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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.

Logarithmic condition numbers of a tail-acquisition requirement quantify the fractional response of a positive recording extension to fractional parameter changes. For $L=\tau \ell$ with $\ell =\log (A\tau /\epsilon )>0$, define the signed sensitivity ${\kappa }_x=\partial \log L/\partial \log x$. ThenTheir absolute values are componentwise relative condition numbers. Acquisition-time sensitivity to tail-model parameters gives the corresponding dimensional derivatives and uncertainty propagation.

These numbers depend on which coordinates are held fixed. If $M=A\tau$ denotes the initial tail-energy allowance, then $L=\tau \log (M/\epsilon )$. In coordinates $(M,\tau ,\epsilon )$, the decay-time sensitivity at fixed $M$ is exactly one, rather than $1+1/\ell$. This is the same physical formula: holding $M$ fixed while changing $\tau$ also changes $A$. Transforming the parameter covariance consistently leaves the propagated output uncertainty unchanged.^[ref]

An original boundary example takes $A\tau /\epsilon =1.01$ and $\tau =20\mathrm{ns}$. The required extension is only $L\approx 0.1990\mathrm{ns}$, but ${\kappa }_A\approx 100.50$. Increasing $A$ by $1\%$ multiplies the ratio by $1.01$, giving $1.0201=(1.01{)}^2$ and doubling the required extension exactly. Its relative change is large because the original extension was very small.

This does not mean the absolute requirement is singular there. In the active branch $\partial L/\partial \log (A\tau /\epsilon )=\tau$ remains finite. At and below the threshold, the physical extension is zero, so $\log L$ is undefined. A logarithmic tail-duration change at the zero-extension boundary must instead use the piecewise requirement or absolute changes. Large relative conditioning alone is not evidence of a large absolute recording-time risk.

Finally, conditioning describes the stated envelope model, not its experimental validity. The optical power tail must be distinguished from detector response, which NIST measured independently in its pulse calibration procedure.^[ref] A well-conditioned calculation based on an invalid optical envelope still gives an unjustified acquisition guarantee.