Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers

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Abstract

We describe in detail a software implementation of a current version of the T-matrix method for computing light scattering by polydisperse, randomly oriented, rotationally symmetric particles. The FORTRAN T-matrix codes are publicly available on the World Wide Web at http://www.giss.nasa.gov/∼crmin. We give all necessary formulas, describe input and output parameters, discuss numerical aspects of T-matrix computations, demonstrate the capabilities and limitations of the codes, and discuss the performance of the codes in comparison with other available numerical approaches.

Introduction

The T-matrix method is a powerful exact technique for computing light scattering by nonspherical particles based on numerically solving Maxwell’s equations. Although the method is, potentially, applicable to any particle shape, most practical implementations of the technique pertain to bodies of revolution. The method was initially developed by Waterman[1]and has been significantly improved as described in Refs. 2–6. Specifically, Refs. 4 and 6 extend the method to much larger size parameters and aspect ratios, Ref. 2 presents an efficient analytical procedure for computing the scattering properties of randomly oriented particles, Ref. 3 describes an automatic convergence procedure convenient in massive computer calculations for particle polydispersions, and Ref. 5 presents benchmark T-matrix computations for particles with non-smooth surfaces (finite circular cylinders). A general review of the T-matrix method can be found in Ref. 7.

In this paper we provide a detailed description of modern T-matrix FORTRAN codes which incorporate all recent developments, are publicly available on the World Wide Web, and are, apparently, the most efficient and powerful tool for accurately computing light scattering by randomly oriented rotationally symmetric particles. For the first time, we collect in one place all necessary formulas, discuss numerical aspects for T-matrix computations, describe the input and output parameters, and demonstrate the capabilities and limitations of the codes. The paper is intended to serve as a detailed user guide to a versatile tool suitable for a wide range of practical applications. We specifically target the users who are interested in practical applications of the T-matrix method rather than in details of its mathematical formulation.

Section snippets

Basic definitions

The single scattering of light by a small-volume element dv consisting of randomly oriented, rotationally symmetric, independently scattering particles is completely described by the ensemble-averaged extinction, Cext, and scattering, Csca, cross sections per particle and the dimensionless Stokes scattering matrix8, 9F(Θ)=a1(Θ)b1(Θ)00b1(Θ)a2(Θ)0000a3(Θ)b2(Θ)00−b2(Θ)a4(Θ),where Θ is the scattering angle, i.e., the angle between the incident and scattered beams. The scattering matrix describes

T-matrix ansatz

Consider scattering of a plane electromagnetic wave by a single nonspherical particle in a fixed orientation with respect to the reference frame. In the framework of the T-matrix approach, the incident and the scattered fields are expanded in vector spherical functions Mmn and Nmn as follows:[19]Einc(R)=n=1nmaxm=-nn[amnRgMmn(kR)+bmnRgNmn(kR)],Esca(R)=n=1nmaxm=-nn[pmnMmn(kR)+qmnNmn(kR)],|R|>r0,where r0 is the radius of a circumscribing sphere of the scattering particle, and the origin of the

Numerical calculation of the t matrix

Explicit closed-form analytical equations for computing the T-matrix elements for rotationally symmetric particles are given in Sec. 3.5.4 of Ref. 19. Although in principle the size of the T matrix is infinite [nmax=∞ in , , , ], in practical computer calculations the T matrix must be truncated to a finite size. The convergence size of the T matrix is determined by increasing nmax in unit steps until the optical cross sections and the expansion coefficients αl1 to βl2 converge within some

Particle shapes and sizes

Although the T matrix codes described below can be easily tuned to any rotationally symmetric particle having a plane of symmetry perpendicular to the axis of rotation, the current versions of the codes are directly applicable to spheroids, finite circular cylinders, and even-order Chebyshev particles. Spheroids are formed by rotating an ellipse about its minor (oblate spheroid) or major (prolate spheroid) axis (Fig. 1). Their shape in the spherical coordinate system is described by the equation

Averaging over size distribution

To average the optical cross sections and the expansions coefficients in , , , , , over a size distribution, one must evaluate numerically the following integrals:Csca=r1r2drn(r)Csca(r),Cext=r1r2drn(r)Cext(r),αli=1Cscar1r2drn(r)Csca(r)αli(r),i=1,…,4,βli=1Cscar1r2drn(r)Csca(r)βli(r),i=1,2,where n(r)dr is the fraction of particles with equivalent-sphere radii between r and r+dr, and r1 and r2 are the minimal and the maximal equivalent-sphere radii in the size distribution. The distribution

Computer codes

Two FORTRAN T-matrix codes for computing light scattering by polydisperse, randomly oriented, rotationally symmetric particles are available on the World Wide Web at http://www.giss.nasa.gov/∼crmim. The first code uses only double-precision floating point variables, while the second one computes the T matrix elements using extended-precision floating point variables. The extended precision code is slower than the double-precision code, especially on supercomputers, but allows computations for

Examples

Table 1 shows the values of the maximal convergent size parameter xs=2πrs/λ for DDELT=0.001 in extended-precision T-matrix computations for monodisperse oblate spheroids with refractive index 1.311 and axial ratios a/b varying from 1.5 to 20. Note that the maximal size parameter xa=2πa/λ measured along the major semi-axis a can be significantly larger than the maximal equivalent-sphere size parameter, especially for highly flattened spheroids. Table 1 also shows the respective values of the

Conclusions

Predecessors of the T-matrix codes described in this paper have been extensively used by members of the scattering community since 1993. Multiple communications with the users provided a positive feedback which helped us to improve the codes and suggested the idea of publishing a detailed user guide collecting in one place all key formulas and describing the basics of the approach, numerical aspects of computer calculations, and the capabilities and limitations of the technique. We hope,

Acknowledgements

We thank many users of the T-matrix codes for their positive feedback and N. T. Zakharova for programming support and help with graphics. This work was supported by the NASA EOS program and grant No. N00014-96-1-G020 from the Naval Research Laboratory.

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