:orphan:

Gallery
=======

Figures of the curated data, models and validation, generated from committed records
when their example scripts run during the documentation build. The scientific
chapters in the :doc:`/user-guide/index` explain the conventions, model choices and
uncertainties. Use the :doc:`/the-data/sources` catalogue to find each source record.

Comparison curves labelled ``luna_physdata`` are the formulas recorded in that
source, not a claim about current Luna.jl. The record identifies the ``modal-fixed``
branch and PR #425, including argon's commit ``56cc3ab``; it does not give a single
commit for every formula. Reference-state conventions are retained per formula.
See :doc:`/generated/sources/luna_physdata` and :doc:`/models` before interpreting
these comparisons.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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The curated data
----------------

Every dataset of each gas against the fitted model, where the data cover the spectrum,
and how independent datasets check one another.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="One row per gas, one tick per curated value that bears on the linear dispersion, coloured by what it is: a source&#x27;s own measurement, a calculation, or a value reproduced from a source outside the library. Static values (the polarizability at zero frequency) sit at the right edge. The gaps are where new data would help most.">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_coverage_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_coverage`

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      <div class="sphx-glr-thumbnail-title">Wavelength coverage of the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Two kinds of measurement constrain dispersion without giving a refractive index (optmat.gas.crosschecks). Shelton&#x27;s phase-matching density ratios tie the near-infrared dispersion \alpha(2\omega) - \alpha(\omega) of two gases together to about 0.1 %; Börzsönyi et al. measured the group-delay dispersion at 800 nm.">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_crosschecks_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_crosschecks`

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      <div class="sphx-glr-thumbnail-title">Cross-checks: phase-matching ratios and group-delay dispersion</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct argon dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Bideau-Mehu et al.&#x27;s formula.">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_ar_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_ar`

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      <div class="sphx-glr-thumbnail-title">Argon: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct methane dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from He et al.&#x27;s formula (Eq. 10).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_ch4_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_ch4`

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      <div class="sphx-glr-thumbnail-title">Methane: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct carbon dioxide dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Old, Gentili &amp; Peck&#x27;s formula.">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_co2_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_co2`

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      <div class="sphx-glr-thumbnail-title">Carbon dioxide: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct deuterium dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. No published formula for this gas is implemented yet, so there is no deviation panel.">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_d2_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_d2`

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      <div class="sphx-glr-thumbnail-title">Deuterium: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct hydrogen dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Peck &amp; Huang&#x27;s two-term formula (Eq. 7).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_h2_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_h2`

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      <div class="sphx-glr-thumbnail-title">Hydrogen: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct helium dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Mansfield &amp; Peck&#x27;s two-pole fit to their own theoretical values (Eq. 8, corrected), which is smooth down to the first resonance.">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_he_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_he`

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      <div class="sphx-glr-thumbnail-title">Helium: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct krypton dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Bideau-Mehu et al.&#x27;s formula (known to lie 0.1-0.7 % below the paper&#x27;s own column).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_kr_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_kr`

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      <div class="sphx-glr-thumbnail-title">Krypton: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct nitrogen dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Peck &amp; Khanna&#x27;s formula at 0 °C (Eq. 4).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_n2_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_n2`

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      <div class="sphx-glr-thumbnail-title">Nitrogen: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct nitrous oxide dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from He et al.&#x27;s formula (Eq. 9).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_n2o_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_n2o`

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      <div class="sphx-glr-thumbnail-title">Nitrous oxide: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct neon dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Bideau-Mehu et al.&#x27;s formula (corrected).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_ne_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_ne`

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      <div class="sphx-glr-thumbnail-title">Neon: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct oxygen dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Kren&#x27;s formula, derived from air (Kren 2011).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_o2_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_o2`

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      <div class="sphx-glr-thumbnail-title">Oxygen: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct sulphur hexafluoride dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from He et al.&#x27;s formula (Eq. 8).">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_sf6_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_sf6`

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      <div class="sphx-glr-thumbnail-title">Sulphur hexafluoride: the curated data</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every direct xenon dataset (a refractivity, polarizability or molar refractivity) turned into the refractivity of the real gas at 0 °C and 1 atm, with the published formulas (solid), the recorded luna_physdata formulas (dashed) and refractiveindex.info&#x27;s entries (dotted), each over the range its source supports. The lower panel shows the deviations from Bideau-Mehu et al.&#x27;s formula (corrected), valid on both sides of the 146.9 nm line.">

