Spinning and orbiting motion of particles in vortex beams with circular or radial polarizations

Manman Li; Shaohui Yan; Baoli Yao; Yansheng Liang; Peng Zhang

doi:10.1364/OE.24.020604

1. Introduction

It is well known that optical momentum and angular momentum (AM) are the main dynamical properties of electromagnetic waves, which manifest themselves and have been widely used in light-matter interactions [1], including optical manipulations [2], laser cooling [3], and opto-mechanical systems [4]. When light is scattered by a particle, the transfers of momentum and angular momentum from light to particle generate a radiation force and torque on the particle [5–9]. The optical AM of an electromagnetic wave includes a spin angular momentum (SAM) and an orbital angular momentum (OAM) [10]. The SAM results in the rotation of particle around its own axis, while the OAM causes rotation of particle around the optical beam axis. These light-induced rotations have attracted increasing interests and attentions [11]. For example, circularly polarized beams and cylindrical vector beams are used to spin absorbing particles [12, 13], and optical vortex (OV) beams are employed to rotate metal particles [14, 15]. Nevertheless, in most previous works, the induced rotation direction is along the beam propagation axis [13–17]. The latest research show that transverse SAM appears in various structured fields: evanescent waves [18], interference fields [19], and focused beams [20]. Transverse spinning of particles in plasmonic fields (i.e., evanescent waves) [21] and interference fields [22] have been implemented. However, the effect of transverse spin in focusing fields still needs more attention and much deeper investigation. This paper will show some theoretical results on this effect.

An optical beam with circular polarization is believed to carry a SAM orientated approximately along the propagation direction, i.e., longitudinal SAM. As a result, it is hard to realize transverse spinning of particle in such a beam. However, under tight focusing, the focusing fields of such beams contain appreciable longitudinal components and may potentially carry transverse SAM. For example, tightly focusing a radially polarized beam leads to a strong longitudinal electric field [23], causing the formation of a very strong transverse SAM. Optical vortex beams carrying OAM [24] can drive particles to orbit around the optical beam axis. Most of the previous studies on OV-induced rotation are focused on the orbital rotation of particles [15–17, 25, 26]. Actually, there also exists induced spinning motion of some particles in highly focused OV beams.

In this paper, we propose to realize spinning and orbiting motion of absorptive particles by highly focused OV beams with circular or radial polarizations. In order to understand the rotating properties of the particles, we firstly study the distributions of the focusing fields. Then, we calculate the optical forces and torques on particles based on T-matrix method, and analyze the physical mechanisms of rotation systematically. Furthermore, the influences of the characteristics of the particles and the OV beams on the rotational dynamics are investigated in detail. Some interesting results of spinning and orbiting motion in tightly focused OV beams will be shown and discussed.

2. Theory

Consider a dielectric spherical particle of radius a and index of refraction n₂ illuminated by an arbitrary electromagnetic wave with the fields E_inc and H_inc. The incident wave is scattered by the particle, yielding the scattered fields E_sca and H_sca outside the particle. Assuming that E_sca and H_sca are known, then the time-averaged optical forces 〈F〉 and torques 〈Γ_s〉 on the particle can be calculated by integrating the Maxwell stress tensor over a surface S enclosing the particle

〈 F 〉 = 〈 \oint_{S} \hat{n} \cdot \bar{T} d s 〉,

〈 Γ_{s} 〉 = - 〈 \oint_{S} \hat{n} \cdot (\bar{T} \times r) d s 〉,

here n̂ is the outwardly directed normal unit vector to the surface, r is the position vector, and T̅ is the Maxwell stress tensor

\bar{T} = ε_{1} E E + μ_{1} H H - \frac{1}{2} (ε_{1} E^{2} + μ_{1} H^{2}) \bar{I},

with ε₁ and μ₁ denoting the permittivity and permeability of the medium surrounding the particle, E ( = E_inc + E_sca) and H ( = H_inc + H_sca) representing the total fields outside the particle, and I̅ is the unit dyadic. So far, there still remains the question of how to determine the scattered fields. Here we employ the T-matrix method by Waterman [27] to treat the scattering problem. For the case of isotropic spherical particles, discussed here, the T-matrix method conforms with the well-known Mie scattering method. In the T-matrix method, the incident and scattered fields are expanded as a series of suitable vector spherical wave functions (VSWFs),

