Superposition Of Bessel Beams Research Articles

High-efficiency generation of long-distance, tunable, high-order nondiffracting beams

Nondiffracting beams (NDBs) have presented significant utility across various fields for their unique properties of self-healing, anti-diffraction, and high-localized intensity distribution. We present a versatile and flexible method for generating high-order nondiffracting beams predicated on the Fourier transformation of polymorphic beams produced by the free lenses with tunable shapes. Based on the tunability of the digital free lenses, we demonstrate the experimental generation of various long-distance nondiffracting beams, including Bessel beams, polymorphic generalized nondiffracting beams, tilted nondiffracting beams, asymmetric nondiffracting beams, and specially structured beams generated by the superposition of Bessel beams. Our method achieves efficiency of up to about seven times compared with complex beam shaping methods. The generated NDBs exhibit characteristics of extended propagation distance and high-quality intensity profiles consistent with the theoretical predictions. The proposed method is anticipated to find applications in laser processing, optical manipulation, and other fields.

Evolution of C-point singularities and polarization coverage of Poincaré–Bessel beam in self-healing process

As a vector version of scalar Bessel beams, Poincaré–Bessel beams (PBBs) have attracted a great deal of attention due to their non-diffracting and self-healing properties as well as the presence of polarization singularities. Previous studies of PBBs have focused on cases that consist of a superposition of Bessel beams in orthogonal circular polarization states; here, we present a theoretical and experimental study of PBBs for which the polarization states are taken to be linear, which we call a linear PBB. Using a mode transformation of a full Poincaré beam constructed from linear polarization states, we observe the linear PBB as providing an in-principle infinite number of covers of the Poincaré sphere in the transverse plane and with an infinite number of C-points with positive and negative topological indices. We also study the dynamics of C-point singularities in a linear PBB in the process of self-healing after being obstructed by an obstacle, providing insight into “Hilbert Hotel” style evolution of singularities in light beams. The present study can be useful for imaging in the presence of depolarizing surroundings, studying turbulent atmospheric channels, and exploring the rich mathematical concepts of transfinite numbers.

Deep learning-assisted light sheet holography.

In a novel approach to layer-based holography, we propose a machine learning-assisted light sheet holography-an optimized holography technique which projects a target scene onto sheets of light along the longitudinal planes (i.e. planes perpendicular to the plane of the hologram). Using a convolutional neural network in conjunction with superposition of Bessel beams, we generate high-definition images which can be stacked in parallel onto longitudinal planes with very high fidelity. Our holography system provides high axial resolution and excellent control over the light intensity along the optical path, which is suitable for augmented reality and/or virtual reality applications.

Exact analytical solutions for micrometer structured vortex beams with applications to optical tweezers

Bessel beams are widely known by their non-diffracting properties, i.e. beams that sustain their transverse profile along the propagation. It is also well known that by a suitable superposition of Bessel beams of the same frequency, but different longitudinal wave numbers, amplitudes and phases, it is possible to obtain a resulting beam whose longitudinal intensity pattern can be shaped along the propagation z-axis. Such approach, named Frozen Wave method, can be implemented through discrete or continuous superposition, the latter being more appropriate for obtaining spatially structured beams in micrometer domains. In previous studies, authors constructed analytical solutions for micrometer zero-order (i.e., null topological charge) Frozen Wave and, thereafter, of order one and two, by making use of a topological charge raising operator. In this paper, we present an analytical closed solution for the general case where the topological charge raising operator is applied an arbitrary number of times to a micrometer zero-order Frozen Wave beam, thus generating one of arbitrary topological charge. We also apply the new solutions to model optical tweezers (in Rayleigh regime) considering different light polarizations.

