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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]

The pedestal energy fraction is the fraction of an optical pulse's laser pulse energy assigned to its optical pulse pedestal. Specify the decomposition and integration interval before quoting it. For a nonnegative additive radiant power model $P(t)=P_m(t)+P_p(t)$,The assumptions $0\le P_p\le P$ imply $0\le f_p\le 1$ by integration. A pedestal fit is a model of the measured radiant power; it is not automatically a unique physical separation into optical fields. Overlapping fields with optical coherence can have wave interference terms, so their separate field energies cannot generally be added as though those terms were absent.

The distinction from optical pulse temporal contrast is quantitative. Use the stipulated rectangular model of optical pulse pedestal: a main contribution has height $P_0$ and duration $\tau$, while a centered pedestal has height $\alpha P_0$ and duration $T\ge \tau$. For $0<\alpha <1$, the total peak is $(1+\alpha )P_0$ and the power full width at half maximum remains $\tau$. Define peak-to-pedestal contrast $C=(1+\alpha )/\alpha$. ThenHalf the laser pulse energy is in the modeled pedestal when $T=(C-1)\tau$. A large peak-to-pedestal ratio therefore does not by itself imply a small energy contribution.

For an original comparison, fix $C=1001$ and the same $P_0,\tau$ in both traces. With $T=10\tau$, the pedestal fraction is $10/1010\approx 0.00990$; with $T=1000\tau$, it is $1/2$. Thus the same contrast and power FWHM can coexist with about1% or50% pedestal energy. At fixed finite $C$, increasing the pedestal duration can make $f_p$ approach one without changing that FWHM.

Optical energy outside a time gate is a related observable that avoids assigning a pedestal underneath the main peak. For a specified gate $G$ containing that peak,In the rectangular example with $G$ equal to the main interval, $f_{\mathrm{out}}=\alpha (T-\tau )/(\tau +\alpha T)$. At $\alpha =10^{-3}$ and $T=1000\tau$, this is $0.4995$, while $f_p=0.5000$. The difference is pedestal power inside the gate, which the outside-gate measurement does not label separately.

The analysis must also specify its temporal coverage and detector baseline subtraction. Subtracting actual weak light as though it were detector background changes the inferred laser pulse energy. A short detection time gate omits outlying light instead. Neither operation is justified solely by knowing the peak-to-pedestal contrast. The Sequoia technical sheet illustrates why temporal coverage and resolving power are separate instrument properties: it lists scan range, scan resolution and optical temporal resolution independently.^[ref]

A pulse-energy interval from incomplete tail measurements combines measured energy with bounds on optical power outside the acquisition window. It describes a set of compatible total energies, not automatically a standard uncertainty or confidence interval. The pulse-energy bounds from tail duration and pedestal height article treats a particular peak-and-width model; a window-based construction need not assume that pulse shape.

Suppose true radiant energy inside a measured window $G$ lies in a justified interval $[L,H]$, with $0\le L\le H$. Partition the unobserved times into regions of durations ${\Delta }_j$ with independently established nonnegative radiant power caps $M_j$. ThenFor an original measurement, let the observed-window energy lie between $48$ and $52\mathrm{nJ}$. An unobserved early region lasting $2\mathrm{ns}$ at most $0.50\mathrm{W}$ contributes at most $1\mathrm{nJ}$; a late region lasting $20\mathrm{ns}$ at most $0.20\mathrm{W}$ contributes at most $4\mathrm{nJ}$. The total interval is therefore $48$ to $57\mathrm{nJ}$. For these assumptions the fraction of optical energy outside a time gate equal to $G$ is at most $5/(48+5)\approx 9.43\%$.

A finite support cutoff is sufficient but not necessary for a finite bound. Bounding unobserved optical energy with an exponential tail model uses a validated cap $P(t)\le P_b{e}^{-(t-b)/\tau }$ after the window endpoint $b$, which integrates to $P_b\tau$. Without a justified duration or integrable envelope, one below-threshold reading cannot bound all later energy.

Establish the caps with characterized detector sensitivity, baseline and temporal coverage; do not identify detector ringing directly with optical tails. NIST's response analysis explicitly models the electrical trace as a convolution of the optical waveform with detector response.^[ref] The interval above follows from chosen optical bounds, not from an uncorrected finite-bandwidth detector trace.