Helicoidal Beams Research Articles

Chirality effects in panto-cylindrical structures

In this paper, we apply a numerical integration strategy recently developed for determining the deformation shapes of structures constituted by Cosserat rods, to predict the behavior of panto-cylinders. Panto-cylinders have, as microstructure, a set of two families of helicoidal beams interconnected by perfect or elastic joints. The pivot’s free rotation axis is, in the reference configuration, orthogonal to the cylindrical surfaces spanned by the beams. We perform a series of numerical simulations looking for the mechanical parameters which exalt the chirality effects in the structure. For the performed compression, extensions, shear, and torsion tests, we find chiral deformation patterns with a dependence on the type of joint and its length.

Multi-objective optimization of snap-through instability of helicoidal composite imperfect beams using Bernstein polynomials method

This paper deals with the multi-objective optimization problems of snap-through instability of geometrically imperfect simply supported helicoidal composite beam with nonlinear orientation schemes. The objective function is to maximize the load required to initiate snap-through and minimize the instability region. It is assumed that the curved bioinspired beam is embedded on nonlinear elastic foundation and is subjected to different types of lateral loads. The presented formulation is based on Euler-Bernoulli theory, nonlinear higher order strain, and imperfect curvature. Bernstein polynomials (BPs) method is employed to solve the nonlinear nonhomogeneous fourth order integro-differential governing equation. Developed model can considered the effect of imperfection, lamination scheme, and elastic foundation on the nonlinear deflection and snap-through behaviors of helicoidal composite structures. Analytical closed form formula of the nonlinear load-deflection relationship is developed. The analytical relation is validated by comparing it with exact solution and numerical results available in literature. Multi-objective particle swarm optimization (MOPSO) algorithm is employed to determine Pareto-optimal set for limiting load and region of instability. In these analyses, optimal lamination parameter, amplitude of imperfection and elastic foundation constants are found for helicoidal composite beam, and the results are presented as 2D Pareto-optimal design points.

Nonlinear Static Stability of Imperfect Bio-Inspired Helicoidal Composite Beams

The objective of this manuscript is to develop, for the first time, a mathematical model for the prediction of buckling, postbuckling, and nonlinear bending of imperfect bio-inspired helicoidal composite beams with nonlinear rotation angle. The equilibrium nonlinear integrodifferential equations of imperfect (curved) helicoidal composite beams are derived from the Euler–Bernoulli kinematic assumption. The differential integral quadrature method (DIQM) and Newton-iterative method are employed to evaluate the response of imperfect helicoidal composite beams. Following the validation of the proposed model, numerical studies are performed to quantify the effect of rotation angle, imperfection amplitude, and foundation stiffness on postbuckling and bending behaviors of helicoidal composite beams. The perfect beam buckles through a pitchfork bifurcation. However, the imperfect beam snaps through the buckling type. The critical buckling load increases with the increasing value of elastic foundation constants. However, the nonlinear foundation constant has no effect in the case of perfect beams. The present model can be exploited in the analysis of bio-inspired structure, which has a failure similar to a metal and low interlaminar shear stress, and is used extensively in numerous engineering applications.

Exact Solution of Nonlinear Behaviors of Imperfect Bioinspired Helicoidal Composite Beams Resting on Elastic Foundations

This paper presents exact solutions for the nonlinear bending problem, the buckling loads, and postbuckling configurations of a perfect and an imperfect bioinspired helicoidal composite beam with a linear rotation angle. The beam is embedded on an elastic medium, which is modeled by two elastic foundation parameters. The nonlinear integro-differential governing equation of the system is derived based on the Euler–Bernoulli beam hypothesis, von Kármán nonlinear strain, and initial curvature. The Laplace transform and its inversion are directly applied to solve the nonlinear integro-differential governing equations. The nonlinear bending deflections under point and uniform loads are derived. Closed-form formulas of critical buckling loads, as well as nonlinear postbuckling responses of perfect and imperfect beams are deduced in detail. The proposed model is validated with previous works. In the numerical results section, the effects of the rotation angle, amplitude of initial imperfection, elastic foundation constants, and boundary conditions on the nonlinear bending, critical buckling loads, and postbuckling configurations are discussed. The proposed model can be utilized in the analysis of bio-inspired beam structures that are used in many energy-absorption applications.

