Axisymmetric Beam Research Articles

Kernel Density Estimators for Axisymmetric Particle Beams

Bright beams are commonly represented by sampled data in the numerical algorithms used to simulate their properties. However, in these calculations and the analyses of their outputs, the beam’s density is sometimes required and must be calculated from the samples. Axisymmetric beams, which possess a rotational symmetry and are naturally expressed in polar coordinates, pose a particular challenge to density estimators. The area element in polar coordinates shrinks as the radius becomes small, and weighting the samples to account for their reduced frequency may cause unwelcome artifacts. In this work, we derive analytical expressions for two kernel density estimators, which solve these problems in the spatial coordinates and in the transverse phase space. We show how the kernels can be found by averaging the Gaussian kernel in Cartesian coordinates over the polar angle and demonstrate their use on test problems. These results show that particle beam symmetries can be taken advantage of in density estimation while avoiding artifacts.

A Null-Filled Axisymmetric Secant-Square-Shaped Beam Antenna for Unmanned Aerial Vehicles

In this letter, an antenna with a null-filled axisymmetric secant-square-shaped beam for the unmanned aerial vehicle (UAV) is proposed. The proposed antenna consists of four monopoles with sequentially rotated feed (SRF) and a coupling-fed grooved circular patch. The radiation power ratio of the monopoles and the patch is used to shape the axisymmetric beam. The effect of the ground is also investigated to operate positively on beam shaping. A prototype was fabricated to validate the method and the performance of the proposed antenna. The gain curves of the proposed antenna coincide well with the secant-squared (sec <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) curve from θ = 0° to θ = 60° in all azimuthal angles with root mean square error (RMSE) below 1.35 dB in 5.5 GHz - 5.7 GHz.

Low-Profile Conical Beam Array Antenna Based on Concentric Annular Planar Inverted-F Antenna Elements

A planar conical beam array antenna with low profile, high gain, and low side lobe level (SLL) is proposed in this letter. The array is composed of three concentric annular planar inverted-F antenna (APIFA) elements, each of which is excited by uniform distributed feeding points with the same excitation amplitude and phase and thus obtain an axisymmetric conical beam. Shorting pins and cambered slots are employed on the APIFA element to realize impedance matching for a given number of feeding points. Synthesis of conical beam for the array can be flexibly realized by adjusting the excitation amplitudes and phases of the elements. A prototype with a low profile of 0.11 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is designed, fabricated, and measured. Measured results show that the prototype achieves 11.2% (4.71 ∼ 5.27 GHz) -10-dB impedance bandwidth. Within the impedance bandwidth, a stable gain of 11.3 to 12.7 dBi, a stable beam-pointing angle (BPA) of about 15°, and a low SLL of less than -12.5dB is obtained, indicating good conical beam performance.

New submodels describing attenuation of the ultrasonic beams in 3-d cubic nonlinear medium in the absence of dissipation

For a three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, we studied the attenuation of ultrasonic beams after the formation of shock fronts. Earlier, in the work of one of the authors for this model, submodels were obtained and studied, described by non-stationary solutions, invariant with respect to some three-dimensional subgroups of the main ten-parameter group of the differential equation that defines this model. In our paper we obtained and studied four non-stationary submodels of this model that are invariant with respect to four-parameter subgroups of the main group of this differential equation. These submodels are new submodels and have not been previously noted in the literature. They are given by invariant solutions of rank 1. Among these 4 submodels, 2 submodels describe axisymmetric ultrasonic beams, the remaining 2 describe one-dimensional ultrasonic beams. The search for invariant solutions that define these 4 submodels is reduced to solving of nonlinear integral equations, the implicit solutions of which are obtained in the form of nonlinear algebraic equations containing transcendental functions. These submodels are used to study the propagation of ultrasonic beams, for which either the acoustic pressure and its rate of change or the acoustic pressure and its gradient are given at the initial time at a fixed point. Conditions are obtained that ensure the existence and uniqueness of solutions to boundary value problems describing these processes. This makes it possible to correctly carry out numerical calculations in the study of these processes. As a result of the numerical solution of these boundary value problems for some values of the parameters characterizing these processes, pressure distribution graphs were obtained. In all cases, ultrasonic beams are weaken monotonically with time and completely fade away in a finite time. At each point, the time of complete attenuation of ultrasonic beams is found.The obtained and studied new submodels are another step towards the creation of a database of physically significant submodels of the three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, which describes the attenuation of ultrasonic beams after the formation of shock fronts.

