 Differential inequality for an exponential tail envelope

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]