Physical Basis: Mie Theory

The main model has been updated to run using miepython for the Mie theory computation. Miepython is an validated python package written by Scott Prahl. Both miepython and the equivalent functions written by the authors of PPOPS (contained in validation/mie_python.py) produce nearly identical results. mie_modules.py is retained for illustrative and testing purposes but is not used in the model.

mie_modules.py — Mie Scattering Utilities

This module implements numerical routines for computing Mie scattering by spherical particles.
It provides functions for calculating:

  • Mie coefficients $a_n$ and $b_n$

  • Angular functions $pi_n(u)$ and $tau_n(u)$

  • Scattering amplitude functions $S_1(theta)$ and $S_2(theta)$

These quantities follow standard formulations from electromagnetic scattering theory and are commonly used in aerosol physics, atmospheric science, and optical particle characterization.

Overview

Functions Included

Function

Description

mie_ab(m, x)

Computes the complex Mie coefficients $a_n$ and $b_n$ describing electric and magnetic multipole contributions.

mie_pt(u, n_max)

Computes angular functions $pi_n(u)$ and $tau_n(u)$ derived from associated Legendre polynomials.

mie_s12(m, x, u)

Computes the scattering amplitude functions $S_1$ and $S_2$ for s- and p-polarized scattered light.

These are the fundamental components needed to compute phase functions, scattering efficiencies, polarization ratios, and full intensity distributions from Mie theory.

Dependencies

import numpy as np
from scipy.special import spherical_jn, spherical_yn

Function Documentation

mie_ab(m: complex, x: float) -> np.ndarray

Compute the Mie coefficients $a_n$ and $b_n$, which describe how a spherical particle scatters light.

Parameters:

  • ``m`` (complex)
    Complex refractive index of the particle, $m = n + ik$.

  • ``x`` (float)
    Size parameter, defined as, $x = frac{pi D}{lambda}$

Returns:

A 2×N NumPy array:

  • a_n — electric multipole coefficients

  • b_n — magnetic multipole coefficients

Key Implementation Notes:

  • Determines the required order $n_text{max}$ via the approximation: $n_{max} approx 2 + x + 4x^{1/3}$

  • Uses Riccati–Bessel functions: $psi_n(x) = x j_n(x), quad chi_n(x) = -x y_n(x)$

  • Computes the logarithmic derivative $D_n(z)$ using backward recurrence.

  • Constructs the Mie coefficients from: $a_n = frac{(frac{D_n}{m} + frac{n}{x})psi_n - psi_{n-1}} {(frac{D_n}{m} + frac{n}{x})xi_n - xi_{n-1}}$ with an analogous expression for $b_n$.

mie_pt(u: float, n_max: int) -> np.ndarray

Compute angular functions $pi_n(u)$ and $tau_n(u)$, where:

  • $u = costheta$

  • $theta$ is the scattering angle.

These functions capture the angular dependence of scattered light.

Parameters:

  • ``u`` (float) — cosine of scattering angle

  • ``n_max`` (int) — maximum multipole order

Returns:

A 2×N array: [π_n(u), τ_n(u)] .

Internal Notes:

  • Initializes dipole (n=1) and quadrupole (n=2) terms.

  • Higher-order terms computed via recurrence: $pi_n = frac{(2n-1)u,pi_{n-1} - n,pi_{n-2}}{n-1}, quad tau_n = n u pi_n - (n+1)pi_{n-1}$

mie_s12(m: complex, x: float, u: float) -> np.ndarray

Compute the Mie scattering amplitude functions $S_1$ and $S_2$, which fully characterize scattering for perpendicular and parallel polarization states.

Parameters:

  • ``m`` (complex) — refractive index

  • ``x`` (float) — size parameter

  • ``u`` (float) — $cos(theta)$

Returns:
A 2-element array: [S1, S2] .

Computation Outline:

  1. Determine n_max using the standard approximation.

  2. Compute Mie coefficients a_n, b_n .

  3. Compute angular functions π_n(u), τ_n(u) .

  4. Apply the weighting: $frac{2n + 1}{n(n+1)}$

  5. Form the scattering amplitudes: $S_1 = sum_n left(a_n pi_n + b_n tau_nright), quad S_2 = sum_n left(a_n tau_n + b_n pi_nright)$

These can be further used to compute intensity patterns: $I(theta) propto lvert S_1 rvert ^2 quad text{or} quad lvert S_2 rvert ^2$

References

  • Bohren, C. F. & Huffman, D. R. (1983)
    Absorption and Scattering of Light by Small Particles.

  • C. Mätzler (2002)
    MATLAB codes for Mie scattering, IAP Research Report.