Optical forces induced behavior of a particle in a non ... · Optical forces induced behavior of a...

$: Optical forces induced behavior of a particle in a non ... · Optical forces induced behavior of a particle in a non-diffracting vortex beam Martin Siler,ˇ ∗ Petr Jakl, Oto Brzobohat´y,$
Optical forces induced behavior of aparticle in a non-diffracting vortex beam

Martin Siler,∗ Petr Jakl, Oto Brzobohaty, and Pavel ZemanekInstitute of Scientific Instruments of the ASCR, v.v.i., Academy of Sciences of the Czech

Republic, Kralovopolska 147, 612 64 Brno, Czech Republic∗[email protected]

Abstract: An interaction between a light field with complex field spatialdistribution and a micro-particle leads to forces that drag the particle inspace and may confine it in a stable position or a trajectory. The particlebehavior is determined by its size with respect to the characteristic lengthof the spatially periodic or symmetric light field distribution. We studytheoretically and experimentally the behavior of a microparticle near thecenter of an optical vortex beam in a plane perpendicular to the beampropagation. We show that such particle may be stably trapped either in adark spot on the vortex beam axis, or in one of two points placed off theoptical axis. It may also circulate along a trajectory having its radius smalleror equal to the radius of the first bright vortex ring.

© 2012 Optical Society of America

OCIS codes: (140.3300) Laser beam shaping; (170.4520) Optical confinement and manipula-tion; (350.4855) Optical tweezers or optical manipulation.

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#174096 - $15.00 USD Received 10 Aug 2012; revised 30 Sep 2012; accepted 1 Oct 2012; published 9 Oct 2012(C) 2012 OSA 22 October 2012 / Vol. 20, No. 22 / OPTICS EXPRESS 24304

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1. Introduction

An optical force arises from the interaction between light and a microparticle as the resultof light scattering by the microparticle. The optical force is studied and employed within theframework of optical micromanipulation techniques and has been utilized, for example, in thefollowing instruments: optical tweezers [1], holographic optical tweezers [2, 3], Raman tweez-


ers [4], optical cell sorters [5], optical stretchers [6], or optical pikotenzometers [7, 8]. Theabove mentioned applications mainly use the transfer of the linear momentum of light fromlaser beams to an object. In the case of more complex spatial light distributions the particlebehavior is strongly determined by its size with respect to the characteristic field pattern. Thisphenomena is sometimes called the size effect and has been observed in various field geome-tries. The particle behavior in the standing wave represents the simplest case [9]. Here theparticle 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 neg-ligible. Such strong dependence of the optical force on the particle size has been employedin various methods of passive optical sorting of microparticles in one- and two-dimensionaloptical lattices [10–14].

Except the linear momentum, light can posses also spin and orbital angular momentum [15].Spin angular momentum is associated with the polarization of light [16] and its change, forexample due to the light transmittance through a birefringent microobject, results in a torquerotating the microobject around its axis [17,18]. It has been pointed out that there exists the or-bital angular momentum which is associated with the spatial field distribution in optical vortexbeams [15, 19]. The transfer of the angular momentum from a vortex laser beam to an objectand 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 withinthe scope of the ray optics [28, 29]. More complex theoretical approaches, such as general-ized Lorenz-Mie theory [30–32], finite-difference time-domain approach [33], finite elementmethod [34], or multidipole approximation of the particle [35], are used less frequently but dueto their wider range of applicability enable to express optical forces and torques acting upona dielectric or metal particle of sizes comparable to the laser wavelength and illuminated bybeams of structured spatial field distribution (e.g. optical vortex beams). The experimental veri-fication of the transfer of angular momentum upon a particle has been demonstrated by rotationof an absorbing particle in the dark center of the vortex beam [36, 37] or orbiting of micropar-ticles around the beam axis in the high-intensity ring of the vortex beam [19, 38–52]. Eventhough 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. Outsidethe optical domain the transfer of angular momentum from acoustic waves upon a macroscopicobject led to its rotation, too [53].

