ndispers

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ndispers is a Python package for calculating refractive index dispersion of various crystals and glasses used in the field of nonlinear/ultrafast optics. It is based on Sellmeier equations and thermo-optic coefficients (dn/dT) reported in literature.

You can easily compute

• Refractive index
• Group delay
• Group velocity
• Group index
• Group velocity dispersion
• Third-order dispersion
• Walk-off angles
• dn/dT
• d²n/dT²

as a function of

1. Wavelength of light
2. Polar (theta) or azimuthal (phi) angle of the wavevector with respect to the dielectric principal axes of an anisotropic crystal
3. Temperature of the medium
4. Polarization of light (ordinary or extraordinary ray)

The crystals also have nonlinear-optics methods:

• Phase mismatch, dk
• Phase-matching angles
• Phase-matching factor, sinc²(dk·L/2)
• Effective nonlinear coefficient, d_eff, with the tensor components scaled to the working wavelengths by Miller's rule (every non-centrosymmetric crystal in the package)

Installation

Requires Python 3.9 or later.

pip install ndispers

The only dependency is numpy: the dispersion formulas are written as sympy expressions, but the functions the package evaluates are generated from them ahead of time and shipped. To work with the expressions themselves (n_expr and friends), or to evaluate your own subclass of a medium, add sympy with pip install ndispers[sym].

Quick start

For a guided tour with plots — from inspecting a crystal's Sellmeier equation to phase-matching curves — see the tutorial notebook, readable directly on GitHub. The essentials:

Make an object of a β-BBO crystal:

>>> import ndispers as nd
>>> bbo = nd.media.crystals.BetaBBO_Eimerl1987()

To compute a refractive index:

>>> bbo.n(0.532, 0, 25, pol='o')
1.674884049110459
>>> bbo.n(0.532, 3.1416/2, 25, pol='e')
1.5554658787539917

where the four arguments are, respectively,

1. wavelength (in micrometer),
2. theta angle (in radian),
3. temperature (in degree Celsius),
4. polarization (pol='o' or 'e', ordinary or extraordinary ray; default 'o').

At theta = 0 (propagation along the optic axis) the o-ray and e-ray indices coincide. Scalar input returns a float. Each argument also accepts a numpy array, returning an array of the same shape:

>>> import numpy as np
>>> bbo.n(np.arange(0.2, 1.5, 0.2), 0, 25, pol='o')
array([1.89001202, 1.69328828, 1.66985875, 1.6612891 , 1.65633946,
       1.65252664, 1.64903624])

The other dispersion methods (GD, GV, ng, GVD, TOD, woa_theta, woa_phi, dndT, dndT2) take the same arguments:

>>> bbo.GVD(0.8, 0, 25, pol='o')   # fs^2/mm
71.86403019943364

Optically isotropic media — the glasses and the cubic crystal CaF₂ — take only wavelength and temperature, since there is no angle or polarization to specify:

>>> silica = nd.media.glasses.FusedSilica()
>>> silica.n(1.064, 20)
1.4495857898590634

To look into the material information, its Sellmeier equation and the literature the coefficients come from, use help(bbo) — or simply bbo? in IPython/Jupyter. bbo.constants returns the coefficient values themselves.

Phase matching

Phase-matching angles for sum-frequency generation are solved directly. For Type-I SHG of 1064 nm in β-BBO at 25 °C:

>>> bbo.pmAngles_sfg(1.064, 1.064, 25, deg=True)
{'wl3': 0.532,
 'ooe': {'theta': [22.884169498625802], 'phi': None},
 'eeo': {'theta': [], 'phi': None},
 'oee': {'theta': [32.56045545648089], 'phi': None},
 'eoe': {'theta': [32.56045545648089], 'phi': None},
 'eoo': {'theta': [], 'phi': None},
 'oeo': {'theta': [], 'phi': None}}