Simplified Stress Design of Helicoidal HSS Beams

To facilitate the initial stress design of three-dimensional (3D) curved hollow structural steel (HSS) steel (i.e., helicoidal) beams and rapid early-stage topology decisions, a series of design charts and closed-form equations are presented that incorporate the effects of coupled forces and helicoidal geometry on stresses. These design aids were motivated by structures with extreme curves, featuring nonprismatic structural systems and nonplanar facades, which increasingly require the design of helicoidal elements. To establish these design aids, a converged beam model was developed for parametric studies of 71 different cross sections and a range of plan angles, lead angles, and beam radii. The results were curve-fitted, providing simplified stress utilization design equations expressed in terms of beam cross section (Sx and Sy). Results for the design equations and simplified design charts were within 1.5% and 5% of detailed finite-element analysis (FEA), respectively. The design aids were compared to two-dimensional (2D) projection methods and 3D helicoidal slab equations showing that the projection methods were able to be nonconservative by up to 34%, whereas the design aids were overconservative by up to 5% compared with the detailed FEA.

Acoustic manipulation and actuation of bubbles in complex environments: Beyond the Bjerknes force

Micron-sized gas bubbles are notoriously difficult to isolate, handle and remotely control. Their large buoyancy in common liquids will usually force them to rise and burst at any gas/liquid interface or remain trapped against a solid boundary until dissolution. While bubble stability issues against dissolution have found numerous practical workarounds, the challenge remains at isolating and maneuvering a single bubble in free space to, for instance, perform precise single bubble dynamics experiments with applied ultrasound or to use them as active carriers for a specific payload deliverable on demand. Here we demonstrate that single-beam acoustical tweezers [D. Baresch et al, Phys. Rev. Lett., 116, (2016)] can trap and manipulate in 3D a single bubble with the radiation pressure of helicoidal ultrasonic beams. Contrary to the situation where bubbles are trapped in the antinodes of a standing wave, the trapping vortex beam does not require oscillating volume changes of the bubble to generate a force, i.e., the trapping mechanisms cannot be explained in terms of Bjerknes forces. Viscous boundary layer effects and large bubble stability will be discussed. Opportunities for our manipulation technique to probe the high speed dynamics and rheology of particle-laden armored bubbles will also be presented [V. Poulichet et al., PNAS, 112, (2015)].

Nonreciprocal Transverse Photonic Spin and Magnetization-Induced Electromagnetic Spin-Orbit Coupling

We present a formulation of electromagnetic spin-orbit coupling in magneto-optic media, and propose an alternative source of spin-orbit coupling to non-paraxial optics vortices. Our treatment puts forth a formulation of nonreciprocal transverse-spin angular-momentum-density shifts for evanescent waves in magneto-optic waveguide media. It shows that magnetization-induced electromagnetic spin-orbit coupling is possible, and that it leads to unequal spin to orbital angular momentum conversion in magneto-optic media evanescent waves in opposite propagation-directions. Generation of free-space helicoidal beams based on this conversion is shown to be spin-helicity- and magnetization-dependent. We show that transverse-spin to orbital angular momentum coupling into magneto-optic waveguide media engenders spin-helicity-dependent unidirectional propagation. This unidirectional effect produces different orbital angular momenta in opposite directions upon excitation-spin-helicity reversals.