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  .. image:: /auto_gallery/data/images/thumb/sphx_glr_plot_data_xe_thumb.png
    :alt:

  :doc:`/auto_gallery/data/plot_data_xe`

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      <div class="sphx-glr-thumbnail-title">Xenon: the curated data</div>
    </div>


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    </div>

The dispersion models
---------------------

Each gas's fitted model of the refractive index against the data it rests on, and the
noble gases in the extreme ultraviolet.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="The refitted CO2 model (OEP 3) from 0.25 to 12 µm at 1 atm and 296 K. Top: n - 1. Bottom: the group-velocity dispersion&#x27;s driver, |D_2| = |d^2(\omega\alpha)/d\omega^2|, proportional to \beta_2 in the dilute limit, with its statistical one-sigma uncertainty and the representation bound of the infrared terms. The shaded regions are the gaps between the qualified intervals, where the gas absorbs or the bands&#x27; Gaussian rules do not meet the tolerance. The model is not to be used there, and warns.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_co2_windows_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_co2_windows`

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      <div class="sphx-glr-thumbnail-title">CO2 from the ultraviolet to its infrared windows</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Each molecule&#x27;s compact model (OEP 2) keeps its fit&#x27;s electronic terms and replaces its vibrational poles, thousands of them for CO2 and N2O, by three to five effective ones. Top: the reduction&#x27;s residuals, compact minus parent, in units of the parent&#x27;s statistical one-sigma, for the polarizability and the dispersion derivatives D_m = d^m(\omega\alpha)/d\omega^m (D_2 drives the group-velocity dispersion, D_3 the third-order dispersion), over each compact model&#x27;s interval. The tolerance is 0.1 (dashed). Bottom: the infrared terms&#x27; share of D_2, which the compact poles have to carry: below 0.1 % in the ultraviolet, 1-17 % at 800 nm, and at the long-wavelength end more than the electronic part, so that \beta_2 changes sign inside the interval of every molecule but CH4 (CO at 1969 nm, CO2 at 1315 nm, N2O at 1465 nm, SF6 at 1271 nm), where the share diverges.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_compact_models_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_compact_models`

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      <div class="sphx-glr-thumbnail-title">The compact models against their parents</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The complex refractive index of helium at 1 bar and 20 °C from 20 to 120 nm, from its constrained oscillator-strength distribution (``euv/He``): Samson et al.&#x27;s measured photoionisation cross section, Henke&#x27;s scattering factors above 120 eV and the NIST Rydberg lines. Above the ionisation threshold (50.4 nm) \mathrm{Re}\,n - 1 is negative and the absorption index \mathrm{Im}\,n appears; between the first line (58.4 nm) and the threshold the Rydberg lines make the index singular. For comparison: the index from Henke&#x27;s f_1 and f_2 (dotted, unreliable below about 30 eV by Henke et al.&#x27;s own account), Liu&#x27;s ab initio absorption (squares) and the ab initio fit fit/He below the first resonance (dashed).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_euv_he_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_euv_he`

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      <div class="sphx-glr-thumbnail-title">Helium in the extreme ultraviolet</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The complex refractive index at 1 bar and 20 °C, from 20 nm to beyond the first resonance, from each gas&#x27;s constrained oscillator-strength distribution (``euv/Ne``, euv/Ar, euv/Kr, euv/Xe): Chan et al.&#x27;s oscillator strengths, Henke&#x27;s scattering factors at high energy. Shaded: the region of the discrete lines, where the index is singular and the model carries no line widths. Dashed: the fitted model below the first resonance.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_euv_heavy_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_euv_heavy`

