This note gives a short introduction to the reprint of the article "Numerical methods in electromagnetic scattering theory" by Kahnert, M (JQSRT 2003:79-80:775-824). Some of the most important developments in the field since the publication of this article are briefly reviewed. A list of typos that have been identified in the original article is given in the appendix. (C) 2009 Elsevier Ltd. All rights reserved.

The effect of inhomogeneous mineralogical composition on the optical properties of mineral dust particles is investigated. More specifically, spheres composed of a non-absorbing mineral with multiple spherical hematite inclusions are considered. The size of the particles, the number of inclusions, and the hematite volume fraction are varied, and the differential and integral optical properties are compared to those computed for homogeneous spheres. The effective refractive index of the homogeneous spheres is obtained (i) by use of four conventional effective medium approximations; and (ii) by freely varying the real and imaginary parts of the refractive index until a best-fit of the scattering matrix elements is achieved for all scattering angles and particle sizes. Among the integral radiometric observables, the single scattering albedo is most sensitive to particle inhomogeneity, while the extinction and scattering efficiency and the asymmetry parameter are rather insensitive. The phase function, the degree of linear polarisation, the linear depolarisation, and, indeed, all elements of the scattering matrix are strongly modulated by particle inhomogeneity. None of the effective medium approaches, not even the best-fit method, are able to reproduce the single scattering albedo and the scattering matrix elements over the entire range of particle sizes. (C) 2014 Elsevier Ltd. All rights reserved.

Numerical solution methods for electromagnetic scattering by non-spherical particles comprise a variety of different techniques, which can be traced back to different assumptions and solution strategies applied to the macroscopic Maxwell equations. One can distinguish between time- and frequency-domain methods; further, one can divide numerical techniques into finite-difference methods (which are based on approximating the differential operators), separation-of-variables methods (which are based on expanding the solution in a complete set of functions, thus approximating the fields), and volume integral-equation methods (which are usually solved by discretisation of the target volume and invoking the long-wave approximation in each volume cell). While existing reviews of the topic often tend to have a target audience of program developers and expert users, this tutorial review is intended to accommodate the needs of practitioners as well as novices to the field. The required conciseness is achieved by limiting the presentation to a selection of illustrative methods, and by omitting many technical details that are not essential at a first exposure to the subject. On the other hand, the theoretical basis of numerical methods is explained with little compromises in mathematical rigour; the rationale is that a good grasp of numerical light scattering methods is best achieved by understanding their foundation in Maxwell's theory. (C) 2015 Elsevier Ltd. All rights reserved.

A T-matrix code tailored to non-axisymmetric particles with finite symmetries is described. The code exploits geometric symmetries of particles by use of group theoretical methods. Commutation relations of the T-matrix are implemented for reducing CPU-time requirements. Irreducible representations of finite groups are employed for alleviating ill-conditioning problems in numerical computations. Further, an iterative T-matrix method for particles with small-scale surface perturbations is implemented. The code can compute both differential and integrated optical properties of particles in,either fixed or random orientation. Methods for testing the convergence and correctness of the computational results are discussed. The package also includes a database of pre-computed group-character tables, as well as an interface to the GAP programming language for computational group theory. The code can be downloaded at http://www.rss.chalmers.se/similar to kahnert/Tsym.html. (C) 2013 Elsevier Ltd. All rights reserved.