E_{i n c} (r) = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [a_{m n} M_{m n}^{1} (k r) + b_{m n} N_{m n}^{1} (k r)],

E_{s c a} (r) = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [p_{m n} M_{m n}^{3} (k r) + q_{m n} N_{m n}^{3} (k r)],

where r is the coordinate of observation point, k is the wave number in the surrounding medium; (a_mn, b_mn) and (p_mn, q_mn) are the incident and scattered coefficients, respectively;

M_{m n}^{1 � 3} (k r)

and

N_{m n}^{1 � 3} (k r)

are VSWFs of the first and third kinds [28].

By applying the boundary conditions on the surface of the particle, the scattered coefficients (p_mn, q_mn) are found to be related to the incident coefficients (a_mn, b_mn), in matrix notation, as [28],

[\begin{matrix} p_{m n} \\ q_{m n} \end{matrix}] = [\begin{matrix} T_{m n m' n'}^{11} & 0 \\ 0 & T_{m n m' n'}^{22} \end{matrix}] [\begin{matrix} a_{m' n'} \\ b_{m' n'} \end{matrix}] .

Here the first matrix of the right side of Eq. (6) is the T-matrix, which is diagonal since we are considering a spherical particle.

Substituting the expressions (4) and (5) for the fields into the integrals (1) and (2), it is found that the forces 〈F〉 and torques 〈Γ_s〉 can be expressed in a sum of the coefficients (a_mn, b_mn) and (p_mn, q_mn) in quadratic form [29]. Note that in evaluating the integral (2), the origin of the coordinate system is overlapped with the center of the particle. Since the particle is of spherical shape, isotropic and homogenous, the torques thus obtained may be regarded as the intrinsic torques or spin torques on the particle. This is why the subscript s is attached to the torque notation in the Eq. (2). In actual optical trapping, if the equilibrium trapping position of particle is not on the optical axis, the particle experienced an orbital torque, is extrinsic and dependent on the position of the particle. Denoting r_o the position vector of particle, the orbital torque 〈Γ_o〉 is clearly 〈Γ_o〉 = r_o × 〈F〉. In most situations, we are particularly interested in the z-component of orbital torque, which is expressed as

〈 Γ_{o, z} 〉 = ρ_{o} 〈 F_{ϕ} 〉,

where ρ_o is radial distance of the particle from the optical axis, <F_ϕ> is the time-averaged azimuthal force.

In optical trapping, the field incident on particles is actually the focused field of some input illumination focused by a high numerical aperture (NA) objective lens. According to Richards and Wolf [30], the focused field near the focus can be expressed as an integral of plane waves of the form of

E_{i n c} (r) = \frac{- i k f}{2 π} \int_{0}^{θ_{m a x}} \int_{0}^{2 π} A (θ, ϕ) \exp (i k \cdot r) s i n θ d ϕ d θ .

Here f is the focal length, θ_max is the maximal converging angle given by the NA and k is the wave number in the image space where the particle is located; k and r designate the wave vector of plane waves and the position vector of observation point; A(θ, ϕ) stands for the apodized field. The apodized field is related to the input field A₀(θ, ϕ) at the entrance pupil according to the following transform rule [31, 32]

A (θ, ϕ) = {(\cos θ)}^{1 / 2} [\begin{matrix} e_{θ} & 0 \\ 0 & e_{ϕ} \end{matrix}] (\begin{matrix} A_{0 ρ} \\ A_{0 ϕ} \end{matrix}),

where (e_θ, e_ϕ) are the respective unit vectors in the θ and ϕ directions, and (A₀_ρ, A₀_ϕ) are the radial and azimuthal components of the input field A₀(θ, ϕ). In principle, the torques on trapped particles are a result of the transfer of optical AM of the incident fields to the particles. It is known that the vortex phase factor exp(imϕ) and the polarization helix of optical fields are manifestation of optical AM. In view of this, we put the input field in a form as A₀(θ, ϕ) = ul(θ)exp(imϕ), where u is the polarization vector, l(θ) is the amplitude function, and m is the topological charge of the vortex phase. Further we assume a constant amplitude l(θ) = l₀ when 0 ≤ θ ≤ θ_max and l(θ) = 0 otherwise, where l₀ is a constant factor dependent on the power of the incident light. For the polarization vector u, we focused on two cases: left-handed circular polarization u =(e_x + ie_y) /

\sqrt{2}

and radial polarization u = e_ρ.

