Solids: conventions and models#

Glasses, crystals and metals share one subpackage, optmat.solid (TC-318). A solid has no gas state: its optical response is a property of the material, its grade and its temperature. This chapter defines the conventions and common complex-index interface. Start with the solid example in Quick start, then follow Fused silica, Crystalline silicon, The window crystals or The metals for model choices and evidence. The source catalogue and The curated data format provide the recorded citations, sample conditions and corrections.

Conventions#

  • The complex index:

    • n + ik, with k >= 0 for absorption; the intensity falls as exp(-4 pi k z / lambda).

    • The relative permittivity is epsilon = (n + ik)^2, with Im epsilon >= 0.

    • This is the convention of Palik’s handbook and of refractiveindex.info.

    • Henke’s n = 1 - delta - i beta (X-ray tables) has the opposite sign of the imaginary part.

  • Wavelengths are vacuum wavelengths, in m at every interface.

  • Absolute indices. Every model returns the absolute index (relative to vacuum). Minimum-deviation refractometry in room air gives the index relative to that air, about 2.7e-4 lower. A dataset or formula says which with index_medium (vacuum or air, with the air’s temperature and pressure); a formula relative to air is multiplied by the index of that air, from Edlen’s dry-air formula at its temperature and pressure (optmat.conditions.air_refractivity(); TC-398). Malitson’s and Rodney & Spindler’s indices are relative to air (TC-399), and so are Dodge’s. Daimon & Masumura measured CaF2 in nitrogen: a formula says index_medium = "nitrogen", with the nitrogen’s temperature and pressure, and is multiplied by the index of optmat’s N2 model at the ideal-gas density (TC-545).

  • Frequencies: omega = 2 pi c / lambda, in rad/s.

  • Photon energies: E = hbar omega / e, in eV (CODATA 2018 through scipy).

  • Temperature: a solid dataset records its sample temperature as parameters.temperature_k when the source states it. A model states the temperature it refers to.

  • Grade: a glass’s grade (Corning 7980, Suprasil 1, Lithosil Q2, …) is recorded per dataset. Grades of fused silica differ in OH and metal impurities and in thermal history. This moves the ultraviolet absorption edge and the 2.7 µm OH band, and n at the level of 1e-5.

  • Anisotropic crystals: a dataset of a uniaxial or biaxial crystal names its axis (o, e; n_p, n_m, n_g or x, y, z), and so does a model.

  • Films differ from bulk material (density, stoichiometry, porosity). Their data carry form = "film", and a bulk model uses them only as stated in its register entry.

The materials are registered in data/materials.toml (optmat.materials); see The curated data, Materials.

The model interface#

Every model of a solid (optmat.solid.models.base.Model) gives, at vacuum wavelengths:

  • permittivity(wavelength): epsilon;

  • dpermittivity_domega(wavelength): the exact derivative d epsilon / d omega;

  • refractive_index(wavelength): n + ik, computed so that a small k keeps its full relative precision (a plain complex square root returns k = 0 below about 1e-16 n);

  • dindex_domega, group_index and gvd.

A model warns (ModelRangeWarning) outside the range its source supports. This matches what a propagation code needs of a bulk medium or a capillary cladding: epsilon(omega) and its derivative, a name, a citation and a range.

The models of a material, and the recommended one with the reason, come from optmat.solid.models.registry (models, model, default_model, default_reason; data/solid/defaults.toml). An anisotropic crystal’s recommendation is per axis, default_model("MgF2", "o"), and asking without the axis is an error (TC-546). A lookup by name reads only what the model needs: a published formula reads its own source record (published_model in optmat.solid.models.formulas), not every record with formulas, so a propagation code pays a fraction of a second.

Published formulas#

A source record’s [[formulas]] entry for a solid becomes a model when it has one of these forms (optmat.solid.models.formulas):

Form

Formula

Terms

n2_sellmeier

n^2 - 1 = A + sum_i B_i lambda^2 / (lambda^2 - C_i), lambda in um

A (optional), B, C (um^2)

n2_sellmeier_wavelength

the same, with the poles printed as wavelengths, C_i = L_i^2

B, L (um)

n2_sellmeier_temperature

n^2 - 1 = sum_i S_i(T) lambda^2 / (lambda^2 - lambda_i(T)^2), S_i and lambda_i polynomials in T (K)

S1, S2, … and L1, L2, … (lowest power first), with reference_temperature_k and temperature_range_k

udm

Franta et al.’s universal dispersion model (below)

parameter_file

The coefficients are the printed strings, corrected where the record says why, as for the gases.

