An interaction between a light field with complex field spatial distribution and a micro-particle leads to forces that drag the particle in space and may confine it in a stable position or a trajectory. The particle behavior is determined by its size with respect to the characteristic length of the spatially periodic or symmetric light field distribution. We study theoretically and experimentally the behavior of a microparticle near the center of an optical vortex beam in a plane perpendicular to the beam propagation. We show that such particle may be stably trapped either in a dark spot on the vortex beam axis, or in one of two points placed off the optical axis. It may also circulate along a trajectory having its radius smaller or equal to the radius of the first bright vortex ring.
© 2012 Optical Society of America
An optical force arises from the interaction between light and a microparticle as the result of light scattering by the microparticle. The optical force is studied and employed within the framework of optical micromanipulation techniques and has been utilized, for example, in the following instruments: optical tweezers , holographic optical tweezers [2, 3], Raman tweezers , optical cell sorters , optical stretchers , or optical pikotenzometers [7, 8]. The above mentioned applications mainly use the transfer of the linear momentum of light from laser beams to an object. In the case of more complex spatial light distributions the particle behavior is strongly determined by its size with respect to the characteristic field pattern. This phenomena is sometimes called the size effect and has been observed in various field geometries. The particle behavior in the standing wave represents the simplest case . Here the particle is pushed with its center to the intensity maximum or minimum depending on its size. However, particles of particular sizes are not pushed at all and the overall optical force is negligible. Such strong dependence of the optical force on the particle size has been employed in various methods of passive optical sorting of microparticles in one- and two-dimensional optical lattices [10–14].
Except the linear momentum, light can posses also spin and orbital angular momentum . Spin angular momentum is associated with the polarization of light  and its change, for example due to the light transmittance through a birefringent microobject, results in a torque rotating the microobject around its axis [17,18]. It has been pointed out that there exists the orbital angular momentum which is associated with the spatial field distribution in optical vortex beams [15, 19]. The transfer of the angular momentum from a vortex laser beam to an object and its subsequent behavior has been mainly investigated theoretically at the level of atoms, nanoparticles and molecules [20–27] or particles much larger that the light wavelength within the scope of the ray optics [28, 29]. More complex theoretical approaches, such as generalized Lorenz-Mie theory [30–32], finite-difference time-domain approach , finite element method , or multidipole approximation of the particle , are used less frequently but due to their wider range of applicability enable to express optical forces and torques acting upon a dielectric or metal particle of sizes comparable to the laser wavelength and illuminated by beams of structured spatial field distribution (e.g. optical vortex beams). The experimental verification of the transfer of angular momentum upon a particle has been demonstrated by rotation of an absorbing particle in the dark center of the vortex beam [36, 37] or orbiting of microparticles around the beam axis in the high-intensity ring of the vortex beam [19, 38–52]. Even though in these cases the sizes of such particles were comparable to the laser beam wavelength, no quantitative comparison between the theory and the experiment has been performed. Outside the optical domain the transfer of angular momentum from acoustic waves upon a macroscopic object led to its rotation, too .
Majority of examples referenced above focused either on experimental or theoretical aspects of angular momentum transfer. In this paper, we combine both theoretical and experimental approaches, we present a parametric theoretical study of the behavior of a single particle in the high-order Bessel beam and we compare the theoretical predictions with experimental observations. As the principle novelty, we identified three different regimes of particle behavior: the particle can orbit along the high intensity ring of the vortex beam, it can be stably localized at the dark spot placed on the vortex beam axis or at one of two points placed off the optical axis.
2. Beam description and calculation of the optical forces
In this paper we assume an ideal high-order Bessel beam generated behind an axicon that is illuminated with a plane wave [54–60]. In the case of a plane wave linearly polarized along the x-axis the vector electric field of such Bessel beam (BB) can be described as [61, 62]:63]
The power carried by the innermost high intensity core of the BB for m = 0, ±1, ±2,... can be expressed in the paraxial case  asEq. (4) is compensated by opposite signs of terms Jm−1(σm) and Jm+1(σm). Therefore, the power carried by the central ring of the BB of topological charge m = 1 is 1.53 × bigger than the power carried by the zero-order BB core. In the case of higher topological charges we obtain P2/P0 = 1.95, P3/P0 = 2.32 etc.
