Spherical Vector Wave Functions Research Articles

Scattering properties of dual Bessel beams on chiral layered particle

The paper investigates the scattering analytical solution of arbitrary direction double zero-order Bessel beams (ZOBBs) incident on a non-uniform layered chiral sphere. Using the generalized Lorenz-Mie theory (GLMT), the electromagnetic field expansion expression of the double ZOBBs is obtained through the vector superposition of the spherical vector wave functions (SVWFs). Analytical solutions of the scattering on chiral layered particles by dual Bessel beams are derived based on the boundary condition. The iterative coefficient equations are constructed to obtain the expansion coefficients of the SVWFs in each region of the multi-layer chiral sphere. The variation of radar cross section (RCS) with angular distribution is numerically simulated, and the scattering results of layered chiral spheres degenerated into isotropic layered spheres are compared with the literature, which verifies the correctness of the theory and program. The effects of incident angle, polarization angle, cone angle and chiral parameters on the scattering characteristics of far region and internal field of the layered chiral particle are analyzed in detail. The theory and results can offer significant assistance in studying the optical properties of non-uniform chiral particles and complex biological cells irradiated by multiple beams and may also provide important practical value in the micromanipulation of multi-layered biological structures.

The Weyl expansion for the scalar and vector spherical wave functions

The Weyl expansion technique, also known as the angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. The scalar spherical waves are the solutions of the homogeneous Helmholtz equation and therefore have direct relation to the scalar multipole fields. This paper gives the Weyl expansion of multipole fields, scalar and vector, of any degree and order for spherical wave functions. The expressions are given in closed form for the scalar, , and vector, , , multipole fields, evaluated across a plane orthogonal to any given direction. In the case of scalar spherical multipoles, the spherical gradient operator has been used, while for the vector spherical multipoles, the vector spherical wave operator has been constructed.

Configurations of dipoles, moving charges and currents for dynamic spherical multipoles

Configurations of dipoles, moving point charges and current loops are presented for the dynamic spherical multipoles of the electric and magnetic type, whose fields are the vector spherical wave functions (VSWFs). The configurations produce the VSWFs of a specific degree and order in the limit that they become infinitely small. We find frequency independent configurations for up to degree n = 5 or higher in some cases, which exceeds the current range of configurations in the literature for up to n = 3, and our configurations for n = 3 are simpler. We also investigate the link with anapoles and toroidal multipoles.

Description and reconstruction of typical structured light beams with vector spherical wave functions.

It is well known that the generalized Lorenz-Mie theory (GLMT) is a rigorous analytical method for dealing with the interaction between light beams and spherical particles, which involves the description and reconstruction of the light beams with vector spherical wave functions (VSWFs). In this paper, a detailed study on the description and reconstruction of the typical structured light beams with VSWFs is reported. We first systematically derive the so-called beam shape coefficients (BSCs) of typical structured light beams, including the fundamental Gaussian beam, Hermite-Gaussian beam, Laguerre-Gaussian beam, Bessel beam, and Airy beam, with the aid of the angular spectrum decomposition method. Then based on the derived BSCs, we reconstruct these structured light beams using VSWFs and compare the results of the reconstructed beams with those of the original beams. Our results will be useful in the study of the interaction of typical structured light beams with spherical particles in the framework of GLMT.

Another look at addition theorems for vector spherical wavefunctions

Addition theorems for vector spherical wavefunctions are widely used, but they are complicated. A critical survey of methods for their derivation is offered. Then, one derivation is given in detail, combining elementary techniques with known results for scalar addition theorems.

Scattering of elastic waves by a sphere with orthorhombic anisotropy and application to polycrystalline material characterization

Scattering of elastic waves by an anisotropic sphere with orthorhombic symmetry inside an isotropic medium is studied and applied to characterization of polycrystalline materials with anisotropic grains. For a single sphere the waves in the isotropic surrounding are expanded in the spherical vector wave functions. Inside the sphere, the elastodynamic equations are first transformed to spherical coordinates and the displacement field is expanded in terms of the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere give recursion relations among the expansion coefficients in the power series. The boundary conditions on the sphere then determine the relation among the scattered wave expansion coefficients and those of the incident wave, expressed as the transition (T) matrix. For low frequencies the elements of the T matrix are obtained in explicit form. According to the theory of Foldy the T matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials, explicitly for low frequencies. Comparisons of the present method with previously published results and recent FEM results show a good correspondence for low frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods.