The use of group theoretical methods can substantially reduce numerical ill-conditioning problems in T-matrix computations. There are specific problems related to obtaining the irreducible characters of high-order symmetry groups and to the construction of a transformation from the basis of vector spherical wave functions to the irreducible basis of high-order symmetry groups. These problems are addressed, and numerical solutions are discussed and tested. An important application of the method is non-convex particles perturbed with high-order polynomials. Such morphologies can serve as models for particles with small-scale surface roughness, such as mineral aerosols, atmospheric ice particles with rimed surfaces, and various types of cosmic dust particles. The method is tested for high-order 3D-Chebyshev particles, and the performance of the method is gauged by comparing the results to computations based on iteratively solving a Lippmann-Schwinger T-matrix equation. The latter method trades ill-conditioning problems for potential slow-convergence problems, and it is rather specific, as it is tailored to particles with small-scale surface roughness. The group theoretical method is general and not plagued by slow-convergence problems. The comparison of results shows that both methods achieve a comparable numerical stability. This suggests that for particles with high-order symmetries the group-theoretical approach is able to overcome the illconditioning problems. Remaining numerical limitations are likely to be associated with loss-of-precision problems in the numerical evaluation of the surface integrals. (C) 2012 Elsevier Ltd. All rights reserved.

The size-dependence of the linear depolarisation ratio of mineral dust aerosols is investigated. Laboratory measurements on 131 different aerosol samples with varying size distributions and mineralogical compositions are fitted with a homogeneous spheroid model. A minimum-bias and minimum-variance fit of the data is obtained for prolate model particles with a refractive index of 1.525+0.001i and an aspect ratio of 0.87. The model error is analysed by varying the input parameters to the light-scattering computations. It is found that the scattering of the measurements about the model can mainly be explained by variation of the morphology and dielectric properties, and to a much lesser extent by variation in the geometric standard deviation of the size distribution. The modelling of the data is extended by using size-shape distributions of spheroids. The results indicate that there is some freedom in choosing the best-fit weights of the shape-distribution of spheroids, which could potentially be useful when extending the model to multiple wavelengths, or to including additional optical parameters other than depolarisation. However, it is also found that the most reasonable fits of the data are obtained by mildly aspherical prolate and oblate spheroids, which limits the freedom of adjusting the best-fit weights. (C) 2020 The Authors. Published by Elsevier Ltd.

The error caused by the uncertainty in the refractive index in the determination of the asymmetry parameter g is studied for a variety of mineral dust aerosol samples at two different optical wavelengths. Lorenz-Mie computations for spherical model particles are compared with results based on laboratory-measured phase functions in conjunction with a commonly used extrapolation method. The difference between the g-value based on measurements and the g-value based on Lorenz-Mie simulations is generally on the same order of magnitude as the error caused by the uncertainty in the refractive index m. For larger effective radii the error in g related to the use of spherical model particles is even larger than that related to the uncertainty in in. This indicates that the use of spherical model particles can be among the major error sources in the determination of the asymmetry parameter of dust aerosols. (c) 2005 Elsevier Ltd. All rights reserved.

A method is developed based on the variational data-analysis formalism to combine laboratory-measured scattering phase functions with forward-scattering phase function computations based on independent size distribution (SD) measurements. The algorithm yields an optimal estimate of the true phase function of the system that is not only based on the measurements and the computational results but also on all available information of the error variances and, if applicable, error covariances of the measured and computed phase functions. The high flexibility of the method is demonstrated by applying it to phase functions of feldspar and fly ash aerosols. Further, the algorithm is employed to determine the asymmetry parameter g of nine different mineral aerosol samples at two different optical wavelengths, and to assess the relative importance of different error sources in the determination of g. It is found that the use of spherical model particles in simulations of g can result in errors on the same order of magnitude as the uncertainty of the refractive index. The use of spherical model particles in computations of forward scattering, however, is found to be only a minor error source. (c) 2006 Elsevier Ltd. All rights reserved.

This review paper provides an overview over model geometries for computing light scattering by small particles. The emphasis is on atmospheric optics, although much of this review will also be relevant to neighbouring fields, in particular to astronomy. Various morphological particle properties are discussed, such as overall nonsphericity, pristine shapes, aggregation, and different forms of inhomogeneity, e.g. porous and compact inhomogeneous morphologies, as well as encapsulated aggregates. Models employed to reproduce the optical properties of complex particles range from strongly simplified to highly realistic and morphologically sophisticated model geometries. Besides reviewing the most recent literature, we discuss the idea behind models of varying degree of complexity with regard to the intended use of the models. Applications range from fundamental studies of light scattering processes to routine applications of particle optics look-up tables in operational modelling systems. (C) 2014 Elsevier Ltd. All rights reserved.