In coincidence with our experimental work, we determine the value of the electric field intensity E0(α) from the power P0,core carried by the central core of the zero-order BB. This approach provides a direct link between the parameters of the idealized BB used for the calculations and the experimental realizations of such beams because P0,core and ρ0 can be measured experimentally. The typical experimental value corresponds to P0,core = 5 mW that we use to obtain the value of E0 for all the calculations presented in this paper independently on their core radius ρ0 and topological charges m ≥ 1. This approach is based on the way how we generated experimentally the vortex beams of different topological charges using the spatial light modulator. First we generated zero-order BB and then we switched to high-order BBs.
Using Eq. (1) and following the same approach as in Ref.  we present analytical formulas for optical forces acting upon a particle much smaller than the trapping wavelength (i.e. induced dipole or Rayleigh particle):62], respectively, ε0 is the permittivity of vacuum, ε1 is the relative permittivity of the surrounding medium and we skipped krρ in Bessel functions. Even though these equations are of limited validity they provide insight into the behavior of tiny particles. Terms related to α′ give rise to gradient force that due to a3 dependence represents the leading force acting upon tiny particles. Terms related to α″ correspond to scattering force. It depends on the particle radius as a6 and, therefore, it becomes more pronounced for larger particles (within the validity of this Rayleigh approximation). Terms with P‖ reflects the non-paraxiality of the beam and their influence increases for larger α0 (i.e. for narrower BB core ρ0). Nice example of the competition between the gradient and scattering force can be shown for Fϕ. Neglecting all terms with P‖ we end up with only scattering azimuthal force that evidently causes particle orbiting. However, due to a6 dependence this force is weak for smaller particles and can be overcome by gradient force (the first term in Eq. (6). The sine term in gradient force leads to two azimuthal equilibrium positions placed off the beam axis, as will be discussed below.
Optical forces acting upon a spherical dielectric particle of any size we calculated using the generalized Lorenz-Mie theory [63–68]. To speed up the numerical calculations presented in the following section we utilized the results of Taylor  and we expressed the scattered field coefficients Aln and Bln analytically for the vortex BB . It has shortened the computation time in the Matlab environment by two orders of magnitude.
3. Numerical results
We considered polystyrene particles of refractive index n2 = 1.59 and radii a in the range 10 nm – 1.5 μm surrounded by water (refractive index n1 = 1.334). Besides water we also considered air as the surrounding medium (n1 = 1) because optical manipulation in air offers much lower friction and probably represents upcoming direction of further development of this technique. The radial and axial optical forces were calculated at a single axial position of the particle considering BB beam core radii ρ0 in the range 0.3 − 1.5μm and topological charges m = 1, 2, 3, 4, 5, 10. Figure 1 shows the force acting upon a polystyrene particle of radius 1 μm placed into the BB having the following core radii ρ0 = 1, 1.25, 5μm. The radial component of the force is depicted in Fig. 1(a) as a function of the radial coordinates ρ along parallel (solid) and perpendicular (dashed) direction to the beam polarization. It can be seen that the particle center is stably trapped at the bright part of the vortex fringe (ρ/ρ1 = 1) only for the widest core radius (ρ0 = 5μm), i.e. when a ≪ ρ1. Moreover, the red curve in Fig. 1(b) shows that the azimuthal force is almost constant and, thus, the particle will orbit along the vortex ring (regime R1). In the case of narrower beam core radius (ρ0 = 1.25μm), i.e. the particle radius is still smaller but comparable to ρ1, the particle is trapped off-axis at the radial distance ρ > 0, however, this distance is smaller than the vortex beam radius and the azimuthal component of the optical force changes its sign. Therefore, the particle does not orbit along the vortex ring but is stably trapped at certain azimuthal position (regime R2). If the beam core radius is even narrower (ρ0 = 1μm), the particle overlaps the first bright vortex fringe of radius ρ1 and it is trapped with its center at the vortex beam axis, i.e. ρ = 0 (regime R3). In this case the azimuthal force is equal to zero and thus no curve associated to this regime is plotted in Fig. 1(b). Furthermore, Fig. 1(a) demonstrates, that as the beam core radius decreases (i.e. α0 increases), the non-paraxial vectorial properties of the optical vortex beam become apparent and the radial force component differs along various azimuthal directions (parallel and perpendicular to the beam polarization).
