Showing papers on "Bessel function published in 2005"
TL;DR: In this article, the theoretical foundation of the Bessel beam is described and various experiments that make use of Bessel beams are discussed: these cover a wide range of fields including non-linear optics, where the intense central core of the bessel beam has attracted interest; short pulse non-diffracting fields; atom optics, and optical manipulation where the reconstruction properties of the beam enable new effects to be observed that cannot be seen with Gaussian beams.
Abstract: Diffraction is a cornerstone of optical physics and has implications for the design of all optical systems. The paper discusses the so-called 'non-diffracting' light field, commonly known as the Bessel beam. Approximations to such beams can be experimentally realized using a range of different means. The theoretical foundation of these beams is described and then various experiments that make use of Bessel beams are discussed: these cover a wide range of fields including non-linear optics, where the intense central core of the Bessel beam has attracted interest; short pulse non-diffracting fields; atom optics, where the narrow non-diffracting features of the Bessel beam are able to act as atomic guides and atomic confinement devices and optical manipulation, where the reconstruction properties of the beam enable new effects to be observed that cannot be seen with Gaussian beams. The intensity profile of the Bessel beam may offer routes to investigating statistical physics as well as new techniques for the...
1,173 citations
TL;DR: The model applies to Bose-Einstein condensates and to optical media with saturable nonlinearity, suggesting new ways of making stable three-dimensional solitons and "light bullets" of an arbitrary size.
Abstract: We investigate the existence and stability of three-dimensional solitons supported by cylindrical Bessel lattices in self-focusing media. If the lattice strength exceeds a threshold value, we show numerically, and using the variational approximation, that the solitons are stable within one or two intervals of values of their norm. In the latter case, the Hamiltonian versus norm diagram has a swallowtail shape with three cuspidal points. The model applies to Bose-Einstein condensates and to optical media with saturable nonlinearity, suggesting new ways of making stable three-dimensional solitons and "light bullets" of an arbitrary size.
103 citations
TL;DR: In this article, Bessel beams are studied within the general framework of quantum optics and the two modes of the electromagnetic field are quantized and the basic dynamical operators are identified.
Abstract: Bessel beams are studied within the general framework of quantum optics. The two modes of the electromagnetic field are quantized and the basic dynamical operators are identified. As we show explicitly, the operators that are usually associated with linear momentum, orbital angular momentum, and spin do not satisfy the algebra of the translation and rotation group. Nevertheless, we identify some components of these operators that represent observable quantities in an appropriate basis, thus characterizing the quantum numbers of Bessel photons. Some physical consequences of these results are discussed.
96 citations
TL;DR: In this article, a rigorous classical analytic frequency domain model of con?ned optical wave propagation along 2D bent slab waveguides and curved dielectric interfaces is investigated, based on a piecewise ansatz for bend mode profiles in terms of Bessel and Hankel functions.
Abstract: A rigorous classical analytic frequency domain model of con?ned optical wave propagation along 2D bent slab waveguides and curved dielectric interfaces is investigated, based on a piecewise ansatz for bend mode profiles in terms of Bessel and Hankel functions This approach provides a clear picture of the behaviour of bend modes, concerning their decay for large radial arguments or effects of varying bend radius Fast and accurate routines are required to evaluate Bessel functions with large complex orders and large arguments Our implementation enabled detailed studies of bent waveguide properties, including higher order bend modes and whispering gallery modes, their interference patterns, and issues related to bend mode normalization and orthogonality properties
95 citations
TL;DR: In this article, the authors present a new method based on an eigenvalue problem for a matrix operator equivalent to that of the integral operator, which gives the values of these functions on the entire real line and is computationally more efficient.
76 citations
Book•
28 Dec 2005
TL;DR: The Euler Gamma Function Integral Solutions Expansion in Basis Functions Airy Phase Integral Methods II Bessel Weber-Hermite Whittaker and Watson Inhomogeneous Differential Equations The Riemann Zeta Function Boundary Layer Problems as mentioned in this paper.
