Showing papers on "Bessel function published in 2021"

Modified frequency-Bessel transform method for dispersion imaging of Rayleigh waves from ambient seismic noise

Abstract: Ambient noise surface wave methods have gained much attention among geophysical and civil engineering communities because of their capability of determining near-surface shear wave velocities in highly populated urban areas. Higher mode information of surface waves is important in dispersion curve inversion for shear wave velocity structure. The frequency–Bessel (F-J) transform method is an effective tool for multimode surface wave extraction, which has been applied to multiscale investigations of the Earth structure. The measured dispersion energy with the F-J method, however, would usually be contaminated by a type of ‘crossed’ artefacts at high frequencies, which are caused by spatial aliasing and bidirectional velocity scan of dispersion analysis methods. The ‘crossed’ artefacts usually cross and smear the true dispersion energy in the frequency–velocity domain. We propose a modified F-J (MFJ) transform method in which the Bessel function is replaced by the Hankel function for dispersion analysis of empirical Green's function. The MFJ method performs a unidirectional velocity scanning on the outgoing wave to avoid the ‘crossed’ artefacts. Synthetic and real-world examples demonstrate the effectiveness of the proposed MFJ method in improving the accuracy of Rayleigh wave multimode dispersion measurements.

Numerical approximations to the nonlinear fractional-order Logistic population model with fractional-order Bessel and Legendre bases

Abstract: The main aim of this manuscript is to obtain the approximate solutions of the nonlinear Logistic equation of fractional order by developing a collocation approach based on the fractional-order Bessel and Legendre functions. The main characteristic of these polynomial approximation techniques is that they transform the governing differential equation into a system of algebraic equations, thus the computational efforts will be greatly reduced. Our secondary aim is to show a comparative investigation on the use of these fractional-order polynomials and to examine their utilities to solve the model problem. Numerical experiments are carried out to demonstrate the validity and applicability of the presented techniques and comparisons are made with methods available in the standard literature. The methods perform very well in terms of efficiency and simplicity to solve this population model especially when the Legendre bases are utilized.

Bessel beams: a unified and extended perspective

Abstract: We present a unified and extended perspective of Bessel beams, irrespective of their orbital angular momentum (OAM)—zero, integer or noninteger—and mode—scalar or vectorial, and LSE/LSM or TE/TM in the latter case. The unification is based on the integral superposition of constituent waves along the angular-spectrum cone of the beam, and enables us to describe, compute, relate, and implement all Bessel beams, and even other types of beams, in a universal fashion. We first establish the integral superposition theory. Then, we demonstrate the existence of noninteger-OAM TE/TM Bessel beams, compare the LSE/LSM and TE/TM modes, and establish useful mathematical relations between them. We also provide an original description of the position of the noninteger-OAM singularity in terms of the initial phase of the constituent waves. Finally, we introduce a general technique for generating Bessel beams using an adequate superposition of properly tuned sources. This global perspective and theoretical extension may be useful in applications such as spectroscopy, microscopy, and optical/quantum force manipulations.

Generation and Conversion Dynamics of Dual Bessel Beams with a Photonic Spin-Dependent Dielectric Metasurface

Abstract: A Bessel beam has the properties of propagation invariance and a self-healing effect, leading to a variety of interesting phenomena and applications. Recently, as a planar diffractive element with miniaturized size, metasurfaces are widely employed to manipulate light in the subwavelength region, including generating a Bessel beam. However, such a metasurface-generated Bessel beam allows output light with no tunable functions. Here, with the interplay of the geometric phase and the dynamic phase, we propose a method to generate and allow conversion from any orthogonal polarizations to independent Bessel beams with a single-layer dielectric metasurface. The simulation results indicate that the arbitrary conversion between different Bessel beams is related to the spin-dependent orbit motion caused by the tight-focusing effect, leading to the singularity of the spot. This physical mechanism is well studied and the theoretical model for revealing the dependence of different incident polarization on the conversion dynamics is presented. Our approach paves a way for efficient generation and multifunctional applications, ranging from high-numerical-aperture devices to compact nanophotonic platforms for spin-dependent structured beams.

Non-spreading Bessel spatiotemporal optical vortices

Abstract: Non-spreading nature of Bessel spatiotemporal wavepackets is theoretically and experimentally investigated and orders of magnitude improvement in the spatiotemporal spreading has been demonstrated. The spatiotemporal confinement provided by the Bessel spatiotemporal wavepacket is further exploited to transport transverse orbital angular momentum through embedding spatiotemporal optical vortex into the Bessel spatiotemporal wavepacket, constructing a new type of wavepacket: Bessel spatiotemporal optical vortex. Both numerical and experimental results demonstrate that spatiotemporal vortex structure can be well maintained and confined through much longer propagation. High order spatiotemporal optical vortices can also be better confined in the spatiotemporal domain and prevented from further breaking up, overcoming a potential major obstacle for future applications of spatiotemporal vortex.