On Saint-Venant's Problem for Helicoidal Beams

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

Parameterization of the scattering and radiation forces and torques on spheres in terms of complex partial wave s-functions: Applications and interpretation

A wide range of physical responses of spheres in acoustic beams, including radiation forces and torques, are closely related to properties of the far field acoustic scattering. See for example the short reviews in [Marston, J. Acoust. Soc. Am. 125, 3539–3547 (2009)] for radiation forces and [Zhang and Marston, J. Acoust. Soc. Am. 129, 1679–1680 (2011)] for torques. When evaluating these properties it can be advantageous to retain the notation of resonant scattering theory from prior work for plane-wave illumination. That notation involves the complex partial wave, s-function such that the nth partial wave, the amplitude, is proportional to (s-1) and s is unimodular in the absence of absorption. Some applications of this parameterization to the interpretation and understanding of the scattering by, and radiation forces and torques on, spheres in acoustic beams will be examined. These include examples of forces and torques associated with helicoidal beams and simple methods for modulating the torque. This f...

Angular spectrum representations of focused acoustic beams and a comparison with the complex source point description

One approach to describing the wave fields of focused acoustic beams is to use an angular spectrum representation. This has the advantage of facilitating a simple evaluation of the scattering by a sphere placed at the focal point of the beam in terms of the scattering by a superposition of Bessel beam components. This was recently demonstrated for a quasi-Gaussian beam [Marston, J. Acoust. Soc. Am. 129, 1773-1782 (2011)]. This approach also has the advantage of separating the propagating spectral components from the evanescent spectral components [Marston, J. Acoust. Soc. Am. (submitted)]. While, for most practical acoustic sources, the evanescent spectral components may be neglected, the spectral representation is helpful for explaining why in principle evanescent spectral components can be important in the production of extremely tightly focused beams. A tightly focused beam with a Gaussian profile at the focal plane is not as easily describable using the more widely investigated complex source point description of the focused wave field. In addition to a Gaussian beam, the propagating-component angular spectrum representation is easily applied to the lowest radial-order Laguerre-Gauss helicoidal acoustic beam in the usual case where the beam is not tightly focused. [Work supported by ONR.]

Geometrical interpretation of negative radiation forces of acoustical Bessel beams on spheres

Various researchers have predicted situations where the acoustical or optical radiation force on a sphere centered on a Bessel beam is opposite the direction of beam propagation. We develop the analogy between acoustical and optical radiation forces of arbitrary-order helicoidal and ordinary Bessel beams to gain insight into negative radiation forces. The radiation force is expressed in terms of the asymmetry of the scattered field, the scattered power, the absorbed power, and the conic angle of the Bessel beam and is related to the partial-wave coefficients for the scattering. Negative forces only occur when the scattering into the backward hemisphere is suppressed relative to the scattering into the forward hemisphere. Absorbed power degrades negative radiation forces.

Acoustic beam interaction with a rigid sphere: The case of a first-order non-diffracting Bessel trigonometric beam

Mathematical expressions for the acoustic scattering, instantaneous (linear), and time-averaged (nonlinear) forces resulting from the interaction of a new type of Bessel beam, termed here a first-order non-diffracting Bessel trigonometric beam (FOBTB) with a sphere, are derived. The beam is termed “trigonometric” because of the dependence of its phase on the cosine function. The FOBTB is regarded as a superposition of two equi-amplitude first-order Bessel vortex (helicoidal) beams having a unit positive and negative order (known also as topological charge), respectively. The FOBTB is non-diffracting, possesses an axial null, a geometric phase, and has an azimuthal phase that depends on cos( ϕ± ϕ 0), where ϕ 0 is an initial arbitrary phase angle. Beam rotation around its wave propagation axis can be achieved by varying ϕ 0. The 3D directivity patterns are computed, and the resulting modifications of the scattering are illustrated for a rigid sphere centered on the beam's axis and immersed in water. Moreover, the backward and forward acoustic scattering by a sphere vanish for all frequencies. The present paper will shed light on the novel scattering properties of an acoustical FOBTB by a sphere that may be useful in particle manipulation and entrapment, non-destructive/medical imaging, and may be extended to other potentially useful applications in optics and electromagnetism.

