TransitionMatrices.jl

The transition matrix method, or T-Matrix method, is one of the most powerful and widely used tools for rigorously computing electromagnetic scattering by single and compounded particles. TransitionMatrices.jl is a generic, arbitrary-precision Julia implementation focused on this method.

📖 Documentation

Installation

TransitionMatrices.jl requires Julia ≥ 1.10. In the Julia REPL's package mode (press ]):

pkg> add TransitionMatrices

To track the development version, or if the package is not yet in your registry, add it by URL:

pkg> add https://github.com/JuliaRemoteSensing/TransitionMatrices.jl

Quick start

using TransitionMatrices

# A prolate spheroid: semi-axes a = 1, c = 2, relative refractive index m = 1.5 + 0.01im
s = Spheroid(1.0, 2.0, 1.5 + 0.01im)

# Auto-converged (classic EBCM) T-matrix at wavelength λ = 2π
𝐓 = transition_matrix(s, 2π)

# Orientation-averaged far-field observables
Qsca = scattering_cross_section(𝐓, 2π)
Qext = extinction_cross_section(𝐓, 2π)
ω    = albedo(𝐓)
g    = asymmetry_parameter(𝐓, 2π)

Features

• Calculate the T-Matrix of various types of scatterers
  • Homogeneous spheres (via bhmie)
  • Coated spheres (via bhcoat)
  • Homogeneous axisymmetric shapes (via EBCM and IITM)
    • Spheroids
    • Cylinders
    • Chebyshev particles
  • Arbitrary shapes (via IITM)
    • Prisms
• Calculate far-field scattering properties using the T-Matrix
  • Cross sections and single scattering albedo (SSA)
  • Amplitude scattering matrix
  • Phase matrix
  • Scattering matrix
• Compute Jacobians through the linearization framework
  • Numerical automatic differentiation for user-defined scalar workflows via ForwardDiff.jl
  • Analytical Mie linearization for size, refractive-index, and wavelength variables
  • Analytical EBCM slices for spheroids, cylinders, and Chebyshev particles
  • Analytical fixed-geometry IITM material/wavelength slices for axisymmetric, n-fold, and arbitrary-shape solvers

Compared to existing packages, TransitionMatrices.jl is special in that it is generic and supports various floating-point types, e.g.:

• Float64 and BigFloat from Base
• Double64 from DoubleFloats.jl
• Float128 from Quadmath.jl
• Arb and Acb from Arblib.jl

By using higher-precision floating-point types, the maximum size parameter that can be handled is greatly improved.

The precision types Double64, Float128, ComplexF128, Arb, and Acb are re-exported by TransitionMatrices.jl and can be directly used after using TransitionMatrices.

The 0.5 compatibility line uses Quadmath.jl 1.x and Wigxjpf.jl 0.3.x.

How to cite

If you use TransitionMatrices.jl in your research, please cite it — a CITATION.bib is provided in this repository. Please also cite the original publication(s) for the specific method you use; see Methods & references below.

Methods & references

Each solver and numerical trick follows the published literature below; the in-source docstrings and comments point to the specific equations used.

T-Matrix framework & far-field conventions

• P. C. Waterman, Symmetry, unitarity, and geometry in electromagnetic scattering, Phys. Rev. D 3, 825–839 (1971) — the null-field / EBCM origin of the T-Matrix method.
• M. I. Mishchenko, L. D. Travis & A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, Cambridge University Press (2002) — the amplitude / phase / scattering matrices, cross sections, asymmetry parameter, T-Matrix rotation, analytic random-orientation average, and the Wigner-d recursions (the pervasive Eq. (x.y) references throughout the source).

Mie & coated spheres (bhmie, bhcoat)

• C. F. Bohren & D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley (1983) — the bhmie and bhcoat algorithms. The Mie T-Matrix uses Mishchenko et al. (2002), Eqs. (5.42)–(5.44).

EBCM for axisymmetric shapes (spheroids, cylinders, Chebyshev particles)

• P. C. Waterman (1971), above.
• M. I. Mishchenko & L. D. Travis, Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers, JQSRT 60, 309–324 (1998) — the automatic convergence procedure (routine_mishchenko: the choice of nₘₐₓ and the Gauss division Ng), which the axisymmetric assembly is translated from.

Numerically stable EBCM — the F⁺ formulation (stable = true)

• W. R. C. Somerville, B. Auguié & E. C. Le Ru, Severe loss of precision in calculations of T-matrix integrals, JQSRT 113, 524 (2012) — the catastrophic-cancellation diagnosis.
• W. R. C. Somerville, B. Auguié & E. C. Le Ru, JQSRT 123, 153 (2013), doi:10.1016/j.jqsrt.2012.07.017 — the cancellation-free F⁺_{nk}(s,x) projection (their Eq. 45–62, Table 2) used for high-aspect-ratio spheroids.

