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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

IMPACT FACTOR 2018: 1.005

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Volume 16, Issue 1

# Experimental investigation on optical vortex tweezers for microbubble trapping

Xiaoming Zhou
• College of Information Science and Engineering, Fujian Key Laboratory of Light Propagation and Transformation, Huaqiao University, Xiamen, Fujian 361021, China
• Other articles by this author:
/ Ziyang Chen
• College of Information Science and Engineering, Fujian Key Laboratory of Light Propagation and Transformation, Huaqiao University, Xiamen, Fujian 361021, China
• Other articles by this author:
/ Zetian Liu
• College of Information Science and Engineering, Fujian Key Laboratory of Light Propagation and Transformation, Huaqiao University, Xiamen, Fujian 361021, China
• Other articles by this author:
/ Jixiong Pu
• Corresponding author
• College of Information Science and Engineering, Fujian Key Laboratory of Light Propagation and Transformation, Huaqiao University, Xiamen, Fujian 361021, China
• Email
• Other articles by this author:
Published Online: 2018-07-09 | DOI: https://doi.org/10.1515/phys-2018-0052

## Abstract

In this paper, we investigated the microbubble trapping using optical vortex tweezers. It is shown that the microbubble can be trapped by the vortex optical tweezers, in which the trapping light beam is of vortex beam. We studied a relationship between the transverse capture gradient force and the topological charges of the illuminating vortex beam. For objective lenses with numerical aperture of 1.25 (100×), the force measurement was performed by the power spectral density (PSD) roll-off method. It was shown that transverse trapping forces of vortex optical tweezers increase with the increment of the laser power for fixed topological charge. Whereas, the increase in the topological charges of vortex beam for the same laser power results in the decrease of the transverse trapping forces. The experimental results demonstrated that the laser mode (topological charge) has significant influence on the lateral trapping force.

## 1 Introduction

Ashkin et al. demonstrated that micro-particles can be trapped in three dimensions by laser beams, indicating that laser beams are a very powerful tool in particle trapping and manipulation [1]. Since this pioneer work, optical trapping, which is known as optical tweezers, has attracted much attention. Studies showed that the trapping force depends on particle diameter, polarization direction, depth of the optical trap and numerical aperture [2, 3, 4] In the traditional Gaussian beam-based optical tweezers, some special beams have been introduced to trap particles as well. Allen et al. found that beams with a helical phase structure, i.e. vortex beams, carry orbital angular momentum (OAM) [5]. Jixiong Pu et al. used the structured beams, generated from Devil’s vortex-lens (DVL), to trap microspheres [6]. The OAM of vortex beams can be transferred to particles and make them rotate. The influence of amplitude profile, polarization and laser power on this rotation has been investigated [7, 8, 9, 10, 11]. Owing to the phase singularity, the intensity center of vortex beams is a central dark core. It has been indicated that particles with refractive index lower than the ambient can be trapped by beams of dark core. Therefore, some studies have focused on trapping the particles with refractive index lower than the ambient by optical vortex tweezers [12, 13, 14, 15]. For vortex optical tweezers trapping the particles with refractive index lower than the ambient, to the best of our knowledge, there have been no experimental studies on the relationship between the transverse gradient force and the topological charges of vortex beams. The aim of the study is to experimentally investigate this relationship. In this paper, we used the power spectral density (PSD) method [16, 17, 18, 19, 20] to demonstrate the relationship between the transverse gradient force and the topological charge of the vortex beam.

## 2 Theory of force calibration

There have been several papers studying the optical tweezers for trapping particles using vortex beams. Ng et al. theoretically studied optical trapping by an optical vortex beam. They found that the trap stiffness of cylindrically symmetric optical vortex has real and imaginary numbers [21]. In our experiment, no rotation was observed in our experiment. Thus, the trap stiffness is always real. Therefore, the optical tweezers can be modeled by Hooke’s law [22].