Majority of examples referenced above focused either on experimental or theoretical aspectsof angular momentum transfer. In this paper, we combine both theoretical and experimentalapproaches, we present a parametric theoretical study of the behavior of a single particle in thehigh-order Bessel beam and we compare the theoretical predictions with experimental obser-vations. As the principle novelty, we identified three different regimes of particle behavior: theparticle can orbit along the high intensity ring of the vortex beam, it can be stably localized atthe 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 isilluminated 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]:

E(ρ ,φ ,z) = E0(α0)eikzcosα0 (−i)m eimφ

×({

Jm(krρ)+12

[Jm+2(krρ)e2iφ + Jm−2(krρ)e−2iφ ]P⊥

}ex

+12i

[Jm+2(krρ)e2iφ − Jm−2(krρ)e−2iφ ]P⊥ ey

− i[Jm+1(krρ)eiφ − Jm−1(krρ)e−iφ ]P‖ ez

), (1)

where P⊥ =1− cosα0

1+ cosα0, P‖ =

sinα0

1+ cosα0, (2)

and ex,y,z are the base vectors along x, y, z Cartesian coordinate axes, ρ =√

x2 + y2 is radialdistance from beam axis, φ is the azimuthal angle, α0 is a semiapex angle of a cone alongwhich the BB is formed, k and kr = k sin(α0) are the wave vector and its projection into radialdirection, respectively, and m is the topological charge of the optical vortex beam. The zero-order BB corresponds to m = 0. For practical reasons we define the radius ρm of the vortexbeam with |m|> 0 using the radius of the first maximum of the intensity in the radial direction,obtained from dJm(krρm)/dρm = 0. The vortex beam radius ρm can be related to the radius ρ0

of the core of the zero-order BB which is defined as the radius of the first intensity minimumin the radial direction and obtained from J0(krρ0) = 0. For the topological charges up to 5 weobtain ρ1 = 0.7656ρ0, ρ2 = 1.27ρ0, ρ3 = 1.747ρ0, ρ4 = 2.2112ρ0, ρ5 = 2.6678ρ0, where [63]

ρ0 =2.4048

kr=

2.4048k sin(α0)

. (3)

In the following parametric studies we assume the zero-order BB core radii in the range ρ0 =0.3−1.5 μm.

The power carried by the innermost high intensity core of the BB for m = 0,±1,±2, ... canbe expressed in the paraxial case [63] as

Pm,core �−π kE20 ρ2

0

2ω0 μ0

σ2m

σ20

Jm−1(σm)Jm+1(σm), (4)

where ω0 is the light angular frequency, μ0 is the vacuum permeability, σm is the first off-axisroot of the Bessel function of the m-th order (e.g. σ0 = 2.4048, σ1 = 3.8317, σ2 = 5.1356, etc.).The minus sign in Eq. (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 topologicalcharges 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 in-tensity E0(α) from the power P0,core carried by the central core of the zero-order BB. Thisapproach provides a direct link between the parameters of the idealized BB used for the calcu-lations and the experimental realizations of such beams because P0,core and ρ0 can be measuredexperimentally. The typical experimental value corresponds to P0,core = 5 mW that we use toobtain the value of E0 for all the calculations presented in this paper independently on theircore radius ρ0 and topological charges m ≥ 1. This approach is based on the way how we gen-erated experimentally the vortex beams of different topological charges using the spatial lightmodulator. 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. [62] we present analytical formulasfor optical forces acting upon a particle much smaller than the trapping wavelength (i.e. induced


dipole or Rayleigh particle):

2Fr

ε0ε1= α ′krE

20

{12

Jm (Jm−1 − Jm+1)(

1−P2‖)− 1

2P2‖ (Jm+1Jm+2 − Jm−1Jm−2)

+14

P4‖ [Jm+2 (Jm+1 − Jm+3)+ Jm−2 (Jm+3 − Jm−1)]

+14

P2‖ [(3Jm−2 −3Jm + Jm+2)(Jm−1 − Jm+1)+ Jm (Jm−3 − Jm+3)]cos2φ

}

+α ′′krE20

14

P2‖ {(Jm+1 + Jm−1)(Jm+2 + Jm−2 − Jm)− Jm (Jm+3 + Jm−3)}sin2φ , (5)

