Showing papers on "Bessel function published in 2012"

Tables of Integral Transforms

Abstract: In this chapter, we provide a set of short tables of integral transforms of the functions that are either cited in the text or are in most common use in mathematical, physical, and engineering applications. For exhaustive lists of integral transforms, the reader is referred to Erdelyi et al. (Tables of Integral Transforms, Vols. 1 and 2, 1954), Campbell and Foster (Fourier Integrals for Practical Applications, 1948), Ditkin and Prudnikov (Integral Transforms and Operational Calculus, 1965), Doetsch (Introduction to the Theory and Applications of the Laplace Transformation, 1970), Marichev (1983), Debnath (1995), Debnath and Bhatta (Integral Transforms and Their Applications, 2nd edition, 2007), Oberhettinger (Tables of Bessel Transforms, 1972).

Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation.

Abstract: In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevwaves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions betweensolitarywavesandcnoidalwavesareobtainedthroughthelocalizationprocedureofnonlocalsymmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlev´ e II solutions.

Generation of Propagating Bessel Beams Using Leaky-Wave Modes

Abstract: The generation of Bessel beams using a leaky radial waveguide is presented. The radial waveguide consists of a capacitive sheet over a ground plane. It supports an azimuthally invariant leaky-wave mode whose normal electric-field component is a truncated, zeroth-order Bessel function. The annular spectrum and nondiffractive extent of the Bessel beam is clearly linked to the complex wavenumber of the leaky-wave mode. The fields inside the radial waveguide are derived using classical vector potential techniques. A vector approach is employed to avoid paraxial approximations of earlier works and the associated limitations on shaping the Bessel beam. Design rules are provided to synthesize a desired propagating Bessel beam. A simple coaxial feed is proposed for the radial waveguide and its input impedance is derived analytically. The analytical results are also validated numerically. The proposed structure and design procedure can be used for generating arbitrary zeroth-order propagating Bessel beams at microwave and millimeter-wave frequencies.

Generation of Propagating Bessel Beams Using Leaky-Wave Modes: Experimental Validation

Abstract: We present the experimental generation of Bessel beams using a leaky radial waveguide. The radial waveguide consists of a capacitive sheet over a ground plane. The capacitive sheet is composed of patch elements printed on both sides of a dielectric substrate. The radial waveguide is coaxially fed and supports an azimuthally invariant leaky-wave mode whose normal electric-field component is a truncated, zeroth-order Bessel function. Two prototypes are presented with the same propagation constant and lateral extent, but different attenuation constants. 2D electric field measurements and their respective Fourier transforms validate the operation of the prototypes as Bessel-beam launchers at two frequency bands. Cleaner patterns are achieved by the prototype with lower attenuation constant. The dual-band capability and associated frequency dependent resolution can be useful in near-field planar focusing systems. The proposed structure can be used for generating arbitrary zeroth-order propagating Bessel beams at microwave and millimeter-wave frequencies.

Effects of Topographic Amplification Induced by a U‐Shaped Canyon on Seismic Waves

Abstract: The series solution of wave functions for 2D scattering and diffraction of plane SH (shear horizontal) waves induced by a U‐shaped canyon is proposed herein to account for the topographic effect of such a canyon. The wave function expansion method has been frequently employed to study the topographic effect because it can reveal the physics of the wave scattering and can test the accuracy of other methods. Through a new domain decomposition strategy, the half‐space having a U‐shaped canyon is divided into three subregions. Hence, we defined three cylindrical coordinate systems. In each coordinate system, the wave field satisfying the Helmholtz equation was represented by means of the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then three wave fields are all represented in the same coordinate system using the Graf addition theorem. The unknown coefficients are solved by satisfying the continuity conditions of the auxiliary boundary and the traction‐free boundary conditions on the bottom of the canyon. To show the effects of symmetrical and nonsymmetrical U‐shaped canyons on the surface ground motion, a parametric analysis is carried out in the frequency domain. Surface and subsurface transient responses in the time domain demonstrate the phenomenon of wave propagating and scattering. It is found that a zone of amplification can obviously take place at the bottom of a U‐shaped canyon with nearly vertical walls.

Analytic results for a PT-symmetric optical structure

Abstract: Propagation of light through a medium 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 1, and in general are not the same for left or right incidence. Within the class of optical potentials of the form v(x) = v1cos (2βx) + iv2sin (2βx) the case v2 = v1 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 L2, 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.

A Bessel collocation method for numerical solution of generalized pantograph equations

Abstract: This article is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, we introduce a collocation method based on the Bessel polynomials for the approximate solution of the pantograph equations. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

On Turán type inequalities for modified Bessel functions

Abstract: In this note our aim is to point out that certain inequalities for modified Bessel func- tions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equiva- lent to the corresponding Turan type inequalities for these functions. Moreover, we present some new Turan type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note a conjecture is posed, which may be of interest for further research.

