Showing papers on "Bessel function published in 2011"
01 Jan 2011
TL;DR: A discussion of the most important formulae and construction of plots for some functions of mathematical physics relevant to quantum mechanics can be found in this paper, where the eigenfunctions of the one-dimensional harmonic oscillator and the radial eigenfunction of the harmonics oscillator in three dimensions are discussed.
Abstract: Discussion of the most important formulae and construction of plots for some functions of mathematical physics relevant to quantum mechanics. These functions are Hermite polynomials, Legendre polynomials and Legendre functions, spherical harmonics, Bessel functions and spherical Bessel functions and Laguerre polynomials. Directly related to some of these and also discussed are the eigenfunctions of the one-dimensional harmonic oscillator and the radial eigenfunctions of the harmonics oscillator in three dimensions and of the hydrogen atom.
276 citations
TL;DR: The analogy between acoustical and optical radiation forces of arbitrary-order helicoidal and ordinary Bessel beams is developed to gain insight into negative radiation forces.
Abstract: Various researchers have predicted situations where the acoustical or optical radiation force on a sphere centered on a Bessel beam is opposite the direction of beam propagation. We develop the analogy between acoustical and optical radiation forces of arbitrary-order helicoidal and ordinary Bessel beams to gain insight into negative radiation forces. The radiation force is expressed in terms of the asymmetry of the scattered field, the scattered power, the absorbed power, and the conic angle of the Bessel beam and is related to the partial-wave coefficients for the scattering. Negative forces only occur when the scattering into the backward hemisphere is suppressed relative to the scattering into the forward hemisphere. Absorbed power degrades negative radiation forces.
123 citations
TL;DR: Although the intensity of a vector-vortex BG beam with a topological charge n=1 is nonzero along the axis in the focal plane, the beams with n≠1 show discrete multiple spots which can be useful for optical trapping.
Abstract: We demonstrate that the amplitude of vector-vortex beams has a Bessel–Gauss (BG) distribution through a rigorous vector electromagnetic analysis. We also investigate the intensity profiles in the focal plane of vector-vortex beams that are focused by a high numerical-aperture lens obeying the Helmholtz condition. Although the intensity of a vector-vortex BG beam with a topological charge n=1 is nonzero along the axis in the focal plane, the beams with n≠1 show discrete multiple spots which can be useful for optical trapping.
122 citations
TL;DR: For non-Hermitian Hamiltonians with an isolated degeneracy, a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of Bessel functions as mentioned in this paper.
Abstract: For non-Hermitian Hamiltonians with an isolated degeneracy (‘exceptional point’), a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of Bessel functions. Floquet solutions, returning exactly to their initial states (up to a constant) are found, as well as exact expressions for the adiabatic multipliers when the evolving states are represented as a superposition of eigenstates of the instantaneous Hamiltonian. Adiabatically (i.e. for slow cycles), the multipliers of exponentially subdominant eigenstates can vary wildly, unlike those driven by Hermitian operators, which change little. These variations are explained as an example of the Stokes phenomenon of asymptotics. Improved (superadiabatic) approximations tame the variations of the multipliers but do not eliminate them.
111 citations
[...]
TL;DR: The free Bessel law as discussed by the authors is related to the free Poisson law via the formulae and the Bessel norm. But it is not related to any real probability measure.
Abstract: We introduce and study a remarkable family of real probability measures that we call free Bessel laws. These are related to the free Poisson law via the formulae and . Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.
109 citations
TL;DR: In this paper, a Clenshaw-Curtis-Filon-type method for approximating highly oscillatory Bessel transforms is proposed, which is based on a special Hermite interpolation polynomial at the CCC points that can be efficiently evaluated using O(N log N) operations.