2Fφ

ε0ε1= −2

rα ′E2

0 P2‖

[12

Jm (Jm+2 + Jm−2)− Jm+1Jm−1

]sin2φ

+1r

α ′′E20

{mJ2

m +P2‖[J2

m+1(m+1)+ J2m−1(m−1)

]+

12

P4‖[J2

m+1(m+1)+ J2m−2(m−1)

]

+mP2‖ [Jm (Jm+2 + Jm−2)−2Jm+1Jm−1]cos2φ

}, (6)

2Fz

ε0ε1= α ′′kcosα0E2

0

{J2

m +P2‖(J2

m+1 + J2m−1

)+

12

P4‖(J2

m+2 + J2m−2

)

+ P2‖ [Jm (Jm+2 + Jm−2)−2Jm+1Jm−1]cos2φ

}, (7)

where α ′ and α ′′ denotes the real and imaginary part of the particle polarizability [62], re-spectively, ε0 is the permittivity of vacuum, ε1 is the relative permittivity of the surroundingmedium and we skipped krρ in Bessel functions. Even though these equations are of limitedvalidity they provide insight into the behavior of tiny particles. Terms related to α ′ give rise togradient 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 Rayleighapproximation). Terms with P‖ reflects the non-paraxiality of the beam and their influence in-creases for larger α0 (i.e. for narrower BB core ρ0). Nice example of the competition betweenthe gradient and scattering force can be shown for Fφ . Neglecting all terms with P‖ we endup with only scattering azimuthal force Fφ = ε0ε1α ′′E2

0 mJm/(2r) that evidently causes particleorbiting. However, due to a6 dependence this force is weak for smaller particles and can beovercome by gradient force (the first term in Eq. (6). The sine term in gradient force leads totwo 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 thegeneralized Lorenz-Mie theory [63–68]. To speed up the numerical calculations presented inthe following section we utilized the results of Taylor [69] and we expressed the scattered fieldcoefficients Aln and Bln analytically for the vortex BB [60]. It has shortened the computationtime 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 range10 nm – 1.5 μm surrounded by water (refractive index n1 = 1.334). Besides water we alsoconsidered air as the surrounding medium (n1 = 1) because optical manipulation in air offersmuch lower friction and probably represents upcoming direction of further development ofthis technique. The radial and axial optical forces were calculated at a single axial positionof the particle considering BB beam core radii ρ0 in the range 0.3− 1.5 μm and topologicalcharges 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, 5 μm. The radialcomponent of the force is depicted in Fig. 1(a) as a function of the radial coordinates ρ alongparallel (solid) and perpendicular (dashed) direction to the beam polarization. It can be seenthat the particle center is stably trapped at the bright part of the vortex fringe (ρ/ρ1 = 1) onlyfor the widest core radius (ρ0 = 5 μm), 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 thevortex ring (regime R1). In the case of narrower beam core radius (ρ0 = 1.25 μm), i.e. theparticle radius is still smaller but comparable to ρ1, the particle is trapped off-axis at the radialdistance ρ > 0, however, this distance is smaller than the vortex beam radius and the azimuthalcomponent of the optical force changes its sign. Therefore, the particle does not orbit alongthe vortex ring but is stably trapped at certain azimuthal position (regime R2). If the beamcore radius is even narrower (ρ0 = 1 μm), the particle overlaps the first bright vortex fringeof 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 isplotted 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 becomeapparent and the radial force component differs along various azimuthal directions (paralleland perpendicular to the beam polarization).

All three different regimes of particle’s behavior are summarized in Table 1 and links tosubsequent Figs. 1, 2, 3 are established.

