Spherical Scatterers Research Articles

The Boltzmann-Grad Limit of the Lorentz Gas in a Union of Lattices

The Lorentz gas describes an ensemble of noninteracting point particles in an infinite array of spherical scatterers. In the present paper we consider the case when the scatterer configuration P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document} is a fixed union of (translated) lattices in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document}, and prove that in the limit of low scatterer density, the particle dynamics converges to a random flight process. In the special case when the lattices in P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document} are pairwise incommensurable, this settles a conjecture from Marklof and Strömbergsson (J Stat Phys 155:1072–1086, 2014). The proof is carried out by applying a framework developed in recent work by Marklof and Strömbergsson (Mem AMS 294, 2024), and central parts of our proof are the construction of an admissible marking of the point set P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document}, and the verification of the uniform spherical equidistribution condition required in Marklof and Strömbergsson (Mem AMS 294, 2024). Regarding the random flight process obtained in the low density limit of the Lorentz gas, we prove that it can be reconstructed from the corresponding limiting flight processes arising from the individual commensurability classes of lattices in P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document}. We furthermore prove that the free path lengths of the limit flight process have a distribution with a power law tail, whose exponent depends on the number of commensurability classes in P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document}.

Interference of echo-signals from spherical scatterers located near the seabed

The paper investigates the impact of the seabed on the echo signal from spherical scatterers. The seabed is modeled as a liquid absorbing half-space. The transmitter/receiver is located in the water half-space. The distance between the transmitter/receiver and the scatterer is assumed to be large compared to the wavelengths of acoustic waves in water and the seafloor. Numerical results are obtained for acoustically rigid spherical scatterers of the same radius. Interaction between the scatterers is not taken into account. The echo signal from a single sphere over a wide frequency range is computed using a method proposed by R.H. Hackman and G.S. Sammelmann, with a crucial step being the computation of the scattering coefficients of the sphere. Asymptotic formulae obtained using the saddle-point method are used in the paper to compute these coefficients. The obtained asymptotic expressions for the scattering coefficients of the sphere significantly reduce the number of summands in the formula for the form function of the echo signal.

A tiny Drude scatterer can accurately model a coherent emitter in nanophotonics

Abstract We add a missing element to the set of directly computable scenarios of light-matter-interaction within classical numerical Maxwell solvers, i.e., light scattering from hybrid systems of resonators and individual Fourier-limited emitters. In particular, individual emitters are incorporated as tiny polarizable and resonant spherical scatterers. This emitter model is based on well-known extremal properties of Mie modes. The spherical emitter is made from an artificial Drude metal with ϵ ( ω ) = ϵ b − ω p 2 / ( ω 2 + i Γ ω ) ${\epsilon}(\omega )={{\epsilon}}_{b}-{\omega }_{p}^{2}/({\omega }^{2}+i{\Gamma }\omega )$ . By tuning ϵ b and ω p we adjust the resonance frequency and the Fourier-limited linewidth and by adjusting Γ we may add non-radiative damping or dephasing. This approach automatically reproduces the ideal text book coherent scattering cross-section of Fourier-limited two level quantum systems of σ 0 = 3λ 2/(2πϵ out) which is not possible with typically used Lorentz permittivities which only mimic optical resonances. Further, the emitter’s linewidth adopts to the surrounding optical local density of states (LDOS). To demonstrate this we successfully benchmark our approach with prominent examples from the literature.

Design of reconfigurable Huygens metasurfaces based on Drude-like scatterers operating in the epsilon-negative regime.

In this study, we investigate the feasibility of designing reconfigurable transmitting metasurfaces through the use of Drude-like scatterers with purely electric response. Theoretical and numerical analyses are provided to demonstrate that the response of spherical Drude-like scatterers can be tailored to achieve complete transmission, satisfying a generalized Kerker's condition at half of their plasma frequency. This phenomenon, which arises from the co-excitation of the electric dipole and the electric quadrupole within the scatterer, also exhibits moderate broadband performance. Subsequently, we present the application of these particles as meta-atoms in the design of reconfigurable multipolar Huygens metasurfaces, outlining the technical prerequisites for achieving effective beam-steering capabilities. Finally, we explore a plausible implementation of these low-loss Drude-like scatterers at microwave frequencies using plasma discharges. Our findings propose an alternative avenue for Huygens metasurface designs, distinct from established approaches relying on dipolar meta-atoms or on core-shell geometries. Unlike these conventional methods, our approach fosters seamless integration of reconfigurability strategies in beam-steering devices.

