Showing papers on "Bessel function published in 2014"
TL;DR: In this paper, a general method based on susceptibility tensors is proposed for the synthesis of metasurfaces transforming arbitrary incident waves into arbitrary reflected and transmitted waves, which is inherently vectorial and therefore better suited for full vectorial (beyond paraxial) electromagnetic problems, and is therefore extremely fast.
Abstract: A general method, based on susceptibility tensors, is proposed for the synthesis of metasurfaces transforming arbitrary incident waves into arbitrary reflected and transmitted waves. The proposed method exhibits two advantages: 1)it is inherently vectorial, and therefore better suited for full vectorial (beyond paraxial) electromagnetic problems, 2) it provides closed-form solutions, and is therefore extremely fast. Incidentally, the method reveals that a metasurface is fundamentally capable to transform up to four independent wave triplets (incident, reflected and refracted waves). In addition, the paper provides the closed-form expressions relating the synthesized susceptibilities and the scattering parameters simulated within periodic boundary conditions, which allows one to design the scattering particles realizing the desired susceptibilities. The versatility of the method is illustrated by examples of metasurfaces achieving the following transformations: generalized refraction, reciprocal and non-reciprocal polarization rotation, Bessel vortex beam generation, and orbital angular momentum multiplexing.
211 citations
TL;DR: A new, three-parameter family of diffraction-free asymmetric elegant Bessel modes (aB-modes) with an integer and fractional orbital angular momentum (OAM) and the asymmetry degree of the nonparaxial aB-mode is shown to depend on a real parameter c≥0.
Abstract: We propose a new, three-parameter family of diffraction-free asymmetric elegant Bessel modes (aB-modes) with an integer and fractional orbital angular momentum (OAM). The aB-modes are described by the nth-order Bessel function of the first kind with complex argument. The asymmetry degree of the nonparaxial aB-mode is shown to depend on a real parameter c≥0: when c=0, the aB-mode is identical to a conventional radially symmetric Bessel mode; with increasing c, the aB-mode starts to acquire a crescent form, getting stretched along the vertical axis and shifted along the horizontal axis for c≫1. On the horizontal axis, the aB-modes have a denumerable number of isolated intensity zeros that generate optical vortices with a unit topological charge of opposite sign on opposite sides of 0. At different values of the parameter c, the intensity zeros change their location on the horizontal axis, thus changing the beam’s OAM. An isolated intensity zero on the optical axis generates an optical vortex with topological charge n. The OAM per photon of an aB-mode depends near-linearly on c, being equal to ℏ(n+cI1(2c)/I0(2c)), where ℏ is the Planck constant and In(x) is a modified Bessel function.
101 citations
TL;DR: It is shown that the trade-off between DOF and peak irradiance of Bessel beams depends solely on the Fresnel number N, and the existence of a low-Fresnel-number regime, N<10, in which axicons with Gaussian illumination can generate energy-efficient Bessel beam with a small number of sidelobes is demonstrated.
Abstract: Bessel beams feature a very large depth-of-focus (DOF) compared to conventional focusing schemes, but their central lobe carries only a small fraction of the total beam power, leading to a strongly reduced peak irradiance. This is problematic for power-limited applications, such as optical coherence tomography (OCT) or optical coherence microscopy, as it can result in a prohibitive reduction of the signal-to-noise ratio (SNR). Using scalar diffraction theory, we show that the trade-off between DOF and peak irradiance of Bessel beams depends solely on the Fresnel number N. We demonstrate the existence of a low-Fresnel-number regime, N<10, in which axicons with Gaussian illumination can generate energy-efficient Bessel beams with a small number of sidelobes. In the context of OCT, this translates into DOF enhancements of up to 13× for a SNR penalty below 20 dB, which is confirmed by our experiments. We expect that these findings will enable improved performance of optical systems with extended DOF.
91 citations
17 Feb 2014
TL;DR: In this article, the authors determined the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization and proved that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function.
Abstract: In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon [IM2], the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind.
88 citations
TL;DR: In this paper, simple inequalities for some integrals involving the modified Bessel functions I ν (x ) and K ν(x ) were established and a monotonicity result was obtained for K 0 ( x ).
88 citations
TL;DR: The exact analytical solution to a Goursat PDE system governing the kernels of a backstepping-based boundary control law that stabilizes a constant-coefficient 2×2 system of first-order hyperbolic linear PDEs is found.