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      <div class="sphx-glr-thumbnail-title">Neon, argon, krypton and xenon in the extreme ultraviolet</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The argon model (``fit/Ar``): the 104.8 nm and 106.7 nm resonance lines at their NIST energies and two effective oscillators, with the magnetic susceptibility \chi_0 = -2.30\times10^{-4} a.u. that makes the electric polarizability the refractive one. The absolute level comes from Lesiuk &amp; Jeziorski&#x27;s ab initio Cauchy moments, the dielectric static polarizability (Gaiser &amp; Fellmuth, via Lesiuk) and Egan&#x27;s refractometry at 633 nm, which agree to a few 1e-5; the shape from 468 nm to 2.06 um from Peck &amp; Fisher, whose level is 3.5e-4 high and is fitted free; the vacuum ultraviolet from Bideau-Mehu and Gordon. Squares are validators, not fitted: Zhang&#x27;s comb value at 800 nm (+2.6e-4), Birch at 633 nm, Borzsonyi, Rusch&#x27;s infrared point, and the Lyman-alpha values of Gill and Heddle.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_ar_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_ar`

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      <div class="sphx-glr-thumbnail-title">Argon: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The methane model (``fit/CH4``): two effective electronic oscillators fitted to the Cuthbertsons&#x27; (1920) and Watson &amp; Ramaswamy&#x27;s visible dispersion (level free), Achtermann&#x27;s 633 nm and Burns et al.&#x27;s 458-633 nm values, Zhao et al.&#x27;s 1566 nm value and Shelton&#x27;s N2/CH4 ratios, with HITRAN&#x27;s vibrational bands below 4500 cm^-1 fixed (reduced chi-squared 0.68; TC-797). Shown from 240 nm to 1.4 um, across the absorption gaps; the qualified intervals (240 to 1123 nm and the infrared windows) are listed in /gas_models (TC-797).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_ch4_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_ch4`

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      <div class="sphx-glr-thumbnail-title">Methane: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The CO model (``fit/CO``): two effective electronic oscillators fitted to Smith et al.&#x27;s 167-288 nm interferometry, the Cuthbertsons&#x27; 1920 visible dispersion (level free), Watson &amp; Ramaswamy&#x27;s green line and Zhao et al.&#x27;s 2026 refractometry at 1566 nm, with HITRAN&#x27;s fundamental and first overtone fixed (reduced chi-squared 0.37). Shown over the optical range. The main qualified interval extends from 167 to 2288 nm, with two further infrared windows; see /gas_models (TC-797).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_co_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_co`

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      <div class="sphx-glr-thumbnail-title">Carbon monoxide: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The CO2 model (``fit/CO2``): two effective electronic oscillators fitted to Old, Gentili &amp; Peck (480-1600 nm, level free), the visible data, the 633 and 1542 nm metrology, Hohm&#x27;s 325-633 nm dispersion, Shelton&#x27;s N2/CO2 ratios with Marchetti&#x27;s CO2-laser values as validators, with HITRAN&#x27;s vibrational bands below 7000 cm^-1 fixed (reduced chi-squared 4.01; TC-797). Shown over the model&#x27;s main range, 250 nm to 1.4 um; seven additional qualified infrared intervals extend to 8.941 um, with gaps listed in /gas_models.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_co2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_co2`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Carbon dioxide: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The deuterium model (``fit/D2``) uses Raj et al.&#x27;s (2018) ab initio average for normal deuterium and Shelton&#x27;s D2/H2 and H2/D2 ratios. Published refractometry is shown as independent evidence where identified as a validator. See /gas_models for the current data selection and /accuracy for limitations.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_d2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_d2`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Deuterium: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The hydrogen model (``fit/H2``) is based on Raj et al.&#x27;s (2018) ab initio average for normal hydrogen, with a 0.1 % allowance, the dispersion shapes of Peck and Hung and of Smith et al., and Shelton&#x27;s He/H2 phase-matching ratio. The current model&#x27;s statistical uncertainty and the remaining absolute-scale evidence are summarised in /gas_models and /accuracy.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_h2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_h2`