Abstract: We consider a Clenshaw-Curtis-Filon-type method for highly oscillatory Bessel transforms. It is based on a special Hermite interpolation polynomial at the Clenshaw-Curtis points that can be efficiently evaluated using O(N log N) operations, where N is the number of Clenshaw-Curtis points in the interval of integration. Moreover, we derive corresponding error bounds in terms of the frequency and the approximating polynomial. We then show that this method yields an efficient numerical approximation scheme for a class of Volterra integral equations containing highly oscillatory Bessel kernels (a problem for which standard numerical methods fail), and it also allows the study of the asymptotics of the solutions. Numerical examples are used to illustrate the efficiency and accuracy of the Clenshaw-Curtis-Filon-type method for approximating these highly oscillatory integrals and integral equations.
87 citations
TL;DR: In this article, the singular Weyl-Titchmarsh m-function of perturbed spherical Schrodinger operators (also known as Bessel operators) was investigated under the assumption that the perturbation q(x) satisfies xqx ∈ L1(0, 1).
85 citations
TL;DR: In this paper, the helical flows of second grade fluid between two infinite coaxial circular cylinders are considered and the exact analytic solutions, obtained by employing the Laplace and finite Hankel transforms and presented in series form in term of usual Bessel functions of first and second kind, satisfy both the governing equations and all imposed initial and boundary conditions.
85 citations
TL;DR: In this article, the authors used theta correspondence to analyze local and global Bessel models for GSp 4 proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon factors in the local case, and non-vanishing of central critical L-value in the global case.
Abstract: Methods of theta correspondence are used to analyze local and global Bessel models for GSp 4 proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon factors in the local case, and the non-vanishing of central critical L-value in the global case.
84 citations
TL;DR: In this article, exact solutions of different forms of wave equation in D-dimensional fractional space are provided, which describe the phenomenon of electromagnetic wave propagation in fractional spaces, which can effectively describe the wave propagation phenomenon in fractal media.
Abstract: The wave equation has very important role in many areas of physics. It has a fundamental meaning in classical as well as quantum field theory. With this view, one is strongly motivated to discuss solutions of the wave equation in all possible situations. The wave equation in fractional space can effectively describe the wave propagation phenomenon in fractal media. In this chapter, exact solutions of different forms of wave equation in \(D\)-dimensional fractional space are provided, which describe the phenomenon of electromagnetic wave propagation in fractional space.
81 citations
TL;DR: In this article, the concept of weighted asymmetries is revisited for semi-inclusive deep inelastic scattering, where convolutions of transverse momentum dependent parton distribution functions and fragmentation functions become simple products.
Abstract: The concept of weighted asymmetries is revisited for semi-inclusive deep inelastic scattering. We consider the cross section in Fourier space, conjugate to the outgoing hadron's transverse momentum, where convolutions of transverse momentum dependent parton distribution functions and fragmentation functions become simple products. Individual asymmetric terms in the cross section can be projected out by means of a generalized set of weights involving Bessel functions. Advantages of employing these Bessel weights are that they suppress (divergent) contributions from high transverse momentum and that soft factors cancel in (Bessel-) weighted asymmetries. Also, the resulting compact expressions immediately connect to previous work on evolution equations for transverse momentum dependent parton distribution and fragmentation functions and to quantities accessible in lattice QCD. Bessel weighted asymmetries are thus model independent observables that augment the description and our understanding of correlations of spin and momentum in nucleon structure.
TL;DR: The property of self-healing at the focal plane for both scalar and vector Bessel-Gauss (BG) beams is investigated in the tight focusing condition and recovery of not only intensity but also polarization distribution is observed for an obstructed vector BG beam.
Abstract: The property of self-healing at the focal plane for both scalar and vector Bessel–Gauss (BG) beams is investigated in the tight focusing condition. For the BG beam, which is partially obstructed at the pupil plane, the spatial intensity distribution at the focal plane is well recovered. Furthermore, recovery of not only intensity but also polarization distribution is observed for an obstructed vector BG beam. This self-healing effect for both the intensity and polarization components is recognized even when the half of the beam is obstructed by a semicircular obstacle. The effect of the size of the obstacle on recovery of polarization and intensity distribution is studied. The role of the beam size at the pupil plane is also discussed.