Table 1. Different regimes of particle’s behavior in non-diffracting vortex beam

Regime Particle position Particle behavior Fig. 1 Fig. 2 Fig. 3

R1 Off-axis Orbits red Fig. 2(a,d) rednear high intensity ring

R2 Off-axis Trapped laterally green Fig. 2(b,e) greennear high intensity ring

R3 On-axis Trapped laterally blue Fig. 2(c,f) bluein the dark beam center

The results of calculations in the xy plane are visualized in Fig. 2 for the polystyrene particleof radius a = 1 μm and six different radii of the BB beam core ρ0. The size of the particle iscompared to the radius of the first bright vortex beam fringe and thus the direct link between theparticle 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 butstill being located off the vortex beam axis (Fig. 2(b), regime R2). Two stable positions existhere for the particle, see green curve in Fig. 1 as well. For even thinner core the particle locatesitself 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 gettingmuch larger than the radius of the first bright vortex fringe of radius ρ1, particle’s behaviordescribed 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 threedifferent regimes R1, R2, R3 of the particle behavior. The force calculations were performedfor beam core values ρ0 varied in steps 10, 25 or 50 nm, however, near the borders between


0 1 2 3 4−30

−20

−10

0

10

ρ/ρ1

Fra

d. [p

N]

×10(a)

ρ0 = 1 μm

ρ0 = 1.25 μm

ρ0 = 5 μm

0 π/2 π 3π/2 2π−0.1

0

0.1

0.2

0.3

0.4

φ [rad]

Faz

im. [p

N]

(b)ρ

0 = 1.25 μm

ρ0 = 5 μm

Fig. 1. (a) The optical force acting upon the polystyrene particle of radius 1 μm in the radialdirection. Solid curves show the force along the beam polarization (i.e. x axis) dashedcurves show the force in direction perpendicular to the beam polarization (y axis). Red(regime R1), green (regime R2), and blue (regime R3) curves show the forces for vortexbeam of corresponding core radii ρ0 = 5 μm, ρ0 = 1.25 μm and ρ0 = 1 μm, respectively.Note that the red curve is multiplied by factor 10. (b) The azimuthal optical force actingupon the same particle placed in the stable radial distance r denoted in Fig. 1(a), i.e. r =ρ1 = 0.7656ρ0 = 3.828 μm for beam core radius ρ0 = 5 μm (red curve) and r = 468nm forρ0 = 1.25 μm (green curve).

all regimes the steps were decreased to 5 nm to ensure the proper border placement. We cansee that very small particles (a ≤ 50 nm) are always trapped off-axis in regime R2 which isin agreement with Eq. (6). This is caused by the fact that such particles (basically elementarydipoles) are strongly influenced by the azimuthal optical gradients along the vortex innermostrings that arises from the non-paraxial terms with P⊥ in the beam description. One can also seethat this region gets narrower as the core radius increases (i.e. α0 decreases) and the azimuthalinhomogeneity in optical intensity becomes less pronounced. Once the optical force associatedto the optical angular momentum of the vortex ring (it is related to azimuthal scattering forcein the case of Rayleigh particles) is strong enough for larger particles and such particle startsto circulate along the innermost fringe (regime R1). At certain particle radii a (∼ 0.82ρ0) theparticle 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-axisposition R2. Behavior of particles surrounded by air follows similar trend as those immersedin water, however, the boundaries between the regimes are less smooth due to the to largercontrast in refractive indices of air and particle giving rise to stronger morphological dependentresonances.

Figure 4 compares the off-axial equilibrium positions r of the particle in regimes R1 andR2 relatively to the radius ρ1 of the inner-most bright intensity vortex fringe. Particles muchsmaller then ρ1 are confined at radial positions very close to the first high intensity ring ofthe vortex beam, i.e. r/ρ1 � 1, because they are under the dominant influence of the intensitygradient near the bright region of the vortex fringe. As the radius of the particle increases andapproaches the beam core radius, the particle starts to overlap larger volume of the vortex beamin contrast to only narrow part near the bright vortex fringe, and thus its radial position getscloser to the beam axis. Particles of radius overlapping several vortex fringes are deviated lessfrom 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 orderto find the average angular velocity ω of the particle movement. We have tracked the particlemotion for several orbits and used the Euler formula to solve the equation of motion in the