The results of calculations in the xy plane are visualized in Fig. 2 for the polystyrene particle of radius a = 1μm and six different radii of the BB beam core ρ0. The size of the particle is compared to the radius of the first bright vortex beam fringe and thus the direct link between the particle size and characteristic dimension of the vortex beam can be established for each regime. The particle is initially trapped in the innermost bright fringe while it orbits there (regime R1, see Fig. 2(a)), similarly as in Fig. 1. As the core radius decreases the particle stops orbiting but still being located off the vortex beam axis (Fig. 2(b), regime R2). Two stable positions exist here for the particle, see green curve in Fig. 1 as well. For even thinner core the particle locates itself with its center on the vortex beam axis, i.e. into the dark, light-free region (Fig. 2(c), regime R3). While reducing the core radius even further, i.e. when the particle radius is getting much larger than the radius of the first bright vortex fringe of radius ρ1, particle’s behavior described by these three regimes repeats(see Figs. 2(d–f)).
Figure 3 uses the particle and the BB core radii as the free parameters to distinguish all three different regimes R1, R2, R3 of the particle behavior. The force calculations were performed for beam core values ρ0 varied in steps 10, 25 or 50 nm, however, near the borders between all regimes the steps were decreased to 5 nm to ensure the proper border placement. We can see that very small particles (a ≤ 50 nm) are always trapped off-axis in regime R2 which is in agreement with Eq. (6). This is caused by the fact that such particles (basically elementary dipoles) are strongly influenced by the azimuthal optical gradients along the vortex innermost rings that arises from the non-paraxial terms with P⊥ in the beam description. One can also see that this region gets narrower as the core radius increases (i.e. α0 decreases) and the azimuthal inhomogeneity in optical intensity becomes less pronounced. Once the optical force associated to the optical angular momentum of the vortex ring (it is related to azimuthal scattering force in the case of Rayleigh particles) is strong enough for larger particles and such particle starts to circulate along the innermost fringe (regime R1). At certain particle radii a (∼ 0.82ρ0) the particle either stops in R2 or jumps directly onto the vortex axis into R3. Even bigger particle (a ≳ 1.7ρ0) jumps again off axis either into the orbiting regime R3 or into the stable off-axis position R2. Behavior of particles surrounded by air follows similar trend as those immersed in water, however, the boundaries between the regimes are less smooth due to the to larger contrast in refractive indices of air and particle giving rise to stronger morphological dependent resonances.
Figure 4 compares the off-axial equilibrium positions r of the particle in regimes R1 and R2 relatively to the radius ρ1 of the inner-most bright intensity vortex fringe. Particles much smaller then ρ1 are confined at radial positions very close to the first high intensity ring of the vortex beam, i.e. r/ρ1 ≃ 1, because they are under the dominant influence of the intensity gradient near the bright region of the vortex fringe. As the radius of the particle increases and approaches the beam core radius, the particle starts to overlap larger volume of the vortex beam in contrast to only narrow part near the bright vortex fringe, and thus its radial position gets closer to the beam axis. Particles of radius overlapping several vortex fringes are deviated less from the on-axial position (r/ρ1 ≲ 0.5 in water and air).