We perform a comparative modelling study to investigate how different morphological features influence the optical properties of hematite aerosols. We consider high-order Chebyshev particles as a proxy for aerosol with a small-scale surface roughness, and spheroids as a model for nonspherical aerosols with a smooth boundary surface. The modelling results are compared to those obtained for homogeneous spherical particles. It is found that for hematite particles with an absorption efficiency of order unity the difference in optical properties between spheres and spheroids disappears. For optically softer particles, such as ice particles at far-infrared wavelengths, this effect can be observed for absorption efficiencies lower than unity. The convergence of the optical properties of spheres and spheroids is caused by absorption and quenching of internal resonances inside the particles, which depend both on the imaginary part of the refractive index and on the size parameter, and to some extent on the real part of the refractive index. By contrast, small-scale surface roughness becomes the dominant morphological feature for large particles. This effect is likely to depend on the amplitude of the surface roughness, the relative significance of internal resonances, and possibly on the real part of the refractive index. The extinction cross section is rather insensitive to surface roughness, while the single-scattering albedo, asymmetry parameter, and the Mueller matrix are strongly influenced. Small-scale surface roughness reduces the backscattering cross section by up to a factor of 2-3 as compared to size-equivalent particles with a smooth boundary surface. This can have important implications for the interpretation of lidar backscattering observations. (C) 2011 Elsevier Ltd. All rights reserved.

We compare four different model geometries for particles with small-scale surface roughness. The geometries are based on regular and stochastic surface perturbations, as well as on 2D- and 3D-roughness models. We further compare T-matrix and discrete dipole computations. Particle size parameters of 5 and 50 are considered, as well as refractive indices of 1.6+0.0005i and 3+0.1i. The effect of small-scale surface roughness on the intensity and polarisation of the scattered light strongly depends on the size parameter and refractive index. In general, 2D surface roughness models predict stronger effects than 3D models. Stochastic surface roughness models tend to predict the strongest depolarising effects, while regular surface roughness models can have a stronger effect on the angular distribution of the scattered intensity. Computations with the discrete dipole approximation only cover a limited range of size parameters. T-matrix computations allow us to significantly extend that range, but at the price of restricting the model particles to symmetric surface perturbations with small amplitudes. (C) 2012 Elsevier Ltd. All rights reserved.

We used four different non-spherical particle models to compute optical properties of an arctic ice cloud and to simulate corresponding cloud radiative forcings and fluxes. One important finding is that differences in cloud forcing, downward flux at the surface, and absorbed flux in the atmosphere resulting from the use of the four different ice cloud particle models are comparable to differences in these quantities resulting from changing the surface albedo from 0.4 to 0.8, or by varying the ice water content (IWC) by a factor of 2. These findings show that the use of a suitable non-spherical ice cloud particle model is very important for a realistic assessment of the radiative impact of arctic ice clouds. The differences in radiative broadband fluxes predicted by the four different particle models were found to be caused mainly by differences in the optical depth and the asymmetry parameter. These two parameters were found to have nearly the same impact on the predicted cloud forcing. Computations were performed first by assuming a given vertical profile of the particle number density, then by assuming a given profile of the IWC. In both cases, the differences between the cloud radiative forcings computed with the four different non-spherical particle models were found to be of comparable magnitude. This finding shows that precise knowledge of ice particle number density or particle mass is not sufficient for accurate prediction of ice cloud radiative forcing. It is equally important to employ a non-spherical shape model that accurately reproduces the ice particle's dimension-to-volume ratio and its asymmetry parameter. The hexagonal column/plate model with air-bubble inclusions seems to offer the highest degree of flexibility. (c) 2007 Elsevier Ltd. All rights reserved.

