Spherical Vector Wave Functions Research Articles

Multiple Scattering by Two PEC Spheres Using Translation Addition Theorem

An analysis of multiple scattering by two Perfect Electric Conducting (PEC) spheres using translation Addition Theorem (AT) for spherical vector wave functions is presented. Specifically, the Cruzan formalism is used to represent the AT for spherical harmonics, which introduces the translation coefficients for transformation of spherical harmonics from one coordinate to another. The adoption of these coefficients with the use of two PEC spheres in a near zone region makes the calculation of multiple scattering electric fields very efficient. As an illustration, the mathematical formation using advanced computational approaches was inspected. Then, the generic truncation criteria in the scattered electric field by two PEC spheres was deeply investigated using translation AT. However, the numerical validation was obtained using Comsol simulation software. This approach will allow to evaluate the scattering from macro-structures composed of spherical particles, i.e., biological molecules, clouds of airborne particles, etc. An original and fully general solution to the problem using vector quantities is introduced, and the convergence of the solution in several numerical examples is also demonstrated. This approach takes into account the effect of multiple scattering by two PEC spheres for spherical vector function.

The Funk‐Hecke type relations and ray transforms for the vector spherical wave functions

The Funk‐Hecke type integral relations for vector spherical wave functions are derived. Applying the obtained Funk‐Hecke integral identities, 3D Fourier transforms of vector spherical wave functions and and are obtained. On the basis of the Central Slice Theorem, the ray transforms for these vector functions are evaluated. This was found to be a useful tool in the scattering and the vector tomography problems.

Analysis of radiation force on a uniaxial anisotropic sphere by dual counter-propagating Gaussian beams.

Based on Maxwell's stress tensor and the generalized Lorenz-Mie theory, a theoretical approach is introduced to study the radiation force exerted on a uniaxial anisotropic sphere illuminated by dual counter-propagating (CP) Gaussian beams. The beams propagate with arbitrary direction and are expanded in terms of the spherical vector wave functions (SVWFs) in a particle coordinate system using the coordinate rotation theorem of the SVWFs. The total expansion coefficients of the incident fields are derived by superposition of the vector fields. Using Maxwell stress tensor analysis, the analytical expressions of the radiation force on a homogeneous absorbing uniaxial anisotropic sphere are obtained. The accuracy of the theory is verified by comparing the radiation forces of the anisotropic sphere reduced to the special cases of an isotropic sphere. In order to study the equilibrium state, the effects of beam parameters, particle size parameters, and anisotropy parameters on the radiation force are discussed in detail. Compared with the isotropic particle, the equilibrium status is sensitive to the anisotropic parameters. Moreover, the properties of optical force on a uniaxial anisotropic sphere in a single Gaussian beam trap and Gaussian standing wave trap are compared. It indicates that the CP Gaussian beam trap may more easily capture or confine the anisotropic particle. However, the radiation force exerted on an anisotropic sphere exhibits very different properties when the beams do not propagate along the primary optical axis. The influence of the anisotropic parameter on the radiation force by CP Gaussian beams is different from that of a single Gaussian beam. In summary, even for anisotropic particles, the Gaussian standing wave trap also exhibits significant advantages when compared with the single Gaussian beam trap. The theoretical predictions of radiation forces exerted on a uniaxial anisotropic sphere by dual Gaussian beams provide effective ways to achieve the improvement of optical tweezers as well as the capture, suspension, and high-precision delivery of anisotropic particles.