3.1. Regime R1: orbiting particles
We have calculated and analyzed the particle trajectories for the orbiting regime R1 in order to find the average angular velocity ω of the particle movement. We have tracked the particle motion for several orbits and used the Euler formula to solve the equation of motion in the over-damped case using v = F/γ where F is the optical force acting upon the particle, v is the particle velocity and γ = 6πηa is Stokes drag coefficient, η is the kinematic viscosity of the medium. In most cases the particle orbited with constant speed and we fitted these trajectories by the sinusoidal function x(t) = Asin(ωt) + B in order to obtain the angular velocity ω. The results are shown in Fig. 5. However, in several cases, mostly near the border between orbiting R1 and off-axis stable R2, the particle velocity along the orbit varied significantly and we used the period of the particle orbit T to determine the average angular velocity ω = 2π/T. Figure 5 shows that tiny particles that just started their orbiting move with the lowest angular velocity. ω further increases significantly (note the logarithmic scale of the Fig. 5) for the small radii of the beam core due to the higher optical intensity in the first bright fringe of these beams. However, this high angular velocity sustains even for larger particles that switched back to the rotation after being located on the vortex axis. Figure 5(b) stresses that due to the lower friction in air the particles orbit here with angular velocities two orders of magnitude higher comparing to motion in water under the same conditions.
3.2. Regime R2: off-axis lateral confinement
The particle placed in regime R2 may be located in one of two locations placed symmetrically with respect to the vortex axis. The angle Φ between radius vector of the first position and the x axis (corresponding to the direction of the beam polarization) is depicted in Fig. 6. Smaller particles are trapped at positions that are perpendicular to the beam polarization, however as the angular force increases with particle radius, Φ increases until the particle starts its orbiting in regime R1. For particles of radius larger than the core of the beam abrupt changes of Φ appear for water. In contrast particles of all sized surrounded by air are confined mainly at positions with Φ ≃ 90°.
3.3. Axial force
In all regimes mentioned above the particle is propelled along the beam propagation axis by the axial force Fz. Figure 7 shows the numerical results corresponding to the axial force Fz acting upon the particle when it is laterally localized (regimes R2 and R3). In the case of orbiting regime R1 the average axial force is shown. The results show that the particle is axially more strongly propelled if it is settled in regimes R3 or R2. For larger particles the force is stronger and less dependent on the particle positions because the particle overlaps several vortex beam fringes.
3.4. Higher topological charges
Figure 8 demonstrates the behavior of a polystyrene particle (a = 1μm) placed into the vortex beams of m = 1, 2, 3, 5, 10 and corresponding BB core radii ρ0 in the range 0.3 – 1.5 μm. The phase curves identifying all three regimes are depicted in Fig. 8(a). Obviously, the orbiting regime (R1) is preferable for higher topological charges while the off-axis stable regime (R2) vanishes there. Since with increasing m and fixed ρ0 the radius of the first high-intensity ring increases, we have observed no trapping at the vortex core for m = 10 for the selected particle radius.
The normalized radial distance ρ/ρm of the particle center from the vortex axis is depicted in Fig. 8(b) for the off-axis regimes R1 and R2, where ρm is the radius of the inner-most bright vortex fringe. The particle orbits or is radially fixed mainly at a distance lower than the corresponding ρm. This distance slowly decreases under the conditions when the particle tends to move to the axial position (R3). However, for vortex beams of higher topological charge having small core radius (ρ0 < 0.75 μm) the particle may orbit along a trajectory having radius even larger than ρm. This is caused by the fact that the distance between the outer intensity fringes decreases for higher topological charges and the particle covers more light intensity while being settled between the first and the second inner-most intensity fringe.
