Showing papers on "Bessel function published in 2008"

Elliptic integral evaluations of Bessel moments and applications

Abstract: We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for cn,k :=

Sturm-Liouville theory and its applications

Abstract: Inner Product Space- The Sturm-Liouville Theory- Fourier Series- Orthogonal Polynomials- Bessel Functions- The Fourier Transformation- The Laplace Transformation

Generation of multiple Bessel beams for a biophotonics workstation

Abstract: We present a simple method using an axicon and spatial light modulator to create multiple parallel Bessel beams and precisely control their individual positions in three dimensions. This technique is tested as an alternative to classical holographic beam shaping commonly used now in optical tweezers. Various applications of precise control of multiple Bessel beams are demonstrated within a single microscope giving rise to new methods for three-dimensional positional control of trapped particles or active sorting of micro-objects as well as “focus-free” photoporation of living cells. Overall this concept is termed a ‘biophotonics workstation’ where users may readily trap, sort and porate material using Bessel light modes in a microscope.

Nondiffracting vortex beams with continuous orbital angular momentum order dependence

Abstract: We introduce a new class of nondiffracting fractional vortex beams that connect Bessel beams of successive order in a smooth transition. The new fractional-order beams preserve the same nondiffracting feature of Bessel beams of integer order, and their orbital angular momentum per photon can take any value in a continuous range. The propagation of the more physically realizable fractional-order Bessel–Gauss beams, i.e. fractional Bessel beams apodized by a Gaussian transmittance, through general ABCD optical systems is studied in detail and is complemented by the experimental generation of several instances of fractional beams which in turn confirms our theoretical predictions.

Titchmarsh–Weyl m -functions for second-order Sturm–Liouville problems with two singular endpoints

Abstract: In this paper we consider some cases of Sturm–Liouville problems with two singular endpoints at x = 0 and x = ∞ which have a simple spectrum, and show that the simplicity of the spectrum can be built into the definition of a Titchmarsh–Weyl m -function from which the eigenfunction expansion can be constructed. The use of initial conditions at a point interior to the interval (0,∞) is avoided in favor of Frobenius solutions near the regular singular point x = 0. In contrast to the classical theory associated with a regular left endpoint, the growth behaviour of the associated spectral functions can be on the order of λβ for any β ∈ (0,∞). Application of the theory to the Bessel equation on (0,∞) and to the radial part of the separated hydrogen atom on (0,∞) is given. In the case of the hydrogen atom a single Titchmarsh–Weyl m -function is obtained which completely describes both the discrete negative spectrum and the continuous positive spectrum. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Generalized fractional integration of Bessel function of the first kind

Abstract: Two integral transforms involving the Gauss-hypergeometric function in the kernels are considered. They generalize the classical Riemann–Liouville and Erdelyi–Kober fractional integral operators. Formulas of compositions for such a generalized fractional integrals with Bessel function of the first kind are proved. Special cases of cosine and sine functions are given. The results are established in terms of generalized Wright function and generalized hypergeometric series. Corresponding assertions for Riemann–Liouville and Erdelyi–Kober fractional integral transforms are presented.

Exact and geometrical optics energy trajectories in twisted beams

Abstract: Energy trajectories, that is, integral curves of the Poynting (current) vector, are calculated for scalar Bessel and Laguerre–Gauss beams carrying orbital angular momentum. The trajectories for the exact waves are helices, winding on cylinders for Bessel beams and hyperboloidal surfaces for Laguerre–Gauss beams. In the geometrical optics approximations, the trajectories for both types of beam are overlapping families of straight skew rays lying on hyperboloidal surfaces; the envelopes of the hyperboloids are the caustics: a cylinder for Bessel beams and two hyperboloids for Laguerre–Gauss beams.