Multimode multiple wave scattering in suspensions of solid particles in viscous liquids: part 2: numerical validation

In the first paper in this series, we presented asymptotic results for the effective wavenumber of the coherent longitudinal waves propagating through a system of pseudo-randomly distributed spherical elastic scatterers in a viscous medium. The analysis was based on multi-modal multiple scattering theory to account for both longitudinal and shear waves in the viscous embedding medium, arising from wave conversions at the scatterers surface, and identified asymptotic results in the long wavelength limit of the longitudinal waves. In this second paper, we present numerical validation of the various asymptotic approximations presented previously, including truncation of the number of partial wave orders and the use of the low concentration expansion of the dispersion relation. We explore the important contributions of wave conversion and of correlations in particle positions to the effective attenuation and to the frequency of the dipolar resonance of the scatterers. A comparison of the experimental results obtained with water containing sub-micrometric silica beads shows the very good validity of the model, particularly in the vicinity of the dipolar resonance.

Interference of Echo Signals from Spherical Scatterers Located Near the Bottom

Interference of Echo Signals from Spherical Scatterers Located Near the Bottom

Propagation of elastic waves in correlated dispersions of resonant scatterers.

The propagation of coherent longitudinal and transverse waves in random distributions of spherical scatterers embedded in an elastic matrix is studied. The investigated frequency range is the vicinity of the resonance frequencies of the translational and rotational motion of the spheres forced by the waves, where strong dispersion and attenuation are predicted. A technique for making samples made of layers of carbide tungsten beads embedded in epoxy resin is presented, which allows control of the scatterers distribution, induce short-range positional correlations, and minimize the anisotropy of samples. Comparison between phase velocity and attenuation measurements and a model based on multiple scattering theory (MST) shows that bulk effective properties accurately described by MST are obtained from three beads layers. Besides, short-range correlations amplify the effect of mechanical resonances on the propagation of longitudinal and transverse coherent waves. As a practical consequence, the use of short-range positional correlations may be used to enhance the attenuation of elastic waves by disordered, locally resonant, elastic metamaterials, and MST globally correctly predicts the effect of short-range positional order on their effective properties.

Determination of upper limit of Guinier approximation on dilute polydisperse system in SAXS

For a dilute polydisperse system in small-angle X-ray scattering (SAXS), Guinier’s law can be used to perform an approximate analysis to determine the mean radius of gyration of scatterers in a sample and extend the scattering curve to the lower angle region. The main idea of the law is that the Guinier plot appears linear in the vicinity of the scattering vector equaling zero. In this short contribution, a differential method is proposed to uniquely determine the upper limit of the effective range of the Guinier approximation on dilute polydisperse systems. The theoretical simulation is mainly performed on the typical spherical scatterers in several dilute polydisperse systems to explore the relationship between the upper limit and the shape and size of the scatterers. The SAXS experiment on a sample of silicon dioxide colloid is carried out to verify the feasibility of the new method.

Normal Mode Solutions of Target Strengths of Solid-filled Spherical Shells and Discussion of Influence Parameters

The normal mode solution for the form function and target strength (TS) of a solid-filled spherical shell is derived. The calculation results of the spherical shell’s acoustic TS are in good agreement with the results of the finite element method (FEM). Based on these normal mode solutions, the influences of parameters such as the material, radius, and thickness of the inner and outer shells on the TS of a solid-filled spherical shell are analyzed. An underwater spherical shell scatterer is designed, which uses room temperature vulcanized (RTV) silicone rubber as a solid filling material and does not contain a suspension structure inside. The scatterer has a good TS enhancement effect.

Kinetic Theory for the Low-Density Lorentz Gas

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including semiconductor physics and radiative transfer. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of the present study is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strömbergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers’ centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

Optical diffraction tomography for assessing single cell models in angular light scattering.

Angularly resolved light scattering (ALS) has become a useful tool for assessing the size and refractive index of biological scatterers at cellular and organelle length scales. Sizing organelle populations with ALS relies on Mie scattering theory models, which require significant assumptions about the object, including spherical scatterers and a homogeneous medium. These assumptions may incur greater error at the single cell level, where there are fewer scatterers to be averaged over. We investigate the validity of these assumptions using 3D refractive index (RI) tomograms measured via optical diffraction tomography (ODT). We compute the angular scattering on digitally manipulated tomograms with increasingly strong model assumptions, including RI-matched immersion media, homogeneous cytosol, and spherical organelles. We also compare the tomogram-computed angular scattering to experimental measurements of angular scattering from the same cells to ensure that the ODT-based approach accurately models angular scattering. We show that enforced RI-matching with the immersion medium and a homogeneous cytosol significantly affects the angular scattering intensity shape, suggesting that these assumptions can reduce the accuracy of size distribution estimates.