A new criterion for self-healing quantification of structured light beams

This paper introduces a new approach to the self-healing quantification of structured light beams. The self-healing is quantitatively determined by defining a self-healing degree (SHD) and a similarity function based on comparing the intensity distributions of the perturbed and unperturbed beams. In addition, the SHD is employed for two other intensity-based methods to compare the methods’ performance. The Bessel beams (BBs) with integer and fractional topological charges (TCs) are examined numerically to verify the method. Further, the effect of superposition of the Bessel and mirrored BBs with respect to the x-axis on the self-healing property is investigated. The perturbation of the beams is applied using a pair of circular masks as a symmetric perturbation. The propagating of perturbed and unperturbed beams is simulated by the angular spectrum method. The obtained quantitative results are confirmed by the intuitive results and also the accuracy of the proposed method is similar to the other used methods. On the other hand, due to using fewer calculations with respect to them and so is less time-consuming (about 57% and 67% reduction in computational time), it can be used as an adequate alternative method. As a result of this method, it is shown that the superposition of BBs with their mirrored ones is an effective factor to improve the self-healing property, in which the SHD of the superposed beams is more than the BBs for each TC. It is also shown that using the fractional TC beams is another advantageous improvement to increase the SHD.

Coherence Properties and Intensity Distribution of a Partially Coherent Lorentz–Gauss Beam Emerging from the Axicon

The propagation of a partially Lorentz–Gauss beam in a uniform-intensity diffractive axicon is studied according to the Huygens–Fresnel principle, the Hermite–Gaussian expansion of a Lorentz function, and using the stationary phase method. We have derived the intensity equation of a partially coherent Lorentz-Gauss beams propagating through uniform-intensity diffractive axicon, and we proved mathematically that it is the superposition of Bessel beams of various orders after emerging from axicon, using Hermite’s function series and the Bessel function integral formulas. The results show that the intensity distribution of the diffracted beam is the intensity pattern evolved from a Lorentz–Gauss shaped spot into a Gaussian-shaped spot at any position on the focal length of the axicon, and the intensity distribution of a partially Lorentz–Gauss beam generated by an axicon becomes uniform by increasing the beam width and more uniform and constant with the larger coherence width.

Rotating azimuthons in dissipative Kerr media excited by superpositions of Bessel beams

We report the existence of persistently rotating azimuthons in media with self-focusing Kerr and absorption nonlinearities. The nonlinear loss is balanced by power influx from the peripheral reservoir stored in slowly decaying tails of the azimuthons. These modes are excited by a superposition of two Bessel beams with opposite vorticities, $\pm s$, and slightly different conicities. The excited azimuthon exhibits opposite vorticity in its center to that of the input Bessel-beam superposition due to spontaneous inversion of the topological charge in the course of the azimuthon formation. Unlike azimuthons in loss-free media, number $N$ of rotating intensity maxima and $s$ are not mutually independent, being related by $N=2s$. The robustness of the rotating azimuthons is enhanced in comparison to similar static dissipative patterns. They can be excited in almost any transparent material, in the range of intensities for which the nonlinear absorption, induced by multiphoton absorption, is relevant. Close to the ionization threshold, the rotating azimuthons reproduce recently observed helical filaments of light in air and CS$_{2}$.

Diffraction of bessel beams on 2D amplitude gratings—a new branch in the talbot effect study

In this paper, an analytical theory for the diffraction of a Bessel beam of arbitrary order Jl(κr) on a 2D amplitude grating is presented. The diffraction pattern behind the grating turned out to be more complicated in comparison with the classical Talbot effect observed under illumination by a plane wave. In particular, the patterns in the main and fractional Talbot planes under certain conditions that are found in the article, are lattices of ring microbeams, the diameters of which depend on the period of the grating, the diameter of the illuminating beam, the number of the Talbot plane, and the topological charge l. For the rings near the optical axis, the latter reproduces l of the illuminating beam. The diffraction patterns observed in experiments on the Novosibirsk free electron laser behind gratings with periods of 1 to 6 mm, illuminated by zero- to second-order Bessel beams at a radiation wavelength λ = 141 µm, are in good agreement with both theoretical predictions and numerical calculations. We emphasize that an analytical theory based on the scalar theory of diffraction perfectly describes resulting diffraction patterns with hole diameters down to 0.25 mm, which are less than 2λ, when, formally speaking, the scalar theory becomes incorrect. Since the Laguerre–Gaussian beams can be represented as a superposition of Bessel beams, results of this paper can be applied to analysis of the Talbot effect with the Laguerre–Gaussian beams. A regular lattice of ring-like vortex microbeams, which can be formed in the Talbot planes, can be applied, for example, to creation of a lattice of optical traps, transmission of orbital angular momentum to elements of micromechanical devices, and to launching surface plasmon polaritons on an array of metal wires by ‘the end-fire coupling technique’.

