= Differential inequality for an exponential tail envelope
{title2=$P^{\prime}\leq-P/\tau$}
A \b[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 $P(b)\leq A$ and
$$
P'(t)\leq-P(t)/\tau
$$
<almost everywhere>, for constant $\tau>0$. The <integrating factor> argument in <bounding unobserved optical energy with an exponential tail model> gives $P(t)\leq A e^{-(t-b)/\tau}$. 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 $k(t)\geq0$, the condition $P'\leq-kP$ gives
$$
P(t)\leq A\exp\!\left(-\int_b^t k(s)\,ds\right).
$$
This follows from the same integrating factor and is a scalar comparison bound. If $k(t)\geq k_{\min}>0$ 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 yields
$$
U_{\rm tail}\leq0.20\left[10(1-e^{-1})+5e^{-1}\right]\,\mathrm{nJ}
\approx1.632\,\mathrm{nJ}.
$$
Using 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.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37]{ref}
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