EBCM for Electromagnetic Modeling of Gyrotropic BoRs

We employ the extended boundary condition method (EBCM) that we properly extend so as to describe gyroelectric and gyromagnetic (i.e., gyrotropic) anisotropy and report on the electromagnetic (EM) complex resonances of magnetooptic (i.e., gyroelectric) bodies of revolution (BoRs), as well as on the complex magnetic plasmon resonances (MPRs) of ferrite (i.e., gyromagnetic) BoRs. The proposed extension is based on an alternative scheme for the expansion of the EM field inside a gyrotropic medium, namely, a discrete eigenfunction (DE) expansion in terms of spherical vector wave functions (SVWFs). This approach provides the transition matrix (namely, T-matrix) that allows not only for the direct computation of the scattered field from the incident one, but also for the determination of the complex resonances of open (i.e., situated in free space) gyrotropic BoR resonators. The EBCM is validated on two levels: first, by calculating the EM scattering from various BoRs, including anisotropic spheroids, cylinders, and rods, and comparing with HFSS commercial software; second, by computing the complex eigenfrequency spectrum of gyroelectric spheroidal resonators and comparing with a recently developed rigorous technique for the EM modeling of anisotropic spheroids.

T-matrix method for calculation of second-harmonic generation in clusters of spherical particles

In this article, we present a T-matrix method for numerical computation of second-harmonic generation from clusters of arbitrarily distributed spherical particles made of centrosymmetric optical materials. The electromagnetic fields at the fundamental and second-harmonic (SH) frequencies are expanded in series of vector spherical wave functions, and the single sphere T-matrix entries are computed by imposing field boundary conditions at the surface of the particles. Different from previous approaches, we compute the SH fields by taking into account both local surface and nonlocal bulk polarization sources, which allows one to accurately describe the generation of SH in arbitrary clusters of spherical particles. Our numerical method can be used to efficiently analyze clusters of spherical particles made of various optical materials, including metallic, dielectric, semiconductor, and polaritonic materials.

Scattering of a non-paraxial Bessel pincer light-sheet by a dielectric sphere of arbitrary size

The scattering of a non-paraxial Bessel pincer light-sheet by a dielectric sphere of arbitrary size is studied in the framework of generalized Lorenz-Mie theory (GLMT). The electrical field of the Bessel pincer light-sheet is expanded using the vector angular spectrum decomposition method (VASDM), and its beam shape coefficients (BSCs) are derived using the method of multipole expansion and vector spherical wave functions (VSWF). By this way, the incident field, scattering field and near-surface field, and absorption and extinction efficiency factors are numerically calculated. Also, the effects of the beam order, scaling parameter, and polarization are mainly discussed. Numerical results represent that the intensity distribution of incident field, scattering field and near-surface field is intensely sensitive to the scaling parameters and the beam order. For the incident field, the bigger the order, the smaller the bending angle, and the closer the focus is to the source. By contrast, for the scattering and total field, the bigger the order, the larger the bending angle, and the farther the focus is to the source. Such results have potential applications in the imaging around steep corners and particle manipulation applications with minimal obstruction by particles.

Light scattering and absorption by two-dimensional arrays of nano and micrometer monodisperse spherical silver particles

The problem of light scattering and absorption is considered for a free-standing two-dimensional array (monolayer) of nano- and micrometer spherical silver particles. To solve this problem for short- and imperfect long-range ordered arrays the statistical theory of multiple scattering of waves is used. The key points of the developed semi-analytical method are described. The method is based on the quasicrystalline approximation and the multipole expansion of electromagnetic fields and the dyadic Green's function in terms of vector spherical wave functions. Results for the angular distribution of scattered light, the coefficients of coherent transmission and reflection, incoherent scattering, and absorption of a monolayer are obtained for a spectral range corresponding to the nanoparticle plasmonic resonance. Some features of light propagation through two-dimensional arrays of particles with different spatial order are investigated and discussed. The results are in close agreement with the existing experimental data for a partially ordered array and the calculations for a regular array obtained under the extended low-energy electron diffraction theory.

