Daniel W. Mackowski

T-matrix computations of light scattering by nonspherical particles: A review

We review the current status of Watermans T-matrix approach which is one of the most powerful and widely used tools for accurately computing light scattering by nonspherical particles, both single and composite, based on directly solving Maxwells equations. Specifically, we discuss the analytical method for computing orientationally-averaged light-scattering characteristics for ensembles of nonspherical particles, the methods for overcoming the numerical instability in calculating the T matrix for single nonspherical particles with large size parameters and/or extreme geometries, and the superposition approach for computing light scattering by composite/aggregated particles. Our discussion is accompanied by multiple numerical examples demonstrating the capabilities of the T-matrix approach and showing effects of nonsphericity of simple convex particles (spheroids) on light scattering.

Calculation of total cross sections of multiple-sphere clusters

A method for calculating the extinction, absorption, and scattering cross sections of clusters of neighboring spheres for both fixed and random orientations is developed. The analysis employs the superposition formulation for radiative interactions among spheres, in which the total field from the cluster is expressed as a superposition of vector spherical harmonic expansions about each of the spheres in the cluster. Through the use of addition theorems a matrix equation for the expansion coefficients is obtained. Further application of addition theorems on the inverse of the coefficient matrix is shown to yield analytical expressions for the orientation-averaged total cross sections of the sphere cluster. Calculations of the cross sections of pairs of spheres and fractal aggregates of several spheres are presented. It is found that a dipole representation of the field in each sphere does not adequately predict the absorption cross section of clusters of small-size-parameter spheres when the spheres are highly conducting. For this situation several multipole orders are required for an accurate calculation of the absorption cross section. In addition, the predicted absorption of sphere clusters can be significantly greater than that estimated from the sum of the isolated-sphere cross sections.

Scattering of light by bispheres with touching and separated components.

We use the T-matrix method as described by Mishchenko and Mackowski [Opt. Lett. 19, 1604 (1994)] to compute light scattering by bispheres in fixed and random orientations extensively. For all our computations the index of refraction is fixed at a value 1.5 + 0.005i, which is close to the refractive index of mineral tropospheric aerosols and was used in previous extensive studies of light scattering by spheroids and Chebyshev particles. For monodisperse bispheres with touching components in a fixed orientation, electromagnetic interactions between the constituent spheres result in a considerably more complicated interference structure in the scattering patterns than that for single monodisperse spheres. However, this increased structure is largely washed out by orientational averaging and results in scattering patterns for randomly oriented bispheres that are close to those for single spheres with size equal to the size of the bisphere components. Unlike other nonspherical particles such as cubes and spheroids, randomly oriented bispheres do not exhibit pronounced enhancement of side scattering and reduction of backscattering and positive polarization at side-scattering angles. Thus the dominant feature of light scattering by randomly oriented bispheres is the single scattering from the component spheres, whereas the effects of cooperative scattering and concavity of the bisphere shape play a minor role. The only distinct manifestations of nonsphericity and cooperative scattering effects for randomly oriented bispheres are the departure of the ratio F(22)/F(11) of the elements of the scattering matrix from unity, the inequality of the ratios F(33)/F(11) and F(44)/F(11), and nonzero linear and circular backscattering depolarization ratios. Our computations for randomly oriented bispheres with separated wavelengthsized components show that the component spheres become essentially independent scatterers at as small a distance between their centers as 4 times their radii.

Multiple scattering by random particulate media: exact 3D results

We use the numerically exact superposition T-matrix method to perform extensive computations of electromagnetic scattering by a 3D volume filled with randomly distributed wavelength-sized particles. These computations are used to simulate and analyze the effect of randomness of particle positions as well as the onset and evolution of various multiple-scattering effects with increasing number of particles in a statistically homogeneous volume of discrete random medium. Our exact results illustrate and substantiate the methodology underlying the microphysical theories of radiative transfer and coherent backscattering. Furthermore, we show that even in densely packed media, the light multiply scattered along strings of widely separated particles still provides a significant contribution to the total scattered signal and thereby makes quite pronounced the classical radiative transfer and coherent backscattering effects.