Universal dispersion curves of a planar waveguide with an exponential graded-index guiding layer and a nonlinear cladding

Abstract: A planar waveguide consisting of three layers is considered. The guiding layer is assumed of exponentially graded index of refraction. The cover layer is a nonlinear material of Kerr-type. The refractive index distribution of the film layer changes as an exponential function from the guiding layer to the substrate. The solutions of Helmholtz equation are found. They are written in terms of three parameters a, b and V. The solutions in the guiding layer and substrate are found as Bessel functions of order V b . The characteristic equation is derived and the dispersion curves are plotted and analyzed. A set of attracting features are found such as there is no cut-off thickness corresponding to a symmetric waveguide structure. The b-values do not exceed unity. This means the dispersion curves refer to guided modes.

DisperNet: An Effective Method of Extracting and Classifying the Dispersion Curves in the Frequency–Bessel Dispersion Spectrum

Abstract: The subsurface shear-wave structure primarily determines the characteristics of the surface-wave dispersion curve theoretically and observationally. Therefore, surface-wave dispersion curve inversion is extensively applied in imaging subsurface shear-wave velocity structures. The frequency–Bessel transform method can effectively extract dispersion spectra of high quality from both ambient seismic noise data and earthquake events data. However, manual picking and semiautomatic methods for dispersion curves lack a unified criterion, which impacts the results of inversion and imaging. In addition, conventional methods are insufficiently efficient; more precisely, a large amount of time is required for curve extraction from vast dispersion spectra, especially in practical applications. Thus, we propose DisperNet, a neural network system, to extract and discriminate the different modes of the dispersion curve. DisperNet consists of two parts: a supervised network for dispersion curve extraction and an unsupervised method for dispersion curve classification. Dispersion spectra from ambient noise and earthquake events are applied in training and validation. A field data test and transfer learning test show that DisperNet can stably and efficiently extract dispersion curves. The results indicate that DisperNet can significantly improve multimode surface-wave imaging.

Cherenkov FEL Reaction With Plasma-Filled Cylindrical Waveguide in Fractional D -Dimensional Space

Abstract: The possibility of generating an electromagnetic (EM) wave by a free-electron laser (FEL) beam from the Cherenkov device to control the cylindrical waveguide’s field attenuation filled with plasma has been investigated by analytical formalism. This new study sheds light on Cherenkov FEL (C-FEL) beam interaction with electrons of inhomogeneous warm plasma to generate an EM wave in fractional dimensional space. The new analysis of traveling and standing waves in terms of Hankel and Bessel functions paves a way for introducing controlled EM wave propagation based on fractional $D$ -dimensional space. It has been found that the C-FEL beam excites the EM wave and enhances the propagation of the electrical field through fractional dimensional space with propagation constant depending on the Langmuir frequency. Within the plasma in the cylindrical waveguide, a TM mode emerges, which contains spatial frequencies with a faster growth rate for traveling waves than standing waves.

Surface wave scattering by multiple flexible fishing cage system

Abstract: A study of the wave dynamics around a multiple cylindrical fishing cage system is carried out under the assumption of linear water wave theory and small-amplitude wave response. The Fourier–Bessel series expansion of the velocity potential is derived for the regions enclosed under the open-water and cage systems and the immediate vicinity. The scattering between the cages is accounted for by employing Graf's addition theorem. The porous flexible cage system is modeled using Darcy's law and the three-dimensional membrane equation. The edges of the cages are moored along their circumferences to balance its position. The unknown coefficients in the potentials are obtained by employing the matched eigenfunction method. In addition, the far-field scattering coefficients for the entire system are obtained by expanding the Bessel and Hankel functions in the plane wave representation form. Numerical results for the hydrodynamic forces, scattering coefficients, and power dissipation are investigated for various cage and wave parameters. The time simulation for the wave scattering from the cage system is investigated. The study reveals that wave loading on the cage system can be significantly reduced by the appropriate spatial arrangement, membrane tension, and porous-effect parameter. Moreover, the far-field results suggest that the cage system can also be used as a breakwater.

Crossing bridges with strong Szegő limit theorem

Abstract: We develop a new technique for computing a class of four-point correlation functions of heavy half-BPS operators in planar $$ \mathcal{N} $$ = 4 SYM theory which admit factorization into a product of two octagon form factors with an arbitrary bridge length. We show that the octagon can be expressed as the Fredholm determinant of the integrable Bessel operator and demonstrate that this representation is very efficient in finding the octagons both at weak and strong coupling. At weak coupling, in the limit when the four half-BPS operators become null separated in a sequential manner, the octagon obeys the Toda lattice equations and can be found in a closed form. At strong coupling, we exploit the strong Szegő limit theorem to derive the leading asymptotic behavior of the octagon and, then, apply the method of differential equations to determine the remaining subleading terms of the strong coupling expansion to any order in the inverse coupling. To achieve this goal, we generalize results available in the literature for the asymptotic behavior of the determinant of the Bessel operator. As a byproduct of our analysis, we formulate a Szegő-Akhiezer-Kac formula for the determinant of the Bessel operator with a Fisher-Hartwig singularity and develop a systematic approach to account for subleading power suppressed contributions.

