Cylindrical Wave Functions Research Articles

An Entire Domain CFVIE-CDSE Method for EM Scattering on Electrically Large Highly Inhomogeneous Gyrotropic Circular Cylinders

A coupled-field volume integral equation (CFVIE) method for electromagnetic (EM) scattering on electrically large, highly inhomogeneous gyrotropic circular cylinders, under normal incidence, is developed in this work. The CFVIEs are solved by the cylindrical Dini series expansion (CDSE) method where the unknown fields are expanded by entire domain orthogonal vectorial basis functions. The main advantage of the present method is that it permits the scatterer to have continuously varying highly inhomogeneous gyrotropic characteristics, i.e., the constitutive parameters of the cylinder can be highly inhomogeneous in both gyroelectric and gyromagnetic tensors. Initially, the 2-D Green's function (GF) is expanded in a tensorial form using the cylindrical vector wave functions (CVWFs). Then, by employing the CDSE for the unknown fields, the 2-D volumetric integrals are carried out analytically, reducing the CFVIEs to a set of algebraic equations. The method is validated by comparisons with the exact solution based on the separation of variables method (SVM) for homogeneous gyroelectric/gyromagnetic cylinders, with HFSS commercial software for three-layered gyroelectric cylinders, as well as with the recently developed hybrid projection method (HPM) for electrically large continuously varying highly inhomogeneous isotropic cylinders. Results for combined gyroelectric-continuously varying highly inhomogeneous isotropic cylinders are presented and discussed.

Gaussian beam propagation through a biaxial anisotropic circular cylinder

Based on the electromagnetic field expansions over appropriate cylindrical vector wave functions, the scattering features of a Gaussian beam by a biaxial anisotropic circular cylinder are analyzed. The unknown expansion coefficients are determined from a system of linear equations derived by using continuous boundary conditions. For a TE and TM polarized Gaussian beam with different propagation directions, the normalized near-surface and internal field intensity distributions are evaluated. The propagation properties are compared and discussed briefly.

Propagation of shaped beam through uniaxially anisotropic chiral slab* *Project supported by the National Natural Science Foundation of China (Grant No. 61771385), the Science Foundation for Distinguished Young Scholars of Shaanxi Province, China (Grant No. 2020JC-42), the Fund from the Science and Technology on Solid-State Laser Laboratory, China (Grant No. 6142404180301), and the Science and Technology Research Plan of Xi’an

A general solution is obtained to a canonical problem of the reflection and refraction of an arbitrary shaped beam by using a uniaxially anisotropic chiral slab. The reflected, internal as well as refracted shaped beams are expanded in terms of cylindrical vector wave functions, and the expansion coefficients are determined by using the boundary conditions and method of moments procedure. As two typical examples, the normalized field intensity distributions are evaluated for a fundamental Gaussian beam and Hermite–Gaussian beam, and some propagation properties, especially the negative refraction phenomenon, are discussed briefly.

Acoustic radiation force on a cylindrical particle near a planar rigid boundary II. – Viscous fluid cylinder example and inherent radiation torque

ObjectiveThis study extends the scope of a previous analysis on the time-averaged acoustic radiation force on a rigid (sound impenetrable) cylinder near a flat boundary [F.G. Mitri, J. Phys. Commun. 2 (2018) 045019] to the case of a viscous compressible fluid (sound penetrable) particle, and determine the time-averaged acoustic radiation torque as well. Motivation and noveltyPrevious analytical formalisms did not consider the case of a sound penetrable cylindrical particle insonified at an arbitrary angle of incidence (in the polar plane) near a reflecting boundary. This work fills this gap, and provides exact expressions and computations for the acoustic radiation force and torque components. MethodThe partial-wave series expansion method, in conjunction with the method of images and the translational addition theorem of cylindrical wave functions are used to derive the analytical expressions for the longitudinal and transverse acoustic radiation force components. Moreover, the emergence of a radiation torque that causes the particle to rotate around its center of mass is computed using an exact partial-wave series expression. Results, key conclusion and some perspectivesAttractive (pulling), repulsive (pushing) and neutral (zero) forces arise depending on the particle-boundary distance, the cylinder size parameter as well as the angle of incidence (in the polar plane) of the insonifying waves. Emphasis is also given on the emergence of an acoustic radiation torque (that vanishes for a rigid or non-viscous circular cylinder). Computations for the axial radiation torque efficiency anticipate the generation of positive radiation torque, its reversal, in addition to a zero efficiency, leading, respectively, to counter-clockwise, clockwise or lack of particle rotation as the angle of the incident waves deviates from normal incidence with respect to the boundary surface. The extension to the case of an elliptical/oval cylinder near a boundary is mentioned, and replies to some misleading and obtuse comments on the paper [F.G. Mitri, Phys. Fluids 28 (2016) 077104] are provided.

