 Unresolved Gaussian pulse-width estimate

An unresolved Gaussian pulse-width estimate occurs when the measured correlation width and characterized response do not establish a positive pulse width with useful precision under the assumed Gaussian model. It is a limitation of the measurement and inference, not evidence that the pulse has zero duration.

In Gaussian pulse-width deconvolution uncertainty, the inferred full width at half maximum isFor $M>R$, a positive central value exists. Whether the data resolve it depends on the uncertainty in $Q$, the response model and their covariance, not just on the sign of the central radicand.

Take original central values $M=61\mathrm{fs}$ and $R=60\mathrm{fs}$. They give $Q=121\mathrm{f}{\mathrm{s}}^2$ and $P\approx 8.98\mathrm{fs}$. Independent standard uncertainties $u(M)=u(R)=2\mathrm{fs}$ give, to first order,This uncertainty is almost three times the central difference. Propagating once more through the square root gives $u(P)\approx 12.7\mathrm{fs}$; that symmetric approximation is poorly suited to the boundary $P=0$. The GUM requires attention to higher-order terms when nonlinearity invalidates the first-order approximation.^[ref]

A simple range calculation makes the difficulty visible without assigning a probability. Suppose only that plausible widths lie in $59\le M\le 63\mathrm{fs}$ and $58\le R\le 62\mathrm{fs}$. These ranges include equal widths, hence $P=0$, while their maximum positive radicand is $63^2-58^2=605\mathrm{f}{\mathrm{s}}^2$, giving $P\approx 20.08\mathrm{fs}$. These stipulated ranges are not a calibrated confidence interval; they demonstrate sensitivity to response uncertainty.

A negative observed $M^2-R^2$ need not imply an imaginary physical duration: noise, incorrect response characterization or a non-Gaussian pulse can invalidate the naive inversion. Conversely, dropping all negative simulated radicands and averaging the rest conditions on a selected subset and biases the inference. A Monte Carlo method must implement a stated input or likelihood model and a justified treatment of the physical boundary.

Report the measured and response widths, their conventions, uncertainties and dependence. An upper limit is meaningful only with its statistical construction and coverage assumptions. Improved response characterization or an independent temporal measurement can resolve the ambiguity; quoting more digits of the positive square root cannot.