A Multi-Spark Electrohydraulic Shock Wave Generator with Adjustable Pressure Field Distribution and Beam Steering Capability.

All clinical shock wave lithotripters produce an axisymmetric acoustic field without accounting for the anatomic features of the kidney or respiratory motion of the patient. This work presents a steerable and adjustable focusing electrohydraulic (SAFE) shock wave generator design with variable beam size and shape. 90 electrohydraulic transducers are mounted concentrically on a spherical basin with adjustable connection to individual transducers. Each transducer consists of 45 3D-printed titanium microelectrodes embedded in epoxy with a tip diameter of 0.3 mm. All the transducers are arranged in 5 concentric rings and sub-divided into 6 sectors. By changing the connections of individual transducers, the focused pressure field produced by the transducer array can be either axisymmetric with a -6 dB focal width of 14.8 mm in diameter, or non-axisymmetric with a long axis of 22.7 mm and a short axis of 15.1 mm. The elongated beam produces a peak positive pressure of 33.7±4.1 MPa and comminution efficiency of 42.2±3.5%, compared to 36.2±0.7 MPa and 28.6±6.1% for axisymmetric beam after 150 pulses at 20 kV. We have demonstrated that the SAFE shock wave generator can produce an elongated non-axisymmetric pressure field with higher stone comminution efficiency. The SAFE shock wave generator may provide a flexible and versatile design to achieve accurate, stable, and safe lithotripsy for kidney stone treatment.

Transverse Particle Trapping Using Finite Bessel Beams Based on Acoustic Metamaterials

We investigate the transverse trapping force acting on a small particle on or off the central axis of zero-order finite Bessel beams. Combining the simulated fields with the Gorkov force potential, the transverse trapping behaviors for small objects are analyzed, and the reversal of the trapping behaviors is recovered when varying the paraxiality parameter of the Bessel beam. The results prove the possibility of using axisymmetric Bessel beams to trap both dense and stiff particles as well as light and soft ones. The particles can be trapped at the maximum central pressure in the main lobe and also the maximum pressure of the other transverse locations. A lossy acoustic metastructure with the ability to decouple the phase and amplitude modulations is used to generate the desired sound field, which is used to trap a foam ball as an example. Our research opens a promising avenue to the design and development of simplified acoustic tweezers.

Stabilization of Axisymmetric Airy Beams by Means of Diffraction and Nonlinearity Management in Two-Dimensional Fractional Nonlinear Schrödinger Equations

The propagation dynamics of two-dimensional (2D) ring-Airy beams is studied in the framework of the fractional Schrödinger equation, which includes saturable or cubic self-focusing or defocusing nonlinearity and Lévy index ((LI) alias for the fractionality) taking values 1≤α≤2. The model applies to light propagation in a chain of optical cavities emulating fractional diffraction. Management is included by making the diffraction and/or nonlinearity coefficients periodic functions of the propagation distance, ζ. The management format with the nonlinearity coefficient decaying as 1/ζ is considered too. These management schemes maintain stable propagation of the ring-Airy beams, which maintain their axial symmetry, in contrast to the symmetry-breaking splitting instability of ring-shaped patterns in 2D Kerr media. The instability driven by supercritical collapse at all values α&lt;2 in the presence of the self-focusing cubic term is eliminated, too, by the means of management.