3. Results and discussion

In what follows, we assume that the input power P = 100 mW and the free space wavelength λ₀ = 1.064μm; the refractive index of image space n₁ = 1.33. The input illumination is focused by an objective lens with NA = 1.26. As mentioned above, the input illumination is set to be circularly polarized vortex beam (CPV) or radially polarized vortex beam (RPV). The intensity distributions of the focused fields along x-axis of the CPV and RPV inputs with topological charge m = 5 or −5 are shown in Fig. 1. The four types of illuminations all give an annular focusing with cylindrical symmetry [31]. However the CPV₅ input produces a larger radius of focusing than the CPV_-5, while the two RPV inputs yield the same focusing pattern with the radius locating between those of CPV inputs. This result lies in the two facts: 1) the conversion of SAM carried by the CPV to OAM under tight focusing leads to a phase factor of exp[i(m + 1)ϕ_s] on the field [33, 34]; 2) the larger (absolute) value of topological charge corresponds to a larger radius of focusing.

Fig. 1 Line scans along the x-axis of intensity distribution for circularly polarized vortex beams (CPV) and radially polarized vortex beams (RPV) with topological charges of m =+ 5 or −5.

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The optical forces and torques experienced by a spherical particle in the focused fields of the above four types of illuminations are calculated. The particle has a radius of a = 0.25μm and a complex refractive index n₂ = 1.59 + 0.01i. For simplicity, the axial equilibrium position is assumed to be in focal plane and only the transverse trapping is considered. Figure 2 shows the radial trapping forces as a function of the position of the particle. It is seen that for each illumination the equilibrium position is overlapped with the circle of intensity maxima of the focused fields. Consequentially, as far as radial forces are concerned, the RPV₅ and RPV_-5 illuminations are equivalent, while the CPV₅ and CPV_-5 illuminations are somewhat different. Having trapped the particle on a circle, the orbital motion is expected to be driven by the azimuthal optical force. To this end, we examine the transverse force distributions in the focal plane for the four illuminations, which are shown in Fig. 3. On the circle of the intensity maxima as well as the equilibrium position, the azimuthal optical force for each illumination exists uniformly along the circle, implying that the particle will orbit around the optical axis. With the topological charge m = 5, the orbital motion is in a clockwise sense around the positive z-axis as shown in Figs. 3(a) and 3(b), while for m = −5 the orbital motion reverses the direction, as shown in Figs. 3(c) and 3(d). Since the sign of the topological charge determines the orientation of the z-component of optical OAM, the results of Fig. 3 are just a reflection of transfer of optical OAM to the particle.

Fig. 2 Radial trapping force curves on the particle under (a) CPV and (b) RPV inputs with topological charges of m =+ 5 or −5.

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Fig. 3 Transverse force distributions experienced by the particle in the focal plane illuminated, respectively, by CPV inputs with topological charges of (a) m = 5 and (c) m = −5, and by RPV inputs with (b) m = 5 and (d) m = −5.

