T-matrix method

A promising method based on an integral equation formulation of the problem of scattering by an arbitrary particle has come into prominence in recent years. It was developed by Waterman, first for a perfect conductor (1965), later for a particle with less restricted optical properties (1971). More recently it has been applied to various scattering problems under the name Extended Boundary Condition Method, although we shall follow Waterman s preference for the designation T-matrix method. Barber and Yeh (1975) have given an alternative derivation of this method. 

Other asymptotic forms consistent with unit Wronskian define different but equally valid Green functions, with different values of the asymptotic coefficient of u>i. In particular, if w k 2 exp i(kr — ln), this determines the outgoing-wave Green function, and the asymptotic coefficient of w is the single-channel F-matrix, F sin ij. This is the basis of the T-matrix method [342, 344], which has been used for electron-molecule scattering calculations [126], It is assumed that Avf is regular at the origin and that Ad vanishes more rapidly than r 2 for r — oo. 

Mishchenko, M.I. and L.D. Travis Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatteiers, J. Quant. Spectrosc. Radiat. Transfer 60 (1998) 309-324. 

Figure 4. Dependences of the extinction A of gold bispheres in water on the number of multipoles included in the single-particle expansions. Calculations by the T-matrix method for randomly oriented bispheres with touching component spheres (hd = 0 ) and those separated by distances hd = 0.5 nm and hd = 1 nm. Particle diameter equals d = 5 nm the wavelength in vacuum equals A = 515 nm (a) and 550 nm (b).

Figure 14. Extinction and scattering spectra for ballistic RF aggregates with different conjugate numbers N = 1-100. All data are averaged over random orientations (T-matrix method) without statistical averaging. Parameters of conjugates are the shell thickness and refractive index s = 2.5 nm, =1.40, respectively, the gold core diameter <7 =15 nm (a,b) and 60 nm (c,d).

To explain these findings, we nse a compnter diffiision-hmited clnster-cluster aggregation model, as described in Section 3.1.1. The optical properties of the aggregates are compnted by the conpled dipole method and by a rigorous GMM and T-matrix methods. The bulk optical constants of metals are modified by the size-limiting effect of nanoparticles [20]. It was shown that a modified version of DDA [58] allows one to explain the shape of the experimental spectra for DLCA aggregates and the dependence of the spectra on the particle size. 

The DLSS method is based on measuring the differential spectra of light scattered at 90° within the wavelength range 350-800 nm. The above qualitative speculations have been confirmed by the cluster T-matrix method generalized to include two-layer spherical cluster particles (see Section 3.2.3). 

Bi L, Yang P Modeling of light scattering by biconcave and deformed red blood ceUs with the invariant imbedding T-matrix method. Journal of biomedical optics 18(5) 055001-1-055001-13, 2013. 

Liu L, Mishchenko MI, Amott WP A smdy of radiative properties of fractal soot aggregates using the superposition t-matrix method, J Quant Spectrosc Radiat Transf 109(15) ... 

The radiation characteristics of axisymmetric spheroidal microorganisms, such a C. reinhardtii (Fig. lA), with major and minor chameten a and b can be predicted numerically using (i) the T-matrix method (Waterman, 1965 Mackowski, 1994 Mishchenko et al., 2002, 1995), (ii) the discrete-dipole approximation (Draine, 1988), and (iii) the finite-difference time-domain method (Liou, 2002). Most often, however, they have been approximated as homogeneous spheres with some equivalent radius r and some effective complex index of refraction nix = n +ikx (Pettier et al., 2005 Berbero u et al., 2007 Dauchet et al., 2015), as discussed in Section 3.6.1. 

The Mie theory [1] and the T-matrix method [4] are very efficient for (multilayered) spheres and axisymmetric particles (with moderate aspect ratios), respectively. Several methods, applicable to particles of arbitrary shapes, have been used in plasmonic simulations the boundary element method (BEM) [5, 6], the DDA [7-9], the finite-difference time-domain method (FDTD) [10, 11], the finite element method (FEM] [12,13], the finite integration technique (FIT) [14] and the null-field method with discrete sources (NFM-DS) [15,16]. There is also quasi-static approximation for spheroids [12], but it is not discussed here. 

Absorption and scattering efficiencies Qabs, <3sea) were calculated using ADDA 0.79 varying the discretization level, characterized by number of dipoles along the x-axis. Reference results were obtained using Mie theory [46], T-matrix method [137], and extrapolation technique combined with the DDA [68] for the spheres, the rod, and the cubes respectively. Some of the results [67] for Qab and Qsca are presented in Fig. 2.3 and Fig. 2.4, respectively. Since the accuracy of these two quantities weakly depends on the size [67] (see also Fig. 2.5), only results for the smaller sphere and cube are shown. 

Yurkin, M. A. and Kahnert, M. (2013) Light scattering by a cube accuracy limits of the discrete dipole approximation and the T-matrix method, J. Quant Spectrosc. Radiat Transfer, http //dx.dol.Org/10.1016/j.jqsrt.2012.10.001. 

The transition matrix relates the expansion coefficients of the incident and scattered fields. The existence of the transition matrix is postulated by the T-Matrix Ansatz and is a consequence of the series expansions of the incident and scattered fields and the linearity of the Maxwell equations. Historically, the transition matrix has been introduced within the null-field method formalism (see [253,256]), and for this reason, the null-field method has often been referred to as the T-matrix method. However, the null-field method is only one among many methods that can be used to compute the transition matrix. The transition matrix can also be derived in the framework of the method of moments [88], the separation of variables method [208], the discrete dipole approximation [151] and the point matching method [181]. Rother et al. [205] foimd a general relation between the surface Green function and the transition matrix for the exterior Maxwell problem, which in principle, allows to compute the transition matrix with the finite-difference technique. 

Symmetry properties of the transition matrix can be derived for specific particle shapes. These symmetry relations can be used to test numerical codes as well as to simplify many equations of the T-matrix method and develop efEcient numerical procedures. In fact, the computer time for the numerical evaluation of the surface integrals (which is the most time consuming part of the T-matrix calculation) can be substantially reduced. The surface integrals are usually computed in spherical coordinates and for a surface defined by... 

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