Orbital angular momentum (OAM) entanglement is investigated in the Bessel-Gauss (BG) basis. Having a readily adjustable radial scale, BG modes provide a more favourable basis for OAM entanglement over Laguerre-Gaussian (LG) modes. The OAM bandwidth in terms of BG modes can be increased by selection of particular radial modes and leads to a flattening of the spectrum. The flattening of the spectrum allows for higher entanglement. We demonstrate increased entanglement in terms of BG modes by performing a Bell-type experiment and violating the appropriate Clauser Horne Shimony Holt (CHSH) inequality. In addition, we reconstruct the quantum state of BG modes entangled in high-dimensions.
The exact analytical solution of Maxwell equations for a Bessel light beam scattered by a sphere is found. Scattered power, stored energy and a generalized Q factor as a function of frequency, the sphere radius, permittivity, and the Bessel beam angle are found. On the base of this solution, modes and pseudo-modes of a dielectric sphere are extracted by calculation of the generalized Q factor. It is shown that an appropriate choice of Bessel beam parameters can provide excitation of a single given mode and an unlimited value of the radiative Q factor of pseudo-modes.
Scalar Bessel beams are derived both via the wave equation and via diffraction theory. While such beams have a group velocity that exceeds the speed of light, this is a manifestation of the "scissors paradox" of special relativty. The signal velocity of a modulated Bessel beam is less than the speed of light. Forms of Bessel beams that satisfy Maxwell's equations are also given.
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 mec…
Contrarily to a common belief, any beam of light possesses to a some extent the ability to "reconstruct itself" after hitting an obstacle. The celebrated Arago spot phenomenon is nothing but a manifestation of this property. In this work we analyze the self-healing mechanism from both a mathematical and a physical point of view, eventually finding a new expression for the minimum reconstruction distance, which is valid for \emph{any} kind of beam, including Gaussian ones. Finally, a witness function that quantify the self-reconstruction capability of a beam is proposed and tested. The results …
We report on a versatile method to compensate the linear attenuation in a medium, independently of its microscopic origin. The method exploits diffraction-limited Bessel beams and tailored on-axis intensity profiles which are generated using a phase-only spatial light modulator. This technique for compensating one of the most fundamental limiting processes in linear optics is shown to be efficient for a wide range of experimental conditions (modifying the refractive index and the attenuation coefficient). Finally, we explain how this method can be advantageously exploited in applications rangi…
Recently, the use of Bessel beams in evaluating the possibility of using them for a new generation of GPR (ground penetrating radar) systems has been considered. Therefore, an analysis of the propagation of Bessel beam in conducting media is worthwhile. We present here an analysis of this type. Specifically, for normal incidence we analyze the propagation of a Bessel beam coming from a perfect dielectric and impinging on a conducting medium, i.e. the propagation of a Bessel beam generated by refracted inhomogeneous waves. The remarkable and unexpected result is that the incident Bessel beam do…
Recently, the use of Bessel beams in evaluating the possibility of using them for a new generation of GPR (ground penetrating radar) systems has been considered. Therefore, an analysis of the propagation of Bessel beam in conducting media is worthwhile. We present here an analysis of this type. Specifically, for normal incidence we analyze the propagation of a Bessel beam coming from a perfect dielectric and impinging on a conducting medium, i.e. the propagation of a Bessel beam generated by refracted inhomogeneous waves. The remarkable and unexpected result is that the incident Bessel beam do…
College of Physics, Hangzhou Dianzi University（杭州电子科技大学物理学院）；State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University（山西大学光电研究所量子光学与量子光学器件国家重点实验室）；Zhejiang Key Laboratory of Quantum State Control and Optical Field Manipulation, Hangzhou Dianzi University（杭州电子科技大学浙江省量子态调控与光场操控重点实验室）
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.
Collapse of a Gaussian beam in self-focusing Kerr media arrested by nonlinear losses may lead to the spontaneous formation of a quasi-stationary nonlinear unbalanced Bessel beam with finite energy, which can propagate without significant distortion over tens of diffraction lengths, and without peak intensity attenuation while the beam power is drastically diminishing.
We present a study of radially and azimuthally polarized Bessel-Gauss beams in both the paraxial and nonparaxial regimes. We discuss the validity of the paraxial approximation and the form of the nonparaxial corrections for Bessel-Gauss beams. We show that, independently from the ratio between the Bessel aperture cone angle $\vartheta_0$ and the Gauss beam divergence $θ_0$, the nonparaxial corrections are always very small and therefore negligible. Explicit expressions for the nonparaxial vector electric field components are also reported.