Modulated helicity for acoustic communications and helicity-selective acoustic receivers.

Helicoidal or vortex beams have an azimuthal phase gradient and an axial null in amplitude. The sign and magnitude of the azimuthal phase gradient determine the helicity and topological charge of the beam and the associated handedness. Beams with a unit-magnitude charge have been generated with a four-sector array, adjacent quadrants being driven with a 90 deg phase offset [B. T. Hefner and P. L. Marston, J. Acoust. Soc. Am. 106, 3313–3316 (1999)]. The present work shows that the helicity can be rapidly modulated with an appropriately timed reversal of the excitation to a single pair of diagonal sectors. This reversal was demonstrated with an electronic commutator that inverts (at a zero crossing of the sine-wave excitation) two of the channels prior to amplification. While the wave helicity may be inferred with processed signals from a detection array, it may also detected with simple superposition of outputs from a four-element detector using appropriately offset elements. Digital communications was demonstrated where signals of the wrong helicity were suppressed. Scattering by symmetric objects on the axis of the beam preserves the helicity of the radiation [P. L. Marston, J. Acoust. Soc. Am. 124, 2905–2910 (2008)]. [Work partially supported by ONR.]

Generalization of helicoidal beams for short pulses

A generalization to the transient regime is developed for waves with a phase singularity of the screw type. These singular waves are commonly called vortices for all kind of waves as, for instance, optical vortex or acoustical vortex. We generalize the definition of vortices to get an azimuthal velocity invariant for all the frequency components contained in the broad spectrum of a short pulse. This generalization leads to a modification of the orbital angular momentum definition. Another generalization is introduced by considering helicoidal waves with a finite number of turns. We demonstrate that, in this last case, the topological charge is no longer an integer. This provides a physical interpretation to vortices of fractional charge that are involved here to take into account the diffraction occurring at both tips of the now finite helical wave front. We show that shortening the pulse implies an angular localization of the wave energy and, as a consequence, a spreading of the angular momentum amplitude due to the uncertainty principle.

Acoustic beams with modulated helicity and a simple helicity-selective acoustic reciever.

Acoustic helicoidal beams have an azimuthal phase gradient and an axial null in amplitude. The sign and magnitude of the azimuthal phase gradient determine the helicity and topological charge of the beam. Beams with a unit-magnitude topological charge have been generated with a four-quadrant sectored array, adjacent quadrants being driven with a 90 deg phase offset [B. T. Hefner and P. L. Marston, J. Acoust. Soc. Am. 106, 3313–3316 (1999)]. The scattering by symmetric targets placed on the axis of such beams preserves the helicity and the axial null [P. L. Marston, J. Acoust. Soc. Am. 124, 2905–2910 (2008); F. G. Mitri, Ann. Phys. 323, 2840–2850 (2008)]. The present work shows how to rapidly alter (or modulate) the helicity with an appropriately timed reversal of the excitation of a single pair of opposite quadrants. A simple four-element helicity-selective detector (using appropriately offset elements) was also demonstrated. These beams may be useful for digital acoustic communication and they have a modulated axial angular momentum flux. They should also generate a modulated acoustic torque on targets. [Work partially supported by ONR.]

Radiation force on spheres in helicoidal Bessel beams modeled using finite elements.

Analysis of the scattering of sound by spheres centered on ordinary and helicoidal (higher-order) Bessel beams [P. L. Marston, J. Acoust. Soc. Am. 124, 2905–2910 (2008)] makes it possible to evaluate the radiation force on spheres centered on the beam in an ideal inviscid fluid. Potentially surprising results include cases where the radiation force is directed opposite the propagation direction of both types of beams [P. L. Marston (submitted for publication)]. For potential biomedical applications such as drug delivery, it would be necessary to know if a small transverse displacement of the sphere from the beam’s axis causes a radiation force that pushes the sphere toward (or away from) the axis of the beam. Three-dimensional finite elements were applied to that problem. To trust FEM calculations of the radiation force with helicoidal beams, it was first necessary to verify that analytical values for the axial force are recovered in the on-axis helicoidal case since only the zero-order beam had been previously studied [D. B. Thiessen and P. L. Marston, J. Acoust. Soc. Am. 122, 3025 (2007)]. Cases were found where displacement of a sphere from the beam axis causes a transverse restoring force. [Supported by ONR and NASA.]