The human factor: results of a small-angle scattering data analysis round robin

A round-robin study has been carried out to estimate the impact of the human element in small-angle scattering data analysis. Four corrected datasets were provided to participants ready for analysis. All datasets were measured on samples containing spherical scatterers, with two datasets in dilute dispersions and two from powders. Most of the 46 participants correctly identified the number of populations in the dilute dispersions, with half of the population mean entries within 1.5% and half of the population width entries within 40%. Due to the added complexity of the structure factor, far fewer people submitted answers on the powder datasets. For those that did, half of the entries for the means and widths were within 44 and 86%, respectively. This round-robin experiment highlights several causes for the discrepancies, for which solutions are proposed.

Mie Scattering for Photonic Devices

AbstractMie scattering is increasingly exploited to manipulate electromagnetic fields to achieve strong resonant enhancement, to obtain perfect absorption of radiation, and to generate polarization or wavelength selectivity and/or sensitivity. In fact, a multitude of photonic applications are arising that benefit from Mie scattering and have already led to the formation of novel image and hologram schemes or to the design of efficient and compact photodetectors and light sources. Here, the Mie scattering theory for spherical scatterers is reviewed, the basics of the field‐decomposition onto the Mie modes are shown, and dielectric and metallic particles are compared. Recent applications for light spectrum control, detection, non‐linear effects enhancement, and emission are reviewed. How a periodic arrangement of Mie‐scatterers can be utilized to create a strong (in the order of 10 to 100) absorption enhancement in otherwise weakly absorbing layers is also demonstrated. The enhancement dependence on the diameter‐to‐wavelength ratio of the scatterer is analyzed, and how the influence of different Mie‐modes can be distinguished in the periodic array by looking at the field components normal to the scatterer surface is discussed.

The method of fundamental solutions for electromagnetic scattering by a perfect conductor in chiral environment

The electromagnetic scattering problem for a time-harmonic plane wave by a perfectly conductive body embedded in a chiral medium is considered. Potentials in terms of the fundamental solution in dyadic form for chiral media are defined. The well-posedness of the scattering problem is studied using defined potentials in terms of the fundamental solution for chiral environment and the boundary integral operators associated with them. An extension of the method of fundamental solutions for electromagnetic scattering in chiral media is applied in order to approximate the solution. The electric scattered field is approximated by a finite linear combination of the dyadic fundamental solutions of the equation governing the problem for singularities placed on an auxiliary surface outside the domain of the problem and for unknown coefficients determined by the boundary conditions. Density properties on the surface of the scatterer are proven for a defined system of functions consisting of the tangential components of the dyadic fundamental solution. These density properties together with the well-posedness of the scattering problem are used to reduce the approximation of the solution to the approximation of the known boundary data. The linear system obtained from the boundary conditions in order to determine the appropriate unknown coefficients of the finite linear combination is presented. Using these results we approximate the far-field patterns and the scattering cross section. Numerical results for the special case of a spherical scatterer and an ellipsoidal scatterer are presented.

Interference of Echo-Signals from Two Buried Spherical Targets

A numerically efficient technique is presented for computing the backscattered fields from two spherical targets embedded in an underwater sediment. The bottom is assumed to be a homogeneous liquid attenuating half-space. The transmitter/receiver is located in a homogeneous water half-space. The distances between the transmitter/receiver and objects of interest are supposed to be large compared to the acoustic wavelengths in water and seabed. In simulations, the spherical scatterers of the same radius are assumed to be acoustically rigid. The interactions between two spheres are not taken into account because of the strong attenuation in the bottom. The scattering from one sphere in a wide frequency range is determined using the Hackman and Sammelmann’s general approach. The arising scattering coefficients of the sphere are evaluated using the steepest descent method. The obtained asymptotic expressions for the scattering coefficients essentially allowed to decrease a number of summands in the formula for the form-function of the backscattered acoustic field.

Optical Transitions and Nonreciprocity in Spatio‐Temporally Periodic Layers of Spherical Particles

AbstractAn extension of the photonic layer multiple scattering methodology to dynamic spherical scatterers, which exhibit a periodic time‐varying response, is presented. The applicability of the method is demonstrated on specific examples of single‐ and bi‐layers of periodically modulated high‐refractive‐index spherical particles arranged on a square lattice. The results provide compelling evidence for strong and tunable inelastic scattering effects under the triple resonance condition, fulfilled for optical transitions between neighboring high‐Q lattice modes of the appropriate symmetry, which originate from multipolar Mie resonances. A consistent interpretation of the underlying mechanisms is provided and potential applications in the design of nonreciprocal devices are discussed.