79 citations
TL;DR: The turbulence-induced transverse phase distortion limits the effectiveness of Bessel and Airy beams for applications requiring propagation over long distances in the turbulent atmosphere.
Abstract: We investigate, through simulation, the modifications to Bessel and Airy beams during propagation through atmospheric turbulence. We find that atmospheric turbulence disrupts the quasi-non-diffracting nature of Bessel and Airy beams when the transverse coherence length (Fried parameter) nears the initial aperture diameter or diagonal, respectively. The turbulence-induced transverse phase distortion limits the effectiveness of Bessel and Airy beams for applications requiring propagation over long distances in the turbulent atmosphere.
79 citations
Dissertation•
01 Jan 2014
TL;DR: Brubaker as discussed by the authors showed that the Bessel function shares many properties with the Macdonald spherical function of G, in particular the properties described in Casselman's 1980 evaluation of that function.
Abstract: Let G be a connected reductive group with a split maximal torus defined over a nonarchimedean local field. I evaluate a matrix coefficient of the unramified principal series of G known as the "Bessel function" at torus elements of dominant coweight. I show that the Bessel function shares many properties with the Macdonald spherical function of G, in particular the properties described in Casselman's 1980 evaluation of that function. The analogy I demonstrate between the Bessel and Macdonald spherical functions extends to an analogy between the spherical Whittaker function, evaluated by Casselman and Shalika in 1980, and a previously unstudied matrix coefficient. Thesis Supervisor: Benjamin Brubaker Title: Associate Professor
63 citations
TL;DR: In this paper, two types of other spherical harmonic gravity fields that bridge the null space of the exterior/interior gravity field expressions by solving Poisson's equation are discussed, referred to as the interior/exterior spherical Bessel gravity fields.
Abstract: Conventional gravity field expressions are derived from Laplace’s equation, the result being the spherical harmonic gravity field. This gravity field is said to be the exterior spherical harmonic gravity field, as its convergence region is outside the Brillouin (i.e., circumscribing) sphere of the body. In contrast, there exists its counterpart called the interior spherical harmonic gravity field for which the convergence region lies within the interior Brillouin sphere that is not the same as the exterior Brillouin sphere. Thus, the exterior spherical harmonic gravity field cannot model the gravitation within the exterior Brillouin sphere except in some special cases, and the interior spherical harmonic gravity field cannot model the gravitation outside the interior Brillouin sphere. In this paper, we will discuss two types of other spherical harmonic gravity fields that bridge the null space of the exterior/interior gravity field expressions by solving Poisson’s equation. These two gravity fields are obtained by assuming the form of Helmholtz’s equation to Poisson’s equation. This method renders the gravitational potentials as functions of spherical Bessel functions and spherical harmonic coefficients. We refer to these gravity fields as the interior/exterior spherical Bessel gravity fields and study their characteristics. The interior spherical Bessel gravity field is investigated in detail for proximity operation purposes around small primitive bodies. Particularly, we apply the theory to asteroids Bennu (formerly 1999 RQ36) and Castalia to quantify its performance around both nearly spheroidal and contact-binary asteroids, respectively. Furthermore, comparisons between the exterior gravity field, interior gravity field, interior spherical Bessel gravity field, and polyhedral gravity field are made and recommendations are given in order to aid planning of proximity operations for future small body missions.
58 citations
TL;DR: The proposed system and accurate analysis of non-diffractive Bessel beams launched by inward waves opens new opportunities for planar, low profile beam generators at microwaves, Terahertz and optics.
Abstract: The focusing capabilities of an inward cylindrical traveling wave aperture distribution and the non-diffractive behaviour of its radiated field are analyzed. The wave dynamics of the infinite aperture radiated field is clearly unveiled by means of closed form expressions, based on incomplete Hankel functions, and their ray interpretation. The non-diffractive behaviour is also confirmed for finite apertures up to a defined limited range. A radial waveguide made by metallic gratings over a ground plane and fed by a coaxial feed is used to validate numerically the analytical results. The proposed system and accurate analysis of non-diffractive Bessel beams launched by inward waves opens new opportunities for planar, low profile beam generators at microwaves, Terahertz and optics.
