Refractive indices for Ne, Ar, and Kr in the VUV, and RDW phase-matching including nonlinear and ionisation terms.

[teodora-grigorova]

VUV Sellmeier models and nonlinear/ionisation RDW phase matching

Summary

This PR tests out some different Sellmeier equations available in the literatutre for the nobel gases Ne, Ar, and Kr, and adds additional options for the refractive index that can be used, particularly with emphasis on the VUV spectral range. It also adds a RDW phase-matching finding function that also includes nonlinear and ionisation contributions.

The main changes are:

• Adds additional literature-based Sellmeier refractive index models for noble gases in the VUV.
• Adds Bideau-Mehu refractive-index models for Ne, Ar, and Kr.
• Adds modified Sellmeier models for Ar and Kr, constructed by combining VUV data from Kingston with VIS/IR behaviour from Börzsönyi.
• Adds RDW phase-matching finding function that can include linear, nonlinear, and ionisation/plasma contributions.

These additions are motivated by the VUV RDW analysis, where the theoretically calculated phase-matched wavelength needs reliable refractive index data until the expected phase-matched wavelength.

Motivation

The existing Börzsönyi refractive-index models are specified over the 400–1000 nm range. For the VUV paper, however, the relevant RDW wavelengths lie in the DUV/VUV, where extrapolation of the refractive index from Börzsönyi might not be guaranteed to match the physics.

The analysis compares several refractive-index models:

• Börzsönyi 2008: existing Luna model, mainly VIS/IR scoped.
• Bideau-Mehu 1981: Sellmeier equations for Ne, Ar, and Kr over the entire VIS/IR range, but importantly also VUV.
• Kingston 1964: VUV data for Ar and Kr.
• JCT-style modified Sellmeier fitting: used here to combine Kingston VUV data with Börzsönyi VIS/IR data.

The goal is to validate the hypothesis that the increase of the refractive index of the gas in the VUV is not limiting the phase-matching of RDW to wavelengths in the VUV.

Details

PhysData.jl

This PR adds new gas symbols and refractive-index models:

• :NeBideauMehu
• :ArBideauMehu
• :KrBideauMehu
• :ArB
• :KrB

The original Börzsönyi models are now available as explicit B variants, while the added Ar/Kr fits that have VUV-extension based on Kingston data plus Börzsönyi VIS/IR points are now set as the default :Ar/:Kr variants. Please see details on the fit construction in the attached notebook.
TestForPullRequestRefIndices.ipynb

Tools.jl

This PR also adds an RDW phase-matching function λRDWfull, based on Novoa et. al., 2015, that can additionally include nonlinear and ionisation/plasma terms to the usual linear phase mismatch term.

There are a few subtleties in this function, to do with the root-finding algorithm.

λlims needs to be set appropriately, otherwide the root-finding algorithm would not return a result (it needs the function to have opposite signs on the two sides of the range).

• For just linear contribution to the phase-matching equation λlims is set to minimum just above the first resonance in the Sellmeier equation for the gas (90 nm for He, 100 nm for Ne, 110 nm for Ar, and 130 nm for Kr), and maximum of the pump wavelength λp minus 10 nm
• When including the ionisation contribution, this lifts the phase-matching curve, creating a second root close to λp. Then the maximum of λlims is modified by first looking for the phase-matching wavelength λlin for just the linear contribution included. Then looking for another root λmax in the range λlin + 50 nm (to compensate for potential ionisation-related red-shift of the phase-matched RDW wavelength) to λp minus 10 nm. Lastly, the RDW phase-matching that includes the ionisation term is searched in the range minimum(λlims) to λmax minus 10 nm.

Analysis summary

The accompanying notebook compares the refractive-index models and resulting capillary dispersion for He, Ne, Ar, and Kr. The key outputs are:

• Refractive index, plotted as n - 1, over the UV/VIS/IR range.
• Capillary group-velocity dispersion β₂, over the UV/VIS/IR range.
• Zoomed-in β₂ in the VUV range.
• RDW phase-matched wavelength as a function of zero-dispersion wavelength. This is calculated with the linear part of the phase-matching curve only.

The analysis uses a 250 µm core-diameter capillary pumped at 800 nm (like the experiment) and compares different refractive index models predictions against the shortest experimentally observed RDW limits used in the VUV analysis.

Key conclusions

• Helium: The difference between the Börzsönyi and JCT helium models in the VUV is the largest of the gases considered. However, this seems to not affect the phase-matching too much.
• Neon: The Börzsönyi and Bideau-Mehu models are quite consistent over the entire range including VUV and VIR/IR, though diverging more very further in the VUV.
• Argon: All the models are pretty much consistent over the entire spectral range, but have increased discpepancy further in the VUV. But in the range where we see the RDW limits, the models are consistent.
• Krypton: Same as argon.
• RDW phase matching: The different refractive-index models shift the predicted VUV RDW wavelength, particularly in the deep VUV. Interestingly, this is a relative small shift, compared to the relatively big difference of β₂ for different refractive index models for some cases (helium vs argon).

Results:

Helium model comparison

Helium refractive index

Helium GVD

Helium GVD VUV zoom

Helium RDW phase matching

Neon model comparison

Neon refractive index

Neon GVD

Neon GVD VUV zoom

Neon RDW phase matching

Argon model comparison

Argon refractive-index comparison

Argon GVD

Argon GVD VUV zoom

Argon RDW phase matching

Krypton model comparison

Krypton refractive-index comparison

Krypton GVD

Krypton GVD VUV zoom

Krypton RDW phase matching

References

• Bideau-Mehu et al., 1981 — https://doi.org/10.1016/0022-4073(81)90057-1
• Börzsönyi et al., 2008 — https://doi.org/10.1364/AO.47.004856
• Ermolov et al., 2015 — https://doi.org/10.1103/PhysRevA.92.033821
• Kingston et. al., 1964 — https://doi.org/10.1364/JOSA.54.001145
• Novoa et. al., 2015 - https://doi.org/10.1103/PhysRevLett.115.033901