Sh-matrix moment separation (prepare_sh, fast (λ, mᵣ) sweeps)

• The separation of the size / material parameters from the particle geometry is the parameter separation of V. G. Farafonov, V. B. Il'in & M. S. Prokopjeva, Light scattering by multilayered nonspherical particles: a set of methods, JQSRT 79–80, 599–626 (2003), doi:10.1016/S0022-4073(02)00310-2; the "Sh-matrix" name and formalism are due to D. Petrov, Yu. Shkuratov, E. Zubko & G. Videen, Sh-matrices method as applied to scattering by particles with layered structure, JQSRT 106, 437–454 (2007), doi:10.1016/j.jqsrt.2007.01.027 (extended in Petrov, Shkuratov & Videen, The Sh-matrices method applied to light scattering by small lenses, JQSRT 110, 1448–1459 (2009), doi:10.1016/j.jqsrt.2009.01.016).
• The term-by-term radial integration reuses the F⁺ projection of Somerville et al. (2013) and the Riccati–Bessel power series of DLMF §10.53.

Invariant Imbedding T-Matrix (IITM) (axisymmetric, N-fold, and arbitrary shapes)

• B. R. Johnson, Invariant imbedding T matrix approach to electromagnetic scattering, Appl. Opt. 27, 4861–4873 (1988) — Eq. (97).
• L. Bi, P. Yang, G. W. Kattawar & M. I. Mishchenko, Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles, JQSRT 116, 169–183 (2013) — Eq. (38).
• A. Doicu & T. Wriedt, The Invariant Imbedding T Matrix Approach, in The Generalized Multipole Technique for Light Scattering (Springer, 2018), doi:10.1007/978-3-319-74890-0_2 — Eq. (2.40).
• B. Sun, L. Bi, P. Yang, M. Kahnert & G. Kattawar, Invariant Imbedding T-matrix Method for Light Scattering by Nonspherical and Inhomogeneous Particles, Elsevier (2019) — Eq. (4.2.36).
• S. Hu (胡帅), Research on the Numerical Computational Models and Application of the Scattering Properties of Nonspherical Atmospheric Particles, PhD dissertation, National University of Defense Technology (2018) — Eq. (5.71).

Far-field observables & orientation averaging

• Mishchenko et al. (2002) — cross sections (Eqs. (5.102), (5.107), (5.140), (5.141)), phase matrix (Eqs. (2.106)–(2.121)), asymmetry parameter (Eq. (4.92)), and the analytic random-orientation T-Matrix (Eq. (5.96)).
• L. Bi & P. Yang, Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method, JQSRT 138, 17–35 (2014), doi:10.1016/j.jqsrt.2014.01.013 — the scattering-matrix expansion coefficients for a general T-Matrix (Eqs. (24)–(74)).

Shapes

• A. Mugnai & W. J. Wiscombe, Scattering from nonspherical Chebyshev particles. 1, Appl. Opt. 25, 1235 (1986) — the Chebyshev-particle definition r(ϑ) = r₀(1 + ε·Tₙ(cos ϑ)).

Linearization & Jacobians

• R. Spurr, J. Wang, J. Zeng & M. I. Mishchenko, Linearized T-matrix and Mie scattering computations, JQSRT (2012).
• F. Xu & A. B. Davis, Derivatives of light scattering properties of a nonspherical particle computed with the T-matrix method, Opt. Lett. 36(22), 4464–4466 (2011), doi:10.1364/OL.36.004464.
• B. Sun, M. Gao, L. Bi & R. Spurr, Analytical Jacobians of single scattering optical properties using the invariant imbedding T-matrix method, Opt. Express 29(6), 9635–9669 (2021), doi:10.1364/OE.421886.
• M. Gao & B. Sun, Improvement and application of linearized invariant imbedding T-matrix scattering method, JQSRT 290, 108322 (2022), doi:10.1016/j.jqsrt.2022.108322.

Numerical kernels (via dependencies)

• Wigner 3-j symbols via Wigxjpf.jl: H. T. Johansson & C. Forssén, Fast and accurate evaluation of Wigner 3j, 6j, and 9j symbols using prime factorization and multiword integer arithmetic, SIAM J. Sci. Comput. 38(1), A376–A384 (2016).
• Gauss–Legendre quadrature via FastGaussQuadrature.jl.