57 citations
TL;DR: In this article, it was shown that one can obtain an asymptotic expression for some special functions with a very explicit error term starting from appropriate upper bounds, such as the Bessel function and the Airy function.
Abstract: We show how one can obtain an asymptotic expression for some special functions with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function and the Airy function In particular, we answer a question raised by Olenko and find a sharp bound on the difference between and its standard asymptotics. We also give a very simple and surprisingly precise approximation for the zeros
TL;DR: In this article, isolated finite interacting quantum systems after an instantaneous perturbation are studied and three scenarios in which the probability for finding the initial state later in time (fidelity) decays nonexponentially, often all the way to saturation.
Abstract: We study isolated finite interacting quantum systems after an instantaneous perturbation and show three scenarios in which the probability for finding the initial state later in time (fidelity) decays nonexponentially, often all the way to saturation. The decays analyzed involve Gaussian, Bessel of the first kind, and cosine squared functions. The Gaussian behavior emerges in systems with two-body interactions in the limit of strong perturbation. The Bessel function, associated with the evolution under full random matrices, is obtained with surprisingly sparse random matrices. The cosine squared behavior, established by the energy-time uncertainty relation, is approached after a local perturbation in space.
TL;DR: In this paper, the radius of convexity for three kinds of normalized Bessel functions of the first kind was determined and the key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel function and their derivatives.
Abstract: In this paper, we determine the radius of convexity for three kinds of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind.
TL;DR: In this article, the radii of starlikeness and convexity for normalized Bessel functions of the first kind were shown to decrease with respect to the parameter of the Bessel function.
Abstract: The radii of $\alpha$-convexity are deduced for three different kind of normalized Bessel functions of the first kind and it is shown that these radii are between the radii of starlikeness and convexity, when $\alpha\in[0,1],$ and they are decreasing with respect to the parameter $\alpha$ The results presented in this paper unify some recent results on the radii of starlikeness and convexity for normalized Bessel functions of the first kind The key tools in the proofs are some interlacing properties of the zeros of some Dini functions and the zeros of Bessel functions of the first kind
TL;DR: In this article, conditions for an attractive force acting in opposite direction of the radiating waves, determined by the choice of the beam's half-cone angle, the size of the radiator, and its distance from a fluid sphere, are established and discussed.
TL;DR: In this article, the authors re-examine the boundary-valued problem of wave scattering and diffraction in elastic half-space from an applied mathematics points of view and redefine the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the halfspace surface.
TL;DR: In this paper, the transformation of paraxial and non-paraxial Bessel beams in a crystal of Iceland spar is considered theoretically and an optical system is designed for investigating the intensity distribution after passage through the crystal.
Abstract: The transformation of paraxial and nonparaxial Bessel beams in a crystal of Iceland spar is considered theoretically. Calculation formulae for matching the thickness of the crystal and the parameters of the incident beam required for complete conversion of a circularly polarized zero-order Bessel beam into a second-order vortex beam are obtained. An optical system was designed for investigating the intensity distribution after passage through the crystal. Experimental and theoretical results are in good agreement.
TL;DR: In this paper, a Bessel beam of de Broglie matter waves was created by the free evolution of a thin toroidal atomic Bose-Einstein condensate (BEC) which has been set into rotational motion.
Abstract: Bessel beams are plane waves with amplitude profiles described by Bessel functions. They are important because they propagate ‘diffraction-free’ and because they can carry orbital angular momentum. Here we report the creation of a Bessel beam of de Broglie matter waves. The Bessel beam is produced by the free evolution of a thin toroidal atomic Bose–Einstein condensate (BEC) which has been set into rotational motion. By attempting to stir it at different rotation rates, we show that the toroidal BEC can only be made to rotate at discrete, equally spaced frequencies, demonstrating that circulation is quantized in atomic BECs. The method used here can be viewed as a form of wavefunction engineering which might be developed to implement cold atom matter wave holography.
23 Jun 2014
TL;DR: In this article, the authors verify the complete monotonicity of the difference and derive an inequality for the first order modified Bessel function of the first kind of the Laurent series.
Abstract: In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-\psi'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind. These results show us some new properties and relations of the exponential, trigamma, the first kind modified Bessel functions and the hypergeometric series. Keywords: property; connection; completely monotonic function; completely monotonic degree; integral representation; difference;exponential function; trigamma function; hypergeometric series; inequality; modified Bessel function.