Electromagnetic Analysis of Anisotropic Impedance Spherical Metasurfaces

We investigate the interaction of electromagnetic (EM) waves with a spherical conformal metasurface, described by a anisotropic surface impedance. To this end, a full-wave method is developed, based on the expansion of the fields in terms of spherical vector wave functions. Appropriate treatment of the anisotropic impedance boundary condition leads to sets of linear equations that allow for the determination of the involved EM fields. The developed method can be used for design and anal- ysis purposes, as well as for the study of metasurface-based EM scattering control. We examine a related application of a cloaking setup and show that anisotropy breaks the spherical symmetry and leads to interesting new effects, such as a polarization-dependent switch function. The presented results suggest that metasurfaces with appropriately engineered impedance tensors may lead to desired scattering responses that otherwise cannot be obtained with their isotropic counterparts.

Analysis of Arbitrary EM Field Exposure to a Multilayer Spherical Head Model Using Spherical Vector Wave Functions

A semi-analytical method is presented for the assessment of induced electromagnetic field inside a multilayer head model exposed to radiated field of an arbitrary source antenna. First the source antenna is simulated by a full-wave software in the absence of the head model to evaluate its radiating characteristics. Then, by sampling of the source radiated fields, its spherical vector wave function (SVWF) amplitudes are evaluated. The well-known translation addition theorem for spherical vector wave functions (SVWFs) is implemented to translate radiating field SVWFs to the local coordinates system of head model. Neglecting the reaction of model on source fields, using boundary conditions on the interfaces of adjacent layers, the unknown SVWF amplitudes of the fields inside each layer as well as those of the scattered field outside the head model are evaluated. Some numerical examples are presented for the verification of the proposed method. The acceptable consistency between the results obtained by the proposed method and full-wave simulations of the problem verifies the authenticity of the proposed method. In comparison to a full-wave numerical method, the proposed method provides an efficient repeatable simulation approach due to the independency of the source and head model analyses.

Scattering of elastic waves by a sphere with cubic anisotropy with application to attenuation in polycrystalline materials

Scattering of elastic waves by an anisotropic sphere with cubic symmetry inside an isotropic medium is studied. The waves in the isotropic surrounding are expanded in the spherical vector wave functions. Inside the sphere, the elastodynamic equations are first transformed to spherical coordinates and the displacement field is expanded in terms of the vector spherical harmonics in the angular directions and a power series in the radial direction. The governing equations inside the sphere give recursion relations among the expansion coefficients in the power series. The boundary conditions on the sphere then determine the expansion coefficients of the scattered wave. This determines the transition ( T ) matrix elements which are calculated explicitly to the leading order for low frequencies. Using the theory of Foldy, the T matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials with cubic symmetry, explicitly for low frequencies and numerically for intermediate frequencies. Numerical comparisons of the present method with previously published results and recent finite element method (FEM) results show a good correspondence for low and intermediate frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods.

Light scattering of a uniform uniaxial anisotropic sphere by an on-axis high-order Bessel vortex beam.

Analytical solutions to the scattering of a uniform uniaxial anisotropic sphere illuminated by an on-axis high-order Bessel vortex beam (HOBVB) are investigated. Using the vector wave theory, the expansion coefficients of the incident HOBVB in terms of the spherical vector wave functions (SVWFs) are obtained. According to the orthogonality of the associated Legendre function and exponential function, more concise expressions of the expansion coefficients are derived. It can reinterpret the incident HOBVB faster compared with the expansion coefficients of double integral forms. The internal fields of a uniform uniaxial anisotropic sphere are proposed in the integrating form of the SVWFs by introducing the Fourier transform. The differences of scattering characteristics of a uniaxial anisotropic sphere illuminated by a zero-order Bessel beam, Gaussian beam, and HOBVB are exhibited. Influences of the topological charge, conical angle, and particle size parameters on the angle distributions of the radar cross section are analyzed in detail. The scattering and extinction efficiencies varied with the particle radius, conical angle, permeability, and dielectric anisotropy are also discussed. The results provide insights into the scattering and light-matter interactions and may find important applications in optical propagation and optical micromanipulation of biological and anisotropic complex particles.

Explicit Conversion Formulas Between Spherical Wave Expansions With Scalar and Vector Expansion Coefficients

Explicit formulas for converting spherical wave expansions with vector basis functions and scalar expansion coefficients into spherical wave expansions with scalar basis functions and vector expansion coefficients and vice versa are presented. The formulas are given in terms of Wigner-3- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula> -symbols. The conversion formulas are derived by spherical harmonics expansions of products of two spherical harmonics and by levering on recurrence relations of the associated Legendre functions. The expansions with vector coefficients are redundant and explicit formulas are given for the linear combination of vector coefficients of which the expanded fields cancel identically to zero. The redundancy in the expansion can be used to find compact expansions with a minimum expansion order. For every vector spherical wave function, two expansions using scalar spherical harmonics are presented: on the one hand, a radial-component free expansion and on the other hand, a minimum-order expansion. The correctness of all expansions is verified with a simple computer code up to a mode order of 65. The agreement of the expanded near and far fields in both types of expansions is better than 14 digits of accuracy for most cases.

