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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]

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.

Optical tail and detector-response ambiguity arises because a measured temporal trace combines the optical waveform with the receiver's dynamics. In a calibrated linear time-invariant system,where $P$ is incident radiant power and $h$ the unit-area detector impulse response. One trace does not generally identify both unknown functions. The parent bounding unobserved optical energy with an exponential tail model gives an impulsive-input example; two finite-width profiles provide another exact ambiguity.

Define $h_{\tau }(t)={\tau }^{-1}{e}^{-t/\tau }$ for $t\ge 0$, and zero otherwise. Consider model A with optical power $P=U{h}_{2\mathrm{ns}}$ and detector response $h_{10\mathrm{ns}}$, and model B with these two time constants exchanged. By commutativity of convolution, both give exactlyTheir optical energies after $20\mathrm{ns}$ are nevertheless $U{e}^{-10}$ and $U{e}^{-2}$. For original chosen $U=100\mathrm{nJ}$, those are about $0.00454$ and $13.53\mathrm{nJ}$. Identical recorded tails therefore coexist with very different true late optical energy.

Known-input calibration of a pulse detector supplies information missing from this inverse problem. NIST measured the detector/cable/oscilloscope response with optical excitation much shorter than the response before calculating waveform distortions.^[ref] The example above is an independent mathematical construction. A fitted electrical decay constant alone cannot establish a global optical-tail envelope.

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.