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 . 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) . Jixiong Pu et al. used the structured beams, generated from Devil’s vortex-lens (DVL), to trap microspheres . 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 . 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 .
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]:
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]:
By solving the Fourier transform of the above equation, the power spectrum of the captured microsphere is expressed as 
where f is the frequency, kb is Boltzmann’s constant, T is the temperature, and the roll-off frequency fc is
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.
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).
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.
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.
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 . 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.
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.
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)
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Published Online: 2018-07-09