Methods & references

Each solver and numerical trick follows the published literature below; the in-source docstrings and comments point to the specific equations used. (This list mirrors the Methods & references section of the repository README.md, which is the authoritative copy.)

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.