## Abstract

The orbital rotation is an important type of motion of trapped particles apart from translation and spin rotation. It could be realized by introducing a transverse offset to the dual-beam fiber-optic trap. The characteristics (e.g. rotation perimeter and frequency) of the orbital rotation have been analyzed in this article. We demonstrate the influences of offset distance, beam waist separation distance, light power, and radius of the microsphere by both experimental and numerical work. The experiment results, i.e. orbital rotation perimeter and frequency as functions of these parameters, are consistent with the theoretical model in the present work. The orbital rotation amplitude and frequency could be exactly controlled by varying these parameters. This controllable orbital rotation can be easily applied to the area where microfluidic mixing is required.

© 2016 Optical Society of America

## Corrections

20 February 2017: A correction was made to the funding section.

## 1. Introduction

Optical tweezers have offered an emerging method of choice for microfluidic actuation and has important applications in the fields of biology, fundamental physics and engineering [1–4]. Controlled translation and spin rotation have traditionally been thought of as the most valuable optical manipulation technique of the trapped objects [5, 6]. The orbital rotation is a novel manipulation technique apart from translation and spin rotation. It actuates the captured particle rotating along a periodic orbit, and will become a topic of general interest to many applications, such as microfluidic mixing, driven machines and cell manipulation [7–9].

H. He and associates first observed the transfer of orbital angular momentum to absorptive particles from a focused linearly polarized Laguerre-Gaussian beam with a helical phase in 1995 [10]. Subsequently, A. T. O’Neil et al. successfully achieved the orbital rotation of the suspended particles driven by the orbital angular momentum carried by Laguerre-Gaussian beam [11].Since then, a number of groups have used this technique to rotate microscopic samples [12–15]. However, for this orbital angular technique, the orbital rotation directions are limited to about the beam axis [16, 17]. Further, the rigorous restricted conditions for beam profile and samples may limit its applications [18, 19].

In 2008, R. Gordon and associates presented that counterpropagating beams with transverse offset might lead to the orbital rotation [20, 21]. Subsequently, they achieved the orbital rotation by combination of fluid drag and optical scattering forces from two angled fibers [22]. Recently, we firstly demonstrated the orbital rotation of a trapped particle by a dual-fiber optical trap with transverse offset [23]. As the offset distance increases, the motion type of the suspended particle starts with capture, then spiral motion, orbital rotation, and eventual escape [24]. This orbital rotation technique is considered to be widely applicable because it is no longer limited by highly controlled beam profile or samples with special optical properties [18, 23, 24].

In this paper, the orbital rotation perimeter and frequency are evaluated by a range of values of parameters, including the offset distance, the beam waist separation distance, the light power, and the radius of microsphere. Our present work discussed the influences of these parameters by experimental and numerical methods, and will contribute to controllable motions of samples in localized environment especially in integrated lab-on-a-chip devices.

## 2. Experiment

The schematic of the dual-beam fiber-optic trap with transverse offset is shown in Fig. 1, where *d* denotes the offset distance. The particle is trapped by two counter-propagating beams emanating from single-mode fibers. When the two counter-propagating beams are misaligned, the trapped particle may rotate along an elliptic orbit orbital in the *xz* plane.

In order to demonstrate the orbital rotation motion, we suspended a polystyrene microsphere (refractive index *n*_{2} = 1.59, radius *r*_{0} = 5 µm) in water by two single-mode fibers (waist radius ω_{0} = 3 μm). Each fiber was coupled with different laser diode source to avoid the generation of coherent interference, both operating at a wavelength of *λ* = 980 nm. The laser powers emitted from two fibers *P* = *P*_{1} = *P*_{2} = 80 mW. The beam waist separation distance *S* = 375 µm. Figure 2 shows the experimental photographs of the dynamic trajectory of the trapped microsphere in a period of time, with the offset distance *d* set as 18 µm.The microsphere rotated along an elliptic orbit, as the red solid curves indicate. The red arrows represent the direction of the motion. The orbital rotation frequency and perimeter are 0.67 Hz and 191 µm, respectively.