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The above calculations show that the particle is to experience an orbital motion, thus obtaining an orbital torque. Since the particle is absorptive, it also actually experiences a spin torque. Table 1 represents a quantitative calculation of the spin and orbital torques on the particle at the equilibrium position when illuminated respectively by the four types of polarized input fields above. It is seen that the x-component of spin torque Γ_s_, _x is equal to the negative of the y-component Γ_s_{, y} for each illumination, indicating that the transverse spin torques are along the positive or negative azimuthal direction. This can be explained by looking at the spin part of the angular momentum of the focusing fields. For time-harmonic fields, the time-averaged SAM density L_s ∝ Im(E × E*) [35]. Under the four types of illuminations above, their focusing fields all exhibit a π/2 phase difference between the radial component and the axial component on the circle of intensity maxima, i.e., the equilibrium position of the particle, while a π phase difference between the azimuthal and axial components [34], leading to a transverse part of the SAM orientated along the azimuthal direction and of course a longitudinal spin component. This fact suggests that the spin of the trapped particle is, in fact, along some direction off the optical axis. In Tab. 1, we also note that under CPV illumination the transverse spin torque as well as the orbital torque Γ_o_, _z reverses the direction as the sign of topological charge is changed. Under RPV illumination, the transverse spin torque, however, remains unchanged against the change of the sign of topological charge, while the longitudinal spin torque Γ_s_, _z and orbital torque Γ_o_, _z experience an inversion. The inversion of the orbital torque is caused directly by the change of the orientation of optical OAM as shown in Fig. 3. As to the spin torque, the orientation of the transverse component of SAM under CPV illumination, as well as the longitudinal component under RPV illumination, is reversed with the change of the sign of topological charge. This phenomenon can also be seen and explained from the corresponding expressions in [34].

Table 1. Spin and orbital torques exerted on the particle at the equilibrium position under CPV and RPV inputs with topological charge of m = 5 or −5.

View Table

To present an institutive picture of the particle’s motion in the focusing fields, we sketch the three-dimensional orientations of the spin and orbital torque of the particle under each illumination, as shown in Fig. 4, where the black arrows represent the spin torque vectors, the black circles around the sphere with arrows denote the directions of the spinning motion (in the counter-clockwise sense), and the white arrows indicate the directions of the orbiting motion. It is seen that the particle when trapped on the circle of intensity maxima will orbit around the optical axis, but suffers different non-axial spinning motion states according to the polarization state and the topological charge sign of the input OV beams.

Fig. 4 Sketch of the spin and orbital motion of the particle illuminated, respectively, by CPV inputs with topological charges of (a) m = 5 and (c) m = −5, and by RPV inputs with (b) m = 5 and (d) m = −5.

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We now examine the dependences of the optical forces and the torques on the particle’s size, the absorptivity (characterized by the imaginary part nʺ of the complex refractive index), and the topological charge value of the OV beams. Figures 5(a1)-5(c1) show the respective azimuthal force, spin and orbital torques at the equilibrium position as a function of the particle’s radius a for a fixed refractive index n₂ = 1.59 + 0.01i under CPV₅ and RPV₅ illuminations. The force and torques increase with increasing size of the particle. We note that the spin torques under CPV₅ and RPV₅ illuminations show different variation tendency. Under the CPV₅ illumination, the longitudinal component is almost equal to the transverse component as a < 0.3μm, thereafter the longitudinal component keeps increasing while the transverse component begins to decrease. Under the RPV₅ illumination, the situation is different. When a < 0.3μm, the transverse component have a greater value than the longitudinal component, while for larger size, a > 0.3μm, the case is contrary. Meanwhile, the longitudinal spin torque under the CPV₅ illumination is larger than that under RPV₅ illumination in all size, and the gap gets smaller and smaller with increasing the size. In calculation, it is found that the equilibrium position of the particle will gradually depart from the circle of intensity maxima and approach to the optical axis with increasing the particle’s radius. The inset in Fig. 5(c1) shows how the radius (R) of stably trapping position changes with the particle’s size. For the particle of radius a < 0.3μm, the stable trapping position is seen to approximately overlap with the circle of intensity maxima so that the orbiting motion as well as the spinning motion is expected observable clearly. The maximum size of the particle that ensures the orbital motion observable is about 1μm for CPV₅ and about 0.9μm for RPV₅. Much larger particles will be trapped at the optical axis and undergo only axial spinning motion.

Fig. 5 The changes of the azimuthal forces (a1)-(a3), the spin torques (b1)-(b3) and the orbital torques (c1)-(c3) exerted on the particle with varying radius a (a1)-(c1), and absorptivity nʺ (a2)-(c2), as well as with the topological charge value m (a3)-(c3) in CPV and RPV inputs. R in the inset of (c1) represents the radius of the equilibrium position of the particle.

