treams_rs.coeffs

Mie and Fresnel coefficients in the helicity basis.

Mirrors treams.coeffs. mie and mie_cyl return one complex128 (2, 2) block per call: rows are the scattered and columns the incident helicity, with index 0 for negative and 1 for positive helicity. Materials are (epsilon, mu, kappa): relative permittivity, relative permeability and chirality parameter, ordered from the innermost layer out to the embedding medium. diff.mie, diff.mie_cyl and diff.fresnel return the same values together with a context that computes gradients.

Differences from treams

• treams' functions are generalized ufuncs that broadcast over leading axes, for example mie(l[:, None], ...). Here mie takes one degree, mie_cyl one order and fresnel one interface: loop over the degrees, orders or interfaces. An array degree or order raises ValueError.
• Invalid layers raise ValueError, for example a zero or decreasing radius; treams raises ZeroDivisionError for a zero radius.

Example:

import numpy as np
from treams_rs import coeffs

# A sphere of size parameter k0 r = 0.5 and permittivity 4 in vacuum.
dipole = coeffs.mie(1, [0.5], [4, 1], [1, 1], [0, 0])
assert dipole.shape == (2, 2)
# Both helicities scatter alike from an achiral sphere.
assert np.isclose(dipole[0, 0], dipole[1, 1])
assert np.isclose(dipole[0, 1], dipole[1, 0])

fresnel

fresnel(ks: ArrayLike, kzs: ArrayLike, zs: ArrayLike) -> NDArray[np.complex128]

Fresnel coefficients of one planar interface between two chiral media.

Mirrors treams.coeffs.fresnel. The interface is the plane z = 0 between the negative side (z < 0) and the positive side (z > 0). The result r[d_out, d_in, p_out, p_in] maps an incident wave of direction d_in and helicity p_in to an outgoing wave of direction d_out and helicity p_out; direction 0 travels up (towards +z) and 1 down. So r[0, 0] transmits from the negative to the positive side, and r[1, 0] reflects back into the negative side.

Args

• ks: Wavenumbers, shape (2, 2): negative then positive side, each as (negative, positive) helicity.
• kzs: z components of the wavevectors, same shape and order.
• zs: Relative impedances of the negative and the positive side, shape (2,).

Returns

complex128 array of shape (2, 2, 2, 2).

Differences from treams

treams broadcasts over leading axes of the inputs; here the inputs describe one interface.

mie

mie(l: int, x: ArrayLike, epsilon: ArrayLike, mu: ArrayLike, kappa: ArrayLike) -> NDArray[np.complex128]

Mie coefficients of a multilayer chiral sphere for one degree l.

Mirrors treams.coeffs.mie. The (2, 2) block maps incident to scattered helicities: the T-matrix entries of degree l, which do not depend on the order m.

Args

• l: Degree, an integer from 1 to 128.
• x: Size parameters k0 * radius of the layers from the inside out, shape (layers,), increasing.
• epsilon: Relative permittivities, shape (layers + 1,): the layers from the inside out, then the embedding medium.
• mu: Relative permeabilities, same shape and order.
• kappa: Chirality parameters, same shape and order.

Returns

complex128 array of shape (2, 2): rows scattered, columns incident helicity.

Differences from treams

treams broadcasts over arrays of degrees and layers; here l is one integer and x one 1-D array. An array l raises ValueError, so loop over the degrees.

mie_cyl

mie_cyl(kz: float, m: int, k0: float, radii: ArrayLike, epsilon: ArrayLike, mu: ArrayLike, kappa: ArrayLike) -> NDArray[np.complex128]

Scattering coefficients of a multilayer chiral cylinder for one order m.

Mirrors treams.coeffs.mie_cyl. The cylinder is infinite along z. The (2, 2) block maps incident to scattered helicities: the T-matrix entries of the axial wavenumber kz and the order m.

Args

• kz: Axial wavenumber, real.
• m: Order, one integer.
• k0: Vacuum wavenumber.
• radii: Radii of the layers from the inside out, shape (layers,), increasing.
• epsilon: Relative permittivities, shape (layers + 1,): the layers from the inside out, then the embedding medium.
• mu: Relative permeabilities, same shape and order.
• kappa: Chirality parameters, same shape and order.

Returns

complex128 array of shape (2, 2): rows scattered, columns incident helicity.

Differences from treams

treams broadcasts over arrays of orders and layers; here m is one integer and radii one 1-D array. An array m raises ValueError, so loop over the orders.