Mixed Boundary Value Problems

Abstract: Overview Examples of Mixed Boundary Value Problem Integral Equations Legendre Polynomials Bessel Functions Historical Background Nobili's Rings Disc Capicator Another Electrostatic Problem Griffith Cracks The Boundary Value Problem of Reissner and Sagoci Steady Rotation of a Circular Disc Separation of Variables Dual Fourier Cosine Series Dual Fourier Sine Series Dual Fourier-Bessel Series Dual Fourier-Legendre Series Triple Fourier Sine Series Transform Methods Dual Fourier Integrals Triple Fourier Integrals Dual Fourier-Bessel Integrals Triple and Higher Fourier-Bessel Integrals Joint Transform Methods The Wiener-Hopf Technique The Wiener-Hopf Technique When the Factorization Contains No Branch Points The Wiener-Hopf Technique When the Factorization Contains Branch Points Green's Function Green's Function with Mixed Boundary Value Conditions Integral Representations Involving Green's Functions Potential Theory Conformal Mapping The Mapping z = w + alog(w) The Mapping tanh[piz/(2b)] = sn(w, k) The Mapping z = w + lambda w2 - 1 The Mapping w = ai(z - a)/(z + a) The Mapping z = 2[w - arctan(w)]/pi The Mapping kw sn(w, kw) = kz sn(Kzz/a, kz) Index

Scattering of a Bessel beam by a sphere: II. Helicoidal case and spherical shell example.

Abstract: In prior work [P. L. Marston, "Scattering of a Bessel beam by a sphere," J. Acoust. Soc. Am. 121, 753-758 (2007)] the partial wave series for the scattering by a sphere centered on a zero-order Bessel beam was derived. The present work extends the analysis of the far-field scattering to the case of a Bessel beam having an angular dependence on phase. The beam considered is an example of a helicoidal beam where "helicoidal" refers to a type of beam that possesses an axial null and has an azimuthal phase gradient. This type of beam is sometimes also referred to as an acoustic vortex. The beam considered here has a phase ramp equal to the azimuthal angle. In agreement with symmetry arguments given previously, the backward scattering and forward scattering vanish for all frequencies. Some of the resulting modifications of the scattering are illustrated for a rigid sphere and an evacuated steel shell in water. For some directions and choices for the frequency, the calculated scattering by the shell increases when shifting to a helicoidal beam illumination.

Family of hypergeometric laser beams

Abstract: We discuss a three-parameter family of paraxial coherent light fields that originate from a complex amplitude composed of the following four cofactors: the Gaussian beam, a logarithmic axicon, a spiral phase plate (angular harmonic), and an amplitude power function with a possible singularity at the origin of coordinates. For such types of beams, the near-field complex amplitude is proportional to the degenerate hypergeometric function, prompting the beams' name--hypergeometric (HyG) beams. When the Gaussian beam is replaced by a plane wave, the above beams change to generalized HyG modes that preserve their structure up to scale upon propagation. The intensity profile of the HyG beams is similar to that of the Bessel modes, forming a set of alternating bright and dark rings. However, the thickness of the rings of the HyG beams decreases with increasing ring number.

On a product of modified Bessel functions

Abstract: Let Iν and Kν denote the modified Bessel functions of the first and second kinds of order ν. In this note we prove that the monotonicity of u → Iν(u)Kν(u) on (0,∞) for all ν ≥ −1/2 is an almost immediate consequence of the corresponding Turán type inequalities for the modified Bessel functions of the first and second kinds of order ν. Moreover, we show that the function u → Iν(u)Kν(u) is strictly completely monotonic on (0,∞) for all ν ∈ [−1/2, 1/2]. At the end of this note, a conjecture is stated. 1. Preliminaries and main results Let Iν and Kν denote, as usual, the modified Bessel functions of the first and second kinds of order ν. Recently, motivated by a problem which arises in biophysics, Penfold et al. [13, Theorem 3.1] proved, in a complicated way, that the product of the modified Bessel functions of the first and second kinds, i.e. u → Pν(u) = Iν(u)Kν(u), is strictly decreasing on (0,∞) for all ν ≥ 0. It is worth mentioning that this result for ν = n ≥ 1, a positive integer, was verified in 1950 by Phillips and Malin [14, Corollary 2.2]. In this note our aim is to show that using the idea of Phillips and Malin, the monotonicity of u → Pν(u) for ν ≥ −1/2 can be verified easily by using the corresponding Turán type inequalities for modified Bessel functions. Moreover, we show that the function u → Iν(u)Kν(u) is strictly completely monotonic on (0,∞) for all ν ∈ [−1/2, 1/2], i.e. for all u > 0, ν ∈ [−1/2, 1/2] and m = 0, 1, 2, . . . , we have (−1) [Iν(u)Kν(u)] > 0. In order to achieve our goal we improve some of the results of Phillips and Malin [14, Eq. 1] concerning bounds for the logarithmic derivatives of the modified Bessel and Hankel functions. Our main result reads as follows: Theorem 1. The following assertions are true: a. the function u → Pν(u) is strictly decreasing on (0,∞) for all ν ≥ −1/2; Received by the editors December 13, 2007. 2000 Mathematics Subject Classification. Primary 33C10, 33C15.