For difference-frequency generation and optical parametric amplification/oscillation the same interaction is entered from the pump side: pmAngles_dfg(wl_p, wl_s, T) gives the angles (and the idler), tuning_dfg(wl_p, angle, T, pol_s, pol_i, pol_p) the signal/idler pairs that phase-match at a fixed angle — one point of an OPO tuning curve — and dk_dfg, pmFactor_dfg, qpm_period_dfg, deff_dfg mirror their SFG counterparts with waves read as (signal, idler, pump). For x-cut KTP pumped at 1064 nm, KTP_zx().tuning_dfg(1.064, np.pi/2, 25, 'o', 'e', 'o') returns the familiar noncritical pair 1571 / 3298 nm; with qpm_period= the same method gives the temperature tuning of a periodically poled crystal.

dk_sfg and pmFactor_sfg give the phase mismatch and the sinc² phase-matching factor for arbitrary angles, and qpm_period_sfg the quasi-phase-matching period — for 1064 nm SHG in 5% MgO:LiNbO₃ along x with all waves extraordinary (d33), MgOLN_Zelmon1997().qpm_period_sfg(1.064, 1.064, np.pi/2, 20, 'e', 'e', 'e') gives the familiar ~7.0 µm.

The effective nonlinear coefficient at that angle, for the φ = 90° cut:

>>> bbo.deff_sfg(1.064, 1.064, np.radians(22.88), np.radians(90), 25, 'o', 'o', 'e')
-1.9937...  # pm/V (the overall sign is a convention); d22 = 2.2 pm/V at 1.064 µm SHG
            # (Shoji et al. 1999), walk-off included
>>> bbo.d_sfg("d22", 0.8, 0.8, 25)
2.3300...   # d22 for 800 nm SHG by Miller's rule

Every non-centrosymmetric crystal has this (KDP, CLBO, LBO, KTP, ...); help(bbo.deff_sfg) and the Conventions page describe the sign and angle conventions, and each crystal's _d_note says where its coefficients come from and how far Miller scaling has been tested for it.

Available media

Every medium is a class in nd.media.crystals or nd.media.glasses; the media catalog lists the class names with each one's Sellmeier equation, validity range and references. Several crystals come in more than one parameterisation, named after the literature or vendor the coefficients come from — pick the source you want to rely on.

Nonlinear optical crystals

Non-centrosymmetric: phase matching, acceptance bandwidths and d_eff are available.

Birefringent optical crystals

Centrosymmetric, so no second-order nonlinearity; dispersion, walk-off and thermo-optics for windows, polarizers and compensators.

Optically isotropic media

One refractive index, no angle or polarization argument: methods take (wl_um, T_degC).

Where a material has several parameterisations they are not interchangeable — each is faithful to its own source, and some sources are better than others. The validation page says which to reach for and why.

For biaxial crystals one class per principal dielectric plane is provided; the angle argument is the one that varies in that plane (φ in xy, θ in yz and zx), and an instance's theta_rad / phi_rad attributes say which. Media whose Sellmeier equation carries no temperature term (α-BBO, calcite, sapphire, quartz, MgF₂, 5% MgO:LiNbO₃, BiBO, the mid-infrared crystals, YAG, N-BK7, LiF, BaF₂, and the rest of the isotropic set) still take the temperature argument for a uniform signature and ignore it; their dndT returns 0.

Parallel processing

Medium objects are picklable, including after use, so they can be passed directly to multiprocessing, concurrent.futures or joblib workers, and stored with pickle/shelve. Lambdified dispersion functions are cached per class, so constructing many instances of the same crystal is cheap.

Documentation

• Browser apps — a refractive-index explorer and a phase-matching calculator, running client-side with nothing to install.
• Tutorial notebook — the basic workflow, from inspecting a crystal's Sellmeier equation to phase-matching curves, viewable directly on GitHub.
• Validation — what has been checked against the literature, with the numbers and their sources, and the caveats the audit turned up.
• ndispers.readthedocs.io — conventions (units, angles, sign conventions) and the media catalog of every crystal and glass with its formula, validity range and references.

Development

git clone https://github.com/akihiko-shimura/ndispers.git
cd ndispers
uv sync
uv run pytest

To run the tutorial notebook, add the notebook group (matplotlib, IPython, JupyterLab, marimo):

uv sync --group notebook
uv run jupyter lab examples/basic_usage.ipynb

Releases are published to PyPI by pushing a version tag; see docs/RELEASING.md.

License

MIT — see LICENSE.