## 3. Numerical methods

For the radius of the microsphere *r*_{0} is much bigger than the wavelength of trapping laser *λ*, we use ray optics approximation to simulate the trajectory of the microsphere. In ray optics approximation, the light beam is decomposed into individual rays. The trapping force acted on the sphere is the sum of contributions of all rays [25].

When a single ray hitting the microsphere, the trapping force generated by this ray can be divided into the scattering force component *d**F** _{s}* and the gradient force component

*d*

*F**, which are given by [26]:*

_{g}_{1}= 1.33.

*dP*is the light power of the ray.

**and**

*s***denote the unit vectors parallel and perpendicular to the ray, respectively.**

*g**q*and

_{s}*q*are the efficiency factors of the optical force, which are given by:

_{g}*α*and

_{i}*α*are the angles of incidence and refraction.

_{r}*R*and

*T*are the Fresnel reflection and transmission coefficients at

*α*. For an unpolarized incident beam, the reflection and transmission coefficients can be calculated by the average of

_{i}*s*and

*p*polarizations. We thus have

In the simulation, the beams emerging from the fibers are considered as two incoherent Gaussian beams. The multiple scattering effects of light between the fibers and the trapped particle are ignored. Thus, the total trapping force *F** _{tot}* can be obtained by summing all the forces calculated for each ray. When moving in the water, the microsphere is also affected by the viscosity resistance:

**is the velocity of the microsphere. The viscosity coefficient**

*v**η*= 0.839 × 10

^{−3}Pa·s at 25 °C. Then the velocity

**can be given by:**

*v*The program designed to simulate the trajectory of the microsphere is based on calculating its position after a small increment of time *Δt*. The next position of the microsphere after *Δt* can be located by the Runge-Kutta method [27]:

Figure 3 shows the simulated trajectory of the microsphere, with all parameters set to be the same as those used in the experiment. The colors and directions of the arrows respectively represent the magnitudes and directions of trapping forces. The red solid curve denotes the trajectory of microsphere. The microsphere eventually rotated along an elliptic orbit, which is consistent with the experimental result.

## 4. Discussion

In this section, the orbital rotation perimeter and frequency are evaluated for a dual-beam fiber-optic trap with transverse offset both experimentally and numerically. The influences of offset distance, beam waist separation distance, light power, and radius of the microsphere are discussed in detail.

#### 4.1 Offset distance

Figure 4 shows the orbital rotation perimeter and frequency versus the offset distance *d* both in experiment and simulation. The other parameters are *P* = 80 mW, *S* = 375 µm, and *r*_{0} = 5 µm. The experimental and simulation results are denoted as black squares and red solid curves, respectively. The measured orbital rotation perimeter and frequency in experiment as a function of offset distance verify the general trend predicted by the theoretical model.

As shown in Fig. 4(a), the orbital rotation perimeter increases with *d* until the microsphere is no longer trapped. The breaking offset is denoted as the double-headed arrow. This can be explained by Fig. 5, where transverse trapping force *F*_{x} along the *x* axis is evaluated for varying d. There are three transverse trapping force zero points of on the *x* axis. One of the zero points is the trap center. The orbital rotation perimeter is proportional to the minor axis, which is determined by the separation distance between the other two zero points. The increasing *d* brings about a larger transverse trap stiffness. As a result, the separation distance between the transverse trapping force zero points increases, leading to a larger orbital rotation perimeter.

The orbital rotation frequency of the microsphere is determined by both orbital rotation perimeter and orbital rotation velocity *v*. According to Eq. (5) orbital rotation velocity *v* is proportional to the magnitude of optical force, which changes slowly with d. Relatively, the orbital rotation perimeter of the microsphere increases more quickly for increased *d*, and therefore the orbital rotation rate decreases.

#### 4.2 Light power

The orbital rotation is also affected by the light power *P* emitting from two fibers. The influence of *P* on the orbital rotation perimeter and frequency is investigated both in experiment and simulation. The results are shown in Fig. 6, where the offset distance of 18 µm, waist separate distance of 375 µm and microsphere radius of 5 µm are fixed. The light power changes the magnitude of trapping force rather than its distribution. As a result, the orbital rotation perimeter is invariable with *P*, as Fig. 6(a) shows. As P increases, the raising of the trapping force magnitude improves the rotation velocity *v*. Therefore the rotation frequency increases monotonically with P, as Fig. 6(b) shows.