Generating Evanescent Bessel Beams Using Near-Field Plates

Abstract: We present a near-field plate that can generate an evanescent Bessel beam. The metallic plate consists of nonperiodic concentric corrugations that surround a coaxially fed aperture. The design procedure for such a device is outlined. The designed plate is simulated using a full-wave electromagnetic solver and is shown to produce an evanescent Bessel beam, thereby verifying its design and operation. The performance of the near-field plate is contrasted against a coaxial probe and a near-field plate designed to produce an Airy focal pattern with the same beamwidth. Such a device, capable of producing evanescent Bessel beams, will find applications in near-field probing/imaging systems, data storage, and biomedical devices.

Axial acoustic radiation torque of a Bessel vortex beam on spherical shells

Abstract: The present paper investigates the interaction of an acoustic Bessel vortex beam centered on a viscoelastic polyethylene sphere and spherical shells filled with air or water immersed in nonviscous water and mercury, and the induced axial acoustic radiation torque (ART) resulting from the transfer of angular momentum. Closed-form series expansions for the axial ART are derived for the case of progressive, standing, and quasistanding waves. The ART is shown to be the result of acoustic absorption inside the particle's material. Numerical predictions shown in the form of two-dimensional (2D) plots illustrate the theory, and reveal new properties related to the ART of Bessel vortex beams. Potential applications are in particle rotation and manipulation. Other applications, such as the characterization of fluids from induced angular accelerations (produced by the ART) and containerless processing, may benefit from the analysis developed here.

Scattering of SH waves induced by a symmetrical V-shaped canyon: a unified analytical solution

Abstract: This paper reports a series solution of wave functions for two-dimensional scattering and diffraction of plane SH waves induced by a symmetrical V-shaped canyon with different shape ratios. A half-space with a symmetrical V-shaped canyon is divided into two sub-regions by using a circular-arc auxiliary boundary. The two sub-regions are represented by global and local cylindrical coordinate systems, respectively. In each coordinate system, the wave field satisfying the Helmholtz equation is represented by the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then, the two wave fields are described in the local coordinate system using the Graf addition theorem. Finally, the unknown coefficients are sought by satisfying the continuity conditions of the auxiliary boundary. To consider the phase characteristics of the wave scattering, a parametric analysis is carried out in the time domain by assuming an incident signal of the Ricker type. Surface and subsurface transient responses demonstrate the characteristics and mechanisms of wave propagating and scattering.

Efficient Analysis of Substrate Integrated Waveguide Devices Using Hybrid Mode Matching Between Cylindrical and Guided Modes

Abstract: In this paper, a new method to efficiently analyze substrate integrated waveguide (SIW) based devices with multiple accessing ports is presented. The problem is considered as a 2-D electromagnetic problem assuming no field variation normal to the dielectric substrate. The incident and scattered fields from each circular cylinder are expanded with cylindrical modes, and the fields in the waveguide ports are expanded using progressive and regressive modal summations. The addition theorems of Bessel and Hankel functions are used to analyze the full-wave behavior of the SIW device. In order to extract the circuital parameters, the hybrid mode-matching between guided and cylindrical modes is done by projecting continuity equations in a circular boundary containing the whole SIW structure over the inner modes of each region. Applying this new technique, it is possible to analyze multiple port devices by solving a set of integrals that can be easily approached analytically or by using the inverse fast Fourier transform, avoiding the use of non efficient numerical methods. It is shown that the new method runs faster than commercial software packages and other techniques recently published.

Nonlinear light-matter interaction with femtosecond high-angle Bessel beams

Abstract: We show that high-angle Bessel beams may significantly reduce nonlinear pulse distortions due, for example, to nonlinear Kerr effects (self-phase-modulation and self-focusing) yet enhance ionization and plasma generation. Holographic reconstruction of Bessel beams in water show intensity clamping at increased intensities and evidence of nontrivial plasma dynamics as the input energy is increased. The solvated electron density increases significantly and the cavitation-induced bubbles are ejected from the focal region indicating a significant excess plasma heating in the Bessel-pulse wake.

Measuring the rotation rates of superpositions of higher-order Bessel beams

Abstract: Experimental measurements are reported of the rotation rates of superpositions of higher-order Bessel beams. Digitally generated phase masks of two annular rings, were imprinted on a spatial light modulator and used to obtain superpositions of higher-order Bessel beams of the same order but of opposite topological charge. Such a superposition field carries on average zero orbital angular momentum, yet exhibits a rotation in the intensity pattern: the resultant field rotates at a constant rate about the optical axis as it propagates. The rotation rates of the generated fields were measured for different orders and for various values of the difference between the wave-vectors of the superimposing beams, and are shown to be in good agreement with that predicted theoretically.

