Spin-controlled orbital motion in tightly focused high-order Laguerre-Gaussian beams

Yongyin Cao; Tongtong Zhu; Haiyi Lv; Weiqiang Ding

doi:10.1364/OE.24.003377

1. Introduction

Since Ashkin’s pioneer work for optical tweezers in the 1970s [1], optical trapping has been a well-known application to manipulate micro- and nano-particles in physics, biology, and other fields [2–4]. Various laser beams were used to form optical traps in the last thirty decades, such as Laguerre-Gaussian (LG) beams [5–7], Bessel beams [8,9], vortex beams [10,11], and polarization-dependent beams [12]. LG beams are even capable of trapping the low-refractive-index particle [13,14]. In addition, they can be used to investigate the spin and orbital angular momentum transfer from the electromagnetic field to particle [15–17]. Many optical rotators have been manipulated in LG beams as well, which could be very helpful to study optical micro-machines [18,19]. These performances of LG beams are relative to the property of the beam, which can carry two kinds of angular momentum: spin and orbital angular momentum, corresponding to the polarization state of the beam and spatial variation of the field, respectively. The spin and orbital angular momentum components along the beam propagation are σħ and lħ per photon, where number l refers to the azimuthal dependence of the field in the term exp(ilφ), while σ indicates the polarization state of the beam. The beam is right and left circularly polarized when $σ = \pm 1$ , and linearly polarized when $σ = 0$ [15,20].

For a dielectric spherical particle trapped in a Gaussian beam, in general, optical force acting on it generally contains gradient force and scattering force, which are connected with the gradient of field intensity and scattering cross section of the particle, respectively. However, Grier et al. [21] has demonstrated that an optical force component arising from phase gradients is also included in the total force in LG beams which can drive particle to translate on an stable orbit around the beam center. Study on orbital motion in LG beams, which can be observed experimentally, is helpful to understand the phase-gradient force. In addition, studing orbit rate of such orbital motion is a way to inverstigate transfer of angular momentum from beam to particle since the particle’s angular momentum, $M = 2 π m υ R^{2},$ is associated with orbit rate υ, where m is the mass of particle and R is the radius of orbital motion. Such orbital motion of objects is not only relevant to orbital angular momentum, but also related to polarization state of light [22]. Spin-orbit conversion might be studied in this way as well. However, the effect of spin of light on orbit rate still needs more attention and much deeper research. This paper will show some theoretical results on this effect.

In this paper, we will take LG beam, a conventional annular beam, as an example to perform such a study, which will be useful to investigate transfer of angular momentum. Orbit rate of dielectric spheres will be studied in detail theoretically. The effect of beam polarization on optical rotation is always neglected in general. However, we’ll reveal that spin of light cannot be neglected in some region, which involves the azimuthal mode of LG beams, the particle’s size and refractive index. Some interesting results of orbital rotation and phase gradient force in tightly focused high-order Laguerre-Gaussian beams will be shown, which might be also much helpful to investigation on optical spin-orbit interaction.

2. Theory

In optical tweezers, optical force exerted on particles are generally considered as a result of transfer of momentum since the light scattering off the particle. Therefore, calculation of the force is actually to solve the light scattering problem. Optical fields can be expanded into incident and scattered field potentials, where the incident and scattered electric fields can be delineated in a discrete basis set of vector spherical wave functions [23]

E_{inc} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} a_{n m} R g M_{n m} (k r) + b_{n m} R g N_{n m} (k r),

E_{scat}^{} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} p_{n m}^{} M_{n m}^{(1)} (k r) + q_{n m}^{} N_{n m}^{(1)} (k r),

where k is the wave number of the beam,

M_{n m}^{(1)}

and

N_{n m}^{(1)}

are vector spherical wave functions,

R g M_{n m}

and

R g N_{n m}

are regular vector spherical wave functions. Cartesian coordinate system centered on the beam focus is chosen for the incident field expansion. Here, the expansion coefficients of the incident field are a₀ and b₀, which are sets of the coefficents a_nm and b_nm respectively. Then the beam coefficients (a and b) in another Cartesian coordinate system in particle frame can be thus obtained using a linear transformation [23–25],

a = R^{- 1} A R a_{0} + R^{- 1} B R b_{0},

b = R^{- 1} A R b_{0} + R^{- 1} B R a_{0},

where R represents the rotation of the beam coefficients, z-axis would thus point along the direction from the beam center to the particle position after this transformation. A and B are the translations of beam coefficients in the rotated coordinate system along the direction from the beam center to the particle position. Given T-matrix of a spherical particle, the expansion coefficients of the scattered field in particle frame can be easily obtained. Then optical force exerted on the object can be calculated using optical tweezers computational toolbox [26].