We propose and experimentally demonstrate a novel experimental scheme to generate high peak power, segmented, smooth, zero-order Bessel beams with tunable range. Illuminating the axicon with hollow Gaussian beams (HGBs) of different orders we have generated Bessel beams of varying range at different positions away from the axicon. The presence of dark core at the center of the HGBs removes the effect of imperfection in the axicon tip. As a result, the entire power of the input beam is transformed into zero-order Bessel beam without any on-axis intensity modulation. We observe the decrease in r…
We present a fully relativistic analysis of Bessel beams revealing some noteworthy features that are not explicit in the standard description. It is shown that there is a reference frame in which the field takes a particularly simple form, the wave appearing to rotate in circles. The concepts of polarization and angular momentum for Bessel beams is also reanalyzed.
We present a ring aperture with independently switchable segments for the three-dimensional control of quasi propagation invariant beams. We demonstrate that our liquid crystal design concept preserves coherence and generates the Bessel beam structure.
Bessel beams are non-diffracting light structures, which can be produced with simple tabletop optical elements such as axicon lenses or ring spatial filters and coherent laser beams. One of their main characteristic is that Bessel beams maintain their spatial characteristics after meters of propagation. In this paper we demonstrate a system and method for generating Bessel beams from amorphous speckle patterns, exploiting adaptive optimization by a spatial light modulator. These speckles are generated by light modes transmitted through a scattering curtain and selected by a ring shaped filter.…
The oblique incidence of a Bessel beam on a dielectric slab with refractive index n1 surrounded by a medium of a refractive index n>n1 may be studied simply by expanding the Bessel beam into a set of plane waves forming the same angle with the axis of the beam. In the present paper we examine a Bessel beam that impinges at oblique incidence onto a layer in such a way that each plane-wave component impinges with an angle larger than the critical angle.
Non-separable superpositions of polarization and spatial modes in Gaussian beams have shown rich patterns of spatially-variable polarization encoded in the transverse mode of the light, with the potential for many applications. In this work we investigate encoding similar types of superpositions in Bessel beams. We find that analogously to what is observed in Poincaré-Gaussian beams, Poincaré-Bessel beams can be produced by Bessel-mode superpositions. They show polarization patterns with disclinations in the orientation of the polarization ellipse and mappings of the Poincaré sphere onto the t…
We experimentally measured the self-healing of the spatially inhomogeneous states of polarization of radial and azimuthal polarized vector Bessel beams. Radial and azimuthal polarized vector Bessel beams were generated via a digital version of Durnin's method, using a spatial light modulator in concert with a liquid crystal $q$-plate. As a proof of principle, their intensities and spatially inhomogeneous states of polarization were measured using Stokes polarimetry as they propagated through two disparate obstructions. It was found, similar to their intensities, the spatially inhomogeneous sta…
We present a method for the realization of radially and azimuthally polarized nonparaxial Bessel beams in a rigorous but simple manner. This result is achieved by using the concept of Hertz vector potential to generate exact vector solutions of Maxwell's equations from scalar Bessel beams. The scalar part of the Hertz potential is built by analogy with the paraxial case as a linear combination of Bessel beams carrying a unit of orbital angular momentum. In this way we are able to obtain spatial and polarization patterns analogous to the ones exhibited by the standard cylindrically polarized pa…
We report on an interferometry-based measurement of the phase and group velocities of optical Bessel beams, providing confirmation of their superluminal character in the non-diffractive region. The measurements were performed in free space with a continuous wave laser and femtosecond pulses for phase and group velocities respectively. The Bessel beams were produced using a conical mirror.
The Bessel-Gaussian modal spectrum, generated in spontaneous parametric down-conversion of a Gaussian pump beam, is considered. This is done by first deriving a general expression for the true probability of detecting specific transverse spatial modes for the pairs of photons generated in a spontaneous parametric down-conversion process. These expressions are applied for Bessel-Gaussian modes in type I phase matching with collinear, degenerate down-converted beams. The result shows that a broad orbital angular momentum spectrum can be obtained for Bessel-Gaussian modes under certain conditions…
The aim of this paper is the study of the propagation of a Bessel beam through two absorbing layers, limited by two different half-spaces. Our approach will be based on the scalar analysis, since this analysis was proved to be an excellent approximation of the vectorial field which describes a Bessel beam.