An efficient algorithm for solving multi-pantograph equation systems

Abstract: In this paper, we present a numerical approach for solving the system of multi-pantograph equations with mixed conditions. This system is usually difficult to solve analytically. By expanding the approximate solutions by means of the Bessel functions of first kind with unknown coefficients, the proposed approach consists of reducing the problem to a linear algebraic equation system. The unknown coefficients of the Bessel functions of first kind are computed using the matrix operations of derivatives together with the collocation method. An error estimation is given. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. All of the numerical computations have been performed on a computer with the aid of a program written in Matlab.

3DEX: a code for fast spherical Fourier-Bessel decomposition of 3D surveys

Abstract: Context. High-precision cosmology requires the analysis of large-scale surveys in 3D spherical coordinates, i.e. spherical Fourier-Bessel decomposition. Current methods are insufficient for future data-sets from wide-field cosmology surveys.

The Lyapunov exponent of products of random $2\times2$ matrices close to the identity

Abstract: We study products of arbitrary random real $2 \times 2$ matrices that are close to the identity matrix. Using the Iwasawa decomposition of $\text{SL}(2,{\mathbb R})$, we identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small. In this regime, the Lyapunov exponent of the product is shown to assume a scaling form. In the general case, the corresponding scaling function is expressed in terms of Gauss' hypergeometric function. A number of particular cases are also considered, where the scaling function of the Lyapunov exponent involves other special functions (Airy, Bessel, Whittaker, elliptic). The general solution thus obtained allows us, among other things, to recover in a unified framework many results known previously from exactly solvable models of one-dimensional disordered systems.

Source amplitudes for active exterior cloaking

Abstract: The active cloak comprises a discrete set of multipole sources that destructively interfere with an incident time harmonic scalar wave to produce zero total field over a finite spatial region. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. The field generated by the active sources vanishes in the infinite region exterior to a set of circles defined by the relative positions of the sources. The results provide a direct solution to the inverse problem of determining the source amplitudes. They also define a broad class of nonradiating discrete sources. (Some figures may appear in colour only in the online journal)

Marichev-Saigo-Maeda fractional integration operators of the Bassel functions

Abstract: In this paper, we apply generalized operators of fractional integration involving Appell’s function F_3 (.) due to Marichev-Saigo-Maeda, to the Bessel function of first kind. The results are expressed in terms of generalized Wright function and hypergeometric functions _pF_q . Special cases involving this function are mentioned. Results given recently by Kilbas and Sebastian follow as special cases of the theorems establish here.

The six-point remainder function to all loop orders in the multi-Regge limit

Abstract: We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric N=4 Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem---the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w*, which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w|->0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log|w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log|w| is given in terms of modified Bessel functions.

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

Abstract: This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.

From stationary annular rings to rotating Bessel beams

Abstract: In this work we use a phase-only spatial light modulator (SLM) to mimic a ring-slit aperture, containing multiple azimuthally varying phases at different radial positions. The optical Fourier transform of such an aperture is currently known and its intensity profile has been shown to rotate along its propagation axis. Here we investigate the near-field of the ring-slit aperture and show, both experimentally and theoretically, that although the near-field possesses similar attributes to its Fourier transform, its intensity profile exhibits no rotation as it propagates.

Neumann series of Bessel functions

Abstract: Recently, Pogany and Suli [Integral representation for Neumann series of Bessel functions, Proc. Amer. Math. Soc. 137(7) (2009), pp. 2363–2368] derived a closed-form integral expression for a Neumann series of Bessel functions. In this note, our aim is to establish another kind of integral representations for the Neumann series of Bessel functions of the first kind J ν.

Study on the propagation parameters of Bessel-Gaussian beams carrying optical vortices through atmospheric turbulence.

Abstract: Based on the integral representation of Bessel function and the extended Huygens–Fresnel principle, an integral expression of the Wigner distribution function (WDF) for partially coherent Bessel–Gaussian beams (PBGBs) propagating through turbulent atmosphere has been obtained. Also, the analytical formulas of the M2-factor for PBGB propagation in such a medium have been derived, which can be applied to cases of different spatial power spectra of the refractive index fluctuations. The performed numerical results reveal that the M2-factor of a PBGB in turbulent atmosphere depends on the beam parameters of the initial input beam, the structure constants of the turbulent atmosphere, and the propagation distance. These results may be useful in long-distance optical communications in free space or in turbulent atmosphere.

Geometric Properties of Generalized Bessel Functions

Abstract: In this work, the generalized Bessel functions with their normal- ization are considered. Various conditions are obtained so that these Bessel functions have certain geometric properties including close-to-convexity (univa- lency), starlikeness and convexity in the unit disc. Results obtained for certain classes are new and for the other classes for which similar results exist in the literature, examples are given to support that these results are better than the existing ones.