Dynamic simulation model used in this paper is also based on hydrodynamic equations. All the particles are considered in low Reynolds number regime, since they will take less than a few microseconds to slow to the flow speed if the exerted forces are removed [27]. Under this condition, the viscous drag can be considered to oppose the optical force, $f_{drag} = - f_{light}$ . Accoding to Stoke’s law $f_{drag} = 3 π η d v,$ additionally, the model for optical rotation of a single sphere can be carried out using the following algorithm, which is based on computing the sphere’s next position a short time interval later. Given the force, position at time t, the equations of motion we solve for a spherical particle are [25,28,29]

v = \frac{f_{light}}{3 π η d},

r (t + d t) = r (t) + v d t,

where η is the drag coefficient of medium. d is the diameter of the sphere. v and r are the velocity and position of the object, respectively. dt is the time interval, which is generally between 10⁻⁶ s and 10⁻⁵ s. In our calculations, a sphere near the beam focus will go into its stable orbit in several milliseconds and rotate with a constant orbit rate. Using the algorithm of dynamic simulation above, one can find the spheres’ steady orbit rate since the orbital motion is periodic. In the other hand, analytical form of orbit rate can also be obtained according to the relationship of linear and angular velocities

υ = \frac{v}{2 π R} = \frac{| f_{o r b i t} |}{6 π^{2} η R d} = \frac{| R \times f_{o r b i t} |}{6 π^{2} η R^{2} d},

where R is the radius of its stable orbit,

f_{o r b i t}

is total optical force acting on the sphere on its stable orbit. Therefore, the orbit rate of a sphere is proportional to

| f_{o r b i t} | / R

, or

| R \times f_{o r b i t} | / R^{2}

associated with optical torque. Nevertheless, Eq. (7) cannot be used for tightly focused LG beams with linear polarization, since the orbit will be no longer circular, that is, both optical force and R are not a constant. Then the value of optical force would not be the same for the sphere located at different positions on the orbit. However, it could be used to approximately estimate the orbit rate of a sphere in a linearly polarized LG beam. The method has been actually validated by comparing to experimental results in Ref [6].

3. Results and discussion

LG beams used in this paper are tightly focused by a high numerical aperture lens (NA = 1.2). The wavelength of incident beam is 1064 nm in free space (and $λ = 800$ nm in water). The power through the focal plane is 100 mW. The propagation of the beam is along z-axis. All the beams here are with nonzero azimuthal mode but with radial node p = 0. The surrounding medium is water, of which the refractive index is 1.33. Transverse electric fields on the focal plane of LG₀₅ beams with right circular polarization (RCP), left circular polarization (LCP) and linear polarization (LP) are shown in Figs. 1(a)-1(c), respectively. One can see that the transverse electric fields aren’t the same as each other, which is induced from the polarization state of light. Axial symmetry of bright ring of LG₀₅ beam with LP is broken. The bright rings for RCP and LCP have different radiuses. Trajectories of a dielectric sphere trapped in LG₀₅ beams with RCP and LCP, respectively, are shown in Figs. 1(d) and 1(e). The refractive index and radius of the sphere is 1.57 and 0.5 µm, respectively. The sphere was rotated periodically on its stable orbit. However, the spin motion isn’t considered for non-absorbing dielectric spheres since the intensity distribution is cylindrically symmetric [30]. For the beams with the same azimuthal mode but with different polarizations, orbit rates of the sphere are different as shown in Fig. 1(f). According to the change of x-component of the sphere’s displacements in LG₀₅ beams with different polarizations, orbit rates can be obtained as 69 Hz, 89 Hz and 81 Hz for RCP, LCP and LP, individually. The orbit rate of a sphere in a LG beam with LP is always between the rates for RCP and LCP. Therefore, orbit rate of an object can be changed by switching polarization state of LG beam. Orbit rate of the object in LG beams with the same azimuthal mode is thus polarization-controlled. Actually we mainly talk about spin effect in this paper. In addition, the orbital motion of the sphere in Figs. 1(d) and 1(e) is anticlockwise, which is determined by the phase gradient of the beam. In the other hand, it can also be observed from the transverse part (x and y components) of Poynting vector of LG₀₅ beams shown in Fig. 2. The direction of orbital motion is consistent with anticlockwise vortexes. Moreover, the difference of radiuses of the vortexes is consistent with that of electric fields on the focal plane.

