Cascaded parabolic‑hyperbolic focusing for stable ultra‑small focal spots

MATLAB was used for data processing, FWHM extraction, Monte Carlo sampling, statistical analysis, kernel-density estimation and figure preparation. The exact software releases should be specified in the final README file.For each focal-field realization, one-dimensional intensity profiles were extracted along the x and y axes through the focal-spot centroid. The two directional widths, FWHMx and FWHMy, were determined from the half-maximum crossings, and the effective focal-spot width was calculated as FWHMeff = sqrt[(FWHMx2 + FWHMy2)/2].

A realization was classified as acceptable when FWHMeff was no greater than 0.90 μm.The static-error dataset includes single-parameter sensitivity scans for three translational errors, Δx, Δy and Δz, measured in micrometres, and two angular errors, θx and θy, measured in microradians. It also includes 10,000 five-degree-of-freedom Monte Carlo trials in which the five errors were sampled independently from uniform distributions within their respective single-parameter tolerance ranges.

Of these trials, 7,122 satisfied the FWHM_eff ≤ 0.90 μm criterion.The dynamic-error dataset contains eight simulation conditions with 200 independent trials per condition, corresponding to 1,600 trials in total. Translational and angular perturbations were sampled independently from zero-mean Gaussian distributions and were parameterized by twice the standard deviation, 2σ. The dataset records the sampled perturbations, directional FWHM values, effective FWHM, and acceptance status for every trial.The numerical data do not represent a chronological time series, and the Monte Carlo trial number is only a simulation index.

Spatial coordinates in the focal plane are measured relative to the focal-spot centroid in micrometres. The temporal coordinate in the spatiotemporal simulations is the delay relative to the pulse centre in femtoseconds rather than calendar time. Wavelengths are expressed in nanometres.

No geographical coverage applies, although the optical design and reported stability measurements relate to the downstream end of the SULF 1-PW beamline in Shanghai, China.