Pulse-to-pulse energy fluctuation (${\sigma }_U$)

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 ${\sigma }_U=\sqrt{\mathbb{E}[(U-\mathbb{E}U{)}^2]}$. 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 $9,11,11,9\mu \mathrm{J}$. Their mean is $10\mu \mathrm{J}$. The sample standard deviation isIts coefficient of variation is 11.55 percent. The root-mean-square deviation of this particular list, using divisor four rather than three, is instead $1\mu \mathrm{J}$ 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.^[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.

Separating detector noise from pulse-energy variance estimates source variability using a model of the readout's own fluctuations. In energy-calibrated units, write $X=U+n$. The exact identity isOnly when the covariance vanishes can a separately estimated noise variance be subtracted directly. Covariance terms are part of measurement-model uncertainty propagation, rather than an optional correction.^[ref]

For an original independent-noise example, suppose the observed energy standard deviation is $0.50\mathrm{nJ}$ and a matched noise-only measurement gives $0.30\mathrm{nJ}$. The inferred source standard deviation is $\sqrt{0.50^2-0.30^2}=0.40\mathrm{nJ}$, not $0.50-0.30=0.20\mathrm{nJ}$. The source contributes $64\%$ of the observed variance in this stipulated model.

A noise-only record should use comparable gain, integration gate and reset conditions. Background subtraction removes an estimated mean, not the random variance remaining around it. Amplifier clipping, unmatched integrator reset or signal-dependent detector noise can invalidate an additive independent-noise model. A negative variance difference is not a physical negative source variance; it can indicate estimation uncertainty or mismatched assumptions.

Splitting pulses between two independently characterized readouts can provide a two-channel covariance estimate of pulse-energy variance under additional noise assumptions. Shared electronics noise and common drift must still be assessed; independence cannot be established just by using two detectors.

A two-channel covariance estimate of pulse-energy variance uses the component shared by paired readouts to estimate source fluctuations. In a stipulated linear model let $X=aU+n_X$ and $Y=bU+n_Y$, with calibrated nonzero gains. If both noise terms are uncorrelated with $U$ and with each other,This follows by expanding the covariance. Correlation assumptions are essential measurement-model inputs; the uncertainty guide retains covariance terms explicitly.^[ref]

For an original example, choose $a=2V/nJ$, $b=4V/nJ$ and measured covariance $0.08{\mathrm{V}}^2$. The inferred source variance is $0.01\mathrm{n}{\mathrm{J}}^2$, giving a standard deviation of $0.10\mathrm{nJ}$. Independent readout-noise variances $0.36$ and $0.64{\mathrm{V}}^2$ would give total variances $0.40$ and $0.80{\mathrm{V}}^2$ while preserving that cross-covariance.

But a shared noise covariance of $0.04{\mathrm{V}}^2$ raises the observed cross-covariance to $0.12{\mathrm{V}}^2$. Ignoring it would infer a source standard deviation of $0.1225\mathrm{nJ}$, about $22.5\%$ too high.

A split-beam experiment records both channels for the same pulses, checks gains and measures dark or independent-input controls for correlated electronics. NIST's calibrated splitting apparatus supplies an example of the optical geometry, not a claim that it used this covariance estimator.^[ref] Matching pulse indices, bandwidth and integration windows is part of the proposed model.