#### 4.3 beam waist separation distance

The beam waist separation distance *S* is also a key factor contributing to the orbital rotation. Figure 7 shows the experimental data and the theoretical results that indicate the influence of *S*. The other parameter values *d* = 16 µm, *P* = 80 mW and *r*_{0} = 5 µm are used. The general trend of the experimental results follows the theoretical curve. For a larger *S*, both the experimental and the simulation results exhibit a decreasing of orbital rotation perimeter. When other parameters are fixed, the total light power intercepting the sphere decreases with increased S. The transverse trap stiffness also decreases, and so does the orbital rotation perimeter. In addition, when S increases, the decreasing of trapping force leads to a slower rotation velocity *v*. The orbital rotation rate is directly proportional to *v* and inversely proportional to orbital rotation perimeter. Figure 7(b) implies incremental value of orbital rotation frequency upon S. Consequently, we can draw a conclusion that the orbital rotation perimeter decreases faster with the increased *S* than *v* does.

#### 4.4 radius of microsphere

Figures 8(a) and 8(b) show the orbital rotation perimeter and frequency versus transverse offset *d* for varying microsphere radius *r*_{0}. The double-headed arrows denote the maximum or minimum value of *d* for the orbital rotation. The parameters not indicated in the legends are fixed at *P* = 80 mW and *S* = 310 µm. When *d* is smaller than the minimum value *d _{min}*, the microsphere will spiral into the trap center. When

*d*is larger than the maximum value

*d*, it will escape [21]. The captured microsphere rotates in the trapping region only when

_{max}*d*< d <

_{min}*d*. Both the experimental and simulation data reveal that the interval [

_{max}*d*,

_{min}*d*] is getting to narrow down with the decreasing of

_{max}*r*

_{0}. For the captured microsphere with a radius of 2 µm, the simulation results indicate that the region will turn to be zero and no orbital rotation will be observed, which is verified by experiment.

As Fig. 8(a) exhibits, the raising of *r*_{0} leads to a smaller rotation perimeter. In order to explain this phenomenon, the transverse trapping force *F*_{x} along the *x* axis is evaluated for varying *r*_{0} in Fig. 9. The total power illuminating at the microsphere increases with *r*_{0}. Thus the maximum value of the transverse force increases. However, the slope of the transverse force also increases along with *r*_{0}. Hence the separation distance between the transverse trapping force zero points decreases, resulting in the decrease of orbital rotation perimeter. Furthermore, the increasing of trapping force improves the rotation velocity *v*. That is, while *r*_{0} increases, the rotation perimeter decreases and the rotation velocity increases. Therefore, the rotation frequency rises with the sphere radius, as shown in Fig. 8(b).

## 5. Conclusion

In this paper, we have analyzed the characteristics of the orbital rotation realized by the dual-beam fiber-optic trap with transverse offset. The microsphere swirls along an elliptic orbit when the offset distance *d* is in an interval [*d _{min}*,

*d*]. The orbital rotation perimeter and frequency are evaluated as functions of the offset distance

_{max}*d*, the beam waist separation distance

*S*, the light power

*P*, and the radius of microsphere

*r*

_{0}both experimentally and numerically. It was found that, when the other parameters are invariable, the orbital rotation perimeter increases with increased

*d*, decreased

*S*, and decreased

*r*

_{0}, while it is invariable with

*P*. The orbital rotation frequency increases with decreased

*d*, increased

*P*, increased

*S*, and increased

*r*

_{0}. The theoretical model results about the orbital rotation perimeter and frequency well agree with the experiments. Both the experimental and simulation results show that the interval [

*d*,

_{min}*d*] is getting to narrow down with the decreasing of

_{max}*r*

_{0}until turn to be zero. Therefore, to achieve the orbital rotation of the captured microsphere in dual-fiber optical trap, the sphere radius must be fairly large.