Axicon optical forces and other kinds of transverse optical forces exerted by off-axis Bessel beams in the Rayleigh regime in the framework of generalized Lorenz-Mie theory

Abstract: In two recent papers, it has been demonstrated that, beside usual scattering and optical forces, new optical forces and terms are exhibited when a Rayleigh particle is illuminated by an off-axis Bessel beam. Namely, (i) axicon optical forces are associated with scattering optical forces and (ii) additional axicon terms are associated with gradient optical forces. These extra-axicon forces and terms are zero when the axicon angle is zero and/or (most often) when an on-axis configuration is considered rather than an off-axis configuration. This study was devoted to longitudinal forces. The present paper is devoted to transverse forces and demonstrates the existence of newly additional axicon forces associated to both the usual scattering and gradient forces. Again, these new extra-axicon forces are zero when the axicon angle is zero and/or when an on-axis configuration is considered rather than an off-axis configuration.

Analysis and Design of Bessel Beam Launchers: Transverse Polarization

Abstract: In this communication, we present a theoretical analysis, design, and fabrication of a limited-diffractive planar Bessel beam launcher, that exhibits a zeroth-order Bessel profile in the transverse electric field component with respect to the ${z}$ -propagation axis. The launcher is designed by synthesizing a finite zeroth-order, first kind Hankel aperture distribution, polarized along a fixed polarization unit vector. The field radiated by such an aperture distribution is derived by following an approximate model based on the geometric theory of diffraction, thus allowing to highlight the relevant wave constituents involved in transverse Bessel beam generation, also including the effect of aperture truncation on the radiated beam. Moreover, the theoretical analysis has been profitably applied to the design of a circular-polarized planar transverse Bessel beam launcher by means of a slotted radial waveguide. A prototype of right-handed circular polarized (RHCP) transverse Bessel beam launcher has then been fabricated at $f = 30$ GHz. The measured transverse electric field component shows a satisfactory agreement with full-wave numerical simulations.

Functional central limit theorems for multivariate Bessel processes in the freezing regime

Abstract: Multivariate Bessel processes (Xt,k)t≥0 describe interacting particle systems of Calogero-Moser-Sutherland type and are related with β-Hermite and β-Laguerre ensembles. They depend on a root system...

Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach

Abstract: Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods are performed to validate our results.

Exponential moments and piecewise thinning for the Bessel point process

Abstract: We obtain exponential moment asymptotics for the Bessel point process. As a direct consequence, we improve on the asymptotics for the expectation and variance of the associated counting function, and establish several central limit theorems. We show that exponential moment asymptotics can also be interpreted as large gap asymptotics, in the case where we apply the operation of a piecewise constant thinning on several consecutive intervals. We believe our results also provide important estimates for later studies of the global rigidity of the Bessel point process.

Particle Trapping in Arbitrary Trajectories Using First-Order Bessel-Like Acoustic Beams

Abstract: In general, singular acoustic beams travel in a straight line and expand during propagation, which limits the scope of their applications. We present a method for generating paraxial zero-order Bessel-like acoustic beams with arbitrary trajectories in water. We then add a vortex phase to obtain a first-order Bessel-like acoustic beam. This beam exhibits a combination of features of vortex, Bessel, and Airy beams. Moreover, it maintains a dark ``hole'' in the center, preserving its angular momentum, and displays resistance to diffraction, self-healing, and self-bending during propagation. Furthermore, we experimentally realize the beam using a three-dimensionally printed phase mask and observe the acoustic field distribution using the Schlieren imaging method. Finally, we realize particle trapping and transport in a curved trajectory in water.

Certain Matrix Riemann–Liouville Fractional Integrals Associated with Functions Involving Generalized Bessel Matrix Polynomials

Abstract: The fractional integrals involving a number of special functions and polynomials have significant importance and applications in diverse areas of science; for example, statistics, applied mathematics, physics, and engineering. In this paper, we aim to introduce a slightly modified matrix of Riemann–Liouville fractional integrals and investigate this matrix of Riemann–Liouville fractional integrals associated with products of certain elementary functions and generalized Bessel matrix polynomials. We also consider this matrix of Riemann–Liouville fractional integrals with a matrix version of the Jacobi polynomials. Furthermore, we point out that a number of Riemann–Liouville fractional integrals associated with a variety of functions and polynomials can be presented, which are presented as problems for further investigations.