We predict that Bessel-like beams of arbitrary integer order can exhibit a tunable self-similar behavior (that take an invariant form under suitable stretching transformations). Specifically, by engineering the amplitude and the phase on the input plane in real space, we show that it is possible to generate higher-order vortex Bessel-like beams with fully controllable radius of the hollow core and maximum intensity during propagation. In addition, using a similar approach, we show that it is also possible to generate zeroth order Bessel-like beams with controllable beam width and maximum inten…
A unified description of the free-space cylindrical vector beams is presented, which is an integral transformation solution to the vector Helmholtz equation and the transversality condition. The amplitude 2-form of the angular spectrum involved in this solution can be arbitrarily chosen. When one of the two elements is zero, we arrive at either transverse-electric or transverse-magnetic beam mode. In the paraxial condition, this solution not only includes the known $J_1$ Bessel-Gaussian vector beam and the axisymmetric Laguerre-Gaussian vector beam that were obtained by solving the paraxial wa…
The effect of the Kerr nonlinearity on linear non-diffractive Bessel beams is investigated analytically and numerically using the nonlinear Schrödinger equation. The nonlinearity is shown to primarily affect the central parts of the Bessel beam, giving rise to radial compression or decompression depending on whether the nonlinearity is focusing or defocusing, respectively. The dynamical properties of Gaussian-truncated Bessel beams are also analysed in the presence of a Kerr nonlinearity. It is found that although a condition for width balance in the root-mean-square sense exists, the beam pro…
We describe a new class of propagation-invariant light beams with Fourier transform given by an eigenfunction of the quantum mechanical pendulum. These beams, whose spectra (restricted to a circle) are doubly-periodic Mathieu functions in azimuth, depend on a field strength parameter. When the parameter is zero, pendulum beams are Bessel beams, and as the parameter approaches infinity, they resemble transversely propagating one-dimensional Gaussian wavepackets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an operator which interpolates between the squared angular momentum ope…
Goos-Haenchen and Imbert-Fedorov shifts are diffractive corrections to geometrical optics that have been extensively studied for a Gaussian beam that is reflected or transmitted by a dielectric interface. Propagating in free space before and after reflection or transmission, such a Gaussian beam spreads due to diffraction. We address here the question how the Goos-Haenchen and Imbert-Fedorov shifts behave for a ``nondiffracting'' Bessel beam.
Bessel beams generated with non-ideal axicons are affected by aberrations. We introduce a method to retrieve the complex amplitude of a Bessel beam from intensity measurements alone, and then use this information to correct the wavefront and intensity profile using a deformable mirror.
In this paper, we show how one can change the stable equilibrium of a particle trapped into an optical tweezer by varying the intensity of superposed Bessel beams with different orders. The gradient forces acting on particles of different radii are determined, and the theoretical results indicates that it is possible to combine Bessel beams in such a way as to manipulate the particle into or out the centre of the beam by exploiting their ring-shaped intensity patterns, without any mechanical displacement of the lasers.
We propose the braiding of optical vortices in a laser beam with more than 2π rotation by superposing Bessel modes with a plane wave. We experimentally demonstrate this by using a Bessel-Gaussian beam and a coaxial Gaussian, and we present measurement of three complete braids. The amount of braiding is fundamentally limited only by the numerical aperture of the system and we discuss how braiding can be controlled experimentally for any number of vortices.
In light filamentation induced by axicon-generated, powerful Bessel beams, the spatial propagation dynamics in the nonlinear medium determines the geometry of the filament channel and hence its potential applications. We show that the observed steady and unsteady Bessel beam propagation regimes can be understood in a unified way from the existence of an attractor and its stability properties. The attractor is identified as the nonlinear unbalanced Bessel beam (NL-UBB) whose inward Hänkel beam amplitude equals the amplitude of the linear Bessel beam that the axicon would generate in linear prop…
We present a unified and extended perspective of Bessel beams, irrespective to their orbital angular momentum (OAM) -- zero, integer or noninteger -- and mode -- scalar or vectorial, and LSE/LSM or TE/TM in the latter case. The unification is based on the integral superposition of constituent waves along the angular-spectrum cone of the beam, and allows to describe, compute, relate, and implement all the Bessel beams, and even other types of beams, in a universal fashion. The paper first establishes the integral superposition theory. Then, it demonstrates the previously unreported existence of…
We present a mathematical formalism describing the propagation of a completely general electromagnetic wave in a birefringent medium. Analytic formulas for the refraction and reflection from a plane interface are obtained. As a particular example, a Bessel beam impinging at an arbitrary angle is analyzed in detail. Some numerical results showing the formation and destruction of optical vortices are presented.