X Wave Radiator Implemented With 3-D Printed Metamaterials

A radiator is presented, capable of generating paraxial X waves over a 50% fractional bandwidth (7.5–12.5 GHz) in its radiative near-field. X waves are localized pulses formed by the superposition of Bessel beams with a common cone angle. Quasiconformal transformation optics, a 2-D variant of transformation optics, is employed to find the inhomogeneous, isotropic dielectric constant profile needed to convert the radiation of a monopole into a paraxial Bessel beam. An impedance matching layer is applied to reduce reflections at the air-dielectric interface of the device. The transformation and impedance matching regions are implemented using rotationally symmetric metamaterial unit cells that yield a spatially varying effective dielectric constant. The resulting design is fabricated through 3-D printing, by combining parts made from three low-loss dielectric filaments. The device is experimentally measured, and shows good agreement with simulations. Its ability to generate paraxial X waves when excited by a broadband pulse is verified.

Bessel beam converter based on c-cut lithium niobate crystal

The transformation of the 0th order Bessel beam into the 2nd order Bessel vortex beam is experimentally investigated. A c-cut lithium niobate crystal with a thickness of 514μm was used to convert the beams. It is shown that for a diffractive axicon with a period of 3.5μm illuminated by semiconductor laser (λ=639 nm), the output beam is a 2nd order Bessel vortex beam. It is shown that for a diffractive axicon with a period of 4μm illuminated by heliumneon laser (λ=632.8 nm), the output beam is a superposition of Bessel beams of the 0th and 2nd orders. The experimental results are in good accordance with mathematical modeling.

On the overall polarisation properties of Poincaré beams

We analyse the polarisation properties of Poincaré beams. We consider different configurations, such as Laguerre–Poincaré (LP), Bessel–Poincaré (BP), and Lambert–Poincaré (LaP) beams. The former considers the well-known cylindrical vector beams and full-Poincaré beams produced by a collinear superposition of two Laguerre–Gauss beams with orthogonal polarisations. For this configuration, we describe the Stokes statistics and overall invariant parameters. Similarly, BP beams are produced by the collinear superposition of Bessel beams with orthogonal polarisations. We describe their properties under propagation and show that they behave as a free-space polarisation attractor transforming elliptical polarisations to linear polarisations. We also propose a novel type of full Poincaré pattern, one which is generated by a Lambert projection of the Poincaré sphere on the transverse plane, and hence we call them LaP. This configuration, contrary to the LP, provides a finite region containing all polarisation states uniformly distributed on the Poincaré sphere.

Metrics for Localized Beams and Pulses

Geometric and energy metrics are here proposed to gauge the confinement properties of localized electromagnetic beams (e.g., Bessel beams) and pulses (e.g., X-waves) in both the nondispersive and dispersive cases. As is well known, the frequency superposition of Bessel beams sharing the same axicon angle leads to the generation of tightly bounded wavepackets, namely X-waves, with remarkable spatio–temporal confinement properties. In the nondispersive case, the axicon angle does not change with frequency, and these features can be predicted by closed-form expressions. Remarkably, it is found that the volumetric confinement of such pulses can be optimized for a specific axicon angle. Since the geometric metrics do not account for the unavoidable presence of energy tails outside the main spot, energy metrics are here introduced and compared with the purely geometric definition. Finally, the dispersive case is investigated, accounting for the typical wavenumber dispersion of radial waveguides. As a result, the axicon angle changes with the frequency, and fundamental limits exist for the maximum theoretical fractional bandwidth. Interestingly, design criteria based on approximate analytical expressions are derived to optimize the confinement, even in the dispersive case.