TL;DR: In this article, the radial components of the electric and magnetic scattering fields are expressed using partial wave series involving the beam-shape coefficients and the scattering coefficients of the sphere, and the results are of some the scattering of importance in applications involving EM HOBVBs by a spherical object.
Abstract: This study investigates the arbitrary scattering of an unpolarized electromagnetic (EM) high-order Bessel vortex (helicoidal) beam (HOBVB) by a homogeneous water sphere in air. The radial components of the electric and magnetic scattering fields are expressed using partial wave series involving the beam-shape coefficients and the scattering coefficients of the sphere. The magnitude of the 3D electric and magnetic scattering directivity plots in the far-field region are evaluated using a numerical integration procedure for cases where the sphere is centered on the beam's axis and shifted off-axially with particular emphasis on the half-conical angle of the wave number components and the order (or helicity) of the beam. Some properties of the EM scattering of an HOBVB by the water sphere are discussed. The results are of some the scattering of importance in applications involving EM HOBVBs by a spherical object.
TL;DR: Theoretical expressions for the incident and scattered field from a rigid immovable sphere are derived in this paper, where the near and far-field acoustic scattering fields are expressed using partial wave series involving the spherical harmonics, the scattering coefficients of the sphere, the half-conical angle of the wave number components of the beam, its order and the beam shape coefficients.
TL;DR: In this article, the seismic response of inhomogeneous soil deposits is explored analytically by means of one-dimensional viscoelastic wave propagation theory, with the excitation defined in terms of vertically propagating harmonic S waves imposed at the base of the system.
TL;DR: It is shown that this fast and efficient new numerical description of zero-order Bessel beams can be used with advantage, for example, in the analysis of optical forces in optical trapping systems for arbitrary optical regimes.
Abstract: Ordinary Bessel beams are described in terms of the generalized Lorenz-Mie theory (GLMT) by adopting, for what is to our knowledge the first time in the literature, the integral localized approximation for computing their beam shape coefficients (BSCs) in the expansion of the electromagnetic fields. Numerical results reveal that the beam shape coefficients calculated in this way can adequately describe a zero-order Bessel beam with insignificant difference when compared to other relative time-consuming methods involving numerical integration over the spherical coordinates of the GLMT coordinate system, or quadratures. We show that this fast and efficient new numerical description of zero-order Bessel beams can be used with advantage, for example, in the analysis of optical forces in optical trapping systems for arbitrary optical regimes.
TL;DR: The main object of as discussed by the authors is to give sufficient conditions for certain families of integral operators, defined here by means of the normalized form of the generalized Bessel functions, to be univalent in the open unit disk.
Abstract: The main object of this paper is to give sufficient conditions for certain families of integral operators, which are defined here by means of the normalized form of the generalized Bessel functions, to be univalent in the open unit disk. In particular cases, we find the corresponding simpler conditions for integral operators involving the Bessel function, the modified Bessel function and the spherical Bessel function.
TL;DR: In this paper, the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal, was established, and Weinberger's method from the corresponding free membrane problem, taking the fundamental modes of the unit ball as trial functions.
Abstract: We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given τ > 0, the free plate eigenvalues ω and eigenfunctions u are determined by the equation ΔΔu − τΔu = ωu together with certain natural boundary conditions. The boundary conditions are complicated but arise naturally from the plate Rayleigh quotient, which contains a Hessian squared term |D
2
u|2. We adapt Weinberger’s method from the corresponding free membrane problem, taking the fundamental modes of the unit ball as trial functions. These solutions are a linear combination of Bessel and modified Bessel functions.
TL;DR: In this paper, the authors consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other, in which if −1<ν<0 the origin is assumed to be reflecting.
Abstract: We consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other, in which if −1<ν<0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function J
ν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.
TL;DR: In this paper, the authors considered the double scaling limit for a model of n non-intersecting squared Bessel processes in the confluent case, where all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at t = 1 at x = 0.
Abstract: We consider the double scaling limit for a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n → ∞ that intersects the hard edge at x = 0 at a critical time t = t*. In a previous paper, the scaling limits for the positions of the paths at time t ≠ t* were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as n → ∞ of the correlation kernel at critical time t* and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix.
