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Abstract
Intracavity optical tweezers have been proposed and demonstrated recently, which allows orders-of-magnitude higher optical confinement with lower-numerical-aperture lens and lower laser power in contrast to the standard optical tweezers. We further investigate its characteristics about the position stability of trapped particles. The dependence of the radial and axial position stability on the laser intensity acting on the particle of 10-µm diameter in intracavity optical tweezers and standard optical tweezers are compared experimentally. Result shows that higher laser intensity can make stronger optical confinement in intracavity optical tweezers under the condition of good trap operation, compared with standard optical tweezers. We demonstrate and analyze the coupling between the particle’s radial and axial motion, and then provide two approaches to reduce it. Our work will benefit the further enhancement of position stability for the trapped particle in intracavity optical tweezers.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the invention of optical tweezers in 1986 [1], as a novel measurement and manipulation tool, optical traps have found many important applications in various research fields including biology [2,3], statistical physics [4–6], macroscopic quantum physics [7,8], nanotechnology [9], and ultrahigh-precise measurement [10,11]. The above-listed applications of optical trapping typically require a high level of control over position, whether it be a measurement, or manipulation [12–14]. Naturally, increasing the laser power can enhance the trapping force and yield strong position control. However, the consequent photothermal side-effect is one limiting factor of the laser intensity used [15]. Thus, one of the most active areas of research in the last several years has been the exploration of routes to obtain high confinement along the three axes per unit laser intensity on the sample [16].
Due to the optical forces, interaction forces between the trapped object and trapping medium, and the object’s inertia, the dynamics of a trapped particle are quite complex. As a result, the application of feedback control to optical trapping is an obvious approach used to improve the system performance and enable new capabilities [17]. The earliest example of feedback control is the stabilization of levitated microparticles with optical forces, as demonstrated by Ashkin and Dziedzic [18]. From then on, advances in position measurement [19,20], position actuators [21], and controller design [22] have brought clear improvements in system performance, including enhanced position stability and wider operational bandwidth. However, a feedback control system usually requires complex hardware and signal processing procedures, and as a result closed-loop bandwidth is limited [23,24].
Kalantarifard et al. first presented and demonstrated intracavity optical tweezers inside an active laser cavity [25]. The particle is trapped inside the cavity of a fiber laser. When the particle tries to escape from the trap region, it scatters less light so the optical loss decreases, the laser power increases and the particle is pulled back. Based on the above-mentioned coupling between the laser power and the particle’s position, Kalantarifard et al. have implemented self-feedback control of the particle’s position. There is no requirement for any detecting and controlling hardware and signal processing procedures. These are distinguishing features different from optical tweezers with feedback controlling system [23,24]. Such systems have obtained a two-order-of-magnitude reduction of the average light intensity at the sample when compared with standard optical tweezers that achieve the same degree of confinement.
In this paper, we further investigate the characteristics of the position stability of the particle held in intracavity optical tweezers. The relationships between the position stability and laser intensity acting on the particle of 10-µm diameter in intracavity optical tweezers and standard optical tweezers were compared experimentally. We analyzed the coupling between the particle’s radial and axial motions which is harmful to axial position stability. Two approaches to compress this coupling effect were ultimately proposed.
2. Experimental setup and result
The experimental setup is shown in Fig. 1, and represents an intracavity optical tweezers system that contains an optical tweezer inside laser cavity. The optical cavity comprises a single-mode Yb-doped fiber (Yb 1200-6/125, nLIGHT, Inc., core diameter of 6 µm, cladding diameter of 125 µm) as gain medium. The pumping laser source is a single-mode fiber-pigtailed diode laser, emitting at a wavelength of λp = 976 nm with a maximal output power of Pmax = 500 mW. A fiberized band-pass filter is used to ensure that the laser wavelength is centered at λ = 1030 nm with a full width at half maximum of 2 nm. The beam is then split by a fiber optical coupler (TW1064R1F2A, Thorlabs, Inc., coupling ratio 99:1). The larger portion is sent to a single-mode fiber-pigtailed collimator (ZC618FC-B, Thorlabs, Inc.), and the smaller portion to a photodiode power meter (PDA50B2, Thorlabs, Inc.). After being reflected by a short-pass dichroic mirror (DMSP1000, Thorlabs, Inc.), the laser is focused on the cuvette by an aspheric lens (RMS10X, Olympus, focal length 10.6 mm, numerical aperture(NA) 0.25) and then collected by a second identical aspheric lens. A cuvette filled with deionized water and polystyrene particles (Unibead polystyrene 6-2-1000, BaseLine, diameter of 10.0µm) is fixed to a three-axis translation system (P15XYZ, Coremorrow). Another short-pass dichroic mirror (DMSP1000, Thorlabs, Inc.) reflects the laser into another collimator. Finally, the laser is coupled back into the fiber by the collimator. An in-line isolator (IO-F-1030, Thorlabs, Inc.) is used to ensure that the light travells within the ring cavity along the arrowhead. We detected the particle’s axial and radial position using two separate CCD cameras (DCU223M, Thorlabs, Inc.) at a sampling rate of ∼120 Hz. A cylindrical coordinate system was established, as shown in Fig. 1, the z-axis of which is along the inverse gravity direction and the origin of which is positioned at the focus point of the lens.