Bayesian inference of axisymmetric plasma equilibrium

We present a Bayesian method for inferring axisymmetric plasma equilibria from the magnetic field and plasma pressure measurements. The method calculates all possible solutions for plasma current and pressure distributions consistent with the measurements and magnetohydrodynamic (MHD) force balance. Toroidal plasma current and magnetic field coils are modelled as a set of axisymmetric current-carrying solid beams. The other parameters such as plasma pressure and poloidal current flux are given as a function of poloidal magnetic flux, which is determined given a 2D current distribution. Plasma pressure and poloidal current flux profiles are modelled as Gaussian processes whose smoothness is optimally chosen based on the principle of Occam’s razor. To find equilibrium solutions, we introduce an MHD force balance constraint at every plasma current beam as a part of the prior knowledge. Given all these physical quantities, predictions calculated by the predictive (forward) models for diagnostics are compared to the observations. The high dimensional complex posterior probability distribution is explored by a new algorithm based on the Gibbs sampling scheme.

Description of the attenuation of invariant ultrasonic beams after formation of the shock fronts in a cubic nonlinear medium in the absence of dissipation

In the framework of a three-dimensional generalized Khokhlov–Zabolotskaya–Kuznetsov model of nonlinear hydroacoustics in a cubic nonlinear medium in the absence of dissipation, the attenuation of ultrasonic beams after the formation of shock fronts is investigated. For the main group of the equation of the investigated model, an optimal system of three-parameter subgroups that do not contain the time transport transformation is constructed. This system contains 68 subgroups. As a result, it is possible to construct 68 nonstationary submodels that are invariant with respect to these subgroups. We have built and studied five submodels. For three submodels, equations are obtained that implicitly determine the acoustic pressure in ultrasonic beams. Namely, for two of them, the pressure is implicitly determined from transcendental equations, and for one submodel, the pressure is implicitly determined from an algebraic equation. Among these five sub-models, two sub-models describe axisymmetric ultrasonic beams, two others describe planar ultrasonic beams, and one describes a one-dimensional ultrasonic beam. Using these submodels, the propagation of ultrasonic beams is investigated, for which either the acoustic pressure and the rate of its change, or the acoustic pressure and its gradient are set at the initial moment of time at a fixed point. The existence and uniqueness of solutions to boundary value problems describing these processes is established. For some values of the parameters characterizing these processes, using a numerical solution, graphs of the distribution of the dynamic component of the acoustic pressure are obtained. From these graphs it follows that the ultrasonic beams monotonically weaken with a time and completely disappear in a finite time.

Generation of random soliton-like beams in a nonlinear fractional Schrödinger equation.

We investigate the generation of random soliton-like beams based on the Kuznetsov-Ma solitons in a nonlinear fractional Schrödinger equation (NLFSE). For Lévy index α = 1, the Kuznetsov-Ma solitons split into two nondiffracting beams during propagation in linear regime. According to the different input positions of the Kuznetsov-Ma solitons, the diffraction-free beams can be divided into three different types: bright-dark, dark-bright and bright-bright beams. In the nonlinear regime, the Kuznetsov-Ma solitons can be evolved into random soliton-like beams due to the collapse. The number of soliton-like beams is related to the nonlinear coefficient and the Lévy index. The bigger the nonlinear coefficient, the more beams generated. Moreover, the peak intensity of soliton-like beams presents a Gaussian distribution under the large nonlinear effect. In practice, the evolution of KM soliton can be realized by a plane wave with a Gaussian perturbation, which can be confirmed that they have the similar dynamics of propagation. In two dimensions, the plane wave with a Gaussian perturbation can be evolved into a bright-dark axisymmetric ring beam in the linear regime. Under the nonlinear modulation, the energy accumulates to the center and finally breaks apart into random beam filaments.

Distributions of lateral earth pressure behind rock-socketed circular diaphragm walls considering radial deflection

The distributions of lateral earth pressure behind rock-socketed circular diaphragm walls are investigated. Two different motion mechanisms of the wall during excavation and their corresponding simplified formulas of radial deflection are considered. Then, combined with the motion modes and radial deflection of circular diaphragm wall during excavation, a semi-analytical solution for the lateral earth pressure by using three-dimension horizontal and vertical stress equilibrium for the differential element of the cohesive surrounding soil is introduced. Furthermore, the applicability of the proposed method is validated by comparison with the results with an axisymmetric beam on a nonlinear elastic foundation method, as well as a field case.

