Spherical Dielectric Particles Research Articles

Electrostatic Interaction of Dielectric Particles in an Electrolytic Solution

On the basis of the Poisson-Boltzmann equation the electrostatic interaction between two charged dielectric spherical particles in a symmetric electrolyte solution is considered. The interaction forces between particles of the same radius under the condition of uniform charge distribution on their surfaces in the absence of an external field have been calculated by the finite element method. The dependence of the electrostatic repulsion forces between the particles on the magnitude of the particle charges and the dielectric permittivities of the particle materials and the surrounding medium has been analyzed.

On the Brownian motion of a colloid trapped in optical tweezers: Experiments and simulations

The trapping potential induced by the interaction of a highly focused laser light with a spherical dielectric particle can be accurately approximated by a parabolic potential. In this work, we revisit experimental and numerical methodologies used to characterize the Brownian motion of a colloidal particle under the influence of a simple harmonic potential produced by optical tweezers. A classic Brownian dynamics simulation is used to model the experimental results, focusing on statistical properties that can be measured by direct visualization of the system using videomicroscopy. This work represents a useful insight into the underlying physics behind the optical tweezers technique, also giving guidelines regarding programming protocols and experimental analysis methodologies, that may be of help for students working with such techniques, as well as for professors teaching undergraduate advanced optics courses.

Enhancement of optical levitation with hyperbolic metamaterials

The tightly focused laser beam in an optical trap has become a useful tool for many recent research areas. The momentum change in the photon-stream path of incident laser beam induces radiation force that enables trapping and manipulating mesoscopic micron-sized objects. In this study, we report the first analytical demonstration of optical trapping and levitation with radiation pressure on a transparent micron-sized spherical object made of hyperbolic metamaterial (HMM). The optical radial and axial forces acting on dielectric and HMM spherical particles are calculated using ray-optics approximation, assuming an optical levitation trapping setup. We compared the net force acting on the two objects, finding that the net radiation force exerted towards HMM particle is enhanced in the axial direction: The optical force enhancement in the HMM particle is more than ~ 8 times stronger compared to the induced force on the conventional dielectric particle with the corresponding material parameters. Besides, a better performance in the radial stabilization is observed for the HMM particle in comparison with the dielectric case, at which some oscillations and unstable saturation locations for the radial stabilization is monitored for TEM00 beam incidence. Furthermore, “zero-force” paths where radial stabilization of the HMM particle exists are also obtained for both TEM00 and TEM01∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$TEM_{01}^{*}$$\\end{document} laser beam incidences. Such phenomenon does not occur for particles of only dielectric and only metal material, which can be considered as another superiority of the proposed HMM particle.

Visualizing the Nanoscopic Field Distribution of Whispering-Gallery Modes in a Dielectric Sphere by Cathodoluminescence.

A spherical dielectric particle can sustain the so-called whispering-gallery modes (WGMs), which can be regarded as circulating electromagnetic waves, resulting in the spatial confinement of light inside the particle. Despite the wide adoption of optical WGMs as a major light confinement mechanism in salient practical applications, direct imaging of the mode fields is still lacking and only partially addressed by simple photography and simulation work. The present study comprehensively covers this research gap by demonstrating the nanoscale optical-field visualization of self-interference of light extracted from excited modes through experimentally obtained photon maps that directly portray the field distributions of the excited eigenmodes. To selectively choose the specific modes at a given light emission detection angle and resonance wavelength, we use cathodoluminescence-based scanning transmission electron microscopy supplemented with angle-, polarization-, and wavelength-resolved capabilities. Equipped with semi-analytical simulation tools, the internal field distributions of the whispering-gallery modes reveal that radiation emitted by a spherical resonator at a given resonance frequency is composed of the interference between multiple modes, with one or more of them being comparatively dominant, leading to a resulting distribution featuring complex patterns that explicitly depend on the detection angle and polarization. Direct visualization of the internal fields inside resonators enables a comprehensive understanding of WGMs that can shed light on the design of nanophotonic applications.

Optical radiation force on a dielectric sphere by a polarized Airy beam.

The optical radiation force acting on a homogeneous and lossless dielectric spherical particle by a polarized Airy beam is analyzed in terms of the generalized Lorenz-Mie theory. The transverse and longitudinal radiation force components are theoretically evaluated and numerically simulated, emphasizing the transverse scale ω0, attenuation parameter γ, and polarization of the incident Airy beam versus the size parameter ka of the sphere. These results reveal that a polarized Airy beam can trap the dielectric sphere in its main caustic or sidelobes of the beam by the optical transverse force and be guided along the parabolic trajectory of the longitudinal optical force. Moreover, γ and ω0 of the Airy beams and ka of the dielectric sphere can affect the amplitude and distribution of the optical force components. This research may be helpful for the development of Airy optical tweezers in applications involving particle manipulation, optical levitation, particle sorting, and other emergent areas.