Analysis of radiation force on a uniaxial anisotropic sphere by dual zero-order Bessel beams

Based on the generalized Lorenz–Mie theory (GLMT), a theoretical approach is introduced to study the radiation force exerted on a uniaxial anisotropic sphere illuminated by dual zero-order Bessel beams (ZOBBs). The beams propagate with arbitrary direction, and are expanded in terms of the spherical vector wave functions (SVWFs) in the particle coordinate system using the coordinate rotation theorem. The total expansion coefficients of the incident fields are derived by superposition of the vector fields. Applying the Maxwell’s stress tensor, the analytical expressions of the radiation forces on a homogeneous absorbing uniaxial anisotropic sphere are derived. Comparing the radiation forces of dual ZOBBs with those results of dual ZOBBs simplified into single ZOBBs and dual plane waves, the accuracy and correctness of the theory are verified. In order to study the equilibrium state, the effects of beam parameters, particle size parameters, and anisotropy parameters on the radiation forces are discussed in detail. Comparing 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 single ZOBB trap and zero-order Bessel standing wave trap are compared. The theory and results of this paper are hopeful to provide an effective theoretical basis for the study of optical micromanipulation of biological and anisotropic complex particles by multi-beams.

An Analytical Solution of the Multiple Scattering from a Buried Medium Coated Conducting Sphere

Based on the image method and addition theorem of spherical vector wave functions, an analytical solution of the multiple scattering by a buried medium-coated conducting sphere is proposed in this paper. An iterative process to obtain the scattered electromagnetic field is discussed on the basis of the continuous boundary condition in the plane boundary, the medium inner and the outer surface of a coated conducting sphere, respectively. Applying an image method and the addition theorem of spherical vector wave functions, the scattering electromagnetic fields by the plane in a local coordinate system can be transformed into the globe coordinate, and they can be regarded as the next incident electromagnetic fields to the buried medium-coated conducting sphere. This process does not end until the scattering electrical field on the plane boundary is accurate enough. Numerical results are given and compared with commercial software FEKO, they coincide enough; the calculation times of the present method are very short compared to those of the software FEKO, and some discussion is given at the end of this paper.

Scattering of arbitrary-shaped optical polarized beams by a PEMC sphere

The generalized Lorenz-Mie theory (GLMT) for a laser beam of arbitrary shape illuminating a perfect electromagnetic conductor (PEMC) sphere is presented. Making use of the vector angular spectrum decomposition method (VASDM), and the multipole expansion method (MEM), in which any beam is expressed in terms of vector spherical wave functions (VSWFs) and beam shape coefficients (BSCs), the BSCs are obtained and expressed by a vector angular spectrum. The incident and scattering waves are expanded using VSWFs. A relation between the coefficients of the scattering fields and the BSCs of the incident beam is established considering the appropriate boundary conditions at the surface of the PEMC sphere. Expressions for the extinction, scattering, and absorption cross-sections are derived and computed. By isolating the co-polarization and cross-polarization components of the scattering coefficients, individual expressions of these physical quantities are also provided. Considering the vector Bessel and Airy polarized beams as examples, the total electric field intensity, scattering cross-section and its co-polarized and cross-polarized components are numerically computed and discussed. The effects of beam polarization, size parameter of the sphere, the admittance M, and beam parameters are analyzed. It has been found that the intensity and distribution of the total (incident + scattered) electric field intensity in the transverse (yz) plane depends on the incident beam characteristics and its polarization states. As the dimensionless size ka increases, the amplitude of the scattering cross-section increases and then decreases as ka increases. As the admittance parameter M changes, there are different characteristics if the incident beam is different. Nonetheless, there is a common behavior that the trend of the co-polarized component of the scattering cross-section decreases first and then increases regardless of the beam and its polarization state. When it comes to the trend of the cross-polarized component versus M, it is opposite to that of the co-polarized component. A few resonance peaks are manifested in the plots when the incident beam is radially or azimuthally polarized. The results may have promising applications in scattering, particle manipulation, optical trapping, and other related researches dealing with a sphere exhibiting rotary polarization.