−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x / ρ0

y /

ρ 0

(a)Regime 1

1

2

3

4

5

6

7

x 10−3

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x / ρ0

y /

ρ 0

(b)Regime 2

0.02

0.04

0.06

0.08

0.1

0.12

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x / ρ0

y /

ρ 0

(c)Regime 3

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x / ρ0

y /

ρ 0

(d)Regime 1

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x / ρ0

y /

ρ 0

(e)Regime 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−2 −1 0 1 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x / ρ0

y /

ρ 0

(f)Regime 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 2. Optical forces and trajectories of a particle of radius 1 μm placed in an opticalvortex beam of topological charge m = 1 having various BB core radii (a: ρ0 = 5 μm, b:ρ0 = 1.25 μm, c: ρ0 = 1 μm, d: ρ0 = 0.58 μm, e: ρ0 = 0.5 μm, f: ρ0 = 0.45 μm). Thebackground pseudo-color plot shows the electric field intensity |E|2 normalized relativeto the maximal intensity in (f). The magenta curves denote the deterministic trajectoriesof a particle (i.e. without considering the Brownian motion) starting at different locationsand following the particle motion towards an equilibrium point or a stable orbit. The bluecircle depicts the particle edge, its center is shown by the blue dot. The black and cyancontour represents the zero forces in the radial and azimuthal directions, respectively. If themagenta trajectories follow the black curve (zero radial force), see (a) and (d), this blackcurve forms a set of equilibrium positions and the particle orbits along the black curve (theparticle center is drawn just in one selected position). If black and cyan curves intersectthere exist equilibrium positions of the particle off the vortex axis, see (b,d). One of suchpossible stable positions of the particle center is denoted by the full blue dot. In other cases,see (c,f), the particle is trapped with its center on the vortex beam axis.

over-damped case using v = F/γ where F is the optical force acting upon the particle, v is theparticle velocity and γ = 6πηa is Stokes drag coefficient, η is the kinematic viscosity of themedium. In most cases the particle orbited with constant speed and we fitted these trajectoriesby the sinusoidal function x(t) = Asin(ωt)+B in order to obtain the angular velocity ω . Theresults are shown in Fig. 5. However, in several cases, mostly near the border between orbitingR1 and off-axis stable R2, the particle velocity along the orbit varied significantly and we usedthe period of the particle orbit T to determine the average angular velocity ω = 2π/T . Figure5 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 radiiof 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 therotation after being located on the vortex axis. Figure 5(b) stresses that due to the lower frictionin air the particles orbit here with angular velocities two orders of magnitude higher comparingto 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 symmetricallywith respect to the vortex axis. The angle Φ between radius vector of the first position and the


0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

regime 1

regime 2

regime 3

a [μm]

ρ0 [μ

m]

(a)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

regime 1regime 2

regime 3

a [μm]

ρ0 [μ

m]

(b)

Fig. 3. Phase map summarizing three different regimes of behavior of polystyrene particlesas a function of the particle radii a and the BB core radii ρ0 in the optical vortex beam oftopological charge m = 1. The particle orbits along the circular trajectories (red areas, R1,it corresponds to cases in Fig. 2(a,d)), settles in one of two off-axis positions (green areas,R2, similar to cases in Fig. 2(b,e)) or in the dark center of the vortex beam (blue areas, R3,it corresponds to cases in Fig. 2(c,f)). The particle is surrounded either by water (a) or air(b).

a [μm]

ρ0 [μ

m]

(a)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

r /

ρ 1

0.2

0.4

0.6

0.8

1

a [μm]

ρ0 [μ

m]

(b)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

r /

ρ 1

0.2

0.4

0.6

0.8

Fig. 4. Stable radial distance of the particle r from the beam axis in the regimes R1 andR2 plotted relatively to the radius ρ1 of the optical vortex beam having topological chargem = 1. The particle is immersed either in water (a) or air (b). The black curves show theborders between all regimes (see Fig. 3).

a [μm]

ρ0 [μ

m]

(a)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

ω [s

−1 ]

0.01

0.1

1

10

100

103

a [μm]

ρ0 [μ

m]

(b)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

ω [s

−1 ]

0.1

1

10

100

103

104

105

Fig. 5. The angular velocity ω (see the text) of the orbiting particle along a circular trajec-tory (R1) in water (a) or air (b). The black curves show the borders between all regimes.