$F=−kixi,$(1)

where ki is the spring constant (trap stiffness) of the trap and xi is the displacement from the centre of the trap. The trap stiffness can be measured using the PSD method based on the OTKBFM-CAL (Thorlabs) software. According to PSD method, the equation of Brownian motion for particles in the optical trap can be described by [20, 23]:

$md2xi/dt2+βdxi/dt+kixi(t)=F(t),$(2)

where m is the mass of the particle, β=6 π η a is the drag coefficient, ηis the fluid viscosity, and a is the radius of the particle. F(t) is a random force, xi(t) is the displacement at time t, dxi/dt is the velocity, and d2xi/dt2 is the acceleration in the xi direction. When the inertial force can be neglected compared with the viscous force, Eq. (2) can be described by a simplified equation as follows [16, 20, 23]:

$βdxi/dt+kixi(t)=F(t).$(3)

By solving the Fourier transform of the above equation, the power spectrum of the captured microsphere is expressed as [16]

$X2(f)=kbT/π2β(f2+fc2),$(4)

where f is the frequency, kb is Boltzmann’s constant, T is the temperature, and the roll-off frequency fc is

$fc=ki/2πβ.$(5)

In the experiment, the dependence of the power spectrum X2(f) on frequency f can be measured, and the roll-off frequency fc can be determined by Eq. (4). The transverse trap stiffness ki can be obtained by substituting the calculated roll-off frequency into Eq. (5) [16, 20].

## 3 Experimental results

The experimental setup of optical vortex tweezers was shown in Figure 1. The laser source was a single-mode laser diode with a maximum output power of P = 340 mW of a wavelength of 975 nm. Spiral phase plates (SPP) were used to transform the incident Gaussian beam into a vortex one. The topological charge of the vortex beam was determined by the phase structure of the SPP. The expanded vortex beam was focused by an objective with NA of 1.25. The particles could be trapped near the focus of the objective if the gradient force was larger than that of scattering and absorption force.

Figure 1

Schematic of the experimental setup. L-lens, M-reflection mirror, OB-objective lens, DM-dichroic mirror, LDC-laser diode and controller, BE-beam expander, F1(IR)-IR filters, F2(ND)-neutral density filters, CCD-charge coupled device, QPD-quadrant position detector, spp-Spiral phase plates, LED-Light-Emitting Diode.

In the PSD method, the quadrant position detector (QPD) needed a high bandwidth to record the particle Brownian motion information. To do the measurements with broad bandwidth and high-resolution, a QPD was placed in the conjugate back-focal plane of the condenser lens (Nikon 10X Air Condenser, numerical aperture of 0.25). The signal generated by the QPD was sensitive to the relative displacement of the trapping particles from the centre of the optical trap, which could be used to determine the position, stiffness, and force of the optical tweezers. The experimental data (voltage signal) from the QPD was loaded into the mechanical acquisition module OTKBFM-CAL (Thorlabs Products). The position calibration and the PSD could be obtained by the software. The work principle of the software is described as follows:

The voltage signal measured in the experiment is V(t) = ξxi(t), ξ is constant. The displacement signal V(t) was measured by QDP. Then we got V(t)-t curve from the software. According to V2(f) = ξ2X2(f)=ξ2kbT/π2β(f2+f2c) and Eq. (4), PSD function can be obtained by performing calculation with software, which is shown in Figure 2. According to Figure 2, we could get the roll-off frequency fc (corner frequency) by Lorentzian fitting PSD function, which is the abscissa value of the left blue line. Finally, the trap stiffness could be obtained by substituting fc into the Eq. (5).

Figure 2

PSD as a function of frequency, for finding the roll-off frequency fc, being 2.3923 × 10 Hz

In the experiment, we focused on the transverse forces of vortex optical tweezers acting on microbubbles (F-30VS, Matsumoto Yushi-Seiyaku). The microbubble was an alkane sphere (n3 = 1.4) surrounded by a transparent dielectric spherical shell (n2 = 1.6). The radius of the microbubbles ranges between 3 to 7 μm, and that of alkane sphere ranges between 2 to 5 μm. Compared with other types of microbubbles, this type of microbubble was found to be more stable. The influence of the temperature on the shape of the microbubble could be neglected as the non-recoverable thermal expansion occurs only when the sample was heated to 80°C. In the experiment, the power of the laser was chosen to be 100 mW, which could heat the sample by approximately 1° C.