29 Jan 2014
TL;DR: In this article, the authors derived the solution of generalized kinetic equations of fractional order involving the Wright generalized Bessel function or Bessel-Mitland function in terms of K 4 -function introduced by Sharma.
Abstract: The object of the present paper is to derive the solution of generalized kinetic equations of fractional order involving the Wright generalized Bessel function or Bessel–Mitland function. The obtained results in this paper, imply the known results of Chaurasia and Pandey [24] more precisely in terms of K 4 –function introduced by Sharma [12] believed to be new. Special case, involving the F- function is considered.
TL;DR: The results show that the physics underlying the self-healing mechanism can be entirely explained in terms of the propagation of plane waves with radial wave vectors lying on a ring.
Abstract: Bessel beams’ great importance in optics lies in that these propagate without spreading and can reconstruct themselves behind an obstruction placed across their path. However, a rigorous wave-optics explanation of the latter property is missing. In this work, we study the reconstruction mechanism by means of a wave-optics description. We obtain expressions for the minimum distance beyond the obstruction at which the beam reconstructs itself, which are in close agreement with the traditional one determined from geometrical optics. Our results show that the physics underlying the self-healing mechanism can be entirely explained in terms of the propagation of plane waves with radial wave vectors lying on a ring.
TL;DR: This work presents a more generalized asymmetric Bessel mode in which the parameter c is a complex constant, and provides an effective way to control the beam's asymmetry and orientation, which may find potential applications in light-sheet microscopy and optical manipulation.
Abstract: Recently, V. V. Kotlyar et al. [Opt. Lett.39, 2395 (2014)] have theoretically proposed a novel kind of three-parameter diffraction-free beam with a crescent profile, namely, the asymmetric Bessel (aB) beam. The asymmetry degree of such nonparaxial modes was shown to depend on a nonnegative real parameter c. We present a more generalized asymmetric Bessel mode in which the parameter c is a complex constant. This parameter controls not only the asymmetry degree of the mode but also the orientation of the optical crescent, and affects the energy distribution and orbital angular momentum (OAM) of the beam. As a proof of concept, the high-quality generation of asymmetric Bessel-Gauss beams was demonstrated with the super-pixel method using a digital micromirror device (DMD). We investigated the near-field properties as well as the far field features of such beams, and the experimental observations were in good agreement with the theoretical predictions. Additionally, we provided an effective way to control the beam’s asymmetry and orientation, which may find potential applications in light-sheet microscopy and optical manipulation.
TL;DR: In this article, the authors deduce sufficient conditions for the close-to-convexity of Bessel, Struve and Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function ${}_1F_2}.
Abstract: In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function ${}_1F_2$. The key tool in our proofs is a result of Shah and Trimble about transcendental entire functions with univalent derivatives. Moreover, a known result of P\'olya on entire functions, the infinite product representations and some results on zeros of Bessel, Struve and Lommel functions of the first kind are used in order to achieve the main results of the paper.
TL;DR: Different sets of range-conditioned, modified cylindrical functions, each associated with nonoverlapped subdomains of (numerical) evaluation to allow for stable computations under any range of physical parameters are introduced.
TL;DR: In this paper, an active elastodynamic cloak destructively interferes with an incident time harmonic in-plane (coupled compressional/shear) elastic wave to produce zero total elastic field over a finite spatial region.
Abstract: An active elastodynamic cloak destructively interferes with an incident time harmonic in-plane (coupled compressional/shear) elastic wave to produce zero total elastic field over a finite spatial region. A method is described which explicitly predicts the source amplitudes of the active field. 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. Importantly, the active field generated by the sources vanishes in the far-field. In practice the infinite summations are clearly required to be truncated and the accuracy of cloaking is studied when the truncation parameter is modified.
TL;DR: Simulation and experimental results have good agreements, which jointly show the formation of the diffraction-free surface waves in the microwave band.
Abstract: We propose a method to design and realize planar Bessel lens using artificial metasurfaces to produce diffraction-free surface waves. The planar Bessel lens is composed of two sublenses: a half Maxwell fisheye lens which can shape the surface cylindrical waves to surface plane waves, and an inhomogeneous flat lens which can convert the surface plane waves into approximate diffraction-free surface waves in a diamond-shaped focusing area. Through the planar Bessel lens, a point source on the metasurface directly radiates the diffraction-free surface waves. In realization, we construct the inhomogeneous metasurfaces by subwavelength metallic patches printed on a grounded dielectric substrate. Simulation and experimental results have good agreements, which jointly show the formation of the diffraction-free surface waves in the microwave band.