A surface integral formulation is developed for the T matrix of a homogenous and isotropic particle of arbitrary shape, which employs scalar basis functions represented by the translation matrix elements of the vector spherical wave functions. The formulation begins with the volume integral equation for scattering by the particle, which is transformed so that the vector and dyadic components in the equation are replaced with associated dipole and multipole level scalar harmonic wave functions. The approach leads to a volume integral formulation for the T matrix, which can be extended, by the use of Green's identities, to the surface integral formulation. The result is shown to be equivalent to the traditional surface integral formulas based on the VSWF basis. (C) 2013 Elsevier Ltd. All rights reserved.

The use of simplified particle shapes for modeling scattering by irregularly shaped mineral-dust particles is studied using polyhedral prisms and spheroids as model particles. Simulated phase matrices averaged over shape and size distributions at wavelength 633 nm are compared with a laboratory-measured phase matrix of feldspar particles with known size distribution with effective radius of 1.0 mu m. When an equi-probable shape distribution is assumed, prisms and oblate spheroids agree with measurements to a similar degree, whereas prolate spheroids perform markedly better. Both spheroids and prisms perform much better than spheres. When ail automatic fitting method is applied for finding optimal shape distributions, it is found that the most elongated spheroids are most important for good fits, whereas nearly-spherical spheroids are generally of very little importance. The phase matrices for the different polyhedral prisms, on the other hand, are found to be similar, thus their shape-averaged phase matrices are insensitive to the shape distribution assumed. For spheroids, a simple parameterization for the shape distribution, where weights increase with increasing departure from spherical shape, is proposed and tested. This parameterization improves the fit of most phase matrix elements attained with an equi-probable shape distribution, and it performs particularly well for reproducing the measured phase function. (c) 2006 Elsevier Ltd. All rights reserved.

We address the question if and how observations of scattered intensity and polarisation can be employed for retrieving particle shape information beyond a simple classification into spherical and nonspherical particles. To this end, we perform several numerical experiments, in which we attempt to retrieve shape information of complex particles with a simple nonspherical particle model based on homogeneous spheroids. The discrete dipole approximation is used to compute reference phase matrices for a cube, a Gaussian random sphere, and a porous oblate and prolate spheroid as a function of size parameter. Phase matrices for the model particles, homogeneous spheroids, are computed with the T-matrix method. By assuming that the refractive index and the size distribution is known, an optimal shape distribution of model particles is sought that best matches the reference phase matrix. Both the goodness of fit and the optimal shape distribution are analysed. It is found that the phase matrices of cubes and Gaussian random spheres are well reproduced by the spheroidal particle model, while the porous spheroids prove to be challenging. The "retrieved" shape distributions, however, do not correlate well with the shape of the target particle even when the phase matrix is closely reproduced. Rather, they tend to exaggerate the aspect ratio and always include multiple spheroids. A most likely explanation why spheroids succeed in mimicking phase matrices of more irregularly shaped particles, even if their shape distributions display little similarity to those of the target particles, is that by varying the spheroids' aspect ratio one covers a large range of different phase matrices. This often makes it possible to find a shape distribution of spheroids that matches the phase matrix of more complex particles. (C) 2011 Elsevier Ltd. All rights reserved.