Nonlinear Hyperbolic Waves in Relativistic Gases of Massive Particles with Synge Energy

Abstract: In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler’s relativistic system when the constitutive equation for energy is that of Synge for a monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the classical to the ultra-relativistic regime. They involve modified Bessel functions of the second kind and this makes Euler’s relativistic system rather complex. Based on delicate estimates of the Bessel functions, we prove: (i) a limit on the speed of sound of $$1{/}\sqrt{3}$$ times the speed of light (which a fortiori implies subluminality, that is causality), (ii) the genuine non-linearity of the acoustic waves, (iii) the compatibility of Rankine–Hugoniot relations with the second law of thermodynamics (entropy growth through all Lax shocks), and (iv) the unique resolvability of the initial value problem of Riemann (if we include the possibility of vacuum as in the non-relativistic context).

On the Domains of Bessel Operators

Abstract: We consider the Schrodinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ , often called the Bessel operator We assume that m is complex We study the domains of various closed homogeneous realizations of the Bessel operator In particular, we prove that the domain of its minimal realization for $$|\mathrm{Re}(m)|<1$$ and of its unique closed realization for $$\mathrm{Re}(m)>1$$ coincide with the minimal second-order Sobolev space On the other hand, if $$\mathrm{Re}(m)=1$$ the minimal second-order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator The properties of Bessel operators are compared with the properties of the corresponding bilinear forms

Generalized Bessel Quasilinearization Technique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order

Abstract: The ultimate goal of this study is to develop a numerically effective approximation technique to acquire numerical solutions of the integer and fractional-order Bratu and the singular Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary conditions were considered and the fractional derivative being considered in the Liouville–Caputo sense. In the direct approach, the generalized Bessel matrix method based on collocation points was utilized to convert the model problems into a nonlinear fundamental matrix equation. Then, the technique of quasilinearization was employed to tackle the nonlinearity that arose in our considered model problems. Consequently, the quasilinearization method was utilized to transform the original nonlinear problems into a sequence of linear equations, while the generalized Bessel collocation scheme was employed to solve the resulting linear equations iteratively. In particular, to convert the Neumann initial or boundary condition into a matrix form, a fast algorithm for computing the derivative of the basis functions is presented. The error analysis of the quasilinear approach is also discussed. The effectiveness of the present linearized approach is illustrated through several simulations with some test examples. Comparisons with existing well-known schemes revealed that the presented technique is an easy-to-implement method while being very effective and convenient for the nonlinear Bratu and Lane–Emden equations.

Rayleigh limit of generalized Lorenz-Mie theory for on-axis beams and its relationship with the dipole theory of forces. Part I: Non dark axisymmetric beams of the first kind, with the example of Gaussian beams

Abstract: The Rayleigh limit of the generalized Lorenz-Mie theory (GLMT) has been recently examined in the case of off-axis circularly symmetric Bessel beams, thereafter in the case of on-axis circularly symmetric Bessel beams, the on-axis case providing an easier framework for the understanding of the optical forces exerted in the Rayleigh limit of GLMT. This work is here extended to the case of non dark on-axis axisymmetric beams of the first kind This encompasses the case of zeroth-order circularly symmetric Bessel beams and the case of localized models of Gaussian beams. Three kinds of optical forces are exhibited, namely traditional gradient and scattering forces, plus another kind of forces which is for the time being denoted as non standard forces. The relationship between the Rayleigh limit of GLMT and the dipole theory of forces is discussed, to the best of our present understanding.

Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials

Abstract: Feng Qi abstract: In the paper, by virtue of the Faá di Bruno formula and identities for the Bell polynomials of the second kind, the author simplifies coefficients in a family of ordinary differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials.

Modified wavelet method for solving fractional variational problems

Abstract: In this article, a newly modified Bessel wavelet method for solving fractional variational problems is considered. The modified operational matrix of integration based on Bessel wavelet functions i...

On Behavior Laplace Integral Operators with Generalized Bessel Matrix Polynomials and Related Functions

Abstract: Recently, the applications and importance of integral transforms (or operators) with special functions and polynomials have received more attention in various fields like fractional analysis, survival analysis, physics, statistics, and engendering. In this article, we aim to introduce a number of Laplace and inverse Laplace integral transforms of functions involving the generalized and reverse generalized Bessel matrix polynomials. In addition, the current outcomes are yielded to more outcomes in the modern theory of integral transforms.

A new product formula involving Bessel functions

Abstract: In this paper, we consider the normalized Bessel function of index α>−12, we find an integral representation of the term xnjα+n(x)jα(y). This allows us to establish a product formula for the genera...