Abstract: Dominant Balance Exact Solutions Complex Variables Local Approximate Solutions Phase Integral Methods I Perturbation Theory Asymptotic Evaluation of Integrals The Euler Gamma Function Integral Solutions Expansion in Basis Functions Airy Phase Integral Methods II Bessel Weber-Hermite Whittaker and Watson Inhomogeneous Differential Equations The Riemann Zeta Function Boundary Layer Problems.
59 citations
TL;DR: In this paper, the Faber-Krahn inequality was used for the uniqueness of the inverse scattering problem in two-dimensional acoustics for one incident plane wave and one wavenumber.
Abstract: In this paper, the problem of uniqueness concerning the inverse scattering problem in two-dimensional acoustics for one incident plane wave and one wavenumber is considered. Using the fact that the optimal lower estimate for the eigenvalues of the Laplacian for a domain is given by the Faber–Krahn inequality, which relates the area of the domain to the first eigenvalue of a disc of equal area, it is proved that the uniqueness holds under the restriction that the possible scatterers do not deviate 'too much' in area. Also an improvement of the results due to Colton and Sleeman (1983 IMA J. Appl. Math. 31 253–9) is presented, based on the a priori information that the unknown scatterers lie inside a given ball and that the far field is known for a finite number of incident plane waves. The main advantage of this work is that it provides uniqueness for the half number of the needed incoming waves in Colton and Sleeman (1983 IMA J. Appl. Math. 31 253–9). For the case of one incoming plane wave uniqueness is satisfied if the scatterers are contained in a ball of radius R such that kR < t10 4.4939, where t10 is the first root of the spherical Bessel function of first order j1(x). The result of local uniqueness is applied to a class of star-shaped scatterers which are smooth perturbations of discs with common centre in for one incident plane-wave direction. Numerical implementations are presented for smooth perturbations of discs.
58 citations
TL;DR: In this paper, the authors examined the creeping wave propagation wavenumbers, modal impedance, and field behavior on a dielectric coated circular cylinder and compared the cylinder's pole waves with the leaky waves and surface waves that occur on a flat, grounded dielectoric slab, to understand how a coated cylinder behaves like a flat slab when the cylinder radius is large.
Abstract: In this paper, we examine the creeping wave propagation wavenumbers, modal impedance, and field behavior on a dielectric coated circular cylinder. The physical interpretation is assisted by comparing the cylinder's pole waves with the leaky waves and surface waves that occur on a flat, grounded dielectric slab. The propagation wavenumbers and modal impedance are computed in the complex wavenumber plane. The cylinder propagation wavenumbers come from a transcendental equation involving Hankel functions, which are entire functions of complex order, whereas for the slab, branch-point singularities are present. This difference is examined, so that one can better understand how a coated cylinder behaves like a flat slab, when the cylinder radius is large. It is found that for the cylinder, the Stokes line for the asymptotic expansion of the Hankel function plays a role that is similar to the planar slab branch cut.
57 citations
TL;DR: In this article, the authors considered the planar random motion of a particle that moves with constant finite speed c and changes its direction 0 with uniform law in [0, 27r] and derived the explicit probability law f(x, y, t) of (X(t), Y(t)), t > 0.
Abstract: We consider the planar random motion of a particle that moves with constant finite speed c and, at Poisson-distributed times, changes its direction 0 with uniform law in [0, 27r). This model represents the natural two-dimensional counterpart of the wellknown Goldstein-Kac telegraph process. For the particle's position (X (t), Y(t)), t > 0, we obtain the explicit conditional distribution when the number of changes of direction is fixed. From this, we derive the explicit probability law f(x, y, t) of (X(t), Y(t)) and show that the density p(x, y, t) of its absolutely continuous component is the fundamental solution to the planar wave equation with damping. We also show that, under the usual Kac condition on the velocity c and the intensity X of the Poisson process, the density p tends to the transition density of planar Brownian motion. Some discussions concerning the probabilistic structure of wave diffusion with damping are presented and some applications of the model are sketched.