TL;DR: A practical matrix method, which is based on collocation points, is presented to find approximate solutions of high-order linear Volterra integro-differential equations (VIDEs) under the mixed conditions in terms of Bessel polynomials.
Abstract: In this study, a practical matrix method, which is based on collocation points, is presented to find approximate solutions of high-order linear Volterra integro-differential equations (VIDEs) under the mixed conditions in terms of Bessel polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on the computer using a program written in MATLAB v7.6.0 (R2008a).
TL;DR: The method allows >99% conversion efficiency, wide or narrow conversion bandwidth, and accurate control of the number of rings in the beam, and is a versatile tool for tuning depth of focus out of fiber tips, with potential uses in endoscopic microscopy.
Abstract: We experimentally generate n=0 Bessel beams via higher-order cladding mode excitation with a long period fiber grating. Our method allows >99% conversion efficiency, wide or narrow conversion bandwidth, and accurate control of the number of rings in the beam. This latter property is equivalent to tuning the beam cone angle and allows for control of width and propagation distance of the center spot. We generate Bessel-like beams from LP0,5 to LP0,15 cladding modes and measure their propagation-invariant characteristics as a function of mode order, which match numerical simulations and a simple geometric model. This yields a versatile tool for tuning depth of focus out of fiber tips, with potential uses in endoscopic microscopy.
TL;DR: In this paper, a novel method is devised for Bessel beam generation in the microwave regime, where the beam is decomposed in terms of a number of guided transverse electric modes of a metallic waveguide.
Abstract: A novel method is devised for Bessel beams generation in the microwave regime. The beam is decomposed in terms of a number of guided transverse electric modes of a metallic waveguide. Modal expansion coefficients are computed from the modal power orthogonality relation. Excitation is achieved by means of a number of inserted coaxial loop antennas, whose currents are calculated from the excitation coefficients of the guided modes. The efficiency of the method is evaluated and its feasibility is discussed. Obtained results can be utilized to practically realize microwave Bessel beam launchers.
TL;DR: Stemming from Maxwell's vector equations and the Lorenz gauge condition, a full vector wave analysis for the electric and magnetic fields is presented, of particular importance in the study of EM wave scattering of a high-order Bessel vortex beam of fractional type α by particles.
Abstract: The scalar wave theory of nondiffracting electromagnetic (EM) high-order Bessel vortex beams of fractional type α has been recently explored, and their novel features and promising applications have been revealed However, complete characterization of the properties for this new type of beam requires a vector analysis to determine the fields' components in space because scalar wave theory is inadequate to describe such beams, especially when the central spot is comparable to the wavelength (k(r)/k≈1, where k(r) is the radial component of the wavenumber k) Stemming from Maxwell's vector equations and the Lorenz gauge condition, a full vector wave analysis for the electric and magnetic fields is presented The results are of particular importance in the study of EM wave scattering of a high-order Bessel vortex beam of fractional type α by particles
TL;DR: In this article, a generalized Hardy-Ramanujan "circle method" was proposed to derive asymptotic series expansions for the products of modular forms and mock theta functions.
Abstract: We develop a generalized version of the Hardy-Ramanujan "circle method" in order to derive asymptotic series expansions for the products of modular forms and mock theta functions. Classical asymptotic methods (including the circle method) do not work in this situation because such products are not modular, and in fact, the "error integrals" that occur in the transformations of the mock theta functions can (and often do) make a significant contribution to the asymptotic series. The resulting series include principal part integrals of Bessel functions, whereby the main asymptotic term can also be identified. To illustrate the application of our method, we calculate the asymptotic series expansion for the number of partitions without sequences. Andrews showed that the generating function for such partitions is the product of the third order mock theta function $\chi$ and a (modular) infinite product series. The resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the mock theta function component appear in the exponential main term.
TL;DR: In this article, it has been shown that the invisibility is not quite exact, in amplitude or phase, in the case of optical potentials of the form v = v(x), where v is the length of the medium in the direction of propagation.
