1. Introduction

Optical angular momentum (OAM) and associated studies with optical vortices are areas of current widespread interest. Inclined wavefronts that imply an azimuthal component to the Poynting vector thus resulting in OAM of a light wavefield are well known. Typical forms of these wavefronts have embedded vortices such as those in Laguerre-Gaussian [1] or Bessel-Gaussian beams [2]. Light fields with OAM are of importance in quantum information processing [3], fundamental light-matter interactions [4], atomic selection rules [5] and vortex propagation [6], amongst other areas of research. Typically, these fields are characterised by an azimuthally-dependent term of the form exp(imφ), where m is an integer that denotes the number of intertwined helices within the field. This helical term, in turn, determines the OAM content of the wavefield.

Optical micromanipulation has been the most powerful experimental technique to elucidate the underlying physics of such light fields: in 1995, transfer of OAM by absorption was directly observed for the first time [7] and subsequently comparisons made between the spin and orbital angular momenta. Such studies have been advanced in recent years with the identification of the exact intrinsic and extrinsic nature of optical angular momentum [8, 9] and how this manifests itself in the behaviour of an off-axis particle in a circularly symmetric light field [10] measuring the local angular momentum density. More recently, several technical improvements to the transfer of OAM in the context of optical tweezers have been proposed [11]. OAM is now encroaching into new emergent areas including microfluidics where such light fields have potential applications for micropumps and directing particle motion [12, 13] where there is a requirement to direct particles along arbitrary trajectories thus suggesting the use of asymmetric light fields possessing OAM. Recently, spatial light modulators (SLMs) [13] and acousto-optical deflectors (AODs) [14] have been used to tailor specific lightfields with dynamically reconfigurable intensity distributions in mesoscopic systems, including shaped vortices for micromanipulation.

The use of fundamental beams in micromanipulation experiments results in the basic understanding of the physical processes by means of which light fields interact with material particles since the associated electromagnetic field of the beam can be accurately described in a closed, analytical form since the exact field distribution is known a priori. To date however, all experimental micromanipulation studies of fundamental beams possesing OAM using fundamental beams have been restricted to circularly symmetric transverse patterns with azimuthally uniform OAM densities, such as Laguerre-Gauss beams [7, 15] and Bessel beams [10, 16]. A particular class of fundamental beams is that of nondiffracting beams, whose transverse intensity profile remains unchanged as the beam propagates in free space for a considerable distance.

In this letter, we show the first experimental demonstration that Helical Mathieu (HM) beams possess OAM. We also present the first experimental results of particle dynamics in the wave-field of a nondiffracting HM beam with azimuthally asymmetric OAM density and intensity profile. In particular, we look at the controlled rotational motion of trapped microparticles within the elliptical lobes of the wavefield and study, for the first time, the interplay of two different phenomena, namely, the controlable rotation induced by transfer of OAM and the optical confinement that results from the gradient force. We also observe the simultaneous balance of these contributions and the Stokes drag force in the motion of particles within the trap and demonstrate a quantitative analysis of particle trajectories under the influence of the forces involved.

2. Helical Mathieu beams

HM beams are fundamental nondiffracting beams which are solutions of Helmholtz equation in elliptical cylindrical coordinates (ξ, η, z). Here ξ and η are the radial and angular elliptical coordinates and z is the propagation axis[17]. The beams are mathematically described by a linear superposition of products of radial and angular Mathieu functions. For a monochromatic, linearly polarized HM beam of order m propagating in the z direction, the field is given by

where Je_m(η) and Jo_m(η) are the even and odd radial Mathieu functions and cem(η), se_m(η) are the even and odd angular Mathieu functions. C_m(q) and S_m(q) are weighting constants that depend on q, a continuous parameter that determines the ellipticity of HM beams. The transverse intensity profile I(ξ, η, z) is characterised by a set of confocal elliptic rings of varying intensity[17, 18]. HM beams also have a linear array of vortices with unitary topological charge distributed along the interfocal line of its elliptical rings [18], associated to a transverse phase gradient along the angular coordinate which accounts for its OAM content [19].