0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

a [μm]

ρ0 [μ

m]

(a)

Φ [d

eg]

20

40

60

80

100

120

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

a [μm]

ρ0 [μ

m]

(b)

Φ [d

eg]

20

40

60

80

100

120

140

160

Fig. 6. Azimuthal position Φ of the particle trapped in one of the stable positions placedoff the vortex axis. The angle Φ shows angular position of the first trap with respect tothe polarization of the incident beam directed along the x-axis. The second trap is locatedsymmetrically with respect to the vortex axis, the particle is immersed either in water (a)or in air (b).

a [μm]

ρ0 [μ

m]

(a)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

Fz [p

N]

2

4

6

8

10

12

14

a [μm]

ρ0 [μ

m]

(b)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

Fz [p

N]

5

10

15

20

Fig. 7. The axial force Fz pushing the particle along the beam propagation axis for ambientwater (a) or air (b). The average force along the particle orbit (in regime R1) or the force atthe particle’s lateral stable position (off-axial in R2 or on-axial in R3) is shown. The blackcurves denote the borders between regimes.

x axis (corresponding to the direction of the beam polarization) is depicted in Fig. 6. Smallerparticles are trapped at positions that are perpendicular to the beam polarization, however as theangular force increases with particle radius, Φ increases until the particle starts its orbiting inregime R1. For particles of radius larger than the core of the beam abrupt changes of Φ appearfor water. In contrast particles of all sized surrounded by air are confined mainly at positionswith Φ � 90◦.

3.3. Axial force

In all regimes mentioned above the particle is propelled along the beam propagation axis by theaxial force Fz. Figure 7 shows the numerical results corresponding to the axial force Fz actingupon the particle when it is laterally localized (regimes R2 and R3). In the case of orbitingregime R1 the average axial force is shown. The results show that the particle is axially morestrongly propelled if it is settled in regimes R3 or R2. For larger particles the force is strongerand less dependent on the particle positions because the particle overlaps several vortex beamfringes.

3.4. Higher topological charges

Figure 8 demonstrates the behavior of a polystyrene particle (a = 1 μm) placed into the vortexbeams of m = 1,2,3,5,10 and corresponding BB core radii ρ0 in the range 0.3 – 1.5 μm. Thephase curves identifying all three regimes are depicted in Fig. 8(a). Obviously, the orbiting


0.5 0.75 1 1.25 1.5

R1R2R3R1R2R3R1R2R3R1R2R3R1R2R3

ρ0 [μm]

(a)

regi

me

m = 1m = 2m = 3m = 5m = 10

0.2

0.4

0.6

0.8

1

(b)

ρ/ρ m

0.5 0.75 1 1.25 1.510

−1

100

101

102

ρ0 [μm]

ω [s

−1 ]

(c)

Fig. 8. (a) Phase map showing three different regimes R1, R2, and R3 of the particle be-havior in the optical vortex beam of the topological charges m = 1,2,3,5,10 for a particlehaving radius a = 1 μm immersed in water. (b) The stable radial distance r of this particlefrom the axis of the vortex beam (in R1 and R2) plotted relatively to the radius of the in-nermost bright vortex fringe ρm, m = 1, 2, 3, 5 and 10. (c) Angular velocity of the particlemotion in regime R1.

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 ringincreases, we have observed no trapping at the vortex core for m = 10 for the selected particleradius.

The normalized radial distance ρ/ρm of the particle center from the vortex axis is depictedin Fig. 8(b) for the off-axis regimes R1 and R2, where ρm is the radius of the inner-most brightvortex fringe. The particle orbits or is radially fixed mainly at a distance lower than the cor-responding ρm. This distance slowly decreases under the conditions when the particle tends tomove to the axial position (R3). However, for vortex beams of higher topological charge havingsmall core radius (ρ0 < 0.75 μm) the particle may orbit along a trajectory having radius evenlarger than ρm. This is caused by the fact that the distance between the outer intensity fringesdecreases for higher topological charges and the particle covers more light intensity while beingsettled 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 de-creases for increasing topological charge and constant ρ0. This result is slightly against thewell accepted idea that optical vortex beams of higher topological charge produces faster angu-lar motion due to larger orbital angular momentum. However, since we keep the power in theBB core constant (P0,core = 5 mW), optical intensity in the ring of the high-order BB decreasesas m raises due to the increased ring radius and, therefore, the density of the orbital angularmomentum decreases. For example, if we kept the radius of the innermost bright fringe con-stant (e.g. ρ1 = ρ2 = 1.15 μm) for different topological charges, we obtain ρ0 = 1.5 μm andρ0 = 0.9 μm corresponding to m = 1 and m = 2, respectively. Comparing with Fig. 8(c) onecan see that for these values of the beam cores ρ0, the particle orbits with much higher angularvelocity for topological charge m = 2. This trend gets even more pronounced for higher vortextopological 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-likeblazed 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