Airy–Gaussian vortex beams in the fractional nonlinear-Schrödinger medium

We address the propagation of vortex beams with the circular Airy–Gaussian shape in a ( 2 + 1 )-dimensional optical waveguide modeled by the fractional nonlinear Schrödinger equation. Systematic analysis of autofocusing of the beams reveals a strongly non-monotonous dependence of peak intensity in the focal plane on the corresponding Lévy index α , with a strong maximum at α ∼ 1.4 . Effects of the nonlinearity strength, the ratio of widths of the Airy and Gaussian factors in the input, as well as the beams’ vorticity on the autofocusing dynamics are explored. In particular, multiple autofocusing events occur if the nonlinearity is strong enough. Under the action of azimuthal modulational instability, an axisymmetric beam may split into a set of separating bright spots. In the case of strong fractality (for α close to one), the nonlinear beams self-trap, after the first instance of autofocusing, into a breathing vortical quasi-soliton. Radiation forces induced by the beams’ field are considered too, and a capture position for a probe nanoparticle is thus identified.

Generalized Lorenz-Mie theory in the analysis of longitudinal photophoresis of arbitrary-index particles: On-axis axisymmetric beams of the first kind

The uniform plane wave is the reference solution of Maxwell’s equations when it comes to rigorous theoretical investigations of photophoretic forces from the Mie theory. Here, we address the theory of photophoresis for a more general class of on-axis axisymmetric beams of the first kind. To do so, the generalized Lorenz-Mie theory (GLMT) is introduced for the first time in the calculation of analytic and closed-form expressions for the asymmetry factor assuming arbitrary-index micro-sized spheres of arbitrary size parameters. The behavior of the asymmetry factor is illustrated for non-volatile particles illuminated by Gaussian and circularly symmetric zero-order Bessel beams. The theory here presented is valid for arbitrary size parameters and is a first attempt towards the incorporation of real optical wave fields into photophoresis of absorbing particles using the GLMT for arbitrary-shaped beams.

Theoretical prediction of photophoretic force on a dielectric sphere illuminated by a circularly symmetric high-order Bessel beam: on-axis case.

Compared to the experimental progresses made in the optical trapping of aerosol particles in gaseous media by means of photophoretic forces, the theoretical analysis of photophoretic forces is less developed, the underlying mechanisms being yet not fully understood. In this paper, theoretical derivations of photopheresis of a dielectric sphere in gaseous media illuminated by a circularly symmetric Bessel beam of arbitrary order is presented within the framework of generalized Lorenz-Mie theory. An analytic and closed-form formula for the asymmetry factor, which ultimately determines the sense of direction of photophoretic force, is provided. The influences of particle size, absorptivity of the particle, half-cone angle, beam order of the Bessel beam on the asymmetry factor are explored in detail. The method proposed in this paper can be applied to a wider class of axisymmetric beams carrying nonzero topological charges.

Rayleigh limit of generalized Lorenz-Mie theory for on-axis beams and its relationship with the dipole theory of forces. Part II: Non-dark axisymmetric beams of the second kind and dark axisymmetric beams, including a review

A recent work has been devoted to the study of the Rayleigh limit of generalized Lorenz-Mie theory for on-axis beams in the case of non dark axisymmetric beams of the first kind. The present work complements this previous work by studying the case of non-dark axisymmetric beams of the second kind and of dark axisymmetric beams. This paper being presumably the last one of a series devoted to the Rayleigh limit of the generalized Lorenz-Mie theory in the case of lossless particles, it is complemented by a mini-review allowing one to gain an overview of the issue.

Geometrized Models of Continuous Axisymmetric and Plane-Symmetric Relativistic Electron Beams

Geometrized Models of Continuous Axisymmetric and Plane-Symmetric Relativistic Electron Beams

Poynting vector and beam shape coefficients: On new families of symmetries (non-dark axisymmetric beams of the second kind and dark axisymmetric beams)

On the one hand, some electromagnetic symmetries of laser beams are reflected in symmetry properties of the Poynting vector. On the other hand, these symmetries are in turn embedded in symmetries relating the beam shape coefficients which encode the structure of laser beams. The present paper extends the beam shape coefficient symmetries of laser beams, already uncovered, to new families. Examples and counter-examples are furthermore provided to illustrate these symmetries and formal laboratories are proposed as a way to discover still more symmetries.