We placed the sample between two plastic slides with built-in channels. To avoid excessively concentrating the sample, we mixed it with deionized water to a dilution ratio of 1:10. The sample was placed near the focus of the objective, and the plastic slides were placed on a displacement stage with a differential regulator (MAX300 NanoMax 3-Axis Flexure Stage, Thorlabs). The displacement stage provided a 4 mm (0.16 inch) coarse stroke and 300 μm fine adjustment stroke, with coarse adjuster 10 μm graduated scale, and the micro adjuster 1 μm graduated scale. To trap particles, the light beam should be tightly focused, and the tightly focused pattern was appropriately adjusted with a NanoMax 3-Axis flexure stage. We showed that the particle in the sample could be trapped and moved by the optical tweezers, as shown in Figure 3. The trapped particle was indicated by the arrow of Figure 3.

Figure 3

(a) A free particle; (b), (c), and (d) a captured particle (the arrow indicates the captured particle; the others are uncaptured microbubbles as a reference)

We wanted to study the influence of the topological charge of the incident light beams on the optical trapping force. The incident vortex beams with topological charge of l = 1, 2 and 3 were generated for trapping microbubbles. The transverse stiffness of optical vortex tweezers was calculated by PSD. For a vortex beam with fixed topological charge, we performed measurements ten times. Owing to that the experimental data are uncertain, we use Box-plot to express the discrete distribution of the experimental data, which is given in Figure 4. It was shown that from the experimental measurements that the transverse stiffness decreases with the increasing topological charge of the vortex beams. This was consistent with theoretical simulations [24, 25]. This result could be understood by the intensity distribution of the vortex beams. The intensity pattern of a tightly focused vortex beam was a dark core, surrounding by a bright annulus. The size of the dark core increases with the increasing topological charge of the vortex beam. Therefore, the magnitude of the intensity of the bright annulus decreased, which led to a lower stiffness.

Figure 4

Relationship between the transverse stiffness and the topological charge. The power of the laser is 6.073 mW

In a Gaussian beam-based optical tweezers, the trapping was influenced by the power of the incident laser. In general, the trapping force increased with a higher laser power [2]. We also investigated the influence of the laser power on the optical vortex tweezers. We took the vortex beam with topological charge of l = 3 as an example. The influence of the laser power on the transverse stiffness was presented in Figure 5. It was found that the increasing laser power results in larger transverse stiffness. This result was similar to that of Gaussian beam-based optical tweezers.

Figure 5

The transverse stiffness as a function of the incident laser power. The topological charge of the vortex beam was chosen as l = 3.

## 4 Conclusions

We had shown that the particles could be trapped and moved in an optical vortex tweezers, which was achieved by introducing a vortex beam as the incident beam. The effect of the topological charge and the power of the incident beam on the transverse stiffness was investigated. A high stiffness could be obtained by vortex beam with lower topological charge and stronger power. This finding was particularly significant for research on transverse trapping forces based on vortex beam.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61575070, 11674111, and 61505059) and the Promotion Program for Young and Middle-Aged Teachers in Science and Technology Research of Huaqiao University (ZQN-PY209)

## References

• [1]

Ashkin A., Dziedzic J.M., Bjorkholm J.E., Chu S., Observation of a single-beam gradient force optical trap for dielectric particles, Opt. Lett., 1986, 11(5), 288.

• [2]

Wright W.H., Sonek G.J., Berns M.W., Parametric study of the forces on microspheres held by optical tweezers, Appl. Opt., 1994, 33(9), 1735-48.

• [3]

Zhong M., Zhou J., Wu J., Li Y., Determination of the axial stiffness of an optical trap with information entropy signals, Proceedings of SPIE, Int. Soc. Opt. Eng., 2009, 7507, 75070J-75070J-7. Google Scholar

• [4]

Liang Y.Z., Ou Q.R., Bo L., Liang R.Q., Improvement of transverse trapping efficiency of optical tweezers, Chinese Phys. Lett., 2008, 25(6), 2300-2302.

• [5]

Allen L., Beijersbergen M.W., Spreeuw R.J., Woerdman J.P., Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes, Phys. Rev. A Atom. Mol. Opt. Phys., 1992, 45(11), 8185.