TL;DR: Nonlinear Bessel vortices are shown to be sufficiently intense to generate a ring-shaped filamentary ionized channel in the medium which is foreseen as opening the way to novel applications in laser material processing of transparent dielectrics.
Abstract: We present a new type of ring-shaped filaments featured by stationary nonlinear high-order Bessel solutions to the laser beam propagation equation. Two different regimes are identified by direct numerical simulations of the nonlinear propagation of axicon focused Gaussian beams carrying helicity in a Kerr medium with multiphoton absorption: the stable nonlinear propagation regime corresponds to a slow beam reshaping into one of the stationary nonlinear high-order Bessel solutions, called nonlinear Bessel vortices. The region of existence of nonlinear Bessel vortices is found semi-analytically. The influence of the Kerr nonlinearity and nonlinear losses on the beam shape is presented. Direct numerical simulations highlight the role of attractors played by nonlinear Bessel vortices in the stable propagation regime. Large input powers or small cone angles lead to the unstable propagation regime where nonlinear Bessel vortices break up into an helical multiple filament pattern or a more irregular structure. Nonlinear Bessel vortices are shown to be sufficiently intense to generate a ring-shaped filamentary ionized channel in the medium which is foreseen as opening the way to novel applications in laser material processing of transparent dielectrics.
TL;DR: The fidelity of the detection method is very high, with modal cross-talk below 5%, even for high orbital angular momentum carrying fields with long propagation ranges, and it is used to observe the modal spectrum changes during the self-reconstruction process of Bessel beams after encountering an obstruction.
Abstract: We propose a simple method for the detection of Bessel beams with arbitrary radial and azimuthal indices, and then demonstrate it in an all-digital setup with a spatial light modulator. We confirm that the fidelity of the detection method is very high, with modal cross-talk below 5%, even for high orbital angular momentum carrying fields with long propagation ranges. To illustrate the versatility of the approach we use it to observe the modal spectrum changes during the self-reconstruction process of Bessel beams after encountering an obstruction, as well as to characterize modal distortions of Bessel beams propagating through atmospheric turbulence.
TL;DR: The extended optical theorem is generalized for scalar acoustical beams of arbitrary character with any angle of incidence interacting with an object of arbitrary geometric shape and size placed randomly in the beam's path with any scattering angle.
Abstract: The extended optical theorem is generalized for scalar acoustical beams of arbitrary character with any angle of incidence interacting with an object of arbitrary geometric shape and size, and placed randomly in the beam's path with any scattering angle. Analytical expressions for the extinction, absorption, and scattering cross sections are derived, and the connections with the axial (i.e., along the direction of wave propagation) torque and radiation force calculations are discussed. As examples to illustrate the analysis for a viscoelastic object, the extinction, absorption, and scattering cross sections are provided for an infinite plane progressive wave, infinite nondiffracting Bessel beams, a zero-order spherical quasi-Gaussian beam, and a Bessel-Gauss vortex beam emanating from a finite circular aperture, which reduces to a finite high-order Bessel beam, a finite zero-order Bessel beam, and a finite piston radiator vibrating uniformly with appropriate selection of beam parameters. The similarity with the asymptotic quantum inelastic cross sections is also mentioned.
TL;DR: The analysis of the behavior of several physical parameters of mean-level optical radiation shows that the shape stability of a vortex Bessel beam increases with the topological charge of this beam during its propagation in a turbulent atmosphere.
Abstract: Transformation of vortex Bessel beams during propagation in turbulent atmosphere is theoretically analyzed. Deforming influence of the random inhomogeneity of the turbulent medium on propagation of diffraction-free beams leads to disappearance of their invariant properties. In the given research, features of evolution of the spatial structure of distribution of mean intensity of vortex Bessel beams in turbulent atmosphere are analyzed. A quantitative criterion of possibility of carrying over of a dark central domain by vortex Bessel beams in a turbulent atmosphere is derived. The analysis of the behavior of several physical parameters of mean-level optical radiation shows that the shape stability of a vortex Bessel beam increases with the topological charge of this beam during its propagation in a turbulent atmosphere.