Maximum distributions of bridges of noncolliding Brownian paths.

Abstract: One-dimensional Brownian motion starting from the origin at time t=0 , conditioned to return to the origin at time t=1 and to stay positive during time interval 0

Nonparaxial TM and TE beams in free space

Abstract: Expressions for the fields of TM and TE laser beams in free space that are rigorous solutions to Maxwell's equations are given in a closed form. The electric and the magnetic fields are both expressed in terms of nonparaxial elegant Laguerre-Gaussian beams that are exact solutions of the Helmholtz equation. These solutions involve well-known functions, such as spherical Bessel and associated Legendre functions. Radially and azimuthally polarized beams of arbitrary order are considered, and the lowest-order radially polarized beam (TM(01) beam) is investigated in detail.

Nonparaxial elegant Laguerre-Gaussian beams.

Abstract: Closed-form nonparaxial expressions for optical beams are useful to calculate the fields produced by tightly focused laser beams. Such expressions for elegant Laguerre-Gaussian (eLG) beams that are exact solutions of the Helmholtz equation are introduced. These solutions are expressed as linear combinations of a finite number of analytic functions that involve spherical Bessel functions and associated Legendre functions of complex arguments. In the paraxial limit, the expressions proposed have the property to reduce to the well-known eLG beams.

Subluminal wave bullets: Exact localized subluminal solutions to the wave equations

Abstract: In this work it is shown how to obtain, in a simple way, localized (non-diffractive) subluminal pulses as exact analytic solutions to the wave equations. These new ideal subluminal solutions, which propagate without distortion in any homogeneous linear media, are herein obtained for arbitrarily chosen frequencies and bandwidths, avoiding in particular any recourse to the non-causal components so frequently plaguing the previously known localized waves. The new solutions are suitable superpositions of ---zeroth-order, in general--- Bessel beams, which can be performed either by integrating with respect to (w.r.t.) the angular frequency omega, or by integrating w.r.t. the longitudinal wavenumber k_z: Both methods are expounded in this paper. The first one appears to be powerful enough; we study the second method as well, however, since it allows dealing even with the limiting case of zero-speed solutions (and furnishes a new way, in terms of continuous spectra, for obtaining the so-called "Frozen Waves", so promising also from the point of view of applications). We briefly treat the case, moreover, of non axially-symmetric solutions, in terms of higher order Bessel beams. At last, particular attention is paid to the role of Special Relativity, and to the fact that the localized waves are expected to be transformed one into the other by suitable Lorentz Transformations. The analogous pulses with intrinsic finite energy, or merely truncated, will be constructed in another paper. In this work we fix our attention especially on electromagnetism and optics: but results of the present kind are valid whenever an essential role is played by a wave-equation (like in acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). [PACS nos.: 03.50.De; 03.30.+p; 03.50.+p; 41.20.Jb; 41.85.-p; 42.25.-p; 42.25.Fx; 43.20.+g; 43.20.Ks; 46.40.-f; 46.40.Cd; 47.35.Rs; 52.35.Lv ].

Random Motions at Finite Speed in Higher Dimensions

Abstract: We present a general method of studying the transport process \(\bold X(t)\) , t≥0, in the Euclidean space ℝm, m≥2, based on the analysis of the integral transforms of its distributions. We show that the joint characteristic functions of \(\bold X(t)\) are connected with each other by a convolution-type recurrent relation. This enables us to prove that the characteristic function (Fourier transform) of \(\bold X(t)\) in any dimension m≥2 satisfies a convolution-type Volterra integral equation of second kind. We give its solution and obtain the characteristic function of \(\bold X(t)\) in terms of the multiple convolutions of the kernel of the equation with itself. An explicit form of the Laplace transform of the characteristic function in any dimension is given. The complete solution of the problem of finding the initial conditions for the governing partial differential equations, is given.

Phase matching with pulsed Bessel beams for high-order harmonic generation

Abstract: We propose a different approach to obtain phase-matched generation of high-order harmonics based on the use of pulsed Bessel beams as pump pulses. By means of the 'coherence map' technique, we show that it is possible to maximize the generation of a chosen harmonic of interest by properly adjusting the phase front tilt of the pulsed Bessel beam to compensate the mismatch arising from material and plasma dispersion and atomic phase.