Abstract: Propagation of light through media with a complex refractive index in which gain and loss are engineered to be $PT$ symmetric has many remarkable features. In particular the usual unitarity relations are not satisfied, so that the reflection coefficients can be greater than one, and in general are not the same for left or right incidence. Within the class of optical potentials of the form $v(x)=v_1\cos(2\beta x)+iv_2\sin(2\beta x)$ the case $v_2=v_1$ is of particular interest, as it lies on the boundary of $PT$-symmetry breaking. It has been shown in a recent paper by Lin et al. that in this case one has the property of "unidirectional invisibility", while for propagation in the other direction there is a greatly enhanced reflection coefficient proportional to $L^2$, where $L$ is the length of the medium in the direction of propagation.
For this potential we show how analytic expressions can be obtained for the various transmission and reflection coefficients, which are expressed in a very succinct form in terms of modified Bessel functions. While our numerical results agree very well with those of Lin et al. we find that the invisibility is not quite exact, in amplitude or phase. As a test of our formulas we show that they identically satisfy a modified version of unitarity appropriate for $PT$-symmetric potentials. We also examine how the enhanced transmission comes about for a wave-packet, as opposed to a plane wave, finding that the enhancement now arises through an increase, of $O(L)$, in the pulse length, rather than the amplitude.
TL;DR: For functions that take values in the Clifford algebra, this article studied the Clifford-Fourier transform on R(m) defined with a kernel function K(x, y) := e(i pi/2) Gamma Ye(-i){x,y}, replacing the kernel of the ordinary Fourier transform.
Abstract: For functions that take values in the Clifford algebra, we study the Clifford-Fourier transform on R(m) defined with a kernel function K(x, y) := e(i pi/2) Gamma Ye(-i){x,y}, replacing the kernel e(i ) of the ordinary Fourier transform, where Gamma(y) :=- Sigma(j
27 Jun 2011
TL;DR: In this article, the Bessel equation x2y′′ + xy′ + (x − n)y = 0 has two linearly independent solutions for a fixed value of n.
Abstract: Recall the Bessel equation x2y′′ + xy′ + (x − n)y = 0. For a fixed value of n, this equation has two linearly independent solutions. One of these solutions, that can be obtained using Frobenius’ method, is called a Bessel function of the first kind, and is denoted by Jn(x). This solution is regular at x = 0. The second solution, that is singular at x = 0, is called a Bessel function of the second kind, and is denoted by Yn(x).
TL;DR: In this paper, theoretical calculations as well as numerical simulations are performed for the time-averaged acoustic force and torque on a rigid cylinder of arbitrary size in a fluid with low viscosity, i.e., the acoustic boundary layer is thin compared to the cylinder radius.
Abstract: In this paper, theoretical calculations as well as numerical simulations are performed for the time-averaged acoustic force and torque on a rigid cylinder of arbitrary size in a fluid with low viscosity, i.e., the acoustic boundary layer is thin compared to the cylinder radius. An exact analytical solution and its approximation are proposed in the form of an infinite series including Bessel functions. These solutions can be evaluated easily by a mathematical software package such as mathematica and matlab. Three types of incident waves, plane traveling wave, plane standing wave, and dual orthogonal standing waves, are investigated in detail. It is found that for a small particle, the viscous effects for an incident standing wave may be neglected but those for an incident traveling wave are notable. A nonzero viscous torque is experienced by the rigid cylinder when subjected to dual orthogonal standing waves with a phase shift even when the cylinder is located at equilibrium positions without imposed acoustic forces. Furthermore, numerical simulations are carried out based on the FVM algorithm to verify the proposed theoretical formulas. The theoretical results and the numerical ones agree with each other very well in all the cases considered.
Posted Content•
TL;DR: In this article, Brackx et al. studied the Fourier transform integral kernel of hypercomplex signals and their Fourier transforms from general principles, using four different yet equivalent definitions of the classical Fourier Transform.
Abstract: Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see [F. Brackx et al., The Clifford-Fourier transform. J. Fourier Anal. Appl. 11 (2005), 669--681]). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.