Fig. 1 Transverse electric fields on the focal plane of LG₀₅ beams with (a) RCP, (b) LCP and (c) LP and trajectories of a sphere with radius of 0.5 µm and refractive index of 1.57 in LG₀₅ beams with (d) RCP and (e) LCP, respectively. The starting position is (-λ, -λ, -λ). (f) The change of x-component of the sphere’s displacements as a function of time. The periods are 0.0113 s, 0.0145 s and 0.0124 s for LCP, RCP and LP, individually. Numerical aperture of lens is 1.2. The wavelength of the beams is λ = 800 nm in water.

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Fig. 2 Transverse part (x and y components) of Poynting vector of tightly focused LG₀₅ beams with (a) RCP and (b) LCP, respectively. The numerical aperture of lens is 1.2.

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More calculations were carried out for the spin effect of the beam with different azimuthal modes. Orbit rate of the sphere in LG₀_l beams changes as a function of azimuthal mode index l for RCP and LCP as shown in Fig. 3(a), where the sphere is 0.5 µm sized, and the refractive index is 1.57. One can see that the larger the azimuthal mode index l is, the slower the orbit rate is. The difference between LCP and RCP becomes smaller when the azimuthal mode index l increases. And it will become very small for very large azimuthal mode index l, which could be ignored. However, the difference of them cannot be ignored for the lower order of azimuthal mode, for instance, it’s about 50 Hz for l = 3. Figure 3(b) shows the tangential optical forces of the sphere at its stable orbits in these beams. Difference between the forces for right and left circularly polarized beams becomes smaller as well as the forces when the order of azimuthal mode becomes higher. The optical force mainly consists of phase-gradient force for high-order aizimuthal mode. Optical torque exerted on the sphere shown in Fig. 3(c) is concurrent with optical force according to Eq. (7), since the orbit rate of a sphere is proportional to $| f_{o r b i t} | / R$ , or $| R \times f_{o r b i t} | / R^{2}$ . Considering transfer of angular momentum, moreover, the ratio of angular momentum of sphere $M_{RCP} / M_{LCP}$ converges to 1. The effect of spin becomes weaker since the total angular momentum of light $J_{RCP} / J_{LCP}$ becomes smaller [ Fig. 3(d)] with increase of azimuthal mode.

Fig. 3 (a) Rotation rates, (b) tangential optical forces and (c) torques acting on a sphere on its stable rotation orbits in LG beams with differing azimuthal modes. the refractive index of the sphere is 1.57, of which the radius is 0.5 µm. (d) Comparison of angular momentum of lights with different polarizaiton states and angular momentum of the orbitally moving sphere in these beams.

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Additionally, orbit rate of the sphere in the high-order LG beam (l, RCP) is approximate to that of the sphere in the beam (l + 1, LCP) according to Fig. 3(a). This is due to the coupling of spin and orbital angular momentum, which will be discussed in detail here. The incident field is expanded using far field matching method. The electric field is purely tangential in the far field, thus the spin angular momentum density can be given as [31]

s_{r} = ε_{0} Im (E_{θ} E_{φ}^{*}) / ω .

The polar spherical coordinates are chosen here. The other two components are zero. If we consider a non-paraxial LG beam, the amplitude of electric field in the far field can be written as [23]

U = U_{0} {(2 ψ)}^{l / 2} L_{p}^{l} (2 ψ) \exp (ψ + i l φ) .

with

ψ = - \tan^{2} θ / \tan^{2} θ_{0}

where

L_{p}^{l}

is the generalized Laguerre polynomial, θ₀ is the beam convergence angle, and θ is the angle measured from the z axis. Then we have

s_{r} = ε_{0} U^{2} / ω

for maximum possible spin. With consideration of the rotational symmetry of beam, only the z component,

s_{z} = s_{r} \cos θ

, is non-zero and possibly contributes to the total angular momentum. Therefore, the total spin angular momentum of LG beam can be given by integrating over a hemisphere:

