= Pulse-to-pulse energy fluctuation
{title2=$\sigma_U$}
\b[Pulse-to-pulse energy fluctuation] is variation in the <laser pulse energy> of successive emitted pulses. For a stationary population with finite variance, its <standard deviation> is $\sigma_U=\sqrt{\mathbb E[(U-\mathbb EU)^2]}$. A finite record supplies an estimate, whose convention should be stated rather than being confused with the uncertainty of the record's mean.
For an original four-pulse record, take measured energies $9,11,11,9\,\mathrm{\mu J}$. Their mean is $10\,\mathrm{\mu J}$. The <sample standard deviation> is
$$
s_U=\sqrt{\frac{1^2+1^2+1^2+1^2}{4-1}}\,\mathrm{\mu J}
=1.155\,\mathrm{\mu J}.
$$
Its <coefficient of variation> is 11.55 percent. The root-mean-square deviation of this particular list, using divisor four rather than three, is instead $1\,\mathrm{\mu J}$ or 10 percent. The definition matters when comparing reported stability figures. The familiar $s_U/\sqrt N$ estimate of uncertainty in a mean additionally assumes independent, identically distributed observations; an ordered record alone does not establish that assumption.
A historical NIST stability assessment compared an internal excimer-laser pulse monitor with a pyroelectric laser-energy meter. Its calorimeter integrated a pulse train, so averaging and individual-pulse variation entered different parts of the analysis.<Calibration Service for Laser Power and Energy at 248 nm by Rodney W. Leonhardt>[§6.2, Laser stability, printedp.23]{ref}
Observed variability also includes readout noise. <Separating detector noise from pulse-energy variance> requires an appropriate measurement model and independent information about that noise. Detector saturation, drift and correlated pulse errors are not removed merely by collecting a longer record. An <average power of a laser pulse train> measurement by itself does not resolve this sequence.
Back to article page
