The T-matrix method is widely used for the calculation of scattering by particles of sizes on the order of the illuminating wavelength. Although the extended boundary condition method (EBCM) is the most commonly used technique for calculating the T-matrix, a variety of methods can be used. Because the T-matrix depends only on the properties of the scatterer and the wavelength and not on other properties of the incident field, the T-matrix method is especially well-suited to repeated calculations involving varying illumination. Thus, it can be highly desirable to express the scattering properties of a given arbitrary particle as a T-matrix. However, the standard EBCM is only applicable to homogeneous scatterers, necessitating the use of other, more general methods. We consider some general principles of "hybrid" T-matrix methods - calculating T-matrices using other techniques for calculating scattering - and consider some specific methods. In particular, we discuss the application of time-domain methods which offer the possibility of simultaneous multiple-wavelength or multiple-size calculations

Heckenberg, N. R.

Nieminen, T. A.

Rubinsztein-Dunlop, H.

University of Queensland eSpace

Preprint of:
T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop
“ “Hybrid” T-matrix methods”
pp. 263–266 in T. Wriedt (ed)
Electromagnetic and Light Scattering — Theory and Applications VII
Universita¨t Bremen, Bremen (2003)
Presented at:
7th Conference on Electromagnetic and Light Scattering by Nonspherical Particles: Theory, Measurements,
and Applications
Bremen (2003)
“Hybrid” T-matrix methods
T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop
Centre for Biophotonics and Laser Science, Department of Physics,
The University of Queensland, Brisbane QLD 4072, Australia
Abstract
The T-matrix method is widely used for the calculation of scattering by particles
of sizes on the order of the illuminating wavelength. Although the extended
boundary condition method (EBCM) is the most commonly used technique for
calculating the T-matrix, a variety of methods can be used.
Because the T-matrix depends only on the properties of the scatterer and the
wavelength and not on other properties of the incident field, the T-matrix
method is especially well-suited to repeated calculations involving varying il-
lumination. Thus, it can be highly desirable to express the scattering properties
of a given arbitrary particle as a T-matrix. However, the standard EBCM is only
applicable to homogeneous scatterers, necessitating the use of other, more gen-
eral methods.
We consider some general principles of “hybrid” T-matrix methods—calculating
T-matrices using other techniques for calculating scattering—and consider some
specific methods. In particular, we discuss the application of time-domain meth-
ods which offer the possibility of simultaneous multiple-wavelength or multiple-
size calculations.
1 Introduction
The T-matrix method in wave scattering involves writing the relationship between the wave incident
upon a scatterer, expanded in terms of orthogonal eigenfunctions,
Uinc =
∞
∑
n
anψ
(inc)
n , (1)
where an are the expansion coefficients for the incident wave, and the scattered wave, also expanded
in terms of orthogonal eigenfunctions,
Uscat =
∞
∑
k
pkψ
(scat)
k , (2)
where pk are the expansion coefficients for the scattered wave, is written as a simple matrix equation
pk =
∞
∑
n
Tknan (3)
1
or, in more concise notation,
P = TA (4)
where Tkn are the elements of the T-matrix. The T-matrix method can be used for scalar waves or
vector waves in a variety of geometries, with the only restrictions being that the geometry of the
problem permits expansion of the waves as discrete series in terms of orthogonal eigenfunctions,
that the response of the scatterer to the incident wave is linear, and that the expansion series for the
waves can be truncated at a finite number of terms. In general, one calculates the T-matrix, although
it is conceivable that it might be measured experimentally.
The T-matrix depends only on the properties of the particle—its composition, size, shape, and
orientation—and the wavelength, and is otherwise independent of the incident field. This means
that for any particular particle, the T-matrix only needs to be calculated once, and can then be used
for repeated calculations. This is a significant advantage over many other methods of calculating
scattering where the entire calculation needs to be repeated [1]. Some cases provide even more ef-
ficiency: if the waves are expanded in spherical functions, the averaging of scattering over various
orientations of the particle compared to the direction of the incident wave can be performed analyti-
cally [2].
In the spherical geometry of elastic light scattering by a particle contained entirely within some radius
r0, the eigenfunction expansions of the fields are made in terms of vector spherical wavefunctions
(VSWFs). These wavefunctions are the electric and magnetic multipole fields; the field representa-