Effects of the coupling of dielectric spherical particles on signatures in infrared microspectroscopy

Infrared microspectroscopy is a powerful tool in the analysis of biological samples. However, strong electromagnetic scattering may occur since the wavelength of the incident radiation and the samples may be of comparable size. Based on the Mie theory of single spheres, correction algorithms have been developed to retrieve pure absorbance spectra. Studies of the scattering characteristics of samples of different types, obtained by microspectroscopy, have been performed. However, the detailed, microscopic effects of the coupling of the samples on signatures in spectra, obtained by infrared microspectroscopy, are still not clear. The aim of this paper is to investigate how the coupling of spherical samples influences the spectra. Applying the surface integral equation (SIE) method, we simulate small dielectric spheres, arranged as double-spheres or small arrays of spheres. We find that the coupling of the spheres hardly influences the broad oscillations observed in infrared spectra (the Mie wiggles) unless the radii of the spheres are different or the angle between the direction of the incident radiation and the normal of the plane where the spheres are located is large. Sharp resonance features in the spectra (the Mie ripples) are affected by the coupling of the spheres and this effect depends on the polarization of the incident wave. Experiments are performed to verify our conclusions.

Induced Optical Bistability in Small Metal and Metal Coated Particles with Nonlinear Dielectric Functions

The theoretical and numerical study of the enhancement of the amplitude of the incident electromagnetic radiation in small metal ellipsoidal particles and spherical dielectric particles covered by a metal shell with nonlinear dielectric functions is carried out. If the frequency of the external radiation approaches the frequency of surface plasmons of a metal, the local field in the particle considerably increases. At intense incident electromagnetic fields (laser radiation), it is necessary to consider the nonlinear part of the dielectric functions of a metal and a dielectric. This results in the induced optical bistability (IOB) when one value of the amplitude of the incident field initiates three values of the local field in a particle. The domains of IOB are specified in the case where the radiation wavelength is much larger than the typical size of particles. The range of external fields and the IOB frequency are specified. A decrease of the IOB domain and an increase of the critical fields with increase in the damping of plasmon vibrations are discussed. The results of numerical computations for typical small silver particles are presented graphically.

Wave theory of virtual image [Invited

The super resolution effect with virtual image was discovered about ten years ago using micron-sized transparent spherical dielectric particles. However, within the range of the corresponding size parameters, the simple approximation of geometric optics is not valid. Correct description of the virtual image needs the wave theory. Here we developed a novel theoretical method based on the wave theory of virtual image formation within a transparent dielectric sphere and discussed a few unusual effects arising in the frame of the wave theory.

Charged dielectric spheres interacting in electrolytic solution: A linearized Poisson-Boltzmann equation model.

We present an analytical theory of electrostatic interactions of two spherical dielectric particles of arbitrary radii and dielectric constants, immersed into a polarizable ionic solvent (assuming that the linearized Poisson-Boltzmann framework holds) and bearing arbitrary charge distributions expanded in multipolar terms. The presented development entails a novel two-center re-expansion analytical theory that expands upon and improves the existing ones, bypassing the conventional expansions in modified Bessel functions. On this basis, we develop a specific matrix formalism that facilitates the construction of asymptotic expansions in ascending order of Debye screening terms of potential coefficients, which are then employed to find exact closed-form expressions for the total electrostatic energy. In particular, this work allows us to explicitly and precisely quantify the k-screened terms of the potential coefficients and mutual interaction energy. Specific cases of monopolar and dipolar distributions are described in particular detail. Comprehensive numerical examples and tests of series convergence and the relative balance of leading and higher-order terms of the mutual interaction energy are presented depending on the inter-particle distance and particles' radii. The results of this work find application in soft matter modeling and, in particular, in computational biophysics and colloid science, where the availability of increasingly larger experimental structures at the atomic-level resolution makes numerical treatment challenging and calls for more efficient expressions and an increased range of validity.