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      <div class="sphx-glr-thumbnail-title">Hydrogen: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The helium model (``fit/He``): the 1s2p (58.4 nm) and 1s3p (53.7 nm) lines at their NIST energies and four effective oscillators, with the magnetic susceptibility \chi_0 = -2.12\times10^{-5} a.u. that makes the electric polarizability the refractive one. The backbone is Puchalski et al.&#x27;s ab initio dispersion coefficients of 4He, with every relativistic and QED correction (the static term to 1e-7), and Schmidt et al.&#x27;s microwave static polarizability: the model reproduces both to their uncertainties, so its one-sigma band is below 1 ppm from the infrared to 110 nm and 7 ppm at 76 nm.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_he_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_he`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Helium: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The krypton model (``fit/Kr``): the 116.5 nm and 123.6 nm resonance lines at their NIST energies and two effective oscillators. The only ab initio dispersion in the library (Pawlowski et al.&#x27;s CC3 moments) is not converged, so the fit rests on data: the dielectric static polarizabilities of Orcutt &amp; Cole and of Huot &amp; Bose (via Kumar &amp; Thakkar; 3.6 sigma from the others), Birch&#x27;s and Hohm &amp; Kerl&#x27;s 633 nm values, Burns et al.&#x27;s 458-633 nm molar refractivities, Smith, Huber &amp; Parkinson&#x27;s 28 ultraviolet points (0.06 %), Bideau-Mehu&#x27;s revised (1976) and vacuum-ultraviolet (1981) values and the Cuthbertsons&#x27; visible shape (level free). Leonard&#x27;s absolute values lie 0.2 % low and his recommended one 0.04 % high. The magnetic term (about -25 ppm) is neglected. Chopra &amp; Heddle&#x27;s Lyman-alpha value lies between the two resonance lines, where the sign of n - 1 is not measured.">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_kr_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_kr`

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      <div class="sphx-glr-thumbnail-title">Krypton: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The nitrogen model (``fit/N2``): effective oscillators fitted to Griesmann &amp; Burnett&#x27;s vacuum ultraviolet (level free), Peck &amp; Khanna&#x27;s visible and near infrared, and the 633 nm, microwave, dielectric and 10.57 um values agree to their uncertainties (reduced chi-squared 1.1).">

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  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_n2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_n2`

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      <div class="sphx-glr-thumbnail-title">Nitrogen: the fitted model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The N2O model (``fit/N2O``): two effective electronic oscillators fitted to Hohm&#x27;s 325-633 nm dispersion (1e-5 relative) and the Cuthbertsons&#x27; (1914) visible dispersion (both level free) and the 633 nm metrology of Birch and Egan et al., with HITRAN&#x27;s vibrational bands below 4500 cm^-1 fixed and the Thomas-Reiche-Kuhn sum as a prior (reduced chi-squared 3.01: the two dispersions differ in shape by about 1e-4; TC-797). The main qualified interval is 320-1516 nm, with three additional infrared windows; see /gas_models.">

.. only:: html

  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_n2o_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_n2o`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Nitrous oxide: the fitted model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The neon model (``fit/Ne``): the 73.6 nm and 74.4 nm resonance lines at their NIST energies and two effective oscillators, with the magnetic susceptibility \chi_0 = -8.48\times10^{-5} a.u. The level comes from Lesiuk, Przybytek &amp; Jeziorski&#x27;s ab initio Cauchy moments and Gaiser &amp; Fellmuth&#x27;s 2018 dielectric static polarizability (via Lesiuk; their 2010 value is 16 ppm higher); the shape from the Cuthbertsons&#x27; visible data (level free: their standard state is ambiguous, and it comes out 0.3 % high) and Bideau-Mehu&#x27;s vacuum ultraviolet.">

.. only:: html

  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_ne_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_ne`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Neon: the fitted model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The oxygen model (``fit/O2``): effective oscillators fitted to Smith, Huber &amp; Parkinson&#x27;s ultraviolet above the Schumann-Runge bands, Egan et al.&#x27;s 2026 values at 633 and 1542 nm, Birch at 633 nm, Hohm&#x27;s 325-633 nm dispersion, Swift and Marchetti; Zhang&#x27;s 800 nm value (A band) and Kumar &amp; Thakkar&#x27;s sums are validators.">

.. only:: html

  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_o2_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_o2`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Oxygen: the fitted model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The O3 model (``fit/O3``) absorbs. Its imaginary part is the measured absorption spectrum (Serdyuchenko et al. 2014 at 293 K from 213 nm, Yoshino et al. 1993, Mason et al. 1996 and Ogawa &amp; Cook 1958 to 12 eV), smoothed with a 50 cm^-1 Gaussian. Its real part is that spectrum&#x27;s Kramers-Kronig transform, plus:">

