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The finite-element method (FEM) is increasingly used as a numerical tool to support experimental and theoretical studies of the optical properties of nanoparticles, in contexts such as surface-enhanced spectroscopy, molecular plasmonics, metamaterials, and optical trapping. Here, we investigate the validity of such calculations, focusing in particular on numerically challenging cases involving strong optical (plasmon) resonances and solutions with large electric field gradients and intensities. These are exemplified by elongated metallic nanoparticles and two closely spaced metallic spheres (dimer), where highly localized regions of intense electric field enhancements occur at the tip or in the gap, so-called electromagnetic hot-spots. We assess the accuracy of the FEM solutions by comparing the result to exact analytic solutions based on the T-matrix method for an elongated particle and generalized Mie theory for a dimer. Particular attention is given to the electromagnetic properties that have seldom been studied in this context, notably near-field properties such as surface-field enhancement factors and far-field radiation profiles. We also demonstrate explicitly how the accuracy of the FEM predictions can be inferred from the solution of two problems with different mesh and bounding box parameters. Such a numerical check is crucial in practice as no exact solutions are in general available to compare with. While we chose the commercial software COMSOL to illustrate our results, the methods and conclusions are equally applicable to other FEM implementations. We provide for convenience full details of how to set up these calculations in COMSOL, which we hope will allow readers to easily reproduce them and seamlessly adapt them to their modeling needs. We expect this work will cement the FEM as a reliable method for routine calculation of electromagnetic scattering by nanoparticles. [ABSTRACT FROM AUTHOR]

• Optical properties and radiative forcing of tar ball aggregates, individual tar balls, and volume equivalent spheres are calculated from 350 to 1150 nm using T-Matrix and Lorenz-Mie simulations. • A sensitivity analysis is performed for different refractive indices, and different monomer sizes and numbers. • Single scattering albedo and hemispherical upscatter fraction of tar ball aggregates differ from individual tar balls by up to 77% and more than 100%, respectively. • Over dark surfaces, the top of the atmosphere simple forcing efficiency of tar ball aggregates differs from individual tar balls by up to 53%. Image, graphical abstract Tar balls are frequently found in slightly aged biomass burning plumes. They are spherical in shape, have diameters between ∼100 and 300 nm, are amorphous and composed mostly of oxygen and carbon. Tar balls are light absorbing and considered to be a component of brown carbon. Tar balls have been typically reported and analyzed as individual spheres; however, in a recent study, we reported the presence of significant fractions of fractal-like aggregates made of several tar balls in fire plumes from different geographical locations. Aggregation affects the optical properties of particles; therefore, we use T-Matrix and Lorenz-Mie simulations to explore the effects of aggregation on the tar balls' optical properties in the 350 – 1150 nm wavelength range. We also evaluate the effects of different refractive indices available from the literature, different monomer numbers, and monomer sizes, as these are key factors determining the aggregates optical properties. Furthermore, we estimate the simple forcing efficiency for low and high surface albedos. Aggregates have a single scattering albedo (SSA) higher than that of individual tar balls (ΔSSA 550 nm up to 0.22). The hemispherical upscatter fraction of individual tar balls is more than 100% larger than for tar ball aggregates in many cases. The top of the atmosphere simple forcing efficiency over dark surfaces shows large variabilities with an increase up to ∼53% for tar ball aggregates compared to individual tar balls. These results demonstrate that aggregation of tar balls can have a significant impact on their optical properties and radiative forcing. [ABSTRACT FROM AUTHOR]

• Beam shape coefficients for Laguerre-Gauss beams focused by a lens are discussed. • The method used is the finite series techniques and expressions within this framework are established. • Expressions rely on recurrence equations which are solved. Pursuing a study concerning the evaluation of Beam Shape Coefficients (BSCs) of Laguerre-Gauss beams, after having dealt with the case of beams freely propagating, the present paper establishes finite series expressions to evaluate the BSCs of Laguerre-Gauss beams focused by a lens, which have the known advantage of speeding up computations with respect to the use of numerical quadrature techniques and which may deal with the limitations of localized approximations in the case of beams exhibiting topological charges and/or axicon angles. [ABSTRACT FROM AUTHOR]

• Beam shape coefficients of freely propagating Laguerre–Gauss beams can- not be evaluated speedily enough using quadratures. • Two algorithms to compute freely propagating Laguerre–Gauss finite series beam shape coefficients with similar time complexity are presented. One takes a recursive approach and the other eliminates recursions resulting in an iterative process. • Numerical comparisons between quadrature, localized and finite series computed radial beam shape coefficients are presented as well as their resulting radial electric field profiles. • Deviations from the "intended" field profiles are more pronounced when the beam focus and topological charges are considered. • This work shows that the finite series method can overcome other methods in terms of accuracy or computational cost for freely propagating Laguerre–Gauss beams. In the framework of a series of papers devoted to warnings concerning the evaluation of beam shape coefficients (BSCs) of electromagnetic beams by using a localized approximation, a recent paper provided finite series explicit expressions to evaluate BSCs of paraxial Laguerre–Gauss freely propagating beam. In the present paper, these expressions are numerically implemented and finite series numerical results are compared with results obtained by using quadratures and a localized approximation. These three methods provide a remodeling of the beam, changing the non-Maxwellian paraxial Laguerre–Gauss beam to a beam which perfectly satisfies Maxwell's equations. Comparisons between the three methods are discussed in terms of accuracy and computational time efficiency. [ABSTRACT FROM AUTHOR]

• Simulations assess an approach for optically characterizing fractal aggregates. • Digital holograms of aggregates analyzed by modeling aggregates as spheres. • Criteria for determining the validity of effective-sphere approach presented. We perform simulations to evaluate a recent experimental technique for using in-line holographic microscopy and an effective-sphere model to measure the population-averaged fractal dimension D f of an ensemble of colloidal fractal aggregates. In this technique, models based on Lorenz–Mie scattering by a uniform sphere are fit to digital holograms of a population of fractal aggregates to determine the effective refractive indices n eff and effective radii a eff of the aggregates. A scaling relationship between n eff and a eff based on the Maxwell Garnett effective-medium theory then determines D f. Here we use a multisphere superposition code to calculate the exact holograms produced by aggregates with tunable fractal dimensions D f. We show that n eff and a eff become less sensitive to the aggregate orientation as D f increases. We also show that the Maxwell Garnett scaling relationship correctly determines D f to within 10.5% when multiple scattering is negligible and the population-averaged coefficient of determination ⟨ R 2⟩ p > 0.6, indicating that the holograms are well-described by the effective-sphere model. [ABSTRACT FROM AUTHOR]