Scattering of evanescent wave generated by total reflection

An exact semi-analytical solution to a scattering problem of evanescent wave incident on a particle is obtained. The evanescent wave is generated by total reflection as an electromagnetic plane wave propagates from an optically denser medium to an optically thinner one. The scattered fields by and internal fields within the particle are expanded in terms of spherical vector wave functions, and an integral representation of spherical vector wave functions over cylindrical ones is used to compute the reflection of the scattered fields on the interface between the optically denser and thinner media. By applying the boundary conditions and projection technique, the unknown expansion coefficients of the scattered fields are determined. The scattering cross sections are evaluated, and the scattering characteristics are analyzed briefly.

Scattering by a uniaxially anisotropic chiral object under electromagnetic beam illumination

ABSTRACT We propose a semi-analytical scheme for investigating the electromagnetic beam scattering by a uniaxially anisotropic chiral object. The scattered and internal fields are expanded in terms of appropriate spherical vector wave functions, and the method of moments procedure is employed for determining the unknown expansion coefficients. Normalized differential scattering cross sections are evaluated for a uniaxially anisotropic chiral spheroid and circular cylinder of finite length, respectively illuminated by a Gaussian beam, Hermite-Gaussian beam and zero-order Bessel beam. The scattering properties are analyzed briefly.

Nano‐antenna synthesis for end‐fire and pencil‐beam far‐field radiation patterns using vector spherical wave functions

Two sets of irregularly placed dielectric nano-spheres are synthesised as two nano-antennas. One of them has an end-fire and the other has a pencil-beam far-field radiation pattern. The electrical sizes (radiuses) and relative permittivities of all spheres are chosen to be 0.1 and 4, respectively. An incident plane-wave is illuminated on spheres. The aim is to find the optimal positions of spheres to scatter the incident wave similar to the end-fire and pencil beam radiation patterns. At first, the numbers of spheres of the nano-antennas are determined. Next, a random three-dimensional distribution of spheres is considered. Desired pencil-beam and end-fire patterns besides all scattered and incident waves are expanded to vector spherical wave functions. Therefore, the analysis is performed in the spectral domain of the spherical harmonics. The multiple scattering between spheres is considered. Coefficients of scattered waves are calculated by applying boundary conditions on the surface of all spheres. The aggregate difference of coefficients of total far-field scattering waves and coefficients of the desired far-field pattern is considered as a cost-function for an optimisation method. Particle swarm optimisation tool is used to optimise the positions of spheres to decrease the cost value. This process is done in several iterations. It is shown that by increasing the number of iterations, the value of cost function is decreased. So, this is an efficient method to design antennas including nano-antennas for any arbitrary radiation pattern and polarisation.

Scattering and Absorption of Light by a Monolayer of Spherical Particles under Oblique Illumination

The problem of light scattering and absorption by a monolayer of homogeneous spherical particles at an arbitrary angle of incidence of a plane wave, which determines the phase shift between the averaged fields in the particles, has been solved. The solution is based on the quasi-crystalline approximation of the theory of multiple scattering of waves, the mean-field approximation, and the multipole expansion of the fields and the tensor Green function in terms of vector spherical wave functions. Formulas for determining the angular distribution of the incoherently scattered field intensity, the incoherent scattering, absorption, coherent transmission, and reflection coefficients have been derived. The dependences of these characteristics on the direction of illumination are illustrated using partially ordered monolayers of dielectric and semiconductor particles as an example. It is shown that: (i) under oblique illumination the angular distribution of the scattered radiation in azimuthal scattering angle for any polarization of the incident wave (except for p and s) is asymmetric relative to the plane of incidence, (ii) the maximum of the angular distribution of the radiation transmitted through the monolayer can approach its normal with increasing angle of incidence and/or concentration of particles, (iii) there exist conditions under which the bulk of the scattered radiation is concentrated in the direction opposite to the direction of the incident radiation, and (iv) the dependence of the monolayer absorption coefficient on the illumination angle has extrema whose positions and magnitudes are determined by the polarization of the incident wave.

Scattering characteristics of a metasurface covered chiral sphere.