Reflection and transmission of a Bessel vortex beam by a stratified uniaxial anisotropic slab

The reflection and transmission of a Bessel vortex beam incident on a stratified uniaxial anisotropic slab are analytically solved by use of the expansion of cylindrical vector wave functions (CVWFs). This work is an extension of our previous paper, in which the reflection and transmission of a Bessel vortex beam incident on a homogeneous uniaxial anisotropic slab were studied. A closed form of the expansion coefficients of the reflected and transmitted beams for a stratified slab is derived. Taking a three-layered uniaxial anisotropic slab as an example, the characteristics of the reflected and transmitted beams and its influencing factors are numerically analyzed, including the magnitude profiles of the electric field components, the energy conversion between the two transverse electric components, and the distortion of orbital angular momentum (OAM) states. The results indicate that the effects of the dielectric properties of each layer on the magnitude profiles of the reflected field components are prominent, while those of the transmitted beams are opposite, which could be applied in the design of optical devices. Compared with other polarization modes, the polarization perturbation for the circularly polarized caseis very slight. As for the oblique incidence, an increase of the incident angle can obtain a high-quality reflected beam with fewer derived OAM states, and the lossy media can also do the same. Since the uniaxial anisotropic media has many applications in optical communication, radar detection, microwave device manufacture, etc., and vortex beams are proved to increase the channel capacity of an optical communication systems, these numerical results are of use in optical communications, for example, reducing the crosstalk in optical communication.

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.

Propagation of light beam through a uniaxially anisotropic chiral cylinder

A general scheme is proposed for investigating a canonical problem of light beam scattering by a uniaxially anisotropic chiral cylinder. The light beams within different regions are expanded in terms of appropriate cylindrical vector wave functions, and their expansion coefficients are determined by using the boundary conditions and projection method. For incidence of a fundamental Gaussian beam and Hermite–Gaussian beam, numerical results of normalized field intensity distributions are provided and discussed briefly.

Acoustic radiation force on an elastic cylinder in a Gaussian beam near an impedance boundary

Acoustic radiation force (ARF) is studied by considering an infinite elastic cylinder near an impedance boundary when the cylinder is illuminated by a Gaussian beam. The surrounding fluid is an ideal fluid. Using the method of images and the translation-addition theorem for the cylindrical Bessel function, the resulting sound field including the incident wave, its reflection from the boundary, the scattered wave from the elastic cylinder, and its image are expressed in terms of the cylindrical wave function. Then, we deduce the exact equations of the axial and transverse ARFs. The solutions depend on the cylinder position, cylinder material, beam waist, reflection coefficient, distance from the impedance boundary, and absorption in the cylinder. To analyze the effects of the various factors intuitively, we simulate the radiation force for non-absorbing elastic cylinders made of stainless steel, gold, and beryllium as well as for an absorbing elastic cylinder made of polyethylene, which is a well-known biomedical polymer. The results show that the impedance boundary, cylinder material, absorption in the cylinder, and cylinder position in the Gaussian beam significantly affect the magnitude and direction of the force. Both stable and unstable equilibrium regions are found. Moreover, a larger beam waist broadens the beam domain, corresponding to non-zero axial and transverse ARFs. More importantly, negative ARFs are produced depending on the choice of the various factors. These results are particularly important for designing acoustic manipulation devices operating with Gaussian beams.

