= Gaussian pulse-width deconvolution uncertainty
{c}
{title2=$u(P)$}
\b[Gaussian pulse-width deconvolution uncertainty] is the uncertainty in an inferred pulse duration after subtracting a characterized Gaussian instrument contribution. For the <Gaussian instrument broadening in third-order pulse correlation> model, let $M$ be measured correlation <full width at half maximum>, $R$ the additional response width and $P$ the inferred pulse width:
$$
P=\sqrt{\frac23(M^2-R^2)},\qquad M>R.
$$
These widths must use the same convention. The relation already includes the ideal nonlinear gate; $R$ represents an additional response.
First-order propagation uses the sensitivity coefficients
$$
\frac{\partial P}{\partial M}=\frac{2M}{3P},\qquad
\frac{\partial P}{\partial R}=-\frac{2R}{3P}.
$$
Thus the <standard uncertainty> obeys
$$
u^2(P)\simeq\frac4{9P^2}\left[M^2u^2(M)+R^2u^2(R)-2MR\operatorname{cov}(M,R)\right].
$$
The negative covariance term follows from the subtraction, not a rule that uncertainties should simply be subtracted. The GUM's general law retains covariance and requires considering higher-order terms when nonlinearity is significant.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1.2–5.1.3 and 5.2, sensitivity coefficients, nonlinear limitation and correlated inputs]{ref}
For an original example, $M=120\,\mathrm{fs}$ and $R=60\,\mathrm{fs}$ give $P\approx84.85\,\mathrm{fs}$. With independent $u(M)=u(R)=2\,\mathrm{fs}$, the formula gives $u(P)\approx2.11\,\mathrm{fs}$. If instead the two width estimates have correlation coefficient 0.8 with those same uncertainties, the covariance is $3.2\,\mathrm{fs^2}$ and $u(P)\approx1.26\,\mathrm{fs}$. The assumed relationship between the estimates changes the result.
\Image[/gaussian-pulse-width-uncertainty.png]
{title=Original Gaussian pulse-width inversion and first-order uncertainty curves for a 60 fs response and independent 2 fs width uncertainties}
Near $M=R$, both derivatives grow because $P$ approaches zero. With $M=61\,\mathrm{fs}$, $R=60\,\mathrm{fs}$ and independent 2 fs uncertainties, the central inferred value is only about $8.98\,\mathrm{fs}$ while the first-order uncertainty is about $12.7\,\mathrm{fs}$. This is an <unresolved Gaussian pulse-width estimate> under these measurement conditions, rather than a reliably resolved 9 fs pulse.
The quantity being subtracted is $Q=M^2-R^2$, with approximate variance $4[M^2u^2(M)+R^2u^2(R)-2MR\operatorname{cov}(M,R)]$. If plausible width values include $Q\le0$, linear symmetric error bars on $P$ can be misleading. Report the response assumptions and consider an appropriate nonlinear inference or bound; simply discarding negative radicands would silently change the statistical model. Even a precise inversion remains conditional on the Gaussian response model being physically appropriate.
Back to article page
