treams_rs.misc

Refractive indices, normal wavenumbers, mode selection and Brillouin zones.

Mirrors treams.misc. Mode arguments out and in_ are label arrays of equal length, as in treams: three arrays (degree, order, pol) or four arrays with the position index first. The last array is always the polarization. out labels the destination modes (rows) and in_ the source modes (columns).

Differences from treams

• firstbrillouin1d is a NumPy ufunc and broadcasts over its arguments.
• pickmodes and basischange raise ValueError unless they get three or four label arrays of equal length.

Example:

import numpy as np
from treams_rs import misc

# Helicity indices sqrt(epsilon mu) -/+ kappa of a chiral medium.
assert np.allclose(misc.refractive_index(4, 1, 0.1), [1.9, 2.1])
# Evanescent waves get a positive imaginary normal wavenumber.
assert np.isclose(misc.wave_vec_z(2, 0, 1), np.sqrt(3) * 1j)
picked = misc.pickmodes(([1, 1], [0, 1], [0, 1]), ([1, 1, 1], [0, 0, 1], [0, 1, 1]))
assert picked.tolist() == [[True, False, False], [False, False, True]]

basischange

basischange(out: ArrayLike, in_: ArrayLike | None=None) -> NDArray[np.float64]

Matrix that changes modes between helicity and parity polarizations.

Mirrors treams.misc.basischange. Modes with equal labels apart from the polarization couple with 1 / sqrt(2), except the pair of two pol 0 modes, which couples with -1 / sqrt(2). The matrix is its own inverse.

Args

• out: Destination modes: three or four label arrays of length M, the last one the polarization.
• in_: Source modes in the same form, length N; out when omitted.

Returns

float64 array of shape (M, N).

firstbrillouin1d

firstbrillouin1d(kpar, b, /, out=None, *, where=True)

kpar reduced to the first Brillouin zone (-b/2, b/2] of a lattice with reciprocal pitch b.

Mirrors treams.misc.firstbrillouin1d.

Differences from treams: broadcasts over arrays, where treams takes scalars; ValueError unless b > 0 and both arguments are finite, where treams raises ZeroDivisionError for b = 0, returns NaN for non-finite input and an unreduced value for b < 0.

firstbrillouin2d

firstbrillouin2d(kpar: ArrayLike, b: ArrayLike, n: int=2) -> NDArray[np.float64]

Wavevector equivalent to kpar in the first Brillouin zone of a 2D lattice.

Mirrors treams.misc.firstbrillouin2d. Each of n rounds first subtracts a rough multiple of the reciprocal vectors, then picks the shortest of the result and its 8 neighbours.

Args

• kpar: Wavevector in the lattice plane, shape (2,).
• b: Reciprocal lattice vectors as rows, shape (2, 2).
• n: Number of rounds.

Returns

float64 array of shape (2,).

firstbrillouin3d

firstbrillouin3d(kpar: ArrayLike, b: ArrayLike, n: int=2) -> NDArray[np.float64]

Wavevector equivalent to kpar in the first Brillouin zone of a 3D lattice.

Mirrors treams.misc.firstbrillouin3d. Each of n rounds first subtracts a rough multiple of the reciprocal vectors, then picks the shortest of the result and its 26 neighbours. For skewed reciprocal cells the result need not be the shortest equivalent wavevector, and reducing it again can change it. Use a nearly orthogonal reciprocal basis where the shortest vector matters.

Args

• kpar: Wavevector, shape (3,).
• b: Reciprocal lattice vectors as rows, shape (3, 3).
• n: Number of rounds.

Returns

float64 array of shape (3,).

pickmodes

pickmodes(out: ArrayLike, in_: ArrayLike) -> NDArray[np.bool_]

Selection matrix P: P[i, j] is True where mode out[i] equals mode in_[j].

Mirrors treams.misc.pickmodes.

Args

• out: Destination modes: three or four label arrays of length M, the last one the polarization.
• in_: Source modes in the same form, length N.

Returns

bool array of shape (M, N).

refractive_index

refractive_index(epsilon: ArrayLike=1, mu: ArrayLike=1, kappa: ArrayLike=0) -> NDArray[np.float64 | np.complex128]

Refractive indices sqrt(epsilon mu) - kappa and sqrt(epsilon mu) + kappa.

Mirrors treams.misc.refractive_index. The first index belongs to negative and the second to positive helicity. Each index changes sign where needed, so that its imaginary part is nonnegative.

Args

• epsilon: Relative permittivity.
• mu: Relative permeability.
• kappa: Chirality parameter.

Returns

Array with the broadcast shape of the arguments plus a last axis of length 2: float64 for real indices, complex128 otherwise.

wave_vec_z

wave_vec_z(kx: ArrayLike, ky: ArrayLike, k: ArrayLike) -> NDArray[np.complex128]

Normal wavenumber kz = sqrt(k^2 - kx^2 - ky^2) with Im kz >= 0.

Mirrors treams.misc.wave_vec_z. On this branch evanescent waves exp(i kz z) decay towards +z; kz is 0 at grazing incidence.

Args

• kx: x component of the wavevector.
• ky: y component of the wavevector.
• k: Wavenumber, real or complex.

Returns

complex128 array with the broadcast shape of the arguments.