Electrostatics analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles

An analysis of radiative absorption and scattering by clusters of spheres in the Rayleigh limit is developed with an electrostatics analysis. This approach assumes that the largest dimension of the cluster is significantly smaller than the wavelength of the radiation. The electric field that is incident upon and scattered by the cluster can then be represented by the gradient of a potential that in turn satisfies Laplaces equation. An analytical solution for the potential that exactly satisfies the boundary conditions at the surfaces of the spheres is obtained with a coupled spherical harmonics method. The components of the polarizability tensor and the absorption, scattering, and depolarization factors are obtained from the solution. Calculations are performed on fractallike clusters of spheres, with refractive-index values that are characteristic of carbonaceous soot in the visible and the IR wavelengths. Results indicate that the absorption cross sections of fractal soot clusters can be significantly larger in the mid-IR wavelengths than what is predicted for Rayleigh-limit spheres that have the same total volume. The absorption cross section (relative to a sphere of the same volume) is dependent on the number of spheres in the aggregate for aggregates with up to approximately 100 primary spheres, and for larger aggregates the relative absorption becomes constant. The predicted spectral variation of soot absorption in the visible and the mid-IR wavelengths is shown to agree well with experimental measurements.

Discrete dipole moment method for calculation of the T matrix for nonspherical particles.

A computational method, based on a moment solution to the discrete dipole approximation (DDA) interaction equations, is proposed for calculation of the T matrix of arbitrary-shaped particles. It is shown that the method will automatically provide the conservation-of-energy and origin-invariance properties required of the T matrix. Furthermore, the method is significantly faster than a T-matrix calculation by direct inversion of the DDA equations. Because the method retains the dipole lattice representation of the particle, it can be applied with relative ease to particles with irregular shapes-although in the same respect it will not automatically simplify for axisymmetric particles. Calculations of scattering matrix distributions, in fixed and random orientations, are made for tetrahedron, cylindrical, and prolate spheroid particle shapes and compared with DDA and extended boundary condition method results.

Light scattering by randomly oriented bispheres.

We describe how the T-matrix approach can be used to compute analytically the Stokes scattering matrix for randomly oriented bispheres with touching or separated components. Computations for randomly oriented bispheres with touching components are compared with those for volume-equivalent randomly oriented prolate spheroids with an aspect ratio of 2 and for a single volume-equivalent sphere. We show that cooperative (multiple-scattering) effects can make bispheres more efficient depolarizers than spheroids in the back-scattering direction.

Single scattering by a small volume element

Starting from first principles, we present a detailed analysis of the concept of single scattering of light by a small volume element filled with sparsely and randomly positioned particles. We first derive the formulas of the far-field single-scattering approximation, which treats the volume element as a single scatterer, and discuss its range of applicability, using for illustration exact T-matrix results for randomly oriented two-sphere clusters. Our second approach is to treat the volume element as a small cloud of particles and apply the so-called first-order-scattering approximation. We demonstrate that although the two approaches are based on somewhat different sets of assumptions, they give essentially the same result for the electromagnetic response of a sufficiently distant polarization-sensitive detector.

Computations of scattering matrices of four types of non-spherical particles using diverse methods

Abstract The scattering matrix as a function of scattering angle has been computed for four different homogeneous particles: a prolate spheroid, an oblate spheroid, a finite cylinder and a bisphere with touching components. The directions of the incident and scattered light beams, as well as the rotation axis of each particle, lie in the same plane. The particles considered have the same refractive index, volume and orientation of the rotation axis with respect to the incident light. The computations were performed with the (superposition) T -matrix method, the separation of variables method for spheroids and the discrete-dipole approximation. The results are presented in the form of tables and graphs. Their usefulness as benchmark results and some other aspects are also discussed.

Electromagnetic scattering by randomly oriented bispheres: Comparison of theory and experiment and benchmark calculations

Abstract A good quantitative agreement is found between laboratory measurements of the scattering matrix for a randomly oriented latex bisphere with touching, nearly identical micron-sized components and theoretical computations using the T -matrix method. Our comparison of theory and experiment provides an additional validation of the computational method and also demonstrates that polarization measurements of light scattering can be employed as an accurate particle sizing technique. The T -matrix method is used to tabulate light scattering properties of two different kinds of randomly oriented bispheres with touching and separated components. Because of high accuracy, our computations can serve as benchmarks.