Optical resonance and rainbow scattering of an electromagnetic Airy light-sheet by a dielectric sphere of arbitrary size

The selection of the type of the structured beam is great importance in the study of optical and first-order rainbow scattering by a particle. The Airy light-sheet is a specific type of limited-diffracting beam (in 2D), which has been used in investigations related to radiation forces and torque. Equally important and of practical significance is the study of the electromagnetic/optical resonance and first-order rainbow scattering by a dielectric sphere. The purpose of this paper is to calculate the electromagnetic resonance scattering, energy efficiencies and far-field scattered intensity of a lossless homogeneous dielectric spherical particle of arbitrary size, illuminated by a linearly-polarized non-diffracting electromagnetic Airy light-sheet. A rigorous parametric study is undertaken here to investigate the effect of varying the light-sheet parameters γ, kw0, and its polarization state as well as the size parameter ka of the lossless dielectric sphere on the scattering efficiencies and intensity. The analysis is extended to examine the first-order rainbow scattering phenomenon in the context of Airy light-sheets with different polarizations. Numerical results show that kw0, γ, ka and the state of polarization affect the amplitude and scattering directivity patterns of the efficiencies and intensity. Rainbow scattering arises along the polar scattering angle θ=143°, and is minimally influenced by the different values of the parameters kw0, γ, polarization, and ka. The results are of some practical significance in optical resonance scattering and related applications in optical tweezers, super-resolution imaging, particle characterization, and the measurement of temperature, velocity and size of a liquid jet to name a few applications.

Unified treatment of nonlinear optical force in laser trapping of dielectric particles of varying sizes

Optical trapping using laser tweezer has revolutionized the field of force spectroscopy having enormous applications in biological manipulation. While a number of theories were developed for particles of different sizes to estimate trapping force under continuous-wave excitation, they were not under short pulsed excitation which leads to nonlinear optical force. Here, we present a comparative study of various theories and provide a unified description for laser trapping under femtosecond pulsed excitation. Numerical results show that exact Mie theory (EMT) can provide a precise qualitative and quantitative prediction of trapping force when optical Kerr effect is included. Moreover, we also show how Mie interference phenomena, leading to observation of Fano resonance, are naturally captured within EMT. Thus, our findings pave the way for potential far-reaching applications in the accurate numerical estimation of nonlinear optical force on arbitrary-sized spherical dielectric particles.

Simultaneous Trapping of Two Types of Particles with Focused Elegant Third-Order Hermite-Gaussian Beams.

The focusing properties of elegant third-order Hermite–Gaussian beams (TH3GBs) and the radiation forces exerted on dielectric spherical particles produced by such beams in the Rayleigh scattering regime have been theoretically studied. Numerical results indicate that the elegant TH3GBs can be used to simultaneously trap and manipulate nanosized dielectric spheres with refractive indexes lower than the surrounding medium at the focus and those with refractive indexes larger than the surrounding medium in the focal vicinity. Furthermore, by changing the radius of the beam waist, the transverse trapping range and stiffness at the focal plane can be changed.

Polarization of light scattered by a two-dimensional array of dielectric spherical particles

The polarization state of light scattered by a two-dimensional array of spherical particles is considered based on the statistical approach. The impact of filling factor, particle size, and the type of their spatial order is analyzed for arrays of silicon and silica scatterers. The conditions are formulated to suppress the p and s components of light specularly reflected by the array of any size and concentration of particles. It is shown that at an angle of suppression of the coherent component of the reflected wave, its incoherent component is also suppressed.

Transportation dynamics of dielectric particles with the gradient forces in the field of orthogonal standing laser waves

We develop the theory of transportation and localization of a transparent dielectric spherical particle with the gradient forces in the interference field of orthogonally directed standing laser waves Ez(coskz) and Ex(coskx). It is shown that, when the waves Ez and Ex are coherent, the interference radiation field contains two harmonic components with the periods Λ0=π/k and ΛΔ=π/ksin(π/4). The amplitudes of the gradient force components depend on the ratio of the particle radius R to the modulation periods due to inhomogeneity of radiation in the particle volume and are given by the Bessel functions J3/2(2πR/Λ0) and J3/2(2πR/ΛΔ). We find the critical particle radii R0 and RΔ=2R0 defined by the Bessel functions zeros and corresponding to the vanishing components of the gradient forces. In particular, for the radiation with the wavelength λ0=1.064 μm and a particle in water, the smallest critical radii are R0=0.286 μm and 0.492 μm and RΔ=0.404 μm and 0.696 μm, respectively. For a number of special cases, we obtain the analytical solutions of the Newton equations and the particle trajectories that depend on the ratio of wave intensities and the particle radius. The results can be used to study the dynamics of the “optical assembly” of a two-dimensional particles matrix which behaves as a molecular crystal [Mellor and Bain, Chem. Phys. Chem. 7 (2006) 329–332].