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The dependences of the optical force and the torques on the absorptivity nʺ of the particle under CPV₅ and RPV₅ illuminations are shown in Figs. 5(a2)-5(c2), in which the particle’s radius a = 0.25μm, and the real part of the refractive index is 1.59. It exhibits a simple linear relation, since the absorptivity is proportional to these quantities. The variation tendency of the spin torques on the particle for CPV₅ is significantly larger than that for RPV₅. This is because the CPV beam carries the optical SAM, which transfers to the particle and enhances the spin effect.

The behaviors of the optical force and the torques on the particle as a function of the topological charge value m for CPV and RPV illuminations are shown in Figs. 5(a3)-5(c3), where the particle’s radius a = 0.25μm, and the refractive index n₂ = 1.59 + 0.01i. Since the focusing spot size of the RPV illumination with m = 1 is too small to guarantee an orbital motion for the particle, the azimuthal force and orbital torque for RPV illumination with m = 1 are dropped from Figs. 5(a3) and 5(c3). An interesting phenomenon concerning the spin torque is observed that the longitudinal spin torque under RPV illumination with m = 1 is much stronger than that under CPV illumination, as shown in Fig. 5(b3), the ratio is roughly 0.894 / 0.494 = 1.81. Furthermore, this value for RPV₁ is larger than any other values for all topological charges and polarizations. As we see, the azimuthal force and spin torques are gradually decreasing with the increase of the value of m. This tendency arises from that the radius of the annular focus increases with the charge m increasing, leading to the attenuation of the intensity of the focused ring. While, the orbital torque increases first and then decreases with an optimal topological charge to attain the maximum orbital torque. The values of m corresponding to the maxima of orbital torques are different for CPV and RPV inputs, which are 4 and 7, respectively. This observation can be interpreted from a simple calculation of Eq. (7).

4. Conclusions

In summary, tightly focused optical vortex beams with circular or radial polarizations are used here to study the spinning and orbiting motion of absorptive micro-sized particles. Numerical results show that the particle is trapped on the circle of intensity maxima and orbits around the optical axis while, but also experiences a non-axial spinning motion. The direction of spinning motion is determined by both the sign of topological charge and the polarization state of the vortex beam. For the CPV illumination, the transverse spin torque reverses the direction as the sign of topological charge is changed, but the longitudinal spin torque does not change. Under the RPV illumination, the transverse spin torque, however, remains unchanged against the change of the sign of topological charge, while the longitudinal spin torque experiences an inversion. The direction of the orbiting motion is only controlled by the sign of topological charge. When the topological charge is positive, the orbital torque vector points to the positive z-axis, otherwise, the orbital torque vector points to the negative z-axis. The azimuthal force, spin torque, and orbital torque increase when the particle’s radius and absorptivity increase. With the increase of the topological charge value, the azimuthal force and spin torque will decrease; while the orbital torque will firstly increases and then decreases with an optimal topological charge to attain the maximum orbital torque. These optical trapping properties might be helpful to investigation of optical manipulation, especially for optically induced rotations of micro-particles.

Funding

National Basic Research Program (973 Program) of China (2012CB921900); National Natural Science Foundation of China (NSFC) (11474352 and 11574389).

Acknowledgments

We would like to thank the anonymous reviewers for their exceptionally pertinent and constructive critical comments.

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Torque (pN⋅μm)	Γ_s _, _x	Γ_s _, _y	Γ_s _, _z	Γ_o _, _z
CPV₅	0.111	−0.111	0.111	2.119
CPV_-5	−0.106	0.106	0.100	−1.879
RPV₅	0.030	−0.030	0.028	1.792
RPV_-5	0.030	−0.030	−0.028	−1.792

Spinning and orbiting motion of particles in vortex beams with circular or radial polarizations

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusions

Funding

Acknowledgments

References and Links

Cited By

Figures (5)

Tables (1)

Equations (9)

Optics Express