An analytical solution to investigate the dynamic impact of a moving surface load on a shallowly-buried tunnel

In order to investigate the dynamic impact of a moving surface load on a shallow-buried tunnel, an analytical model of a tunnel embedded in an elastic half-space was proposed. The half-space and the tunnel structure were modeled as visco-elastic media and the moving surface load was simplified as a moving point load on the ground surface. Based on the fundamental solution for the isotropic elastic half-space system in Cartesian and cylindrical coordinates, the dynamic response of a shallowly-buried tunnel in a half-space generated by a moving surface load was obtained. The transformations between plane wave and cylindrical wave functions were used to facilitate the application of boundary conditions at the ground surface and the tunnel interface. It was found that the vibration of the shallowly-buried tunnel increases significantly as the load moving speed increases, and reaches a maximum value at a critical load velocity. The tunnel vibration can be greatly reduced as the buried depth increases, and can satisfy the requirement of vibration specification (ISO 04866-2010) after it exceeds the critical depth. The critical depth increases exponentially with the increase of the moving speed of the surface load.

A general solution for scattering by infinite cylinders in the presence of an interface

A general solution for scattering by multiple parallel infinite cylinders in the presence of a half space interface is presented in this paper. The cylinders are located in either or both half spaces, which are irradiated by a homogeneous or evanescent source wave traversing in a direction inclined from the axes of the cylinders. The theoretical approach is an exact analytical treatment that accounts for depolarization at oblique incidence, near-field scattering between cylinders, interaction with reflected waves from the interface, and coupling of cylinders across the interface. The electromagnetic fields are formulated using Hertz potenials that are expanded in terms of cylindrical wave functions. The unknown expansion coefficients are determined by boundary matching. Analytical formulas are derived for the electromagnetic fields and Poynting vector of scattered radiation in the far field. Application of the present solution to all scenarios of scattering by cylinders in the presence of a half space interface is illustrated analytically and by numerical examples.

Characteristics of electromagnetic wave transmission and reflection from isotropic plasma coated circular nihility cylinder

Theoretical investigation has been carried out to study the scattering and transmission of electromagnetic energy from isotropic plasma coated circular nihility cylinder. The physical modeling of the problem is done in frame work of classical scattering theory. All the fields are expanded in terms of cylindrical vector wave functions (CVWFs). The analytical as well as numerical solution for both Transverse Electric (TE) and Transverse Magnetic (TM) polarizations is presented. The unknown scattering coefficients are computed by implementing the boundary conditions at each interface (free space/plasma and plasma/nihility). The influence of plasma parameters i.e., plasma density (ND) and effective collision frequency (v) on the bistatic echo width is analyzed and it is concluded that the plasma coating can be used to control and tune the scattering amplitude. Further it is also concluded that for the case ω > ωp the scattering amplitude is much less than for the case ω < ωp. Under special conditions, the results of present work compared with the published literature and good agreement is found.

Evaluation of average cross‐polarised scattered field from a PEMC cylinder of the random radius with uniform/normal distribution

The average scattered field from a random perfect electromagnetic conductor (PEMC) is investigated. First, a very small cylinder is considered and analytical expressions for average scattered field are written using a small approximation of cylindrical wave functions. It is shown that co-polarised average scattered field from a very small PEMC cylinder depends on only zeroth-order scattering coefficient, whereas higher-order coefficients also contribute to cross-polarised average scattered field. Moreover, the average scattered field from a very small PEMC cylinder can be written in terms of average scattered field of a very small perfect electric conductor (PEC) cylinder. The second case deals with small and large radii. In this case, it is not possible to write down analytical expressions for the average field, the numerical average is done. For small PEMC cylinder, it has been observed that average co-polarised scattered field is equal to the scattered field from a cylinder with a mean radius. This is not true in case of cross-polarised scattered field and the difference can be seen for most of the scattering angles. In case of large PEC/PEMC cylinders, it is observed that the average scattered field is almost equal to the scattered field from a cylinder with a mean radius.