Scattering of focused beams by spheres: Understanding the high-frequency angular structure

Previous work on scattering of Bessel beams by spheres [P. L. Marston, J. Acoust. Soc. Am. 122, 247–252 (2007)] and recent work expanding linear focused beams in terms of a Bessel beam superposition [T. D. Daniel et al., J. Acoust. Soc. Am. 144, 3076–3083 (2018)] provide a framework for exploring the scattering of focused beams by spheres. In analogy with the work done on the scattering of Gaussian beams by spheres [P. L. Marston, J. Acoust. Soc. Am. 129, 1773–1782 (2011)], the angular scattering pattern of a sphere in a focused beam is calculated from a partial wave series with appropriate weighting factors (also known as beam shape coefficients). The high-frequency pattern for rigid spheres is found to have distinctive shadow boundaries in both the forward and backward scattering hemispheres. The sphere is centered on the beam axis. A physical optics model of the scattering reproduces the location and general shape of the shadow boundaries in both scattering hemispheres, confirming the calculated beam shape coefficients for a spherical cap focused source. [Work supported by ONR.]

Assessing the validity of the localized approximation for discrete superpositions of Bessel beams

This paper analyzes the validity of the integral localized approximation for discrete superpositions of arbitrary order Bessel beams—known as frozen waves. It is motivated by the fact that such an approximation has been adopted in previous works under the framework of the generalized Lorenz–Mie theory for electromagnetic scattering by microparticles when illuminated by paraxial frozen waves. The results presented here give support to those previous works, but they also serve as a warning for future research in the field in the sense that the localized approximation can be of limited validity for nonparaxial laser beams when the axicon angle is not small enough, or when a net topological charge exists.

Microwave generation of X-waves by means of a planar leaky-wave antenna

We analyze and experimentally demonstrate the possibility of generating X-waves at microwave and millimeter-wave frequencies by means of a partially open radial parallel-plate waveguide antenna. The structure is azimuthally symmetric and fed in the center by means of a simple vertical coaxial probe, which excites a cylindrical leaky wave. Radially periodic annular slots etched in the upper metal plate allow the propagation of a backward leaky wave, as required for generating Bessel beams in the near-field region. Since X-waves are polychromatic superpositions of Bessel beams, the wavenumber frequency dispersion of the relevant leaky mode is accounted for by the antenna design. In particular, a dispersion-engineering approach is used to properly select the operating fractional bandwidth for the antenna. Even if the beneficial effect of a large bandwidth is partially neutralized by the dispersive character of the resulting pulse, this being more prominent as the bandwidth increases, the experimental results demonstrate the capability of this simple planar design of generating X-waves in the microwave regime. The antenna can be of interest for the design of next-generation medical imaging devices, for non-destructive evaluations, as well as for wideband near-field secure communications and wireless power transfer systems at microwaves and millimeter waves.

Bessel beam superposition for analyzing off-axial scattering, forces, and torques

Illumination of an arbitrary-order Bessel beam on an off-axial object is expanded as illuminations of a series of Bessel beam components of different orders on an axially centered object [L. Zhang, J. Acoust. Soc. Am. 143(5), 2796–2800 (2018)]. The expansion follows from a parallel-axis relation that is derived as a generization of Graf's addition theorem of Bessel functions. The off-axial Bessel beam expansion enables to understand the off-axial scattering, forces, and torques in terms of superposing that in the on-axial illuminations by the series of beam components. The superposition reveal negative axial radiation force and negative radiation torque in terms of contributions of different beam components in the on-axial configuration. The exact locations of the object for torque direction reversal are analytically identified for a small particle. The superposition also reveals that the off-axial radiation torque on a small particle exists even for higher-order vortex beams or vortices.