56 citations
TL;DR: A novel method for generating both propagating and evanescent Bessel beams is proposed using a pair of distributed Bragg reflectors with a resonant point source on one side of the system and transmission of a resonan point source through a thin film.
Abstract: We propose a novel method for generating both propagating and evanescent Bessel beams. To generate propagating Bessel beams we propose using a pair of distributed Bragg reflectors (DBRs) with a resonant point source on one side of the system. Those modes that couple with the localized modes supported by the DBR system will be selectively transmitted. This is used to produce a single narrow band of transmission in κ space that, combined with the circular symmetry of the system, yields a propagating Bessel beam. We present numerical simulations showing that a propagating Bessel beam with central spot size of ∼0.5λ0 can be maintained for a distance in excess of 3000λ0. To generate evanescent Bessel beams we propose using transmission of a resonant point source through a thin film. A transmission resonance is produced as a result of the multiple scattering occurring between the interfaces. This narrow resonance combined with the circular symmetry of the system corresponds to an evanescent Bessel beam. Because propagating modes are also transmitted, although the evanescent transmission resonance is many orders of magnitude greater than the transmission for the propagating modes, within a certain distance the propagating modes swamp the exponentially decaying evanescent ones. Thus there is only a certain regime in which evanescent Bessel beams dominate. However, within this regime the central spot size of the beam can be made significantly smaller than the wavelength of light used. Thus evanescent Bessel beams may have technical application, in high-density recording for example. We present numerical simulations showing that with a simple glass thin film an evanescent Bessel beam with central spot size of ∼0.34λ0 can be maintained for a distance of 0.14λ0. By choice of different material parameters, the central spot size can be made smaller still.
51 citations
TL;DR: Basic properties of quiescent and rotating multipole-mode solitons supported by axially symmetric Bessel lattices in a medium with defocusing cubic nonlinearity are studied.
Abstract: We study basic properties of quiescent and rotating multipole-mode solitons supported by axially symmetric Bessel lattices in a medium with defocusing cubic nonlinearity. The solitons can be found in different rings of the lattice and are stable when the propagation constant exceeds the critical value, provided that the lattice is deep enough. In a high-power limit the multipole-mode solitons feature a multi-ring structure.
TL;DR: In this paper, the local spectral theory of the relative trace formula for the Waldspurger correspondence has been studied and several regularity theorems for Bessel distributions have been proved.
Abstract: We prove certain identities between Bessel functions attached to irreducible unitary representations of PGL2(R) and Bessel functions attached to irreducible unitary representations of the double cover of SL2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. CONTENTS
TL;DR: In this paper, Havelock's type of expansion theorems are utilized to derive analytical solutions for the radiation or scattering of oblique water waves by a fully extended porous barrier in both the cases of finite and infinite depths of water in two-layer fluid with constant densities.
Abstract: Havelock’s type of expansion theorems, for an integrable function having a single discontinuity point in the domain where it is defined, are utilized to derive analytical solutions for the radiation or scattering of oblique water waves by a fully extended porous barrier in both the cases of finite and infinite depths of water in two-layer fluid with constant densities. Also, complete analytical solutions are obtained for the boundary-value problems dealing with the generation or scattering of axi-symmetric water waves by a system of permeable and impermeable co-axial cylinders. Various results concerning the generation and reflection of the axisymmetric surface or interfacial waves are derived in terms of Bessel functions. The resonance conditions within the trapped region are obtained in various cases. Further, expansions for multipole-line-source oblique-wave potentials are derived for both the cases of finite and infinite depth depending on the existence of the source point in a two-layered fluid.
04 May 2005
TL;DR: The truncated region eigenfunction expansion (Tthis paper ) method was used in this paper to solve the axisymmetric, time harmonic boundary value problem of a coil above a coaxial hole in a plate.