.. only:: html

  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_o3_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_o3`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ozone: the fitted model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The SF6 model (``fit/SF6``): two effective electronic oscillators fitted to Swift&#x27;s, Vukovic&#x27;s, Achtermann&#x27;s and Burns et al.&#x27;s values and Shelton&#x27;s N2/SF6 ratios, with HITRAN&#x27;s measured infrared cross-sections as fixed bands (reduced chi-squared 4.1; the two Shelton sets prefer the bands times 1.036, TC-146). The bands give the gas a zero-dispersion wavelength near 1.2 um. Shown from 230 nm to 5 um.">

.. only:: html

  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_sf6_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_sf6`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Sulphur hexafluoride: the fitted model</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The xenon model (``fit/Xe``): the 146.96 nm and 129.56 nm resonance lines at their NIST energies and two effective oscillators, one of which settles at 121.6 nm, between the strong 119.2 nm and the 125.0 nm lines. The level comes from Huot &amp; Bose&#x27;s dielectric static polarizability (via Kumar &amp; Thakkar), Burns et al.&#x27;s 458-633 nm molar refractivities and Hohm &amp; Kerl&#x27;s 633 nm value; the shape from Abjean et al.&#x27;s and Bideau-Mehu et al.&#x27;s ultraviolet and vacuum ultraviolet, which crosses the 146.96 nm line (140.4 nm is between the two lines), and the Cuthbertsons&#x27; visible data (level free). The fitted strength of the 146.96 nm line, 0.2688, is Bideau-Mehu&#x27;s measured 0.268(8). Leonard&#x27;s recommended value is 0.13 % high. The magnetic term (about -25 ppm) is neglected.">

.. only:: html

  .. image:: /auto_gallery/dispersion/images/thumb/sphx_glr_plot_fit_xe_thumb.png
    :alt:

  :doc:`/auto_gallery/dispersion/plot_fit_xe`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Xenon: the fitted model</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Accuracy and comparisons
------------------------

How well the models are known, and how their predictions depend on the published
formulas, measurement conditions and density conventions.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The quantity a change of dispersion model moves most visibly in a gas-filled hollow capillary is the zero-dispersion wavelength (ZDW), where the gas&#x27;s normal dispersion balances the waveguide&#x27;s anomalous term. For the fundamental EH11 mode of a capillary of core radius a (Marcatili &amp; Schmeltzer, lossless part)">

.. only:: html

  .. image:: /auto_gallery/accuracy/images/thumb/sphx_glr_plot_capillary_zdw_thumb.png
    :alt:

  :doc:`/auto_gallery/accuracy/plot_capillary_zdw`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Capillary zero-dispersion wavelength</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The refractive index of a gas at number density N follows from the per-particle polarizability \alpha through the Lorentz-Lorenz relation. The linear scaling \chi = N\alpha/\varepsilon_0 is its dilute limit. This figure shows the relative error of linear scaling in \chi = n^2 - 1 against pressure, for a polarizability typical of each gas.">

.. only:: html

  .. image:: /auto_gallery/accuracy/images/thumb/sphx_glr_plot_density_scaling_thumb.png
    :alt:

  :doc:`/auto_gallery/accuracy/plot_density_scaling`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Linear density scaling against Lorentz-Lorenz</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="How well each fitted model knows the group-velocity dispersion \beta_2 and the third-order dispersion \beta_3: the relative one-sigma uncertainty propagated from the fit&#x27;s covariance (scaled by the reduced chi-squared where it exceeds one; dispersion_sigma_relative), against wavelength over each model&#x27;s range. In the dilute limit these relative uncertainties do not depend on pressure or temperature.">

.. only:: html

  .. image:: /auto_gallery/accuracy/images/thumb/sphx_glr_plot_dispersion_uncertainty_thumb.png
    :alt:

  :doc:`/auto_gallery/accuracy/plot_dispersion_uncertainty`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Dispersion uncertainty and sensitivity to dataset selection</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The group-velocity dispersion \beta_2 = (\lambda^3/2\pi c^2)\,\mathrm{d}^2n/\mathrm{d}\lambda^2 of each noble gas at 1 bar and 20 °C, from the fitted model, and the ratio to it of every published, legacy and comparison formula over its stated range. The dispersion of a gas-filled hollow fibre is the gas&#x27;s \beta_2 against the waveguide&#x27;s anomalous term, so these ratios move zero-dispersion wavelengths and phase-matching points directly.">