Coaxial superposition of Bessel beams by discretized spiral axicons

BackgroundA diffractive spiral axicon can be used for the generation of a vortex beam with orbital angular momentum. The coaxial superposition of multiple vortices can generate a complex field with off-axis optical vortices. These fields are known as optical vortex lattices. In general, this superposition is done by the use of spatial light modulators. Discretization of the continuous spiral in radial and azimuthal direction introduces additional degrees of freedom and thus, more complex fields are generated.Methods and ResultsHere, we discuss the basic theory for discretized spiral axicons. Then, as an example, we consider a discretized multi-pronged element where radial and azimuthal coordinates are discretized. Simulations of the near-field distribution show the occurrence of additional off-axis vortices with anisotropic character. The number of off-axis vortices depends on the number of discretization steps in azimuthal direction. Theory is confirmed by experiments. The diffractive element used in the experiments was fabricated lithographically. For instance, a Shack-Hartmann sensor was used to measure orbital momentum of on- and off-axis vortices.ConclusionOptical vortex fields can be achieved due to the discretization of the continuous spiral axicon. The resulting field distribution can be seen as superposition of different non-diffracting fundamental vortex modes.

Superposition of Bessel beams: geometrical wavefronts, light rays, caustic, intensity patterns and experimental generation

Recently, it has been shown that if S(r, θ, ϕ) is a two-parameter solution of the eikonal and Laplace equations, then is an exact solution of the scalar wave equation in an isotropic optical medium. In particular, if we take , the resulting solution is a superposition of non-diffracting beams in free space with different cone angle θ for the k vector. In this paper, we analyze the geometrical properties of a particular example, when O(θ, ϕ) = 1 and g(θ,ϕ) = m(ϕ − π/2), which corresponds to the superposition of non-diffracting Bessel beams. That is, we compute its wavefronts, light rays and caustic. The Bessel beam is a special case of a non-diffracting beam, first proposed by Durnin in 1986, the superposition of two Bessel beams have been studied before in order to minimize the size of the central core. In this work, we present a beam resulting from the continuous superposition of Bessel beams with the same order m and different cone angle. To show the behavior of this beam, we simulate the intensity pattern at some different planes perpendicular to the Z and Y axes. Finally, we generate this beam experimentally, and we show that the caustic determines qualitatively the maximum of the corresponding intensity pattern.

Arbitrary Control of Polarization and Intensity Profiles of Diffraction-Attenuation-Resistant Beams along the Propagation Direction

We report on the theory and experimental generation of a class of diffraction-attenuation-resistant beams with state of polarization (SoP) and intensity that can be controlled on demand along the propagation direction. This is achieved by a suitable superposition of Bessel beams, whose parameters are systematically chosen based on closed-form analytic expressions provided by the Frozen Waves (FWs) method. Using an amplitude-only spatial light modulator, we experimentally demonstrate three scenarios. In the first, the SoP of a horizontally polarized beam evolves to radial polarization and is then changed to vertical polarization, with the beam intensity held constant. In the second, we simultaneously control the SoP and the longitudinal intensity profile, which was chosen such that the beam's central ring can be switched-off over predefined space regions, thus generating multiple foci with different SoP and at different intensity levels along the propagation. Finally, the ability to control the SoP while overcoming attenuation inside lossy fluids is shown experimentally for the first time in the literature (to the best of our knowledge). Therefore, we envision our proposed method to be of great interest for many applications, such as optical tweezers, atom guiding, material processing, microscopy, and optical communications.

Axial sub-Fourier focusing of an optical beam.

We demonstrate experimentally the generation of an optical beam having an axial focusing that is narrower than the Fourier limit. The beam is constructed from a superposition of Bessel beams with different longitudinal wave vectors, realizing a super-oscillatory axial intensity distribution. Such beams can be useful for microscopy and for optical particle manipulation.