 Gaussian pulse-width deconvolution uncertainty

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: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 coefficientsThus the standard uncertainty obeysThe 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.^[ref]

For an original example, $M=120\mathrm{fs}$ and $R=60\mathrm{fs}$ give $P\approx 84.85\mathrm{fs}$. With independent $u(M)=u(R)=2\mathrm{fs}$, the formula gives $u(P)\approx 2.11\mathrm{fs}$. If instead the two width estimates have correlation coefficient 0.8 with those same uncertainties, the covariance is $3.2\mathrm{f}{\mathrm{s}}^2$ and $u(P)\approx 1.26\mathrm{fs}$. The assumed relationship between the estimates changes the result.

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 $4[M^2{u}^2(M)+R^2{u}^2(R)-2MR\mathrm{cov}(M,R)]$. If plausible width values include $Q\le 0$, 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.