Theory & conventions

This page summarises the definitions and conventions used throughout TransitionMatrices.jl. They follow Mishchenko, Travis & Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002) — referred to below as MTL — which the in-source docstrings cite by equation number. Full references are listed in Methods & references.

The transition matrix

For a single scatterer embedded in a non-absorbing host medium, both the incident and the scattered electric fields are expanded in vector spherical wave functions (VSWFs). The incident field uses the regular VSWFs $\mathrm{Rg}\mathbf{M}_{mn}, \mathrm{Rg}\mathbf{N}_{mn}$ with expansion coefficients $(a_{mn}, b_{mn})$; the scattered field uses the outgoing VSWFs $\mathbf{M}_{mn}, \mathbf{N}_{mn}$ with coefficients $(p_{mn}, q_{mn})$. Because Maxwell's equations are linear, the two sets of coefficients are linearly related, and the matrix of that relation is the transition matrix $T$ (MTL §5):

\[\begin{bmatrix} p_{mn} \\ q_{mn} \end{bmatrix} = \sum_{n'=1}^{\infty} \sum_{m'=-n'}^{n'} \begin{bmatrix} T^{11}_{mn m'n'} & T^{12}_{mn m'n'} \\ T^{21}_{mn m'n'} & T^{22}_{mn m'n'} \end{bmatrix} \begin{bmatrix} a_{m'n'} \\ b_{m'n'} \end{bmatrix}.\]

The transition matrix depends only on the particle (its shape, size, and refractive index relative to the host) and on the wavelength — not on the incident field. Computing it once therefore gives access to scattering for any incidence direction and polarisation, and to orientation averages in closed form.

Storage

AbstractTransitionMatrix stores $T$ as a six-dimensional array indexed $T_{mn m'n'}^{kl}$: the unprimed $(m, n)$ label the scattered harmonic, the primed $(m', n')$ the incident one, and $k, l \in \{1, 2\}$ select the magnetic ($\mathbf{M}$) and electric ($\mathbf{N}$) parts. The degree runs $1 \le n \le n_{\max}$ and the order $-n \le m \le n$, so a truncation order $n_{\max}$ fixes the matrix size. Concrete subtypes (TransitionMatrix, AxisymmetricTransitionMatrix, MieTransitionMatrix, RandomOrientationTransitionMatrix) exploit symmetry to store and apply $T$ more compactly.

Size parameter and units

Lengths and the wavelength are expressed in the same (arbitrary) unit; only their ratio matters. The wavenumber in the host medium is $k = 2\pi/\lambda$, and the dimensionless size parameter of a scatterer of characteristic radius $r$ is

\[x = k r = \frac{2\pi r}{\lambda}.\]

The refractive index m passed to a shape is relative to the host medium ($m = m_\text{particle}/m_\text{host}$); a complex m carries absorption in its imaginary part. The default wavelength in the API is $\lambda = 2\pi$, for which $k = 1$ and the size parameter equals the radius numerically — convenient for quick experiments.

The truncation order needed for convergence grows with the size parameter (roughly $n_{\max} \gtrsim x + c\,x^{1/3}$), which is why large particles are expensive and eventually require extended precision or the stable formulation; see Choosing a solver.

Orientation

Because $T$ is an operator between VSWF bases, a rotation of the particle by Euler angles $(\alpha, \beta, \gamma)$ acts on it through Wigner $D$-functions (MTL Eq. (5.29)); rotate returns the rotated transition matrix without recomputing it from scratch.

For randomly oriented particles the orientation-averaged transition matrix has a closed form (MTL Eq. (5.96)), implemented by RandomOrientationTransitionMatrix; orientation_average instead integrates numerically against a user-supplied orientation distribution and converges to the analytic result. The orientation-averaged scattering matrix is expanded in generalised spherical functions with coefficients computed by expansion_coefficients.

Far-field observables

All of the following are methods of an AbstractTransitionMatrix (see Post-processing for runnable examples).

Amplitude (Jones) matrix

For a fixed incidence direction $\hat{\mathbf{n}}^{\text{inc}}$ and scattering direction $\hat{\mathbf{n}}^{\text{sca}}$, the $2\times 2$ complex amplitude matrix $\mathbf{S}$ maps the incident to the scattered transverse field (MTL Eqs. (5.11)–(5.17)):

\[\begin{bmatrix} E_\vartheta^{\text{sca}} \\ E_\varphi^{\text{sca}} \end{bmatrix} = \frac{e^{ikr}}{r}\,\mathbf{S}\, \begin{bmatrix} E_\vartheta^{\text{inc}} \\ E_\varphi^{\text{inc}} \end{bmatrix}.\]

amplitude_matrix returns $\mathbf{S}$ for given incidence/scattering angles.

Phase (Mueller) matrix

The real $4\times 4$ phase matrix $\mathbf{Z}$ propagates the Stokes vector and follows from $\mathbf{S}$ (MTL Eqs. (2.106)–(2.121)); phase_matrix builds it from an amplitude matrix.

Cross sections, albedo, asymmetry

The orientation-averaged scattering and extinction cross sections (MTL Eqs. (5.140)/(5.141) and (5.102)/(5.107)) are returned by scattering_cross_section and extinction_cross_section; their difference is the absorption_cross_section. The single-scattering albedo is $\omega = C_\text{sca}/C_\text{ext}$ and the asymmetry_parameter is $g = \langle\cos\Theta\rangle = \alpha_1^1/3$ (MTL Eq. (4.92)), where $\alpha_1^1$ is the first scattering-matrix expansion coefficient.

Scattering matrix

For a macroscopically isotropic and mirror-symmetric ensemble the scattering matrix $\mathbf{F}(\Theta)$ has the block-diagonal structure with up to ten independent elements $(\alpha_1\!-\!\alpha_4, \beta_1\!-\!\beta_6)$; scattering_matrix evaluates it on a grid of scattering angles $\Theta$.

Methods for building $T$

TransitionMatrices.jl provides several engines, all producing the same kind of transition matrix:

Which one to pick, and when to enable stable or raise the precision, is covered in Choosing a solver. The literature behind each method is collected in Methods & references.