Figure 8(c) shows the average angular velocity ω of the particle. One can see, that ω decreases for increasing topological charge and constant ρ0. This result is slightly against the well accepted idea that optical vortex beams of higher topological charge produces faster angular motion due to larger orbital angular momentum. However, since we keep the power in the BB core constant (P0,core = 5 mW), optical intensity in the ring of the high-order BB decreases as m raises due to the increased ring radius and, therefore, the density of the orbital angular momentum decreases. For example, if we kept the radius of the innermost bright fringe constant (e.g. ρ1 = ρ2 = 1.15μm) for different topological charges, we obtain ρ0 = 1.5μm and ρ0 = 0.9μm corresponding to m = 1 and m = 2, respectively. Comparing with Fig. 8(c) one can see that for these values of the beam cores ρ0, the particle orbits with much higher angular velocity for topological charge m = 2. This trend gets even more pronounced for higher vortex topological charges.
4. Experimental beam generation and measurement procedures
The BB of the zero-order we formed behind a lens (i.e. in the Fourier space) using a ring-like blazed phase diffraction grating imposed on the spatial light modulator (SLM, Holoeye LC-2500R). The width of its core was determined by the semiapex angle α0 which was controlled by the radius of the diffraction grating in the form of a ring. The BB of topological charge m was generated by adding an azimuthally linearly increasing phase from 0 to 2mπ to the previous grating. Since the SLM enabled real-time modification of such diffraction grating, it gave us the freedom to generate non-diffracting beam of different orders and also widths. Figure 9 introduces how this key optical element was placed in the experimental setup. The incoming laser beam (Verdi V10, Coherent, λvac = 532 nm) was spatially filtered using the achromatic doublet L1 (f1 = 19 mm, Thorlabs AC127–019–A) and the pinhole of diameter 10 μm (Thorlabs P10C). The beam was collimated by the achromatic doublet L2 (Thorlabs AC254–200–A) of focal length f2 = 200 mm so that the beam fully overlapped the SLM chip. The grating imposed on the SLM diffracts the incident beam and only the first diffraction order was used for subsequent experiments. The zeroth diffraction order was blocked in the focal plane of the lens L3 (Thorlabs AC508–750–A–ML) of focal length 750 mm while the first diffraction order was collimated by lens L4 (Thorlabs AC254–300–A, f4 = 300 mm) and entered the aspheric lens used as the objective (Thorlabs C240TME-A, fO = 8 mm). The lens L4 and the objective served as a telescope that demagnified the width of the BB approximately 40× in the sample space. The sample space was filled with deionized water with dispersed polystyrene microspheres of radius 1 μm (DukeScientific 4K-02) or 2 μm (DukeScientific 4K-04) and illuminated by the BB. The microspheres behavior was observed with the planachromat microscope objective (Olympus PLCN 60×, NA 0.8) and recorded with the fast CCD camera (Basler piA640-210gm). These images were post-processed and the microspheres positions were obtained in nanometer precision (using the calibrated graticule giving 107 nm/pixel).
During the single experimental procedure, the same microsphere was illuminated by the high-order BB and the microsphere positions were recorded. The beam center was found using the zero-order BB that confined the microsphere at the beam center. Both, the high-order and the zero-order BBs corresponded to the same radius of the diffraction ring imposed on the SLM. The radial positions of the particle in high-order and zero-order BBs were recorded over 20 successive measurements. The above described experimental procedure was repeated for BBs of topological charges m = 1 − 5 and for different radii of the diffraction ring imposed on the SLM corresponding to different radii ρ0 of the BB cores in the sample plane.
Further on, the lateral profiles of the beam optical intensity were recorded for each of the above mentioned combinations when a region with no microspheres was illuminated and the notch filter was removed from the imaging path. The measured profiles were fitted with the expected theoretical ones (see Eq. (1)) to find the beam parameters. This procedure was repeated for all studied topological charges.