The generation of isolated attosecond pulses with high efficiency and high beam quality is essential for attosec- ond spectroscopy. We numerically investigate the supercontinuum generation in a neutral rare-gas medium driven by a two-color Bessel-Gauss beam. The results show that an efficient smooth supercontinuum in the plateau is obtained after propagation, and the spatial profile of the generated attosecond pulse is Gaussian-like with the divergence angle of 0.1 degree in the far field. This bright source with high beam quality is beneficial for detecting and controlling the microscopic pro…
We investigate the dynamics of spatiotemporal optical waves with one transverse dimension that are obtained as the intersections of the dispersion cone with a plane. We show that, by appropriate spectral excitations, the three different types of conic sections (elliptic, parabolic, and hyperbolic) can lead to optical waves of the Bessel, Airy, and modified Bessel type, respectively. We find closed form solutions that accurately describe the wave dynamics and unveil their fundamental properties.
Nonlinear Bessel beams in self-defocusing media are found to be the natural, non-diffracting background where vortex solitons can be nested, interact and survive for propagation distances that are one order of magnitude larger than in the usual Gaussian or super-Gaussian backgrounds. The dynamics of the vortex solitons approaches that in the ideal, uniform background, preventing vortex spiraling and decay, which eases vortex steering for applications.
In a simple picture, a Bessel beam is viewed as a transverse standing wave formed in the interference region between incoming and outgoing conical waves. Based on this interpretation we propose an optical resonator that supports modes that are approximations to Bessel-Gauss beams. The Fox-Li algorithm in two transverse dimension is applied to confirm the conlcusion
The propagation of a Bessel beam (or Bessel-X wave) is analyzed on the basis of a vectorial treatment. The electric and magnetic fields are obtained by considering a realistic situation able to generate that kind of scalar field. Specifically, we analyze the field due to a ring-shaped aperture over a metallic screen on which a linearly polarized plane wave impinges. On this basis, and in the far field approximation, we can obtain information about the propagation of energy flux and the velocity of the energy.
Spatially accelerating beams that are solutions to the Maxwell equations may propagate along incomplete circular trajectories, after which diffraction broadening takes over and the beams spread out. Taking these truncated Bessel wave fields to the paraxial limit, some authors sustained that it is recovered the known Airy beams (AiBs). Based on the angular spectrum representation of optical fields, we demonstrated that the paraxial approximation rigorously leads to off-axis focused beams instead of finite-energy AiBs. The latter will arise under the umbrella of a nonparaxial approach following …
Optical beams are solutions to the paraxial wave equation (PWE). In this work we report a new, to our knowledge, optical beam. We solve the PWE by using the angular spectrum of plane waves theory in circular cylindrical coordinates. This lead us to solutions expressed in integral form that can be evaluated in closed-form for half-integer order Bessel functions. We show how to implement numerical simulation and how these results are confirmed by experiment. We believe this approach facilitates the analysis and synthesis of generic optical beams.
An alternative method to generate J0 Bessel beams with controlled spatial partial coherence properties is introduced. Far field diffraction from a discrete number of source points on an annular region is calculated. The average for different diffracted fields produced at several rotation angles is numerically calculated and experimentally detected. Theoretical and experimental results show that for this particular case, J0 Bessel beam is a limit when the number of points tends towards infinity and the associated complex degree of coherence is also a function of the number of points.
