= Acquisition duration from a power-law tail-energy tolerance
{title2=$L=t_0(R^{1/(\alpha-1)}-1)$}
An \b[acquisition duration from a power-law tail-energy tolerance] extends a measurement until the validated envelope's unrecorded energy is below a chosen tolerance. For the model in <power-law tail from a decaying rate envelope>, with $\alpha>1$,
$$
U_{\rm after}(L)\leq\frac{At_0}{\alpha-1}(1+L/t_0)^{1-\alpha}.
$$
The parent article integrates this envelope and inverts the tolerance. Writing $R=At_0/[(\alpha-1)\varepsilon]$, its positive-duration requirement is $L=t_0(R^{1/(\alpha-1)}-1)$ when $R>1$; otherwise no additional duration is required.
In applying the rule, uncertainty in the exponent can dominate the result. With fixed $A,t_0$ and $\varepsilon$, differentiating the active branch gives
$$
\frac{\partial\log(1+L/t_0)}{\partial\alpha}
=-\frac{1+\log R}{(\alpha-1)^2}.
$$
The required duration decreases with a larger exponent. For a guaranteed parameter range, a <validated lower bound on a power-law tail exponent> is therefore more useful than its best-fit value alone. Since $(1+u/t_0)^{-\alpha}$ decreases with $\alpha$ for every $u\geq0$, using the smallest allowed exponent bounds the entire tail, not just one selected sample.
For an original design example, let $A=0.10\,\mathrm W$, $t_0=10\,\mathrm{ns}$ and $\varepsilon=0.005\,\mathrm{nJ}$. A nominal exponent $\alpha=3$ gives initial allowance $0.5\,\mathrm{nJ}$, $R=100$ and $L=90\,\mathrm{ns}$. If the validated exponent range is instead $2\leq\alpha\leq3$, its slowest permitted tail has initial allowance $1.0\,\mathrm{nJ}$ and requires $L=1990\,\mathrm{ns}$. The nominal duration does not guarantee the tolerance over that range. Upper bounds on $A$ and $t_0$ can likewise be used conservatively because increasing either raises the envelope pointwise.
If the allowed exponents reach one or below, these envelope assumptions alone no longer supply a finite all-time energy guarantee. This is a limitation of the bound, not proof that the actual pulse has infinite energy. The model must also hold throughout the unobserved interval: later pulses, persistent backgrounds or an electrical detector tail misidentified as optical power can invalidate it. NIST's independent receiver-response measurement illustrates this last experimental distinction.<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} The power-law examples are original conditional models, not observed decay laws attributed to that apparatus.
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