• [6]

Pu J., Jones P.H., Devil’s lens optical tweezers, Opt. Expr., 2015, 23(7), 8190-9.

• [7]

Volkesepúlveda K., Chávezcerda S., Garcéschávez V., Dholakia K., Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles, J. Opt. Soc. Amer. B, 2004, 21(10), 1749-1757.

• [8]

Kovalev A.A., Kotlyar V.V., Porfirev A.P., Optical trapping and moving of microparticles by using asymmetrical laguerre-gaussian beams, Opt. Lett., 2016, 41(11), 2426.

• [9]

Li M., Yan S., Yao B., Liang Y., Zhang P., Spinning and orbiting motion of particles in vortex beams with circular or radial polarizations, Opt. Expr., 2016, 24(18), 20604-20612.

• [10]

Gao H., Rotation dynamics of yeast cell in vortex optical tweezers, Chin. J. Lasers, 2011, 38(4), 0404002, (in Chinese). Google Scholar

• [11]

O’Neil A.T., Padgett M.J., Axial and lateral trapping efficiency of laguerre–gaussian modes in inverted optical tweezers, Opt. Comm., 2001, 193(1), 45-50.

• [12]

Prentice P.A., Macdonald M.P., Frank T.G., Cuschieri A., Spalding G.C., Sibbett W. et al., Manipulation and filtration of low index particles with holographic laguerre-gaussian optical trap arrays, Opt. Expr., 2004, 12(4), 593-600.

• [13]

Swartzlander G.A., Gahagan K.T., Trapping of low-index microparticles in an optical vortex, J. Opt. Soc. Amer. B, 1998, 15(15), 524-534.

• [14]

Ahluwalia B.S., Løvhaugen P., Hellesø O.G., Waveguide trapping of hollow glass spheres, Opt. Lett., 2011, 36(17), 3347-9.

• [15]

Fury C., Harfield C., Memoli G., Jones P.H., Stride E., Experimental characterisation of holographic optical traps for microbubbles, Proceedings of SPIE - Int. Soc. Opt. Eng., 2014, 9126, 91263L-10, 1117/12.2055889. Google Scholar

• [16]

Sarshar M., Wong W.T., Anvari B., Comparative study of methods to calibrate the stiffness of a single-beam gradient-force optical tweezers over various laser trapping powers, J. Biomed. Opt., 2014, 19(11), 115001.

• [17]

Florin E.L., Pralle A., Ehk S., Jkh H., Photonic force microscope calibration by thermal noise analysis, Appl. Phys. A, 1998, 66(1), S75-S78. Google Scholar

• [18]

Fei P., Yao B.L., Ming L., Yan S.H., Measurement of optical trapping force and stiffness of micro-particles with the drag-force method, J. Optoel. Laser, 2010, 21(1), 78-82, (in Chinese). Google Scholar

• [19]

Neuman K.C., Block S.M., Optical trapping, Rev. Sci. Instr., 2004, 75(9), 2787-2809.

• [20]

Berg-Sørensen K., Flyvbjerg H., Power spectrum analysis for optical tweezers, Rev. Sci. Instr., 2004, 75(3), 594-612.

• [21]

Ng J., Lin Z., Chan C.T., Theory of optical trapping by an optical vortex beam, Phys. Rev. Lett., 2010, 104(10), 103601.

• [22]

Mclaren M., Sidderashaddad E., Forbes A., Accurate measurement of microscopic forces and torques using optical tweezers, S. Afr. J. Sci., 2011, 107(9/10), 67-74.

• [23]

Mazur P., On the theory of brownian motion, Physica, 1959, 25(1-6), 149-162.

• [24]

Wang J., Ren H., Trapping forces of core-shell particles using laguerre-gaussian beams, Chinese J. Lasers, 2015, 42(6), 0608001, (in Chinese). Google Scholar

• [25]

Chai H.S., Wang L.G., Improvement of optical trapping effect by using the focused high-order laguerre–gaussian beams, Micron, 2012, 43(8), 887-892.

Accepted: 2018-05-15

Published Online: 2018-07-09

Citation Information: Open Physics, Volume 16, Issue 1, Pages 383–386, ISSN (Online) 2391-5471,

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