.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_gallery/accuracy/plot_capillary_zdw.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_gallery_accuracy_plot_capillary_zdw.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_gallery_accuracy_plot_capillary_zdw.py:


Capillary zero-dispersion wavelength
====================================

.. _capillary-zero-dispersion-wavelength-the-fits-against-solprop-s-formulas:

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's normal dispersion
balances the waveguide's anomalous term. For the fundamental EH11 mode of a capillary of
core radius :math:`a` (Marcatili & Schmeltzer, lossless part)

.. math:: \beta(\omega) = \sqrt{k^2 n_\text{gas}^2(\omega) - u_{01}^2/a^2},
   \qquad k = \omega/c,

with :math:`u_{01}` the first zero of :math:`J_0`. The ZDW is the root of
:math:`\beta_2 = \mathrm{d}^2\beta/\mathrm{d}\omega^2`. Here :math:`a = 75` µm, the
gas at 20 °C. The recommended models are evaluated at each pressure with the
real-gas density, Lorentz-Lorenz relation and curated refractivity virial. The curves
show how pressure shifts the balance between material and waveguide dispersion.
See :doc:`/accuracy` for uncertainty in the material dispersion. This is the
lossless capillary approximation, not a complete waveguide propagation model.

.. GENERATED FROM PYTHON SOURCE LINES 29-86



.. image-sg:: /auto_gallery/accuracy/images/sphx_glr_plot_capillary_zdw_001.svg
   :alt: EH11, 75 µm core radius, 20 °C
   :srcset: /auto_gallery/accuracy/images/sphx_glr_plot_capillary_zdw_001.svg
   :class: sphx-glr-single-img





.. code-block:: Python

    import warnings

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import brentq
    from scipy.special import jn_zeros

    from optmat.conditions import State
    from optmat.constants import C
    from optmat.exceptions import ModelRangeWarning
    from optmat.gas.models.registry import model

    RADIUS = 75e-6
    U01 = float(jn_zeros(0, 1)[0])
    PRESSURES = np.geomspace(0.1e5, 20e5, 25)
    TEMPERATURE = 293.15

    MODELS = {gas: model(f"fit/{gas}") for gas in ("He", "Ne", "Ar", "Kr", "Xe")}


    def beta2(dispersion, omega, state):
        """Return the EH11 mode's beta_2 (s^2/m) by a centred second difference."""
        step = omega * 1e-3
        points = np.array([omega - step, omega, omega + step])
        with warnings.catch_warnings():
            warnings.simplefilter("ignore", ModelRangeWarning)
            n = 1 + dispersion.refractivity(2 * np.pi * C / points, state)
        k = points / C
        beta = np.sqrt((k * n) ** 2 - (U01 / RADIUS) ** 2)
        return (beta[0] - 2 * beta[1] + beta[2]) / step**2


    def zdw(dispersion, pressure):
        """Return the zero-dispersion wavelength in nm (NaN if none in 150-3000 nm)."""
        state = State(pressure, TEMPERATURE)
        grid = 2 * np.pi * C / np.geomspace(3000e-9, 150e-9, 200)
        values = np.array([beta2(dispersion, w, state) for w in grid])
        change = np.nonzero(np.diff(np.sign(values)))[0]
        if len(change) == 0:
            return np.nan
        i = change[0]
        root = brentq(lambda w: beta2(dispersion, w, state), grid[i], grid[i + 1])
        return 2 * np.pi * C / root * 1e9


    fig, ax = plt.subplots(figsize=(8, 5), layout="constrained")
    COLOURS = ["#2a78d6", "#eb6834", "#1baf7a", "#4a3aa7", "#e87ba4"]
    for colour, (gas, fitted) in zip(COLOURS, MODELS.items(), strict=True):
        wavelengths = np.array([zdw(fitted, p) for p in PRESSURES])
        ax.plot(PRESSURES / 1e5, wavelengths, color=colour, label=gas)
    ax.set_xscale("log")
    ax.set_xlabel("pressure (bar)")
    ax.set_ylabel("zero-dispersion wavelength (nm)")
    ax.legend()
    ax.set_title("EH11, 75 µm core radius, 20 °C")
    ax.grid(which="both", alpha=0.25)
    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.409 seconds)


.. _sphx_glr_download_auto_gallery_accuracy_plot_capillary_zdw.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_capillary_zdw.ipynb <plot_capillary_zdw.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_capillary_zdw.py <plot_capillary_zdw.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_capillary_zdw.zip <plot_capillary_zdw.zip>`