An analysis of the scattering characteristics of a chiral sphere covered with a metasurface is presented. The electromagnetic fields inside and outside of a metasurface covered chiral sphere are expanded in terms of spherical vector wave functions. The influences of chirality and surface impedance of a metasurface upon the scattering cross section are investigated. It is seen that a specific type of metasurface can be used to significantly enhance and diminish the back scattering cross section from a chiral sphere having a fixed value of chirality but different radii. It is also observed that a chiral sphere having chirality close to unity apparently cannot be cloaked using a lossless metasurface as compared to a dielectric sphere, both having the same electromagnetic and geometrical parameters. A comparative study of the scattering characteristics of metasurface covered realistic chiral and low-loss Teflon dielectric spheres is developed. It is found that a metasurface covered realistic chiral sphere can be used to significantly reduce the scattering as compared to scattering from a metasurface covered Teflon sphere at a specific angle, provided that both spheres have the same radii and surface reactance.

An overview of the null-field method. I: Formulation and basic results

In this paper we revisit the fundamentals of the null-field method with discrete sources. We prove the unique solvability of the null-field equations for the total field inside the particle and the internal field outside the particle, at all wavenumbers. For this purpose, we use the equivalence between the null-field and surface integral equations methods. Furthermore, we discuss the completeness property of different systems of discrete sources for a surface field approximation, and derive an infinite set of integral equations for the surface fields in a variety of discrete sources. Finally, we formulate the null-field scheme as an approach aiming to construct an approximate solution to the scattering problem. The way in which we introduce the T matrix of a particle is different from the standard approach relying on the assumption that the system of regular vector spherical wave functions for the interior problem is a basis.

An overview of the null-field method. II: Convergence and numerical stability

In this paper we provide an analysis of the convergence and numerical stability of the null-field method with discrete sources. We show that (i) if the null-field scheme is numerically stable then we can decide whether or not convergence can be achieved; (ii) if the null-field scheme is numerically unstable then we cannot draw any conclusion about the convergence issue; and (iii) the numerical stability is closely related to the property of a tangential system of radiating discrete sources to form a Riesz basis. Our numerical analysis indicates that for prolate spheroids and localized vector spherical wave functions, the null-field scheme is numerically unstable (this system of vector functions does not form a Riesz basis), while for distributed vector spherical wave functions, the numerical instability is not so pronounced (this system of discrete sources almost possesses the property of being a Riesz basis). We also describe an analytical method for computing the surface integrals in the framework of the conventional null-field method with localized vector spherical wave functions which increases the stability of the numerical scheme.

A Multiple-Reflection Solution to Electromagnetic Scattering by a Buried Sphere

Based on the image theory and addition theorem of spherical vector wave functions, a multiple-reflection solution is described for the electromagnetic scattering by a buried sphere. An iterative process is introduced to obtain the scattered field by enforcing the boundary conditions at the plane interface between free space and an isotropic medium and the spherical surface of a dielectric sphere buried in the isotropic medium. Numerical results for the scattered fields are provided and compared with those obtained by the commercial software FEKO. A good agreement is observed, which verifies the presented formulation.

Scattering characteristics of an anisotropic medium sphere

Based on the invariable displacement current of anisotropic media, a theoretical model for electromagnetic reconstructing anisotropic media and a basic method for selecting electromagnetic field are proposed. Taking the gypsum crystal as an example, the analytical solution of bi-axial medium scattering is presented by using the spherical vector wave function of electromagnetic field. The electromagnetic wave scattering of the anisotropic medium spheres are simulated in radiation of electromagnetic waves of different frequency. The electromagnetic analysis of the simulation results shows that the electromagnetic mechanism is consistent with the results of relevant literature. When the anisotropic medium degenerates to the isotropic medium, the results obtained by the model are completely consistent with those obtained by Mie theory.

Nano-antenna synthesis for an arbitrary far-field radiation pattern and polarization based on vector spherical wave functions expansion.

