4.2.4 The T-Matrix Method
Mie theory assumes spherical particles. A technique for computing light scattering
by nonspherical particles is the T-matrix method. It was originally formulated by
P. C. Waterman and is, effectively, an extension to Mie theory. The approach is
based on the Huygens principle which, when applied to the propagation of light
waves, states that every point on a wavefront may be considered a source of
secondary spherical wavelets which spread out in the forward direction at the
speed of light. The new wavefront is the tangential surface to all of these secondary
wavelets. The conceptual idea is that the incident and scattered waves can be
expanded into appropriate vector spherical wave functions and then related using a
transition (or T) matrix. It also has the advantage that it reduces exactly to the Mie
theory solution when the scattering particle is spherical and homogeneous. As noted
by Mishchenko et al. (2002), the T-matrix approach can, in many applications,
surpass other techniques (such as the discrete dipole approximation; Sect. 4.2.7) in
terms of efficiency. It is also possible to control the numerical accuracy so that it can
form a benchmark for particles lacking spherical symmetry.
Codes are available for use
2 with simplified geometries allowing computation of
scattering functions and the relevant cross-sections needed for cometary dust particle
scattering computations. A simplified Java interface is also available (Halder et al.
2014). Meng et al. (2010) compiled a database of T-matrix solutions for ellipsoidal
particles with different axial ratios and an example is shown in Fig. 4.11. The Mie
and T-matrix calculations agree well for spherical particles with the same properties
but the ellipsoidal particle phase function is smoother with less of a dip at intermediate scattering angles.
Fig. 4.11 A Mie theory
phase function compared to
two T-matrix calculations
from the database of Meng
et al. (2010). Solid: Mie
theory x ¼ 10.026,
m ref ¼ (1.6, i0.1). Dashed:
T-matrix for a spherical
particle with identical
properties. Dot-dash:
T-matrix for an ellipsoidal
particle with dimensions in
the ratio 1:1:3.3 but with
similar refractive index
2 https://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html
4.2 Scattering of Light by Dust
295
Mie theory assumes spherical particles. A technique for computing light scattering
by nonspherical particles is the T-matrix method. It was originally formulated by
P. C. Waterman and is, effectively, an extension to Mie theory. The approach is
based on the Huygens principle which, when applied to the propagation of light
waves, states that every point on a wavefront may be considered a source of
secondary spherical wavelets which spread out in the forward direction at the
speed of light. The new wavefront is the tangential surface to all of these secondary
wavelets. The conceptual idea is that the incident and scattered waves can be
expanded into appropriate vector spherical wave functions and then related using a
transition (or T) matrix. It also has the advantage that it reduces exactly to the Mie
theory solution when the scattering particle is spherical and homogeneous. As noted
by Mishchenko et al. (2002), the T-matrix approach can, in many applications,
surpass other techniques (such as the discrete dipole approximation; Sect. 4.2.7) in
terms of efficiency. It is also possible to control the numerical accuracy so that it can
form a benchmark for particles lacking spherical symmetry.
Codes are available for use
2 with simplified geometries allowing computation of
scattering functions and the relevant cross-sections needed for cometary dust particle
scattering computations. A simplified Java interface is also available (Halder et al.
2014). Meng et al. (2010) compiled a database of T-matrix solutions for ellipsoidal
particles with different axial ratios and an example is shown in Fig. 4.11. The Mie
and T-matrix calculations agree well for spherical particles with the same properties
but the ellipsoidal particle phase function is smoother with less of a dip at intermediate scattering angles.
Fig. 4.11 A Mie theory
phase function compared to
two T-matrix calculations
from the database of Meng
et al. (2010). Solid: Mie
theory x ¼ 10.026,
m ref ¼ (1.6, i0.1). Dashed:
T-matrix for a spherical
particle with identical
properties. Dot-dash:
T-matrix for an ellipsoidal
particle with dimensions in
the ratio 1:1:3.3 but with
similar refractive index
2 https://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html
4.2 Scattering of Light by Dust
295