A temperature-dependent formula is evaluated at its reference_temperature_k; at_temperature(T) gives it at another temperature, and warns outside the stated range (Leviton & Frey’s fused silica, 30-300 K: leviton_2008/sellmeier).

Franta’s universal dispersion model#

The universal dispersion model (UDM) of Franta, Nečas and co-workers builds the complex dielectric function of a solid from its excitations (Proc. SPIE 9628, 96281U (2015), franta_2015b):

  • the interband transitions, a symmetric term plus excitonic terms;

  • the higher-energy excitations of the valence electrons;

  • the Urbach tail;

  • Gaussian-broadened absorption on localised states and phonons.

Each contribution’s imaginary part is a simple model of a joint density of states, normalised to unit transition strength (the integral of epsilon_i E over E). Its real part is the Kramers-Kronig transform in closed form. The model is therefore causal and passive by construction, and the transition strengths satisfy the Thomas-Reiche-Kuhn sum rule.

optmat.solid.udm implements it from the paper’s equations. Its parameters are the values of the authors’ parameter files from their program newAD2, which the source records hold (TC-320). It is checked against the authors’ own tables and program, against 50-digit arithmetic, and against a numerical Kramers-Kronig transform (TC-321):

  • above 1 eV it reproduces the published n and k of all seven files to half a unit of their seventh printed digit;

  • below 1 eV the published tables carry the authors’ program’s approximations (at most 2 units of the seventh digit for silica), and optmat’s values are the exact ones.

Two details follow the program rather than the paper, because the program reproduces the published tables and the parameters were fitted with it:

  • the Urbach tail ends at E_h, where the paper’s Eq. (33) defines E_m = (E_g + E_h)/2;

  • the phonon wavenumbers are converted with the program’s constants.

Why a model and not a table. A spline of a tabulated n and k is continuous, but its derivatives are the spline’s, not the material’s. It gives noisy group-velocity dispersion and wrong derivatives near absorption features. Franta’s table is a sampling of this analytic model; evaluating the model itself removes the problem exactly, with no smoothing.

  • Differentiability: the closed forms are analytic between the few band edges where a joint density of states has a kink (E_g, E_h, E_xi, …). There epsilon remains continuous, and its higher derivatives have at most logarithmic singularities, at isolated energies inside absorption bands.

  • Derivatives: d epsilon / dE is exact to rounding (complex-step differentiation).

  • Low energies: far below each band the real part is evaluated from its moment series, which avoids the cancellation that costs the closed forms digits.

Tables of optical constants#

A metal has no single model from the ultraviolet to the far infrared. Its constants are measured piecewise, each range by a different technique: electron energy-loss spectra, ellipsometry, reflectance with a Kramers-Kronig analysis, far-infrared reflectance. A table model (optmat.solid.models.tabulated, named table/<material>) joins curated datasets, one pair per wavelength range, as data/solid/tables/<material>.toml chooses them, each range with its reason (TC-544, TC-645):

material = "Ag"

[[segments]]
real = "babar_2015/..."        # refractive_index, refractive_index_decrement or
imaginary = "babar_2015/..."   # permittivity_real; its extinction_coefficient or
range_nm = [70.5, 12400.0]     # permittivity_imaginary
reason = "..."

  • A segment takes the wavelengths its two datasets share. An x-ray table may print n to some energy and 1 - n (refractive_index_decrement) above it, with k throughout; a permittivity pair is converted to n + ik at the points (TC-645).

  • In the soft x-ray range a segment may instead be Henke et al.’s atomic scattering factors (henke = "<element>", CXRO’s files in data/external/henke/) with a curated atom_density dataset: n + ik = 1 - r_e λ² N (f1 - i f2)/(2π). This independent-atom limit fails near the plasma and absorption edges; no recommended table uses it yet (TC-645).

  • The ranges may not overlap. Nothing is averaged: where two sources cover the same wavelengths, the configuration chooses one and says why.

  • The model interpolates log n and log k against log omega with a monotone piecewise-cubic Hermite interpolant (PCHIP). Unlike a natural cubic spline, it neither overshoots between points nor rings at a join between sources. Its first derivative is continuous and exact for the interpolant, so epsilon and d epsilon / d omega agree.

  • Outside the tabulated range the constants are held at the end points, with a ModelRangeWarning.

Group-velocity dispersion from a table is the interpolant’s, not the material’s: between the points it carries the table’s noise and the interpolant’s piecewise second derivative. For a metal this rarely matters (a metal is a reflector or a cladding, not a propagation medium), but the caveat of the previous section applies.