Analysis of trapping force exerted on multi-layered chiral sphere induced by laser sheet

Theoretical study on optical trapping of multi-layered chiral sphere has attracted more and more attention for its important applications in many frontier scientific fields such as chemical engineering, biomedicine, optical tweezers, micro/nano lithography etc. In order to trap and manipulate chiral multi-layered particles efficiently, the present paper aims at developing the theoretical research of trapping force (TF) exerted on a multi-layered chiral sphere induced by laser sheet which might have great potential to improve the light performance in optical trapping as well as capture, suspension, and high-precision delivery of chiral cells. Here, based on the Generalized Lorenz Mie theory and the completeness of spherical vector wave functions (SVWFs), the electromagnetic field of incident laser sheet are expanded in terms of SVWFs. Accordingly, by introducing the beam scattering theory and the conservation law of electromagnetic momentum (EM), the analysis of TF exerted on multi-layered chiral sphere can be analytically expressed in terms of the incident and scattering coefficients. Taking the chiral cell as an example, the TF induced by laser sheet is simulated numerically. Numerical effects of the varying chirality, polarization states, beam waist width, inner material loss and outmost size on the TF induced by laser sheet are analyzed and compared with those by circular Gaussian beam incidence in detail. It is found that the introduction of chirality parameter may reduce the axial TF exerted on chiral multi-layered cell. Thus, it is more difficult to trap and manipulate stratified chiral cells than to trap general isotropic cells. Also it is shown that the TF of chiral cells can be significantly discriminatory in nature, depending upon both the handedness of the interacting particles and the polarization of the incident light. Thus, an appropriately polarized beam should be considered in trapping chiral cells. For chiral multi-layered cells with small loss in the inner layer, when the inner refractive indices are less than the outmost refractive index, the TF of multi-layered chiral cell becomes stronger with the outmost radius decreasing. Conversely, for the inner refractive indices are greater than the outer refractive index, TF becomes weaker as the outmost radius decreases. Besides, compared with the traditional circular Gaussian beam, the strong convergence of elliptical Gaussian beam can be easier to achieve three-dimensional capture of stratified chiral cells, which may provide a recipe to understand the light interaction with more complex chiral cells with the aid of the analytical approach and could be a promising avenue for the design of optical trapping systems.

EM Scattering by Core-Shell Gyroelectric-Isotropic and Isotropic-Gyroelectric BoRs Using the EBCM

We employ the extended boundary condition method (EBCM) and construct a solution for the problem of electromagnetic (EM) scattering by anisotropic core-shell bodies of revolution (BoRs). In particular, two different core-shell configurations are examined: the gyroelectric-isotropic and the isotropic-gyroelectric setup. To construct the solution, we employ two groups of integral representations (IRs)&#x2014;one group for each configuration solved&#x2014;in conjunction with the discrete eigenfunction (DE) expansion of the fields in terms of spherical vector wave functions (SVWFs) for the gyroelectric regions. We demonstrate the validity and the computational performance of the method by comparisons with the HFSS commercial software for various core-shell setups such as spheroidal, cylindrical, and combined spherical-cylindrical BoRs. We also employ ADDA, a particular version of the discrete dipole approximation (DDA) method, to trace the boundaries of validity of the EBCM. Finally, we present an application of the method to the study of magnetically-tunable spheroidal THz antennas. The method can be used in a variety of potential EM applications including microwaves, functional photonics structures, as well as nanoantenna engineering.

Propagation of double zero-order Bessel beam and its scattering properties to uniaxial anisotropic spheres

Based on the generalized Lorenz Mie theory, the propagation and scattering properties of a uniaxial anisotropic spherical particle illuminated separately by double zero-order Bessel beam with arbitrary propagation direction and polarization direction are studied. The propagation and scattering characteristics are compared with those of a uniaxial anisotropic spherical particle illuminated by a single zero-order Bessel beam. Using the orthogonal relation of the spherical vector wave function and coordinate rotation theorem, the expanded forms of double zero-order Bessel beams with arbitrary propagation direction and polarization direction are derived. The analytical expressions of the expansion coefficients are derived by the integral method. The expansion coefficients of total incident field are obtained through the vector superposition principle. Based on the Fourier transform and tangentially continuous boundary conditions, the internal electromagnetic field of the uniaxial anisotropic sphere is expanded in terms of the spherical vector wave function and the scattering coefficients are derived. By comparing the angular distribution of the radar cross section of the particle illuminated by single and double zero-order Bessel beam when degenerating into plane waves with those results given by the literature, the correctness of the theory and the program in this paper are both verified. The effects of the incidence angle, conic angle and polarization angle on angle distribution of the radar cross section are numerically analyzed. The theoretical and numerical results in this paper are expected to be used to study the scattering properties, particle size analysis and optical trapping for anisotropic particles, biological cells and other particles illuminated by multi-beams.

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.