A proof of a recursion for bessel moments

Abstract: We provide a proof of a conjecture in (Bailey, Borwein, Borwein, Crandall 2007) on the existence and form of linear recursions for moments of powers of the Bessel function~$K_0$.

Hybrid of Method of Moments and Cylindrical Eigenfunction Expansion to Study Substrate Integrated Waveguide Circuits

Abstract: A hybrid method that combines method of moments (MOM) and cylindrical eigenfunction expansion is presented to study a substrate integrated waveguide circuit that consists of metallic and dielectric circular cylinders. The problem is considered as a 2-D electromagnetic problem assuming no field variation normal to the dielectric substrate. The scattered field from each circular cylinder is expanded by cylindrical eigenfunctions, and the equivalent current densities at a waveguide port are expanded using the MOM. After enforcing boundary conditions and applying the additional theorem of Bessel and Hankel functions, a set of linear equations are constructed using the orthogonality of the exponential function. With matrix manipulation and sub-ports combination, the S matrix can then be obtained. Four examples are analyzed by using the hybrid method and verified by using Ansoft Corporation's commercial High Frequency Structure Simulator (HFSS) software package. It is shown that the hybrid method runs faster than HFSS and requires less memory. It is also pointed out that the method can be used to study a circuit with noncircular cylinders.

Two-Sided Optimal Bounds for Green Functions of Half-Spaces for Relativistic α -Stable Process

Abstract: The purpose of this paper is to find optimal estimates for the Green function of a half-space of the relativistic α -stable process with parameter m on ℝ d space This process has an infinitesimal generator of the form mI–(m 2/α I–Δ) α/2, where 0 0, and reduces to the isotropic α-stable process for m=0 Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space As a byproduct we obtain some improvements of the estimates known for bounded sets Our approach combines the recent results obtained in Byczkowski et al (Bessel Potentials, Hitting Distributions and Green Functions (2006) (preprint) http://arxivorg/abs/math/0612176 ), where an explicit integral formula for the m-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic α-stable process For example, for d≥3, it is comparable with the Green function for the isotropic α-stable process, provided that the points are close enough

On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension d = 2(1 − α ), 0 < α < 1

Abstract: The precise choice of the local time at 0 for a Bessel process with dimension d ∈ ]0,2[ plays some role in explicit computations or limiting results involving excursion theory for these processes. Starting from one specific choice, and deriving the main related formulae, it is shown how the various multiplicative constants corresponding to other choices made in the literature enter into these formulae.

Mapping properties of fundamental operators in harmonic analysis related to Bessel operators

Abstract: We prove sharp power-weighted strong type, weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators.

Finite-energy, accelerating Bessel pulses.

Abstract: We numerically investigate the possibility to generate freely accelerating or decelerating pulses. In particular it is shown that acceleration along the propagation direction z may be obtained by a purely spatial modulation of an input Gaussian pulse in the form of finite-energy Bessel pulses with a cone angle that varies along the radial coordinate.We discuss simple practical implementations of such accelerating Bessel beams.

Theory of the acoustic radiation force exerted on a sphere by standing and quasistanding zero-order Bessel beam tweezers of variable half-cone angles

Abstract: A rigorous theory is developed to predict the radiation force (RF) exerted on a sphere immersed in an ideal fluid by a standing or quasistanding zero-order Bessel beam of different half-cone angles. A standing or a quasistanding acoustic field is the result of counter propagating 2 equal or unequal amplitude zero-order Bessel beams, respectively, along the same axis. Each Bessel beam is characterized by its halfcone angle betalscr;lscr=1, 2 of its plane wave components, such that betalscr=0 represents a plane wave. Analytical expressions of RF are derived for a homogeneous viscoelastic sphere chosen as an example. RF calculations for a polyethylene sphere immersed in water are performed. Particularly, the half-cone angle dependency on the RF is analyzed for standing and quasistanding waves. Changing the half-cone angle is equivalent to changing the beamwidth. Potential applications include particle manipulation in microfluidic lab-on-chips as well as in reduced gravity environments.

Recurrence Relations for Strongly q-Log-Convex Polynomials

Abstract: We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.