In summary, the orbital rotation amplitude and frequency could be exactly controlled by varying the offset distance, the beam waist separation distance, the light power, and the radius of microsphere. The orbital rotation technique will become generally applicable to achieving controllable motions of samples in microfluidic systems.

## Funding

National Natural Science Foundation of China (61308058); Open Research Fund (SKLST201507) of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences.

## Acknowledgment

The authors would like to acknowledge the helpful discussion from Xiang Han.

## References and Links

**1. **A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**(4), 156–159 (1970). [CrossRef]

**2. **J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. **77**(1), 205–228 (2008). [CrossRef] [PubMed]

**3. **K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. **75**(9), 2787–2809 (2004). [CrossRef] [PubMed]

**4. **M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. **4**, 1768 (2013). [CrossRef] [PubMed]

**5. **A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. **18**(21), 1867–1869 (1993). [CrossRef] [PubMed]

**6. **B. J. Black, D. Luo, and S. K. Mohanty, “Fiber-optic rotation of micro-scale structures enabled microfluidic actuation and self-scanning two-photon excitation,” Appl. Phys. Lett. **101**(22), 221105 (2012). [CrossRef]

**7. **M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics **5**(6), 343–348 (2011). [CrossRef]

**8. **A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. **88**(5), 053601 (2002). [CrossRef] [PubMed]

**9. **K. Y. Bliokh, J. Dressel, and F. Nori, “Conservation of the spin and orbital angular momenta in electromagnetism,” New J. Phys. **16**(9), 093037 (2014). [CrossRef]

**10. **H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**(5), 826–829 (1995). [CrossRef] [PubMed]

**11. **A. T. O’Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. **185**(1), 139–143 (2000). [CrossRef]

**12. **Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express **17**(25), 23316–23322 (2009). [CrossRef] [PubMed]

**13. **J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B **72**(8), 085130 (2005). [CrossRef]

**14. **V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. **91**(9), 093602 (2003). [CrossRef] [PubMed]

**15. **H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A **75**(6), 063409 (2007). [CrossRef]

**16. **K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. **4**(2), S82–S89 (2002). [CrossRef]

**17. **Z. Hong, J. Zhang, and B. W. Drinkwater, “Observation of orbital angular momentum transfer from bessel-shaped acoustic vortices to diphasic liquid-microparticle mixtures,” Phys. Rev. Lett. **114**(21), 214301 (2015). [CrossRef] [PubMed]

**18. **B. J. Black and S. K. Mohanty, “Fiber-optic spanner,” Opt. Lett. **37**(24), 5030–5032 (2012). [CrossRef] [PubMed]

**19. **J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**(13), 133901 (2003). [CrossRef] [PubMed]

**20. **R. Gordon, M. Kawano, J. T. Blakely, and D. Sinton, “Optohydrodynamic theory of particles in a dual-beam optical trap,” Phys. Rev. B **77**(24), 245125 (2008). [CrossRef]

**21. **M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, “Theory of dielectric micro-sphere dynamics in a dual-beam optical trap,” Opt. Express **16**(13), 9306–9317 (2008). [CrossRef] [PubMed]

**22. **J. T. Blakely, R. Gordon, and D. Sinton, “Flow-dependent optofluidic particle trapping and circulation,” Lab Chip **8**(8), 1350–1356 (2008). [CrossRef] [PubMed]

**23. **G. Xiao, K. Yang, H. Luo, X. Chen, and W. Xiong, “Orbital rotation of trapped particle in a transversely misaligned dual-fiber optical trap,” IEEE Photonics J. **8**(1), 1–8 (2016). [CrossRef]

**24. **X. Chen, G. Xiao, H. Luo, W. Xiong, and K. Yang, “Dynamics analysis of microsphere in a dual-beam fiber-optic trap with transverse offset,” Opt. Express **24**(7), 7575–7584 (2016). [CrossRef] [PubMed]

**25. **E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. **36**(25), 6423–6433 (1997). [CrossRef] [PubMed]

**26. **A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**(2), 569–582 (1992). [CrossRef] [PubMed]

**27. **R. Alexander, “Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s,” SIAM J. Numer. Anal. **14**(6), 1006–1021 (1977). [CrossRef]