It is well established that spherical and nonspherical particles scatter light differently. There are a large number of studies where scattering properties of different nonspherical particles are studied. Here we study to what degree scattering matrices of different nonspherical particles resemble each other, and whether there are significant correlations between morphological similarity and similar single-scattering properties. Altogether 15 different shapes are considered, including both irregular and regular shapes as well as homogeneous and inhomogeneous particles. For all nonspherical particles, orientation- and ensemble-averaged scattering properties are considered, and variability within each ensemble is ignored. The results reveal that different nonspherical shapes have surprisingly similar phase functions. An analysis of the asymmetry parameter reveals that the resemblance is, however, only qualitative: the phase functions are featureless and predominantly flat at side scattering, but they are nevertheless different. The degree of linear polarization for unpolarized incident light shows much larger differences among the shapes, albeit it is much more positive for all nonspherical targets than for Mie spheres. Similar to the phase function, the depolarization ratio tends to be similar among the nonspherical particle types, implying that the strength of depolarization cannot be used as a measure for the type of nonsphericity. In general, it is found that there does not seem to be a clear correlation between particle morphology and scattering properties. (C) 2012 Elsevier Ltd. All rights reserved.

The single-scattering properties of eight black carbon (BC, soot) fractal aggregates, composed of primary spheres from 7 to 600, computed by the geometric-optics surface-wave (GOS) approach coupled with the Rayleigh-Gans-Debye (RGD) adjustment for size parameters smaller than approximately 2, are compared with those determined from the superposition T-matrix method. We show that under the condition of random orientation, the results from GOS/RGD are in general agreement with those from T-matrix in terms of the extinction and absorption cross-sections, the single-scattering co-albedo, and the asymmetry factor. When compared with the specific absorption (m(2)/g) measured in the laboratory, we illustrate that using the observed radii of primary spheres ranging from 3.3 to 25 nm, the theoretical values determined from GOS/RGD for primary sphere numbers of 100-600 are within the range of measured values. The GOS approach can be effectively applied to aggregates composed of a large number of primary spheres (e.g., > 6000) and large size parameters (>> 2) in terms of computational efforts. (C) 2013 Elsevier Ltd. All rights reserved.

The single-scattering properties of Gaussian random spheres are calculated using the discrete dipole approximation. The ensemble of model particles is assumed to be representative for a feldspar dust sample that is characteristic for weakly absorbing irregularly shaped mineral aerosol. The morphology of Gaussian random spheres is modeled based on a statistical shape analysis using microscope images of the dust sample. The size distribution of the dust sample is based on a particle sizing experiment. The refractive index of feldspar is estimated using literature values. All input parameters used in the light scattering simulations are thus obtained in an objective way based on the true properties of the mineral sample. The orientation-averaged and ensemble-averaged scattering matrices and cross sections of the Gaussian random spheres are compared with light scattering simulations using spheroidal shape models which have been shown to be applicable to the feldspar sample. The Gaussian random sphere model and the spheroidal shape model are assessed using the measured scattering matrix of the feldspar dust sample as a reference. Generally, the spheroidal model with strongly elongated prolate and strongly flattened oblate shapes agrees better with the measurement than the Gaussian random sphere model. In contrast, some features that are characteristic for light scattering by truly irregular mineral dust particles are rendered best by the Gaussian random sphere model; these features include the flat shape of the phase function and a minimum in the scattering matrix element F-22/F-11 as a function of the scattering angle. (c) 2005 Elsevier Ltd. All rights reserved.

We simulated light-scattering by small and wavelength-sized cubes with three largely different values of the refractive index using the discrete dipole approximation (DDA) and the T-matrix method. Our main goal was to push the accuracy of both methods to the limit. For the DDA we used an earlier developed extrapolation technique based on simulation results for different levels of discretization. For the T-matrix method we developed a procedure to estimate a confidence range for the simulated value, using results for different values of the truncation index (number of multipoles). In most cases this confidence range was reliable, enclosing the corresponding DDA result. We present benchmark results by both methods, including estimated uncertainties, for selected integral and angle-resolved scattering quantities. Estimated relative uncertainties of the DDA result are unprecedentedly small (from 10(-7) to 10(-3)), while relative differences between the T-matrix and DDA results are larger (from 10(-4) to 0.2) in accordance with estimated T-matrix uncertainties. (C) 2012 Elsevier Ltd. All rights reserved.