tions are multipole expansions (references [1] to [6] discuss these in more depth).
For other geometries, other sets of eigenfunctions, such as cylindrical wavefunctions (for scatterers
of infinite length in one dimension), or a Floquet expansion (planar periodic scatterers), are more ap-
propriate. There is no requirement that the modes into which the incident and scattered fields are ex-
panded be the same, or even similar. For example, the null-field method with discrete sources [7] uses
an expansion in multipoles centred on multiple origins. In all of these cases, the T-matrix method
remains applicable.
2 Non-EBCM calculation of the T-matrix
Methods other than the EBCM can be used to calculate the T-matrix. In general, one would calcu-
late the scattered field, given a particular incident field. The most direct way in which to use this
to produce a T-matrix is to solve the scattering problem when the incident field is equal to a sin-
gle spherical mode – that is, a single VSWF such as Einc(r) = RgM11(kr), Einc(r) = RgN11(kr),
Einc(r) = RgM21(kr), etc, and repeat this for all VSWFs that need to be considered (up to n = Nmax).
The expansion coefficients for the scattered field can be found in each case, if necessary, by using the
orthogonal eigenfunction transform (the generalised Fourier transform), and each scattering calcula-
tion gives a single column of the T-matrix.
Therefore, the calculation of a T-matrix requires that 2Nmax(Nmax + 2) separate scattering problems
are solved. This provides a criterion for deciding whether it is desirable to calculate a T-matrix: if
more than 2Nmax(Nmax + 2) scattering calculations will be performed, then it is more efficient to cal-
culate the T-matrix and use this for the repeated calculations than it is to use the original scattering
method repeatedly. Repeated calculations are expected if orientation averaging is to be carried out,
or if inhomogeneous illumination is to be considered, such as, for example, scattering by focussed
beams, where there are generally 6 degrees of freedom, namely the three-dimensional position of
the scatterer within the beam, and the three-dimensional orientation of the scatterer. Even if only a
modest number of points are considered along each degree of freedom, the total number of scattering
calculations required rapidly becomes very large, and even if the T-matrix takes many hours to cal-
culate, the total time saved by doing so can make an otherwise computationally infeasible problem
tractable.
Volume methods are of interest, since they can readily be used for inhomogeneous or anisotropic
particles. The two most likely candidates are the finite-difference time-domain method (FDTD) [1, 8]
2
and the discrete dipole approximation (DDA). In FDTD, the Maxwell equations are discretised in
space and time, and, beginning from a known initial state, the electric and magnetic fields at each
spatial grid point are calculated for successive steps in time. The number of grid points required
is O(N3max) for three-dimensional scattering, and O(Nmax) time steps required, so FDTD solutions
scale as O(N4max). Therefore, calculation of the T-matrix using FDTD should scale as O(N6max), which
is the same scaling as the EBCM. However, the grid required must be closely spaced compared to
the wavelength, and the space outside the scatterer must also be discretised, making FDTD substan-
tially slower than EBCM, especially for smaller particles. However, FDTD is an extremely general
technique, and has potential as a method for the calculation of T-matrices.
We should add that there is an additional consideration that makes FDTD potentially attractive as
a method for calculating the T-matrix: FDTD does not assume that the incident wave is monochro-
matic. Consider the case when the illumination is a brief pulse with a Gaussian envelope. The
frequency spectrum of the incident wave is Gaussian, and the scattering of a range of frequencies can
be found by taking the Fourier transform of the scattered field [9]. Even if we are not interested in
other than monochromatic illumination, we will frequently be interested in scattering by size distri-
butions of particles. Since varying the frequency for a particular particle is equivalent to varying the
size of the particle for a fixed incident frequency, the T-matrices for a range of particle sizes can be
calculated simultaneously.
The other major volume method for computational scattering, the discrete dipole approximation
(DDA), also known as the coupled-dipole method, has been recently applied to the calculation of
the T-matrix by Mackowski [10], who obtained good results, with reasonable computational effi-
ciency using a moment method to solve the DDA system of equations. There is no need to dis-
cuss his method in detail here, and the interested reader is referred to his recent description of the
method [10].
Lastly we note our recent application of the point-matching method to the calculation of T-