Fig. 9. Experimental setup (description in the text).

by the radius of the diffraction grating in the form of a ring. The BB of topological charge mwas generated by adding an azimuthally linearly increasing phase from 0 to 2mπ to the previousgrating. Since the SLM enabled real-time modification of such diffraction grating, it gave usthe freedom to generate non-diffracting beam of different orders and also widths. Figure 9 in-troduces how this key optical element was placed in the experimental setup. The incoming laserbeam (Verdi V10, Coherent, λvac = 532 nm) was spatially filtered using the achromatic dou-blet L1 ( f1 = 19 mm, Thorlabs AC127–019–A) and the pinhole of diameter 10 μm (ThorlabsP10C). The beam was collimated by the achromatic doublet L2 (Thorlabs AC254–200–A) offocal length f2 = 200 mm so that the beam fully overlapped the SLM chip. The grating imposedon the SLM diffracts the incident beam and only the first diffraction order was used for subse-quent 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 col-limated by lens L4 (Thorlabs AC254–300–A, f4 = 300 mm) and entered the aspheric lens usedas the objective (Thorlabs C240TME-A, fO = 8 mm). The lens L4 and the objective served asa telescope that demagnified the width of the BB approximately 40× in the sample space. Thesample space was filled with deionized water with dispersed polystyrene microspheres of ra-dius 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 (Olym-pus 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 nanometerprecision (using the calibrated graticule giving 107 nm/pixel).

During the single experimental procedure, the same microsphere was illuminated by thehigh-order BB and the microsphere positions were recorded. The beam center was found usingthe zero-order BB that confined the microsphere at the beam center. Both, the high-order andthe zero-order BBs corresponded to the same radius of the diffraction ring imposed on theSLM. The radial positions of the particle in high-order and zero-order BBs were recorded over20 successive measurements. The above described experimental procedure was repeated forBBs of topological charges m = 1−5 and for different radii of the diffraction ring imposed onthe 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 theabove mentioned combinations when a region with no microspheres was illuminated and thenotch filter was removed from the imaging path. The measured profiles were fitted with theexpected theoretical ones (see Eq. (1)) to find the beam parameters. This procedure was repeated


m = 1 m = 2 m = 3 m = 4 m = 5Regime 3 Regime 3 Regime 3 Regime 1 Regime 1

Fig. 10. Examples of particle (radius 2 μm) motion when the BB is changed from thezero-order to high-order with m = 1− 5, ρ0 = 870 nm in all cases. Top row: Theoreticalpredictions of particle trajectories following the conventions of Fig. 2. Bottom row: Exper-imental observations. Background plot shows the measured intensity profile of the vortexbeam. Yellow spots denote the particle trapped in the zero-order BB core and magentacurves show the motion of the particle when its is illuminated by the high-order BB. Thebeams are polarized along horizontal axis.

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 nmwhen the topological charge was changed from m = 0 at the beginning to m = 1− 5. Theyellow spots in the bottom row indicate places where the particle is trapped in the zero-orderBB and the magenta curves denote the trajectories of the particle to its new equilibrium positionwhen the high-order BB is established. The particle remains near the vortex beam axis for thetopological 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 stayon the beam axis (i.e. in the regime R3) for topological charges m = 1−3 while it should orbitaround the bright fringe in regime R1 for topological charges m = 4,5. We have not observedparticle orbiting over the whole circle even though we used the in-situ aberrations correctionmethod [70] 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 indicatesthat we were not able to suppress the aberrations completely and the intensity variation alongthe vortex fringe survived. Such variation probably caused that we have not been able to observethe orbiting of the single trapped particle (regime R1) but we only observed particle motiontowards the vortex fringe followed by its motion less then half a circle along the fringe. Theaberrations became more pronounced for small diameter vortices of low topological charges 1or 2 while their influence decreased for vortices of charges 3 to 5. The main contribution toimperfect aberration corrections comes from the poor surface flatness of the SLM chip [71]