A systematic method is proposed to synthesize a nano-antenna based on theoretical principles. This nano-antenna, which is composed of a set of small dielectric spheres, is designed to have a desired far-field radiation pattern and polarization. The basis of the proposed method is expanding all electromagnetic waves into the series of vector spherical wave functions. First, the forward problem of calculation of scattering from single and multiple spheres is studied. For cases with more than one sphere, a multiple scattering method is implemented to calculate total scattering. Near-field and far-field waves, absorption, extinction, and differential scattering cross sections are calculated for a single sphere with different sizes and permittivities. Moreover, far-field waves for linear arrays of small spheres are analyzed. All results are validated using an electromagnetic simulation software. Next, the problem of inverse scattering begins by considering a three-dimensional arbitrary pattern and polarization. The aim is to find a set of spheres that generates this pattern. Particle swarm optimization and non-iterative spectral-domain forward scattering methods are combined as a novel method to find the optimal positions of the spheres.

Scattering from multiple PEC sphere using Translation Addition Theorems for Spherical Vector Wave Function

In our manuscript, we are reporting the translation criteria of scattering from Perfect Electric Conductor (PEC) sphere along three-dimensional axes using semianalytical approach. The presented scattering model is based on a generalized Lorenz-Mie theory framework and ensemble with the vector translation Addition Theorem (AT) for the Vector Spherical Harmonics (VSH). Applying extended Mie theory on a sphere leads to a set of an unknown coefficients by the use of translation AT. In the literature, there are many authors reporting different sets of the vector translation coefficients, of which we mention those calculated by Stein, Cruzan and, Mackowski in particular. We have selected the Cruzan formulation of the vector translation coefficients for its structure based on the Wigner 3-j function. As an illustration, we want to present numerical examples than simulations. of total scattered field PEC sphere using vector translation AT. We used advanced computational tools and approaches for mathematical modeling of an observation. During our numerical test, we have deeply investigated generic truncation criteria in scattered electric field using translation AT. However, we have been obtained numerical validation by using computational approach.

Gaussian beam scattering by an infinite cylinder with a spherical inclusion

An analytical method of analyzing the electromagnetic scattering of a Gaussian beam by an infinitely long circular cylinder with a spherical inclusion is presented. The fields within different regions are expanded in terms of appropriate cylindrical or spherical vector wave functions. By applying the integral representation of spherical vector wave functions over cylindrical ones, boundary conditions and projection procedure, the unknown expansion coefficients are determined. For a localized beam model, numerical results of the total scattered field and normalized field intensity distribution are shown, and the scattering properties are discussed briefly.

Optical radiation force on a dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet

Based on the generalized Lorenz–Mie theory (GLMT) and the angular spectrum decomposition method (ASDM), we calculate the optical radiation force exerted on a lossless dielectric sphere of arbitrary size illuminated by an Airy light-sheet. The beam shape coefficients (BSCs) of the Airy light-sheet are calculated using the vector angular spectrum decomposition and vector spherical wave functions methods. The optical radiation force acting on the spherical particle is obtained by the integral of Maxwell’s stress tensor. The transverse (Fy) and longitudinal (Fz) forces are numerically computed. Two kinds of polarization (TE and TM) are considered for the Airy light-sheet, and the negative longitudinal optical (pulling) force is particularly emphasized. The influence of the transverse scale parameter w0 and attenuation parameter γ of the Airy light-sheet on the force is discussed. The results of the present theory are verified using the dipole approximation method in which the gradient force has been also computed for a Rayleigh sphere. The numerical results show that when the transverse scale parameter w0 and attenuation parameter γ increase, the transverse and longitudinal forces decrease. Furthermore, the force caustic (i.e., maximum) shifts to the direction of y < 0 as the transverse scale parameter w0 increases. As the dimensionless size parameter of the sphere ka increases (where k is the wavenumber and a is the radius), the resonance peaks of the optical forces become larger. The results of this paper are of practical significance for the development of Airy light-sheet based optical manipulation technologies.