S_{z} = \frac{\int_{0}^{π / 2} {(2 ψ)}^{l / 2} L_{p}^{l} (2 ψ) \exp (ψ) \sin θ \cos θ d θ}{\int_{0}^{π / 2} {(2 ψ)}^{l / 2} L_{p}^{l} (2 ψ) \exp (ψ) \sin θ d θ},

where

S_{z}

is in unit of ħ per photon. Spin angular momentum in ħ per photon for non-paraxial LG₀_l beams is shown in Fig. 4. The spin angular momentum flux is reduced here because of being tightly focused. The maximum possible spin angular momentum is reduced to a little more than 0.5ħ per photon for high-order LG beams. There must be a corresponding increase in orbital angular momentum since the total angular momentum can’t be changed by being tightly focused. In other word, it’s only about half of spin angular momentum coupling with orbital angular momentum. Therefore, orbital angular momentum of LG beam, which results in the orbital motion of the sphere, will increase about 0.5ħ per photon for right-circular polarization and −0.5ħ per photon for left-circular polarization. Then the orbital angular momentum of high-order LG beam (l, RCP) is approximate to that of the beam (l + 1, LCP). Therefore, Spin angular momentum can contribute to both spin and orbital motion of particle.

Fig. 4 Spin angular momentum in ħ per photon for non-paraxial Laguerre-Gaussian beams as a function of azimuthal mode index. The numerical aperture of the objective lens is NA = 1.2. Here the radial mode is p = 0.

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Orbit rates of different spheres in right and left circularly polarized LG₀₅ beams were investigated in Fig. 5. “•” and “◦” are the discrete results of our calculations in Fig. 5(a). The refractive index of spheres here is 1.57. The orbit rate depends on the radiuses of bright rings of LG beams with different modes, the particle’s equilibrium position, optical force and torque exerted on the sphere. So it isn’t odd for the high scatter. The high scatter has also been mentioned in Ref [32]. The dashed and dotted curves are the fitting curves of those discrete data, which are used to make it easy to compare the difference between orbit rate in LG beams with LCP and RCP. The radius is changed from 0.1 to 1 µm. For the small spheres, the difference of orbit rate between LG₀₅ beams with LCP and RCP is very small, which can be ignored. For the large spheres, however, the difference becomes more than 10 Hz, which is larger than 10% of its orbit rate. Figure 5(b) shows the orbit rates of spheres as a function of refractive index for LG₀₅ beams with LCP and RCP. The radius of spheres is 0.5µm. All the refractive indices of the spheres chosen here are larger than that of the surrounding medium. It’s obviously shown that both the orbit rates of spheres in different circularly polarized beams and the difference between them increase when the refractive index increases. The difference is about 30 Hz when the refractive index increases to 1.7. Additionally, the orbit rates of spheres with lower refractive index, which is close to that of the surrounding medium, are very small due to the very weak optical force acting on those spheres. The difference of orbit rates for LCP and RCP can be then ignored.

Fig. 5 Rotation rates of spheres with varying (a) sizes and (b) refractive indices in right and left circularly polarized LG₀₅ beams. (a) The refractive index of the spheres is 1.57. (b) The radius of the spheres is 0.5 µm.

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4. Conclusions

In summary, Laguerre-Gaussian beams with high-order azimuthal mode were used here to study the orbit rate of dielectric spheres. Hydrodynamic effect of surrounding viscous medium was also concerned, in which case the particle in the optical trap can be considered in low Reynolds number regime. Polarization has an effect on the distribution of electromagnetic field, and optical force acting on a microparticle. Then orbit rate of objects located in LG beams with the same azimuthal mode is spin-controlled. Orbit rates of spheres in LG beams with different azimuthal modes were calculated as well as opical forces and torques. Orbit rate, optical force and the effect of spin decrease when the azimuthal mode l increases. When the azimuthal mode index l is larger enough, the difference can be ignored since total angular momentum of light with LCP and RCP becomes much close. In addition, orbit rate of the sphere in the beam (l, RCP) is approximate to that for the sphere in the beam (l + 1, LCP) for high order of azimuthal mode, which results from the spin-orbit coupling of the tightly focused LG beam. It’s about half of spin angular momentum coupling with orbital angular momentum because of being tightly focused. According to the results about orbit rates with different sizes and refractive indices, moreover, the effect of polarization on orbit rate can be ignored for spheres with very small size or low refractive index. However, it’s obvious for large spheres and increases with the increase of refractive index of the sphere. These results would be much helpful to investigation on optical rotation and transfer of spin and orbital angular momentum.

Acknowledgments

The supports of this work by the National Natural Science Foundation of China (Nos. 11404083, and 11474077) and the Program for Innovation Research of Science in Harbin Institute of Technology (Grant Nos. B201407 and A201411) are gratefully acknowledged.

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Spin-controlled orbital motion in tightly focused high-order Laguerre-Gaussian beams

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (11)

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