Invariant submodels describing a propagation of the ultrasonic beams in a cubically nonlinear medium without dissipation after self-focusing

Recently, for high-power ultrasonic beams, it was experimentally found that as a result of self-action, their self-focusing occurs. With self-focusing, a powerful ultrasonic beam is noticeably narrowed, has a nonlinear narrowing, and it is significantly amplified at the focus. A generalization of the three-dimensional Khokhlov–Zabolotskaya–Kuznetsov model in a cubic nonlinear medium in the presence of dissipation with a special nonlinearity coefficient describes the propagation of ultrasonic beams after self-focusing. This work is devoted to the study of submodels of this model without dissipation, which are invariant with respect to four-parameter subgroups of the main group of the equation of this model. These submodels are defined by invariant solutions of rank 0 or 1, which describe one-dimensional, plane and axisymmetric ultrasonic beams. Solutions of rank 0 and some solutions of rank 1 are found explicitly. Some of them, at each fixed moment of time, contain a destroying element in the form of an ultrasonic needle or an ultrasonic knife, which at each fixed moment of time are localized in a limited domain, on the surface of which the acoustic pressure is zero. These submodels describe one-dimensional, plane, or axisymmetric ultrasonic beams. Finding other invariant solutions of rank 1 is reduced to solving nonlinear integral equations. These invariant solutions are used to study the propagation of ultrasonic beams after self-focusing, for which either the acoustic pressure and its gradient, or the acoustic pressure and the rate of its change are given at the initial moment of time at a fixed point. Under certain additional conditions, the existence and uniqueness of the solutions to boundary value problems describing these processes are established. This makes it possible to correctly carry out numerical calculations when studying these processes. The graphs of the pressure distribution obtained as a result of the numerical solution of these boundary value problems for some values of the parameters characterizing the indicated processes are presented. These graphs show that for these submodels in the ultrasonic beams a monotonic increase in acoustic pressure occurs over time.

Rayleigh limit of generalized Lorenz-Mie theory for on-axis beams and its relationship with the dipole theory of forces. Part I: Non dark axisymmetric beams of the first kind, with the example of Gaussian beams

The Rayleigh limit of the generalized Lorenz-Mie theory (GLMT) has been recently examined in the case of off-axis circularly symmetric Bessel beams, thereafter in the case of on-axis circularly symmetric Bessel beams, the on-axis case providing an easier framework for the understanding of the optical forces exerted in the Rayleigh limit of GLMT. This work is here extended to the case of non dark on-axis axisymmetric beams of the first kind This encompasses the case of zeroth-order circularly symmetric Bessel beams and the case of localized models of Gaussian beams. Three kinds of optical forces are exhibited, namely traditional gradient and scattering forces, plus another kind of forces which is for the time being denoted as non standard forces. The relationship between the Rayleigh limit of GLMT and the dipole theory of forces is discussed, to the best of our present understanding.

TM-polarized angle-dispersive metasurface for axisymmetric extension of beam steering angles.

Axisymmetric extension of beam steering angles by metasurfaces is of great interest due to its potential application in extending the angular scan range of existing phased arrays. This angle-multiplexed manipulation functionality requires anti-symmetrically angle-dispersive phase gradient, as well as anti-symmetric angular phase dispersion over continuous incident angles, which is difficult to be implemented by existing metasurfaces. In this work, a series of meta-atoms with asymmetric structures are developed to achieve the required phase response. Based on the asymmetric meta-atoms, an angle-dispersive metasurface for axisymmetric beam steering of transmitted transverse-magnetic (TM) wave is designed, fabricated and tested, whose simulation and experimental results demonstrate the axisymmetric extension capability of beam steering angles from -53° ≤ θi ≤ 53° to almost -90° ≤ θt ≤ 90°.