Abstract: A number of complex problems in eddy current nondestructive evaluation have been solved recently using the truncated region eigenfunction expansion method. The solution of a boundary value problem is commonly obtained by separation of variables. In a particular unbounded coordinate, the solution is usually expressed as an integral form such as a Fourier or Bessel integral. However, by truncating the domain of the problem, a modified solution is obtained in the form of a series expansion instead of an integral. Although one achieves a gain in computation efficiency in this way, the most significant advantage of the approach is the ability to match interface conditions across several boundaries simultaneously and thus obtain analytical solutions to complex problems. We illustrate the approach by solving the axisymmetric, time harmonic boundary value problem of a coil above a coaxial hole in a plate.
TL;DR: In this article, the Schrodinger equation for inverse fourth-and sixth-power potentials reduces to peculiar cases of the double-confluent Heun equation and its Ince's limit, respectively.
Abstract: We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z, while the other is given by a series of modified Bessel functions and converges for ∣z∣>∣z0∣, where z0 denotes a regular singularity. For short, the preceding limit is called Ince’s limit after Ince who have used the same procedure to get the Mathieu equations from the Whittaker-Hill ones. We find as well that, when z0 tends to zero, the Ince limit of the generalized spheroidal wave equation turns out to be the Ince limit of a double-confluent Heun equation, for which solutions are provided. Finally, we show that the Schrodinger equation for inverse fourth- and sixth-power potentials reduces to peculiar cases of the double-confluent Heun equation and its Ince’s limit, respectively.
TL;DR: Based on the relationship between separable solutions of the Helmholtz equation, the authors expanded the fields amplitude associated with Mathieu beams in terms of Bessel beams and derived an approximated analytical expression of the amplitude distribution of Mathieu beam at the output plane of any apertured paraxial ABCD optical system.
TL;DR: In this paper, the authors used three orthogonal polynomial representations, including Hermite polynomials, Laguerre function and Bessel function, to generate a wideband and temporal response of three-dimensional composite structures.
Abstract: The objective of this paper is to generate a wideband and temporal response of three-dimensional composite structures by using a hybrid method that involves generation of early time and low-frequency information. The data in these two separate time and frequency domains are mutually complementary and contain all the necessary information for a sufficient record length. Utilizing a set of orthogonal polynomials, the time domain signal (be it the electric or the magnetic currents or the near/far scattered electromagnetic field) could be expressed in an efficient way as well as the corresponding frequency domain responses. The available data is simultaneously extrapolated in both domains. Computational load for electromagnetic analysis in either domain, time or frequency, can be thus significantly reduced. Three orthogonal polynomial representations including Hermite polynomial, Laguerre function and Bessel function are used in this approach. However, the performance of this new method is sensitive to two important parameters-the scaling factor l/sub 1/ and the expansion order N. It is therefore important to find the optimal parameters to achieve the best performance. A comparison is presented to illustrate that for the classes of problems dealt with, the choice of the Laguerre polynomials has the best performance as illustrated by a typical scattering example from a dielectric hemisphere.
TL;DR: It is shown that for i- DWNTs the coupling is negligible between lowest energy subbands, but it becomes important as the higher subbands become populated, and the elastic mean-free path of i-DWNTs is reduced for increasing energy.
Abstract: We study the asymptotic dynamics of a driven spin-boson system where the environment is formed by a broadened localized mode. Upon exploiting an exact mapping, an equivalent formulation of the problem in terms of a quantum two-state system (qubit) coupled to a harmonic oscillator which is itself Ohmically damped, is found. We calculate the asymptotic population difference of the two states in two complementary parameter regimes. For weak damping and low temperature, a perturbative Floquet-Born-Markovian master equation for the qubit-oscillator system can be solved. We find multi-photon resonances corresponding to transitions in the coupled quantum system and calculate their line-shape analytically. In the complementary parameter regime of strong damping and/or high temperatures, non-perturbative real-time path integral techniques yield analytic results for the resonance line shape. In both regimes, we find very good agreement with exact results obtained from a numerical real-time path-integral approach. Finally, we show for the case of strong detuning between qubit and oscillator that the width of the n-photon resonance scales with the nth Bessel function of the driving strength in the weak-damping regime.