Optical radiation force (per–length) on an electrically conducting elliptical cylinder having a smooth or ribbed surface

The aim of this work is to develop a formal semi-analytical model using the modal expansion method in cylindrical coordinates to calculate the optical/electromagnetic (EM) radiation force-per-length experienced by an infinitely long electrically-conducting elliptical cylinder having a smooth or wavy/corrugated surface in EM plane progressive waves with different polarizations. One of the semi-axes of the elliptical cylinder coincides with the direction of the incident field. The modal matching method is used to determine the scattering coefficients by matrix inversion. Standard cylindrical (Bessel and Hankel) wave functions are used. Simplified expressions leading to exact series expansions for the optical/EM radiation forces assuming either electric (TM) of magnetic (TE) plane wave incidences are provided without any approximations, in addition to integral equations demonstrating the direct relationship of the radiation force with the square of the scattered field magnitude. Numerical computations for the non-dimensional radiation force function are performed for electrically conducting elliptic and circular cylinders having a smooth or ribbed/corrugated surface. Adequate convergence plots confirm the validity and correctness of the method to evaluate the radiation force with no limitation to a particular frequency range (i.e. Rayleigh, Mie or geometrical optics regimes). Emphases are given on the aspect ratio, the non-dimensional size of the cylinder, the corrugation characteristic of its surface, and the polarization of the incident field. The results are particularly relevant in optical tweezers and other related applications in fluid dynamics, where the shape and stability of a cylindrical drop stressed by a uniform external electric/magnetic field are altered. Furthermore, a direct analogy with the acoustical counterpart is discussed.

Analytical solution for calculating vibrations from twin circular tunnels

The calculation of vibrations from a single tunnel embedded in a full- or half-space due to underground moving trains has been well studied in the literature. A common assumption made in the majority of vibration prediction models is to neglect the presence of a neighboring tunnel, however, twin tunnels are a particularly preferred construction for urban underground railways with one as the inbound tunnel and the other as the outbound tunnel. The neglect of the interaction between twin tunnels may lead to inaccuracies of the calculation of vibrations from underground railways. The present paper proposes a novel analytical solution, which takes the multiple scattering effects between the two tunnels into account. The two tunnels are modelled as elastic hollow cylinders and the soil surrounding the tunnels as an elastic, homogeneous full-space containing two cylindrical cavities. The wave field in the full-space with two cavities consists of outgoing waves from each tunnel outside the two scattering surfaces while the wave field in the tunnel wall is a combination of outgoing and regular cylindrical waves. The translation of the outgoing cylindrical wave functions is required to satisfy the boundary conditions at the other tunnel. Numerical results show that there are two critical velocities for a twin tunnel in a full-space system, both of which are around the shear wave velocity of the soil. Through comparison with the single tunnel solution, the addition of a second tunnel at different separation distances and angles has a significant influence on the soil response, especially in the high frequency range, but the response at the tunnel with the source is not much affected.

Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field

The effect of homogeneous electric field on the energy spectrum, wave functions of electron and oscillator strengths of intra-band quantum transitions in a double cylindrical quantum ring (GaAs/Al$_{x}$Ga$_{1-x}$As) is studied within the approximations of effective mass and rectangular potentials. The calculations are performed using the method of expansion of quasiparticle wave function over a complete set of cylindrical wave functions obtained as exact solutions of Schr\"odinger equation for an electron in a nanostructure without electric field. It is shown that the electric field essentially affects the electron localization in the rings of a nanostructure. Herein, the electron energies and oscillator strengths of intra-band quantum transitions non-monotonously depend on the intensity of electric field.

Transmission of electromagnetic beam through a uniaxial anisotropic slab

We present a general theoretical scheme to investigate the propagation characteristics of a focused electromagnetic beam in a uniaxial anisotropic slab. The reflected, internal as well as transmitted waves are expanded in terms of cylindrical vector wave functions, and the unknown expansion coefficients are determined by virtue of the boundary conditions and the projection method. For incidence of a fundamental Gaussian beam, Hermite–Gaussian beam and doughnut mode beam, normalized field intensity distributions are evaluated and discussed briefly.

