T-matrix particle scattering

The T-matrix method relates an incident electromagnetic field to the field scattered by a particle. Both fields are expanded in vector spherical waves; for expansion-coefficient vectors \(\boldsymbol a\) and \(\boldsymbol b\), the linear relation is

\[\boldsymbol b = \boldsymbol T\boldsymbol a.\]

For a given particle and wavelength, the same T-matrix can be used for different illumination and observation directions. The particle model considered here is a homogeneous spheroid or finite circular cylinder.

Particle size and orientation

Volume-equivalent radius is the radius of a sphere with the same volume as the particle. Surface-area-equivalent radius is the radius of a sphere with the same surface area. These definitions agree for a sphere, but generally differ for nonspherical particles.

A fixed-orientation calculation specifies the particle orientation as well as the incident and scattered directions. A random-orientation calculation averages over particle orientations. Averaging over a particle size distribution is an additional operation.

Amplitude and phase matrices

The complex 2 by 2 Jones amplitude matrix acts on electric-field polarization components. The corresponding real 4 by 4 Mueller phase matrix acts on the Stokes vector. Its elements are quadratic combinations of the complex amplitudes. With an amplitude measured in length units, the phase matrix has units of length squared.

For randomly oriented particles of the shapes considered here, the normalized phase matrix in the scattering-plane basis has the form

\[\begin{split}\boldsymbol F(\Theta) = \begin{pmatrix} F_{11} & F_{12} & 0 & 0 \\ F_{12} & F_{22} & 0 & 0 \\ 0 & 0 & F_{33} & F_{34} \\ 0 & 0 & -F_{34} & F_{44} \end{pmatrix},\qquad \int_{4\pi} F_{11}\,d\Omega = 4\pi.\end{split}\]

These matrix elements are dimensionless. Multiplication by \(C_{\mathrm{sca}}/(4\pi)\) gives the differential scattering cross-section matrix, where \(C_{\mathrm{sca}}\) is the scattering cross section. For a size distribution, the cross sections and normalized phase matrix must refer to the same distribution average.

See T-matrix interface for ARTS parameter names, units, and result types.