST measurement procedure supplies an example of characterizing that response independently.^[ref]