TL;DR: In this paper, a theoretical model for the quantized spectrum of spin modes frequencies in cylindrical magnetic dots of radius ranging from the nanometric to the submicrometric scale in the vortex ground state at zero applied magnetic field is presented.
Abstract: A theoretical model for the calculation of the quantized spectrum of spin modes frequencies in cylindrical magnetic dots of radius ranging from the nanometric to the submicrometric scale in the vortex ground state at zero applied magnetic field is presented. The effective field includes both the surface and the volume dynamic magnetostatic and exchange fields. We also show that the core energy affects the spin dynamics. The modes at lower frequencies present as radial eigenvectors Bessel functions of high orders $(m\ensuremath{\geqslant}1)$, while the axially symmetric modes at higher frequencies correspond to zero order Bessel functions.
TL;DR: In this article, the authors discuss the Gaussian beam mode analysis of Bessel beams, eigen-solutions of the wave-equation in cylindrical polar coordinates which neither change form nor spread out as they propagate.
TL;DR: Based on the generalized Collins diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expressions of Bessel-Gaussian beams and QBG beams passing through a paraxial ABCD optical system with an annual aperture are derived as mentioned in this paper.
TL;DR: In this paper, an integro-differential equation is derived for the diffraction of incident surface waves by a floating elastic circular plate, and an algorithm of its numerical solution is proposed.
TL;DR: While the stable Bessel resonator retains a Gaussian radial modulation on the Bessel rings, the unstable configuration exhibits a more uniform amplitude modulation that produces output profiles more similar to ideal Bessel beams.
Abstract: A rigorous analysis of the unstable Bessel resonator with convex output coupler is presented. The Huygens–Fresnel self-consistency equation is solved to extract the first eigenmodes and eigenvalues of the cavity, taking into account the finite apertures of the mirrors. Attention was directed to the dependence of the output transverse profiles; the losses; and the modal-frequency changes on the curvature of the output coupler, the cavity length, and the angle of the axicon. Our analysis revealed that while the stable Bessel resonator retains a Gaussian radial modulation on the Bessel rings, the unstable configuration exhibits a more uniform amplitude modulation that produces output profiles more similar to ideal Bessel beams. The unstable cavity also possesses higher-mode discrimination in favor of the fundamental mode than does the stable configuration.
TL;DR: In this article, the existence of vector Helmholtz-Gauss (vHzG) and vector Laplace Gauss (λG) beam solutions of the Maxwell equations in the paraxial approximation was shown.
Abstract: We demonstrate the existence of vector Helmholtz-Gauss (vHzG) and vector Laplace-Gauss beams that constitute two general families of localized vector beam solutions of the Maxwell equations in the paraxial approximation. The electromagnetic components are determined starting from the scalar solutions of the two-dimensional Helmholtz and Laplace equations, respectively. Special cases of the vHzG beams are TE and TM Gaussian vector beams, nondiffracting vector Bessel beams, polarized Bessel-Gauss beams, modes in cylindrical waveguides and cavities, and scalar Helmholtz-Gauss beams. The general expression of the vHzG beams can be used straightforwardly to obtain vector Mathieu-Gauss and vector parabolic-Gauss beams, which to our knowledge have not yet been reported.
TL;DR: In this paper, the propagation and interaction of hyperelastic cylindrical waves are studied by means of the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships.