The spatial separation of the vortices is also determined by the value of q. For low values of q, the OAM per photon for an HM beam of order m ≥ 1 originates in its m interfocal vortices. Since the phase gradient between any two adjacent vortices is null, the transverse position of the vortices remains constant as the beam propagates. Interestingly, as q decreases, the foci gradually approach the origin and for q = 0, all the vortices concur into a single vortex of topological charge m, resulting in the well-known case of Bessel beams. Clearly, this is a consequence of the symmetry of the elliptical-cylindrical coordinate system, which collapses to circular-cylindrical coordinates when the interfocal distance vanishes. In contrast to Bessel beams however, the OAM density of HM beams is not independent of the azimuthal coordinate but it varies with the elliptic angular coordinate η. For HM beams of the same order, an increase in the ellipticity factor results in the horizontal stretching of the ringed structure of the beam, deviating gradually from circular symmetry. After a critical value q_c, the beam elliptical rings are broken, the transverse profile symmetry becomes rather hyperbolic and the vortex structure becomes more complex compared to its original in-line configuration [19].

3. Experimental Setup

In order to study and characterise the transfer of OAM from an HM beam to dielectric microscopic particles, we have used an optical tweezer setup with a HM beam (m = 7, q = 5) generated with an off-axis blazed phase computer-generated hologram (CGH)[19]. The hologram was backlit by a collimated beam from a linearly polarized 1064 nm Ytterbium fiber laser and the resulting field distribution immediately after the hologram was then focused by means of a converging Fourier lens. An image of the angular spectrum of the HM beam was formed in the +1 diffraction order at the Fourier plane. The remaining orders of diffraction were blocked here by means of an iris diaphragm and a second converging lens was then used to reconstruct the HM beam from the image of its angular spectrum. The beam was then directed downwards and focused tightly into a sample chamber with the particles in solution and the trapping plane was imaged with an inverted microscope (Nikon TE2000E).

The sample chamber was filled with a diluted solution of monodisperse spherical polystyrene particles 3.0 microns in size, suspended in a mixture of anionic, non-anionic and amphoteric surfactants (1% in volume) in D₂O. The solution is intended to circumvent the absorption of IR radiation by using D₂O instead of water and, at the same time, to reduce the viscosity of the medium [20] as seen by the particles moving within the chamber. While the density of the solvent is essentially unchanged by adding a small amount of surfactants, its viscosity is reduced to 61% relative to the value for pure D₂O as assessed by means of a Beral pipet. Interestingly, OAM transfer is only observed effectively in the presence of surfactants, thus an upper bound for the local value of the scattering force in our experiment is the Stokes drag force

for a spherical particle of radius R moving with velocity dr/dt, where r(t) = [x(t),y(t)] immersed within a fluid of viscosity coefficient μ. Because the transverse intensity profile of the HM beam varies in the azimuthal coordinate, there exists a variable OAM transfer as from the beam to a trapped particle as it moves about the trapping plane. This allows for the observation of variations in the terminal velocity of the particles as their motion is influenced differently by both, the gradient force[21]

and also by the Stokes drag force (Eq. 2). Here n is the relative refractive index of the particles in the solution and c the speed of light.

As the beam traversed the sample chamber, the particles in the sample were quickly aligned with the beam elliptical rings as shown in Fig 1(a), and experienced a strong tendency to drift towards the transverse intensity regions of maxima in the transverse plane due to the gradient force. In the longitudinal direction, the upwards force acting on the particles due to buoyancy was counteracted by the radiation pressure of the beam acting downwards within the sample volume. Particles find their longitudinal position of equilibrium immediately below the plane that defines half the maximum propagation distance of the beam z_MAX as defined in Ref. [2]. This was verified by displacing the z = z_MAX/2 plane downwards in the chamber until the particles were pressed against the bottom of the sample cell.

We observed the immediate onset of rotational motion in accordance with the sense of the wavefront inclination of the beam. We verified that this rotational behaviour was indeed due to the OAM of the beam by rotating the CGH by 180 degrees about an axis perpendicular to the direction of propagation [7] thus reversing the rotation direction of the particles. Trapped particles described a curve of ξ = ξ ₀, over a full rotation over η. The observed paths of the trapped particles were consistent with the elliptical orbit predicted by the intensity distribution of the HM beam. Figure 1(b) shows the measured position of one trapped particle sampled at a rate of 4 Hz over a sampling time of T_s = 100 s.