.. only:: html

  .. image:: /auto_gallery/accuracy/images/thumb/sphx_glr_plot_gvd_comparison_thumb.png
    :alt:

  :doc:`/auto_gallery/accuracy/plot_gvd_comparison`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Group-velocity dispersion: the fits against the formulas in use</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A 50/50 helium-argon fill at 294 K, from 1 to 60 bar, at 800 nm. The reference is optmat&#x27;s mixture law (OEP 4): the mixture&#x27;s equation of state for the densities and Lorentz-Lorenz of the whole mixture, with its pair terms. Each curve leaves one piece out and shows the relative change in n - 1 (left) and in the group-velocity dispersion (right):">

.. only:: html

  .. image:: /auto_gallery/accuracy/images/thumb/sphx_glr_plot_mixture_he_ar_thumb.png
    :alt:

  :doc:`/auto_gallery/accuracy/plot_mixture_he_ar`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The dispersion of a helium-argon fill against pressure</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Each fit refitted without each of its datasets in turn (``tools/fit/robustness.py``, stored in data/gas/models/robustness): the change of the polarizability at five wavelengths across the model&#x27;s range, in units of the full fit&#x27;s one-sigma uncertainty. A dataset whose removal moves the model by much more than one sigma determines the model there on its own. For the backbones (the ab initio moments of helium, neon and argon, the dielectric static values) that is by design. Two levels rest on one metrology value each: hydrogen&#x27;s on Achtermann&#x27;s (-870 ppm without it) and oxygen&#x27;s on Birch&#x27;s (-840 ppm), where the band understates the uncertainty of the level (TC-101).">

.. only:: html

  .. image:: /auto_gallery/accuracy/images/thumb/sphx_glr_plot_robustness_thumb.png
    :alt:

  :doc:`/auto_gallery/accuracy/plot_robustness`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Robustness: leaving one dataset out</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

The third-order and Raman response
----------------------------------

The second hyperpolarizability of every process, its accuracy against the frozen
coefficients in use, and the gases' Raman response functions.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="For each gas, gamma / gamma_0 of the fitted model (line, the Kerr process) against the dispersion variable W2 = sum of the squared wavenumbers of the four frequencies. The data of every process are shown against their own W2:">

.. only:: html

  .. image:: /auto_gallery/nonlinear/images/thumb/sphx_glr_plot_nonlinear_thumb.png
    :alt:

  :doc:`/auto_gallery/nonlinear/plot_nonlinear`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The second hyperpolarizability: every process on one curve</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The shaded band is the fitted Kerr gamma&#x27;s propagated statistical one-sigma uncertainty, scaled by the reduced chi-squared where it exceeds one. Bars at 400, 800 and 1800 nm are the working estimates in /nonlinear_accuracy, which include leave-one-out sensitivity and each gas&#x27;s systematic floor. They can be much larger than the covariance alone suggests. These are relative uncertainties of the electronic response, not the total nonlinear response or an error bar on extrapolation. Ozone&#x27;s model starts at 700 nm.">

.. only:: html

  .. image:: /auto_gallery/nonlinear/images/thumb/sphx_glr_plot_nonlinear_accuracy_thumb.png
    :alt:

  :doc:`/auto_gallery/nonlinear/plot_nonlinear_accuracy`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Third-order uncertainty: statistical and working estimates</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The gases&#x27; Raman responses (optmat.gas.raman_response.raman_response), at 295 K.">

.. only:: html

  .. image:: /auto_gallery/nonlinear/images/thumb/sphx_glr_plot_raman_response_gases_thumb.png
    :alt:

  :doc:`/auto_gallery/nonlinear/plot_raman_response_gases`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Raman: the response functions of the gases</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Ionisation
----------