Apart from a lot of fundamental interest, vector Bessel beams are widely used in optical manipulation, material processing, and imaging. However, the existing description of such beams remains fragmentary, especially when their scattering by small particles is considered. We propose a new general classification of all existing vortex Bessel beam types in an isotropic medium based on the superposition of transverse Hertz vector potentials. This theoretical framework contains duality and coordinate rotations as elementary matrix operations and naturally describes all relations between various be…
Bessel beams are becoming a very useful tool in many areas of optics and photonics, because of the invariance of their intensity profile over an extended propagation range. Finite-Difference-Time-Domain (FDTD) approach is widely used for the modeling of the beam interaction with nanostructures. However, the generation of the Bessel beam in this approach is a computationally challenging problem. In this work, we report an approach for the generation of the infinite Bessel beams in three-dimensional FDTD. It is based on the injection of the Bessel solutions of Maxwell's equations from a cylindri…
We experimentally analyze a Bessel beam produced with a conical mirror, paying particular attention to its superluminal and diffraction-free properties. We spatially characterized the beam in the radial and on-axis dimensions, and verified that the central peak does not spread over a propagation distance of 73 cm. In addition, we measured the superluminal phase and group velocities of the beam in free space. Both spatial and temporal measurements show good agreement with the theoretical predictions.
Bessel beams carry orbital angular momentum (OAM). Opening up of the Hilbert space of OAM for information coding makes Bessel beams potential candidates for utility in data transfer and optical communication. A laser bullet is the ultra-short and tightly-focused analogue of a non-diffracting and non-dispersing laser Bessel beam. Here, we show fully analytically that a Bessel-Bessel laser bullet possesses orbital angular momentum. Analytic investigation of the energy, linear momentum, energy flux, and angular momentum, associated with the fields of a Bessel-Bessel bullet, in an under-dense plas…
We show that in addition to well known Bessel, Hermite-Gauss, and Laguerre-Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of rho. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann-Silberstein vector and the Whittaker represe…
We present a new feasible way to flatten the axial intensity oscillations for diffraction of a finite-sized Bessel beam, through designing a cardioid-like hole. The boundary formula of the cardioid-like hole is given analytically. Numerical results by the complete Rayleigh-Sommerfeld method reveal that the Bessel beam propagates stably in a considerably long axial range, after passing through the cardioid-like hole. Compared with the gradually absorbing apodization technique in previous papers, in this paper a hard truncation of the incident Bessel beam is employed at the cardioid-like hole ed…
For propagating and evanescent vector Bessel beams, we study the singularities of the Poynting vector (Poynting singularities), at which the energy flux density turns to zero. Poynting singularities include all the phase singularities and some of polarization ones (L- and C-points). We reveal the existence conditions and positions of singularities, which are located at cylindrical surfaces around the beam axis. We mark the special case of the evanescent Bessel beam in the form of cylindrical standing wave, that is singular at any spatial point.
A new class of nonparaxial accelerating optical waves is introduced. These are beams with a Bessel-like profile that are capable of shifting laterally along fairly arbitrary trajectories as the wave propagates in free space. The concept expands on our previous proposal of paraxial accelerating Bessel-like beams to include beams with subwavelength lobes and/or large trajectory angles. Such waves are produced when the phase at the input plane is engineered so that the interfering ray cones are made to focus along the prespecified path. When the angle of these cones is fixed, the beams possess a …
This paper describes the partial wave expansion and integral representation of Bessel beams in free space and in the presence of dispersion. The expansion of the Bessel beam wavepacket with constant spectrum is obtained as well. Furthermore, the sum of a triple Legendre polynomial product of same order but different argument follows naturally from the partial wave expansion. The integration of all Bessel beams over all conical angles is shown to have a simple series representation, which confirms the equivalence between the results for both expansion and integral representation.
A simple analytical way of creating superpositions of Bessel-Gaussian light beams with knotted nodal lines is proposed. It is based on the equivalence between the paraxial wave equation and the two-dimensional Schrödinger equation for a free particle. The $2D$ Schrödinger propagator is expressed in terms of Bessel functions, which allows to obtain directly superpositions of beams with a desired topology of nodal lines. Four types of knots are constructed in the explicit way: the unknot, the Hopf link, the Borromean rings and the trefoil. It is also shown, using the example of the figure-eight …
In a recent article [R. R. Alfano and D. A. Nolan, Opt. Commun. 361 (2016) 25] the group velocity reduction below the speed of light in the case of certain Bessel beam pulses has been considered and an idea of its application for a natural optical buffer presented. However, the authors treat the problem as if only one type of Bessel pulse existed, no matter how it is generated. The deficiencies of the article stem from not being familiar with an extensive literature on Bessel pulses, in particular, with a couple of papers published much earlier in the J. Opt. Soc. Am. A, which have studied exa…