Optical torque on a Rayleigh particle by photonic jet

The optical torque exerted on a dielectric Rayleigh spherical particle by photonic jet is investigated. In the study of optical torque by dipole approximation, orbital and spin torques are usually considered. Optical Orbital Torque (OOT) is a type of optical torque that causes a particle to rotate along the optical axis, and the rotation of a tiny particle around its center of mass is caused by Optical Spin Torque (OST). The photonic jet is generated by a plane wave illuminating a dielectric Generalized Luneburg Lens (GLLs). The effects of wavelength, focal length and radius of the GLLs on the OOT and OST are analyzed in this paper. The numerical results show that the wavelength and focal length of GLLs greatly affect the optical orbital torque. Optical spin torques contain a large amount of negative torques and are sensitive to the focal length of the GLLs. The results of this paper are expected to provide theoretical support for the manipulation and rotation of the Rayleigh particle.

Resonant binding of dielectric particles to a metal surface without plasmonics

High index dielectric spherical particle supports the high-$Q$ resonant Mie modes that results in a regular series of sharp resonances in the radiation pressure. A presence of perfectly conducting metal surface transforms the Mie modes into the extremely high-$Q$ magnetic bonding or electric anti-bonding modes for close approaching of the sphere to the surface. We show that the electromagnetic plane wave with normal incidence results in repulsive or attractive resonant optical forces relative to metal for excitation of the electric bonding or magnetic anti-bonding resonant modes respectively. A magnitude of resonant optical forces reaches order of one nano Newton of magnitude for micron size of silicon particles and power of light $1mW/\mu m^2$ that exceeds the gravitational force by four orders. However what is the most remarkable there are steady positions for the sphere between pulling and pushing forces that gives rise to resonant binding of the sphere by metal surface. A frequency of mechanical oscillations of particle around the equilibrium positions reaches a magnitude of order MHz.

A Linear Scaling in Accuracy Numerical Method for Computing the Electrostatic Forces in the $N$-Body Dielectric Spheres Problem

This article deals with the efficient and accurate computation of the electrostatic forces between charged, spherical dielectric particles undergoing mutual polarisation. We use the spectral Galerkin boundary integral equation framework developed by Lindgren et al. (J. Comput. Phys. 371 (2018): 712-731) and subsequently analysed in two earlier contributions of the authors to propose a linear scaling in cost algorithm for the computation of the approximate forces. We establish exponential convergence of the method and derive error estimates for the approximate forces that do not explicitly depend on the number of dielectric particles $N$. Consequently, the proposed method requires only $\mathcal{O}(N)$ operations to compute the electrostatic forces acting on $N$ dielectric particles up to any given and fixed relative error.

An integral equation formulation of the N-body dielectric spheres problem. Part I: numerical analysis

In this article, we analyse an integral equation of the second kind that represents the solution of N interacting dielectric spherical particles undergoing mutual polarisation. A traditional analysis can not quantify the scaling of the stability constants- and thus the approximation error- with respect to the number N of involved dielectric spheres. We develop a new a priori error analysis that demonstrates N-independent stability of the continuous and discrete formulations of the integral equation. Consequently, we obtain convergence rates that are independent of N.

An integral equation formulation of the N-body dielectric spheres problem. Part II: complexity analysis

This article is the second in a series of two papers concerning the mathematical study of a boundary integral equation of the second kind that describes the interaction of N dielectric spherical particles undergoing mutual polarisation. The first article presented the numerical analysis of the Galerkin method used to solve this boundary integral equation and derived N-independent convergence rates for the induced surface charges and total electrostatic energy. The current article will focus on computational aspects of the algorithm. We provide a convergence analysis of the iterative method used to solve the underlying linear system and show that the number of liner solver iterations required to obtain a solution is independent of N. Additionally, we present two linear scaling solution strategies for the computation of the approximate induced surface charges. Finally, we consider a series of numerical experiments designed to validate our theoretical results and explore the dependence of the numerical errors and computational cost of solving the underlying linear system on different system parameters.

Theory of optical tweezing of dielectric microspheres in chiral host media and its applications

We report for the first time the theory of optical tweezers of spherical dielectric particles embedded in a chiral medium. We develop a partial-wave (Mie) expansion to calculate the optical force acting on a dielectric microsphere illuminated by a circularly-polarized, highly focused laser beam. When choosing a polarization with the same handedness of the medium, the axial trap stability is improved, thus allowing for tweezing of high-refractive-index particles. When the particle is displaced off-axis by an external force, its equilibrium position is rotated around the optical axis by the mechanical effect of an optical torque. Both the optical torque and the angle of rotation are greatly enhanced in the presence of a chiral host medium when considering radii a few times larger than the wavelength. In this range, the angle of rotation depends strongly on the microsphere radius and the chirality parameter of the host medium, opening the way for a quantitative characterization of both parameters. Measurable angles are predicted even in the case of naturally occurring chiral solutes, allowing for a novel all-optical method to locally probe the chiral response at the nanoscale.