MieLib

MieLib is a Python library for the scattering of waves by a sphere (Mie theory) in optics and in acoustics, and for the vector spherical harmonics.

• mielib.optics: an isotropic sphere, also magnetic, in a magnetic host. Mie coefficients, T-matrix, internal amplitudes and the denominators, whose zeros are the modes of the sphere.
• mielib.acoustics: a fluid sphere, from the Supplemental Material of Phys. Rev. Lett. 123, 183901 (2019). The same functions as in optics.
• Cross sections in both physics: scattering, extinction, absorption, backscattering, radiation pressure, the asymmetry parameter, and the part of each multipole.
• mielib.vsh: the complex vector spherical harmonics $\mathbf{M}$, $\mathbf{N}$, $\mathbf{L}$, the harmonic $\mathbf{X}$ of Jackson, and the real harmonics of Bohren & Huffman.
• mielib.special: spherical Bessel, Hankel and Riccati–Bessel functions with their first and second derivatives.
• mielib.coordinates: spherical and cylindrical coordinates, and the components of vector fields in them.

Installation

From PyPI:

pip install --upgrade mielib

mielib needs Python 3.12 or later and SciPy 1.18 or later.

Quick start

import numpy as np
from mielib import optics

# a sphere with the radius 100 nm and eps = 16 + 0.3i in vacuum
wavelength = np.linspace(500e-9, 1100e-9, 300)
k = 2 * np.pi / wavelength            # wavenumber in the host
q = optics.cross_sections(k, 100e-9, 16 + 0.3j, norm='geom')

q.sca, q.ext, q.abs                   # cross sections divided by pi a^2
q.sca_e[1], q.sca_m[1]                # the electric and the magnetic dipole parts of q.sca

All the Mie functions take the size parameter $x = ka$ in the host and the parameters of the particle relative to the host: $\varepsilon_\mathrm{rel} = \varepsilon_p / \varepsilon_h$ and $\mu_\mathrm{rel} = \mu_p / \mu_h$ in optics, the density $\rho_\mathrm{rel}$ and the compressibility $\beta_\mathrm{rel}$ in acoustics. For example, optics.mie_a(j, x, eps_rel, mu_rel) and acoustics.mie_a(j, x, rho_rel, beta_rel). The harmonics take the indices in the order (j, m), for example vsh.N(j, m, rho, theta, phi, superscript=1).

Theory

The note Vector spherical harmonics and Mie scattering in optics and acoustics: conventions and explicit formulas gives all the definitions and the formulas that mielib uses. Its notation follows the PhD thesis of I. Toftul, Resonant Metaphotonics: From Optical Manipulation to Chiral Metasurfaces (Australian National University, 2026).

The main conventions:

• The time dependence is $e^{-i\omega t}$.

• The electromagnetic field is the pair $(\sqrt{\varepsilon}\mathbf{E}, \sqrt{\mu}\mathbf{H})$. Thus, the electromagnetic duality is

  \sqrt{\varepsilon}\,\mathbf{E} \to \sqrt{\mu}\,\mathbf{H}, \qquad \sqrt{\mu}\,\mathbf{H} \to -\sqrt{\varepsilon}\,\mathbf{E},
  

  and for a sphere it swaps $a_j$ and $b_j$ when $\varepsilon_\mathrm{rel}$ and $\mu_\mathrm{rel}$ are swapped.

• The T-matrix is the outgoing amplitude divided by the regular amplitude of the same multipole. In optics, $T^e_j = -a_j$ and $T^m_j = -b_j$, with the $a_j$ and $b_j$ of Bohren & Huffman. In acoustics, $T_j = a_j$.

Examples

The examples folder has Jupyter notebooks:

• cross_sections.ipynb: the cross sections of a sphere and their multipole parts, in optics and in acoustics.
• fields.ipynb: the fields near a sphere in a plane wave, the electric and the magnetic fields in optics and the pressure and the velocity in acoustics.
• harmonics.ipynb: the vector spherical harmonics, and a plane wave from the harmonics.

The changes between the versions are in CHANGELOG.md.

ToDo

• Search for complex poles of Mie scattering coefficients

Credits

If you use mielib, please cite it:

@misc{mielib,
	author = {Ivan Toftul},
	title = {{MieLib python library}},
	journal = {GitHub},
	year = {2023},
	url = {https://github.com/toftul/mielib},
	doi = {10.5281/zenodo.8008648}
}

The conventions and the theory are from the thesis:

@phdthesis{Toftul2026thesis,
	author = {Ivan Toftul},
	title = {{Resonant Metaphotonics: From Optical Manipulation to Chiral Metasurfaces}},
	school = {Australian National University},
	year = {2026},
	url = {https://hdl.handle.net/1885/733812237}
}