The strong-field ionisation rates of the noble gases and of molecules, against the
Schrödinger calculations and the measured yields.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The rate of Perelomov, Popov and Terent&#x27;ev is that of a short-range potential times a Coulomb factor. This figure compares three analytic forms (optmat.gas.ionisation_rates.coulomb_factor). Here each is compared with time-dependent Schrödinger calculations, with everything else the same: one electron in the m = 0 orbital, Tong, Zhao and Lin&#x27;s coefficient and Tong and Lin&#x27;s static correction, so that all three have the same static limit.">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_ionisation_frequency_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_ionisation_frequency`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: the Coulomb factor against Schrödinger calculations</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Two measurements have an intensity scale that does not rest on a rate model.">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_ionisation_measured_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_ionisation_measured`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: the measured yields of argon, krypton and xenon</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The default ionisation rates (optmat.gas.ionisation_rates.ionisation_rate_model).">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_ionisation_rates_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_ionisation_rates`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: the rates of the noble gases</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Van der Hart&#x27;s R-matrix calculations of the whole atom, with one state of the ion (optmat.gas.ionisation_validation.rmatrix_rates): R-matrix Floquet rates of He, Ne and Ar (2006, read from the image of its Fig. 6), and time-dependent R-matrix rates of Ar (2007, its Table I and the vector drawing of its Fig. 4). Values read from figures follow data rule 8 (TC-699 to TC-701).">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_ionisation_rmatrix_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_ionisation_rmatrix`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: the noble gases against all-electron rates at 390 nm</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The static limit of the ionisation rate (optmat.gas.ionisation_rates) against the curated references (optmat.gas.ionisation_validation.static_rates).">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_ionisation_static_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_ionisation_static`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: the static rates of helium, neon and argon</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="optmat&#x27;s rate of molecules and the forms of the literature (optmat.gas.molecular_ionisation_rates.molecular_rate_model).">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_molecular_ionisation_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_molecular_ionisation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: the rates of molecules</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The default rate of molecules (optmat.gas.molecular_ionisation_rates.molecular_rate_model).">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_molecular_ionisation_default_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_molecular_ionisation_default`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: optmat's rate of molecules</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Wahlstrand et al. (2018) measured the absolute yield of N2 and O2 in the same experiment as that of argon, krypton and xenon: 800 nm, 42 fs, the electron density from the refractive index of the plasma, at a local intensity calibrated through the Kerr effect to 13 %.">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_molecular_ionisation_measured_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_molecular_ionisation_measured`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ionisation: the measured yields of N2 and O2</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The two-photon ionisation cross sections of the models of optmat.gas.two_photon.two_photon_ionisation_model (OEP 6, Phase 1), each the curated reference curve the author approved (TC-943):">

.. only:: html

  .. image:: /auto_gallery/ionisation/images/thumb/sphx_glr_plot_two_photon_thumb.png
    :alt:

  :doc:`/auto_gallery/ionisation/plot_two_photon`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Two-photon ionisation of the noble gases</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Solids
------

Fused silica: its complex index, its n2 and its Raman response.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Fused silica&#x27;s Raman models (optmat.solid.raman_response.silica_raman_response).">

.. only:: html

  .. image:: /auto_gallery/solids/images/thumb/sphx_glr_plot_raman_response_silica_thumb.png
    :alt:

  :doc:`/auto_gallery/solids/plot_raman_response_silica`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Raman: fused silica's response</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Top: the recommended model&#x27;s n and k (``fit/SiO2``: Franta et al.&#x27;s universal dispersion model of Lithosil Q2 refitted to the absolute refractometry of the window, TC-399) from 25 nm to 100 um. The points are Philipp&#x27;s tables in Palik&#x27;s handbook (``philipp_1985``); only Hunter&#x27;s and the Philipp-Lamy EUV points are close to Franta&#x27;s measurement-based model, as TC-324 records.">

.. only:: html

  .. image:: /auto_gallery/solids/images/thumb/sphx_glr_plot_silica_thumb.png
    :alt:

  :doc:`/auto_gallery/solids/plot_silica`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Fused silica: the complex index and n2</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:
   :includehidden:


   /auto_gallery/data/index.rst
   /auto_gallery/dispersion/index.rst
   /auto_gallery/accuracy/index.rst
   /auto_gallery/nonlinear/index.rst
   /auto_gallery/ionisation/index.rst
   /auto_gallery/solids/index.rst