The instantaneous position r(t) = (x(t),y(t)) of the particle was then tracked using 10-minute video samples captured at 15 frames per second by means of a CCD camera. The video samples were processed using dedicated software implemented using LabVIEW in order to extract the position of the particles within the trapping plane. Our tracking routines use dynamic pattern recognition algorithms to compute the position of the centers of individual particles.

 

Fig. 1. (a) Trapped particles in the elliptical rings of the HM beam. (b) Horizontal and vertical position of a single particle in the beam, note the periodic regular motion. Sampling time is 400 s. (c) Variation of the terminal velocity of a single particle in the sample as a function of the polar angle during one full orbital displacement. Velocity minima correspond to intensity maxima. [Media 1]

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In order to extract information about the transfer of OAM from the beam to the particles, we have used a multiple-point moving average method for analyzing the data time series thus isolating the local mean value of the statistical position, in turn decoupling the uncorrelated random displacements associated to Brownian motion from the elliptical motion. Dynamic variables of motion were subsequently calculated by taking successive numerical derivatives of the measured instantaneous position. Here, we present quantitative assessments of the dynamic variables of the particles in motion and explain these in terms of the forces involved in the experiment.

4. Results

Due to the structure of the transverse field of HM beams, trapped particles orbit about the inter-focal line of the elliptical rings of the transverse field of the HM beam, in contrast to previous experiments [9, 10], in which off-axis rotation was observed around a single point in circular orbits. The elliptical orbital motion results from a combination of the confinement of particles within the ellipse due to the gradient force, and to the induction of rotation due to the transfer of OAM. Overall, OAM transfer dominated the orbital dynamics of the particles. However, local variations of the instantaneous angular velocity were observed due to the interplay between two forces. On one side, the optical gradient force increases in magnitude in the proximity of high intensity regions and thus tends to displace the trapped particles to local maxima within the transverse extent of the beam. On the other side, the OAM transfer rate increases as the particles approach regions of varying intensity and are transfered OAM from the angular momentum component in the propagation direction. The dynamics of the particles thus arise from the delicate interplay between the transverse gradient force, the transfer of OAM and the Stokes drag force. Clearly, Brownian motion also contributes to the instantaneous displacements of the trapped particles. This is evidenced in the points of intensity minima, where the confinement is less tight and thus particles deviate more from their elliptic orbit due to random motion.

Since the transverse field of HM beams is not azimuthally symmetric, particles trapped within the beam are expected to increase their angular speed as they approach intensity maxima and subsequently slow down due to the local gradient force. Since the scattering mechanism in turn becomes more significant in high-intensity regions, it competes with the local gradient force and OAM transfer thus propels the trapped particles away from intensity maxima and into an elliptical orbit with a terminal angular velocity determined by the Stokes drag force. The tuning of each of these contributions can be used to configure microfuidic devices that could vary the terminal speed of trapped particles. In the particular case of transparent polystyrene particles, the mechanism of OAM transfer is that of scattering, as absorption at this wavelength is essentially negligible. In this case, the transfer of OAM is proportional to the intensity of the wavefield, this was experimentally observed as an increase in the beam power resulted in a linear variation of the average angular velocity of the particles in the sample (see Fig. 2). Particles typically completed one cycle in 89 s at a beam power of 700 mW, while doubling the power resulted in a velocity increase of nearly 270% yielding a new rotation period of 39 s.

 

Fig. 2. Rotation rates as a function of beam power. The straight line shows the linear fit.

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5. Conclusions

We have demonstrated the transfer of OAM from a HM beam to trapped transparent particles and measured the evolution of the particles under the influence of the azimuthally asymmetric beam profile. We have observed variations in the terminal velocity of individual particles and associated the motion of the particles to the interplay of the optical gradient force, the azimuthally varying OAM transfer and the Stokes drag force. We have also observed the linear increase of the OAM transfer rate with beam power by measuring the average rotation rate of the particles in the beam. The random displacements associated to Brownian dynamics are decoupled from the motion of interest by using multiple-point moving average data processing.

The modulation of the OAM transfer can be achieved by adjusting the ellipticity parameter q of the transverse field. Increasing the order m of the beam, will also increase the net OAM per photon, thus increasing the orbital rotation speed for a given effective scattering rate and beam power, which would be useful for microfluidics applications.