Figure 10 compares the theoretically expected behavior to the experimental observations. The figure shows examples of particles trajectories in the BB corresponding to ρ0 = 870 nm when the topological charge was changed from m = 0 at the beginning to m = 1 − 5. The yellow spots in the bottom row indicate places where the particle is trapped in the zero-order BB and the magenta curves denote the trajectories of the particle to its new equilibrium position when the high-order BB is established. The particle remains near the vortex beam axis for the topological charges m = 1, 2, 3 while it moves to the bright ring for topological charges m = 4, 5. This behavior coincides with the theoretical calculations that predict that the particle should stay on the beam axis (i.e. in the regime R3) for topological charges m = 1 − 3 while it should orbit around the bright fringe in regime R1 for topological charges m = 4, 5. We have not observed particle orbiting over the whole circle even though we used the in-situ aberrations correction method  to approach the vortex beam profile expected theoretically as close as possible. Comparison of the BB intensity profiles shown at the top and bottom rows in Fig. 10 indicates that we were not able to suppress the aberrations completely and the intensity variation along the vortex fringe survived. Such variation probably caused that we have not been able to observe the orbiting of the single trapped particle (regime R1) but we only observed particle motion towards the vortex fringe followed by its motion less then half a circle along the fringe. The aberrations became more pronounced for small diameter vortices of low topological charges 1 or 2 while their influence decreased for vortices of charges 3 to 5. The main contribution to imperfect aberration corrections comes from the poor surface flatness of the SLM chip 
Since we were able to distinguish experimentally whether the particle settled into the vortex dark core (R3) or off the beam axis, we were able to determine the particle distance from the vortex beam axis for different core radii and topological charges. Figure 11 shows the average distance of the particle center from the vortex axis relative to the radius of the innermost vortex fringe. The left and right columns show the experimental data for the polystyrene particles having radius 1 and 2 μm, respectively. The red curves show the theoretical prediction of the particle distance from the vortex axis, similarly to Fig. 8(b).
Due to the imperfect compensation of the beam aberrations discussed above, the coincidence between the experimental results and the theoretical predictions is worse for m = 1 and gets better for m = 2 and a = 1μm. For other parameters the coincidence is slightly worse for lower ρ0 where the radius of the innermost vortex beam ring is in the range of 1 to 3 wavelengths (in water) because small disturbances in the beam propagation cause intensity or phase inhomogeneities that are not considered in the theoretical description of the beam and cause differences between the predicted and the observed particle behavior. Very good coincidence is obtained for the topological charges m = 3, 4, 5 where the actual radius of the vortex ring is much higher than in the previous cases.
The influence of the particle size on optical forces localizing such particle in the spatially structured beam has been already demonstrated for the zero-order BB, standing waves and 2D optical lattices. Our new results demonstrate that optical forces acting upon a particle illuminated by the high-order BB are strongly influenced by particle size with respect to the radius of the innermost bright vortex fringe. Using the generalized Lorenz-Mie theory we have identified three different regimes of particles behavior in the lateral plane that have been summarized in Table 1. The particle either orbits close to the high intensity ring of the vortex beam (regime R1), or it is trapped off the beam axis in one of two stable azimuthal positions (regime R2), or settles with its center in the zero intensity center of the beam (regime R3). We have shown how these regimes depend on the radius of the beam core, radius of the spherical particle, and topological charge of the beam. The smallest particles follow the R2 regime, larger particles switch to R1 and orbits in the vortex beam with certain angular frequency. In both cases the particles’ distance from the beam center is smaller than the radius of the high intensity fringe. With increasing size of the particle with respect to the radius of the innermost vortex fringe the regimes switch between each other and the particles positions, angular frequency or axial optical force are influenced. The regime R1 becomes dominant for higher topological charges under the range of investigated parameters. Experimental investigations have been performed for 24 beam widths and 5 topological charges and the results coincide with the theoretical radial positions of the particles better for wider beams and higher topological charges (m > 2). The worse coincidence was observed for lower topological charges and narrower cores due to imperfect compensation of the beam aberrations.
The authors acknowledge the support from Czech Science Foundation (P205/11/P294), Ministry of Education, Youth and Sports of the Czech Republic (LH12018) together with the European Commission (ALISI No. CZ.1.05/2.1.00/01.0017).
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