Internal and near-surface electromagnetic fields for a chiral cylinder with arbitrary monochromatic illumination

A semi-analytical solution is proposed to the electromagnetic scattering by an infinite chiral circular cylinder with arbitrary monochromatic illumination. The scattered and internal fields are expanded in terms of appropriate cylindrical vector wave functions, and their expansion coefficients are determined by virtue of the boundary conditions and the projection method. As a demonstration of the theoretical procedure, the normalized internal and near-surface field intensity distributions are evaluated for a fundamental Gaussian beam, and the scattering properties are discussed briefly for different chirality parameters.

Fast Analysis of Parallel-Plate Cylindrical Luneberg Lens Antennas Through Dyadic Green’s Functions

An accurate analytic and computationally efficient method based on dyadic Green's function (DGF) is presented for the analysis of parallel-plate cylindrical Luneberg lens antennas. DGFs for electric and magnetic current sources are rigorously derived for an infinite parallel-plate waveguide filled with radially stratified media, which is the ideal model of cylindrical Luneberg lens antennas. The DGF for the electric current source is constituted of cylindrical vector wave functions and their associated coefficients, which is developed from the scattering superposition and the corresponding matching of boundary conditions, respectively. The DGF for the magnetic current source is proved to be directly related to the DGF for the electric current source by applying the second-kind vector Green's theorem. The proposed DGFs and the aperture field method are then applied to analyze the radiation characteristics of parallel-plate cylindrical Luneberg lens antennas with equivalent sources. Numerical results of several large-sized parallel-plate cylindrical Luneberg lens antennas show that the results of the proposed approach are in good agreement with those from commercial software simulations. The proposed approach has much less computational costs and can be used for the fast analysis of large-sized cylindrical Luneberg lens antennas.

Electromagnetic binding and radiation force reversal on a pair of electrically conducting cylinders of arbitrary geometrical cross-section with smooth and corrugated surfaces

The electromagnetic (EM) radiation force-per-length exerted on a pair of electrically-conducting cylindrical particles of circular and non-circular cross-sections is examined using a formal semi-analytical method based on boundary matching in cylindrical coordinates. Initially, the scattering coefficients of the particle pair are determined by imposing suitable boundary conditions leading linear systems of equations computed via matrix inversion and numerical integration procedures. Standard cylindrical (Bessel and Hankel) wave functions are used and closed-form expressions for the dimensionless longitudinal and transverse radiation force functions are evaluated assuming either magnetic (TE) or electric (TM) plane wave incidences. Particle pairs with smooth and corrugated surfaces are considered and numerical computations are performed with emphasis on the distance separating their centers of mass, the angle of incidence of the incident illuminating field and the surface roughness. Adequate convergence plots confirm the validity of the method to evaluate the radiation force functions, and the model is adaptable to any frequency range (i.e. Rayleigh, Mie or geometrical optics regimes). The results can find potential applications in optical tweezers and other related applications in fluid dynamics. In addition, the acoustical analogue is discussed.

Investigating the reduction of cross-polarized Gaussian beam scattering from a PEMC buried cylinder coated with a topological insulator.

An investigation is presented for the reduction of cross-polarized Gaussian beam scattering from a perfect electromagnetic conducting (PEMC) cylinder coated with topological insulator (TI) material and buried in a semi-infinite medium. To calculate the successive reflections of the scattered field from the interface, cylindrical wave functions are written into the spectrum of plane waves. The scattered reflected field is calculated for each plane wave of the spectrum. Near- and far-zone scattered fields are reported for time-reversal symmetry and symmetry-broken TI coating. It is observed that the near-zone scattered field is very helpful to understand why the scattering width (SW) of a cylinder coated with time-reversal symmetry TI material is different from the SW of a cylinder coated with symmetry-broken TI material. The effect of spot size of the Gaussian beam on the scattering pattern is investigated, while the result for a plane wave is served as a reference. It is observed that cross-polarized scattering can be reduced by increasing the thickness of the coating layer. It also decreases as admittance of the PEMC core increases. Moreover, the cross-polarized scattered field becomes significantly small when time-reversal symmetry is broken, irrespective of PEMC or perfect electric conductor (PEC) core.