Abstract: The propagation and interaction of hyperelastic cylindrical waves are studied. Nonlinearity is introduced by means of the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. To analyze wave propagation, an asymptotic representation of the Hankel function of the first order and first kind is used. The second-order analytical solution of the nonlinear wave equation is similar to that for a plane longitudinal wave and is the sum of the first and second harmonics, with the difference that the amplitudes of cylindrical harmonics decrease with the distance traveled by the wave. A primary computer analysis of the distortion of the initial wave profile is carried out for six classical hyperelastic materials. The transformation of the first harmonic of a cylindrical wave into the second one is demonstrated numerically. Three ways of allowing for nonlinearities are compared
TL;DR: A differential theory for solving Maxwell equations in cylindrical coordinates, projecting them onto a Fourier-Bessel basis, and applications of such a method are presented, with a special emphasis on the near-field map inside a hole pierced in a plane metallic film.
Abstract: We present a differential theory for solving Maxwell equations in cylindrical coordinates, projecting them onto a Fourier–Bessel basis. Numerical calculations require the truncation of that basis, so that correct rules of factorization have to be used. The convergence of the method is studied for different cases of dielectric and metallic cylinders of finite length. Applications of such a method are presented, with a special emphasis on the near-field map inside a hole pierced in a plane metallic film.
TL;DR: In this article, the eigenfunction representation of the integrals of the Debye-Wolf formula in terms of Bessel and circular prolate spheroidal functions was obtained.
Abstract: The Debye–Wolf electromagnetic diffraction formula is now routinely used to describe focusing by high numerical aperture optical systems. In this paper we obtain the eigenfunction representation of the integrals of the Debye–Wolf formula in terms of Bessel and circular prolate spheroidal functions. This result offers considerable analytical simplification to the Debye–Wolf formula and it could also be used as a mathematical basis for its inversion. In addition, we show that numerical evaluation of the Debye–Wolf formula, based on the eigenfunction representation of its integrals, is faster and more efficient than direct numerical integration. Our work has applications in a large variety of areas, such as polarised light microscopy, point spread function engineering and micromachining.
TL;DR: Closed form expressions for the first derivatives with respect to the order of the Bessel functions J ν(z), Y ν, I ν (z), I ξ, K ξ(z) were obtained in this article with n = 0, 1, 2.
Abstract: Closed form expressions are obtained for the first derivatives with respect to the order of the Bessel functions J ν(z), Y ν(z), I ν(z), K ν(z); integral Bessel functions Ji ν(z), Yi ν(z), Ki ν(z); and Struve functions Hν(z), Lν(z) at ν = ±n, ν = ±n + 1/2, with n = 0, 1, 2….
TL;DR: This work addresses soliton spiraling in optical lattices induced by multiple coherent Bessel beams and shows that the dynamic nature of such lattices makes it possible for them to drag different soliton structures, setting them into rotation.
Abstract: We address soliton spiraling in optical lattices induced by multiple coherent Bessel beams and show that the dynamic nature of such lattices makes it possible for them to drag different soliton structures, setting them into rotation. We can control the rotation rate by varying the topological charges of lattice-inducing Bessel beams.
TL;DR: In this paper, the radiation from a rectangular waveguide with a perfectly conducting infinite flange is rigorously studied by using the method of the Kobayashi potential (KP), where the fields in the waveguide and half-space are expanded in terms of the wave-guide modes and the Weber-Schafheitlin discontinuous integrals, respectively.
Abstract: The radiation from a rectangular waveguide with a perfectly conducting infinite flange is rigorously studied by using the method of the Kobayashi potential (KP). The fields in the waveguide and half-space are expanded in terms of the waveguide modes and the Weber-Schafheitlin discontinuous integrals, respectively. Continuity of the tangential aperture fields yields matrix equations for the expansion coefficients and the matrix elements consist of double infinite integrals and double infinite series of Bessel functions, which are calculated efficiently by applying the asymptotic approximation of the Bessel function. Numerical results are presented for various physical quantities, such as the aperture admittance, reflection coefficient of the incident wave, and magnitudes of higher-order mode waves, as well as the far-radiation pattern and aperture fields. To verify the validity of our method, the results are compared with other methods and excellent agreement is obtained.