Since we were able to distinguish experimentally whether the particle settled into the vortexdark core (R3) or off the beam axis, we were able to determine the particle distance from thevortex beam axis for different core radii and topological charges. Figure 11 shows the averagedistance of the particle center from the vortex axis relative to the radius of the innermost vortexfringe. The left and right columns show the experimental data for the polystyrene particleshaving radius 1 and 2 μm, respectively. The red curves show the theoretical prediction of theparticle distance from the vortex axis, similarly to Fig. 8(b).

Due to the imperfect compensation of the beam aberrations discussed above, the coincidencebetween the experimental results and the theoretical predictions is worse for m = 1 and gets


0

0.5

1

r/ρ 1

|m| = 1

Particle radius a = 1 μm

0

0.5

1

r/ρ 2

|m| = 2

0

0.5

1r/

ρ 3

|m| = 3

0

0.5

1

r/ρ 4

|m| = 4

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.5

1

r/ρ 5

|m| = 5

ρ0 [μm]

0

0.5

1

r/ρ 1

|m| = 1

Particle radius a = 2 μm

0

0.5

1

r/ρ 2

|m| = 2

0

0.5

1

r/ρ 3

|m| = 3

0

0.5

1

r/ρ 4

|m| = 4

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.5

1

r/ρ 5

|m| = 5

ρ0 [μm]

Fig. 11. Measured stable radial position r of the particle in the vortex beam relative tothe radius of the innermost vortex ring ρm of different corresponding ρ0. Blue (positivetopological charge m> 0) and green (negative topological charge m< 0) points correspondto the measured data, the error-bars indicate 95 % confidence level of the average. The redcurves denote the theoretical prediction for the measured beam parameters. The BBs ofvarious corresponding beam core radius ρ0 and topological charges m = 1−5 (rows fromthe top to the bottom) were generated by the setup shown in Fig. 9. The left and rightcolumn corresponds to polystyrene particle of radii 1 μm and 2 μm, respectively.

better for m = 2 and a = 1 μm. For other parameters the coincidence is slightly worse for lowerρ0 where the radius of the innermost vortex beam ring is in the range of 1 to 3 wavelengths (inwater) because small disturbances in the beam propagation cause intensity or phase inhomo-geneities that are not considered in the theoretical description of the beam and cause differencesbetween the predicted and the observed particle behavior. Very good coincidence is obtainedfor the topological charges m = 3,4,5 where the actual radius of the vortex ring is much higherthan in the previous cases.

5. Conclusion

The influence of the particle size on optical forces localizing such particle in the spatially struc-tured beam has been already demonstrated for the zero-order BB, standing waves and 2D opti-cal lattices. Our new results demonstrate that optical forces acting upon a particle illuminatedby the high-order BB are strongly influenced by particle size with respect to the radius of theinnermost bright vortex fringe. Using the generalized Lorenz-Mie theory we have identifiedthree different regimes of particles behavior in the lateral plane that have been summarized inTable 1. The particle either orbits close to the high intensity ring of the vortex beam (regimeR1), 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 shownhow these regimes depend on the radius of the beam core, radius of the spherical particle, andtopological charge of the beam. The smallest particles follow the R2 regime, larger particlesswitch to R1 and orbits in the vortex beam with certain angular frequency. In both cases theparticles’ 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 fringethe regimes switch between each other and the particles positions, angular frequency or axialoptical force are influenced. The regime R1 becomes dominant for higher topological charges


under the range of investigated parameters. Experimental investigations have been performedfor 24 beam widths and 5 topological charges and the results coincide with the theoretical ra-dial 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 toimperfect compensation of the beam aberrations.

Acknowledgment

The authors acknowledge the support from Czech Science Foundation (P205/11/P294), Min-istry of Education, Youth and Sports of the Czech Republic (LH12018) together with the Euro-pean Commission (ALISI No. CZ.1.05/2.1.00/01.0017).


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