 Acquisition duration from a power-law tail-energy tolerance

An 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 $\alpha >1$,The parent article integrates this envelope and inverts the tolerance. Writing $R=A{t}_0/[(\alpha -1)\epsilon ]$, 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 $\epsilon$, differentiating the active branch givesThe 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 $u\ge 0$, 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 $\epsilon =0.005\mathrm{nJ}$. 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 $2\le \alpha \le 3$, 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.^[ref] The power-law examples are original conditional models, not observed decay laws attributed to that apparatus.