matrices [11], where we showed it to be a feasible alternative to the EBCM for particles devoid of
symmetry. The point-matching method is an attractive candidate since a T-matrix implementation
will generally include routines to calculate VSWFs, making the implementation of a point-matching
T-matrix calculator simple. A wide range of possible point-matching methods exist, such as the
multiple-multipole method (MMP), discrete sources method, etc [12].
3 Time-domain methods
As mentioned above, time-domain methods offer the possibility of simultaneously calculating the
T-matrix for multiple wavelengths. If one can provide an incident field of which the spectral compo-
nents in time and space are single electric or magnetic multipoles, the scattered field expansions can
be found from the scattered field frequency space components. This would directly give a column of
the T-matrices for each frequency component.
On the other hand, it might only be possible to use time-varying plane wave illumination in the
time-domain code. In this case, as a first step, one would obtain a T-matrix relating an incident
plane wave expansion to a scattered multipole expansion. Although the convergence properties of
multipole expansions and plane wave expansion are very different when considering infinite regions,
equivalent accuracy is obtainable with a finite set of components in a limited volume. Sufficient
plane wave components must be used for this be valid. The plane wave expansion can be directly
transformed into the equivalent spherical wave expansion. Presumably, the transformation matrix
for this would be pre-calculated, reducing the conversion of the T-matrix to the desired form to a
matrix-matrix multiplication.
Ideally, one would know beforehand the wavelengths at which the T-matrix is desired, allowing
a suitable envelope w.r.t. time to be chosen for the incident wave. If the variation of refractive
index with wavelength is assumed to be zero, variation of the T-matrix with wavelength is exactly
equivalent to variation of the T-matrix with scatterer size for a fixed wavelength. This may prove to
be a useful technique for scattering by polydisperse particles.
3
4 Conclusion
While the extended boundary condition method (EBCM) is the standard method for the calculation
of the T-matrix of a scatterer, the T-matrix method itself is a very general technique, consisting of the
representation of the scattering properties of a particle as a matrix transforming a truncated discrete
eigenfunction expansion of an incident wave to that of the scattered wave. Accordingly, a diverse
range of computational methods can be used for calculating T-matrices. These are of most value
when considering inhomogeneous scatterers for which the EBCM cannot be used. In particular,
time-domain methods offer the possibility of simultaneous calculation of the T-matrix at multiple
wavelengths or, equivalently, multiple particle sizes.
References
[1] M. I. Mishchenko, J. W. Hovenier, L. D. Travis (eds), Light scattering by nonspherical particles:
theory, measurements, and applications. (Academic Press, San Diego, 2000).
[2] M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles”, J. Opt.
Soc. Am. A 8, 871–882 (1991).
[3] P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering”, Phys. Rev.
D 3, 825–839 (1971).
[4] L. Tsang, J. A. Kong, R. T. Shin, “Theory of microwave remote sensing”, (Wiley, New York, 1985).
[5] V. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, “Quantum theory of angular momen-
tum”, (World Scientific, Singapore, 1988).
[6] B. Brock, “Using vector spherical harmonics to compute antenna mutual impedance from mea-
sured or computed fields”, (Sandia report SAND2000-2217-Revised, Sandia National Laborato-
ries, Albuquerque, 2001).
[7] T. Wriedt, A. Doicu, “Formulations of the extended boundary condition method for three-
dimensional scattering using the method of discrete sources”, J. Mod. Opt. 45, 199–213 (1998).
[8] P. Yang, K. N. Liou, M. I. Mishchenko, B.-C. Gao, “Efficient finite-difference time-domain scheme
for light scattering by dielectric particles: application to aerosols”, Appl. Opt. 39, 3727–3737
(2000).
[9] K. S. Yee, D. Ingham, K. Shlager, “Time-domain extrapolation to the far field based on FDTD
calculations”, IEEE T. Antenn. Propag. 39, 410–413 (1991).
[10] D. W. Mackowski, “Discrete dipole moment method for calculation of the T matrix for non-
spherical particles”, J. Opt. Soc. Am. A 19, 881–893 (2002).
[11] T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg,, “Calculation of the T-matrix: gen-
eral considerations and application of the point-matching method”, J. Quant. Spectrosc. Radiat.
Transfer. 79–80, 1019–1029 (2003).
[12] Th. Wriedt, A. Doicu, Yu. Eremin, “Acoustic and electromagnetic scattering analysis using dis-
crete sources”, (Academic Press, San Diego, 2000).
4


"Hybrid" T-Matrix Methods

http://espace.library.uq.edu.au/view/UQ:9207/tmat_preprint.pdf

"Hybrid" T-Matrix Methods

Abstract

Similar works

Full text

Available Versions

University of Queensland eSpace