Radiation torque on a sphere centered on an acoustic helicoidal (vortex) Bessel beam.

Circularly polarized electromagnetic waves carry axial angular momentum and analysis shows that the axial radiation torque on an illuminated sphere is proportional to the power absorbed by the sphere [P. L. Marston and J. H. Crichton, Phys. Rev. A. 30, 2508–2516 (1984)]. Hefner and Marston [J. Acoust. Soc. Am. 106, 3313–3316 (1999)] proposed that a helicoidal acoustic beam (an acoustic vortex) also carries axial angular momentum and noted that absorption of such a beam should also produce an axial radiation torque. In the present work the acoustic radiation torque on a sphere centered on an acoustic helicoidal Bessel beam is analyzed and the torque is predicted to be proportional to the ratio of the power absorbed as a result of the interaction of sound with the sphere to the acoustic frequency. The torque is also proportional to the topological charge of the beam. Depending on the beam helicity, the torque is parallel or anti-parallel to the beam axis. The analysis uses a relation between the scattering and the partial wave coefficients for a sphere in a helicoidal Bessel beam [P. L. Marston, J. Acoust. Soc. Am. 124, 2905–2910 (2008)]. [This research was supported in part by ONR and NASA.]

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

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

Scattering of a helicoidal Bessel beam by a sphere: The effect of an azimuthal phase gradient.

In prior work [P. L. Marston, J. Acoust. Soc. Am. 121, 753–758 (2007)] the partial wave series for the scattering by a sphere centered on an ordinary Bessel beam was derived. The present work extends the analysis of the far field scattering to the case of a helicoidal Bessel beam having an angular phase ramp equal to the azimuthal angle. Helicoidal beams possess an axial null and have an azimuthal phase gradient. Some of the resulting modifications of the scattering are illustrated for an empty steel shell in water. In agreement with symmetry arguments given previously [B. T. Hefner and P. L. Marston, J. Acoust. Soc. Am. 106, 3313–3316 (1999)] the backward scattering and forward scattering vanish. For some directions the scattering increases when shifting from ordinary to a helicoidal beam illumination. In addition to describing the scattering by acoustic beams, there are potential applications of this analysis to the description of acoustic radiation forces and torques. [Work supported in part by ONR and by NASA.]

Radiation force on spheres in acoustic beams and related aspects of scattering

The close connection between the acoustic radiation force on objects in fluids and the angular distribution of the farfield scattering is especially useful in cases where dissipative effects are weak in the surrounding fluid. This connection also applies to objects illuminated by acoustic beams and has been used to extend the analysis of scattering by spheres in Bessel beams [P. L. Marston, J. Acoust. Soc. Am. 121, 753‐757 (2007); 122, 247‐252 (2007)] to the evaluation of the radiation force on spheres [P. L. Marston, J. Acoust. Soc. Am. 120, 3518‐3524 (2006); 122, 3162‐3165 (2007)]. The quantitative predictions (which include situations of negative radiation force) have been verified by nearfield analysis of the radiation force using the finite element method. We have also examined the radiation force on a sphere for a special case of co‐propagating Bessel beams having unequal beam parameters. The scattering of a higher order Bessel beam by a sphere placed on the axis has also been analyzed. The backscattering in that case vanishes in agreement with prior predictions [B. T. Hefner and P. L. Marston, J. Acoust. Soc. Am. 106, 3313‐3316 (1999)] for a general class of helicoidal acoustic beams. [Supported by NASA and ONR.]