Refractive-index models¶

A Sphere takes its refractive index as a model of the wavelength, from scatrs.refractive. Ensemble.compute evaluates the model once at the requested wavelengths; the values used are in the result’s refractive_index. scatrs has no database of materials: the data or coefficients come from the user.

Conventions¶

\(m = n + ik\) relative to the surrounding medium, with \(k \ge 0\) for absorbing particles (Conventions). Codes and tables that write \(m = n - ik\) need the complex conjugate.
Wavelengths are in the unit of the radii, and the coefficients of Cauchy and Sellmeier are in that unit too. Published coefficients usually assume micrometres: with radii in nanometres, convert them (Cauchy’s \(B\) scales by \(10^6\) and \(C\) by \(10^{12}\); Sellmeier’s \(C_i\) by \(10^6\)).
There is no extrapolation. A model evaluated where it has no valid value (a wavelength outside a table, a Sellmeier law with \(n^2 \le 0\), a callable returning \(k < 0\) or a value that is not finite) raises InvalidInputError naming the wavelength.

Models¶

Model	\(m(\lambda)\)	Parameters
`Constant(m)`	\(m\)	`n`, `k`
`Table(wavelength, m, interpolation="linear")`	\(n\) and \(k\) linear in \(\lambda\) between the nodes; `"log_k"` interpolates \(\ln k\) instead	none
`Cauchy(A, B=0, C=0, k=0)`	\(n = A + B/\lambda^2 + C/\lambda^4\), constant \(k\)	`A`, `B`, `C`, `k`
`Sellmeier(B, C, k=0)`	\(n^2 = 1 + \sum_i B_i\lambda^2/(\lambda^2 - C_i)\), constant \(k\)	`B0`, …, `C0`, …, `k`
`Function(f)`	`f(wavelength)`	none

A number given to Sphere becomes a Constant, and any other callable a Function. An array is not accepted: values of \(m\) mean something only together with their wavelengths, which a Table holds.

A Table needs at least two nodes, at strictly increasing wavelengths. Linear interpolation suits smooth \(n\) and \(k\); absorption that changes by decades between nodes is better interpolated in \(\ln k\) ("log_k", which needs \(k > 0\) at every node). The table is data: it has no parameters.

Each model lists its parameters in parameters, and derivatives(wavelength) gives \(\partial m/\partial\theta\) for each. The derivatives of an ensemble with respect to "m_real" and "m_imag" are those with respect to \(\mathrm{Re}\,m\) and \(\mathrm{Im}\,m\) at each wavelength, whatever model produced \(m\).

Example¶

Fused silica from its Sellmeier coefficients (Malitson, 1965, \(\lambda\) in µm), and an absorbing material from a short table, interpolated in \(\ln k\):

import numpy as np

import scatrs
from scatrs import refractive

silica = refractive.Sellmeier(
    B=[0.6961663, 0.4079426, 0.8974794],
    C=[0.0684043**2, 0.1162414**2, 9.896161**2],
)
dust = refractive.Table(
    [0.3, 0.5, 1.0, 2.0],
    [1.55 + 5e-3j, 1.53 + 1.5e-3j, 1.52 + 8e-4j, 1.50 + 2e-3j],
    interpolation="log_k",
)
wavelength = np.array([0.35, 0.55, 1.6])
print("silica:", silica.evaluate(wavelength).real)
print("dust:  ", dust.evaluate(wavelength))
assert abs(silica.evaluate(0.5893) - 1.4584) < 1e-4  # the sodium D line

r = scatrs.Ensemble(scatrs.Sphere(dust), scatrs.LogNormal(0.5, 1.8)).compute(wavelength)
print("C_ext:", r.c_ext, "µm²; m used:", r.refractive_index)
assert np.array_equal(r.refractive_index, dust.evaluate(wavelength))

silica: [1.47689141 1.45991089 1.443419  ]
dust:   [1.545+0.00370041j 1.529+0.00140861j 1.508+0.00138629j]
C_ext: [3.63928857 3.88869459 4.50411128] µm²; m used: [1.545+0.00370041j 1.529+0.00140861j 1.508+0.00138629j]