Analytical solution for the single scattered radiance of two-layered turbid media in the spatial frequency domain. Part 2: Vector radiative transfer equation

In this work, the analytical solution in the spatial frequency domain based on the vector radiative transfer equation is derived for the single scattered radiance of a scattering medium. A two-layer model with spherical scatterers is assumed as the scattering medium, where the second layer is infinitely extended and there is no refractive index mismatch between the two scattering layers. In contrast, a refractive index mismatch to the surrounding non-scattering medium is considered. The time-resolved and time-independent solutions are given and are verified using a self-implemented electrical field Monte Carlo method. Excellent agreement between the analytical solution and the Monte Carlo method is demonstrated. Possible errors due to discretizations in the Monte Carlo method are reduced, for example, by using the polarized last flight variance reduction method.

A study of optical scattering modelling for mixed-phase polar stratospheric clouds

Abstract. Scattering codes are used to study the optical properties of polar stratospheric clouds (PSCs). Particle backscattering and depolarization coefficients can be computed with available scattering codes once the particle size distribution (PSD) is known and a suitable refractive index is assumed. However, PSCs often appear as external mixtures of supercooled ternary solution (STS) droplets, solid nitric acid trihydrate (NAT) and possibly ice particles, making the assumption of a single refractive index and a single morphology to model the scatterers questionable. Here we consider a set of 15 coincident measurements of PSCs above McMurdo Station, Antarctica, using ground-based lidar, a balloon-borne optical particle counter (OPC) and in situ observations taken by a laser backscattersonde and OPC during four balloon stratospheric flights from Kiruna, Sweden. This unique dataset of microphysical and optical observations allows us to test the performances of optical scattering models when both spherical and aspherical scatterers of different composition and, possibly, shapes are present. We consider particles as STS if their radius is below a certain threshold value Rth and NAT or possibly ice if it is above it. The refractive indices are assumed known from the literature. Mie scattering is used for the STS, assumed spherical. Scattering from NAT particles, considered spheroids of different aspect ratio (AR), is treated with T-matrix results where applicable. The geometric-optics–integral-equation approach is used whenever the particle size parameter is too large to allow for a convergence of the T-matrix method. The parameters Rth and AR of our model have been varied between 0.1 and 2 µm and between 0.3 and 3, respectively, and the calculated backscattering coefficient and depolarization were compared with the observed ones. The best agreement was found for Rth between 0.5 and 0.8 µm and for AR less than 0.55 and greater than 1.5. To further constrain the variability of AR within the identified intervals, we have sought an agreement with the experimental data by varying AR on a case-by-case basis and further optimizing the agreement by a proper choice of AR smaller than 0.55 and greater than 1.5 and Rth within the interval 0.5 and 0.8 µm. The ARs identified in this way cluster around the values 0.5 and 2.5. The comparison of the calculations with the measurements is presented and discussed. The results of this work help to set limits to the variability of the dimensions and asphericity of PSC solid particles, within the limits of applicability of our model based on the T-matrix theory of scattering and on assumptions on a common particle shape in a PSD and a common threshold radius for all the PSDs.

Backscattering enhancement, vanishing extinction, and morphological effects of active scatterers

Using analytical Lorenz–Mie scattering formalism and numerical methods, we analyze the response of active particles to electromagnetic waves. The particles are composed of homogeneous, non-magnetic, and dielectrically isotropic medium. Spherical scatterers and sharp and rounded cubes are treated. The absorption cross section of active particles is negative, thus showing gain in their electromagnetic response. Since the scattering cross section is always positive, their extinction can be either positive, negative, or zero. We construct a five-class categorization of active and passive dielectric particles. We point out the enhanced backscattering phenomenon that active scatterers display and also discuss extinction paradox and optical theorem. Finally, using COMSOL Multiphysics and an in-house method-of-moments code, the effects of the non-sphericity of active scatterers on their electromagnetic response are illustrated.

The SSSS scheme: a method for calculating multiple scattering of electromagnetic radiation by a collection of sparsely spaced spherical scatterers of Mie-scattering size based on first principles.

We present a method for calculating multiple scattering of electromagnetic radiation by a collection of sparsely spaced spherical scatterers (SSSS) of Mie-scattering size based on first principles rather than radiative transfer theory. In this respect, our methodology is conceptually similar to the superposition T-matrix method. However, our implementation, which we call the SSSS scheme, differs in a number of respects. Overall, the SSSS scheme is simpler, it is better suited numerically to sparse spacing, and the computer memory required is only linearly dependent on the total number of scatterers. We suggest that the SSSS scheme would be particularly useful for examining the effects of different spatial configurations of drops within water clouds in Earth's atmosphere and would also be useful in other fields of research in which the exact configuration of a collection of sparsely spaced Mie-sized scatterers is important.