In most optical trapping, high positional stability is desirable. We define the radial and axial stability Sr, Sz, as the inverse positional variances, and define the radial and axial confinement efficiency Cer, Cez as the inverse positional variance per trapping power density It, written as
We trapped a polystyrene particle of 10-µm diameter at different laser power values exerted on it for the cases of the intracavity optical tweezers (seeing in Fig. 1) and a standard optical tweezers (NA = 1.2, λ=980nm) respectively. Figure 2 shows the comparison of their radial and axial confinement efficiencies. Clearly, the minimum trapping power that the standard optical tweezers require to achieve stable trapping is higher than the intracavity optical tweezers.
Furthermore, for the case of the standard optical tweezers, the radial and axial confinement efficiency keeps a constant while the trapping beam intensity increase, see Fig. 2(a) and Fig. 2(b). That is because the inverse radial and axial position variance Sr, Sz is linearly proportional to the trapping power in theory for standard optical tweezers. This means that the optical confinement efficiencies Cer, Cez are independent of the trapping power according to the formula (1).
Fig. 2. Comparison of experimentally measured radial (◊, red) and axial (□, blue) confinement efficiency, Cer and Cez, at different trapping power densities, for standard optical tweezers (a)(b) and intracavity optical tweezers (c)(d).
On contrast, the optical confinement efficiency tends to rise up with the growth of trapping beam intensity for the case of the intracavity optical tweezers as shown in Fig. 2(c) and Fig. 2(d). On one hand, the axial equilibrium position is located far below the focal spot (see Supplementary Fig. 4(b) of Ref. [25]). When the trapping power increases, the axial equilibrium position will be closer to the beam waist of laser, leading to enhanced confinement efficiency. On the other hand, the higher trapping power can achieve stronger nonlinear feedback intensity according to the coupling relation between the laser power and the particle displacement. This may generate larger nonlinear feedback force per unit intensity, resulting in higher optical confinement efficiency. The dependency between the optical confinement efficiencies and the trapping power will get a further in-depth and careful study next.
In addition, we observed that the coupling between the particle’s radial and axial motions, aggravating its axial positional stability. Through the three-axis translation system, we moved the cuvette away from its initial position for 5 µm radially (point a), and returned after 25 s (point b). The laser power and the particle’s positions were recorded at a pump power of 120 mW as shown in Fig. 3, where the origin is located at the initial equilibrium point. When the particle was in the center of the optical trap, the laser operated at low power 0.3 mW. When the particle moved away from its initial position along the radial direction, more of the laser beam reached the collector lens, leading to an increase of the laser power to 43.1 mW. As a result, an increase of the scattering force pushed the particle up approximately 7.3 µm along the axial direction. At the same time, an increased radial restoring force (mainly gradient force) pulled the particle back to its initial radial position. Finally, the laser power decreased, letting the particle fall back toward its initial position in the axial direction. When the particle moved away from the trap center along the opposite radial direction, the similar process would manifest. From Fig. 3, it can also be noticed that the laser power and the particle’s axial offset changed as soon as its radial displacement occurred. Then, the radial position and laser power recovered quickly (approximately 1.6 s), while the particle moved back slowly to its original axial position (approximately 14.2 s), resulting in its worse axial positional stability.
Fig. 3. Experimental results of coupling between and radial motions. Radial (dotted curve) and axial (solid curve) particle displacements (a) and laser power in the cavity (b), when an applied step radial displacement (dashed curve, 5 µm, interval time 25s) acts on the particle.
Fig. 4. Simulation results for coupling between axial and radial motions, with parameters: λ = 1030 nm, the beam waist w0=1 mm, the focal length of objective f = 10.6 mm, NA = 0.25, the diameter of the polystyrene particle d = 10 µm, the density of the particle ρ = 2.2 g/cm2, the refractive index of the particle n = 1.59, and η=0.89×10−3Pa•s. Radial (dotted curve) and axial (solid curve) particle offsets(a) and the laser power (b) in the cavity corresponding to (a), when an applied step radial displacement (dashed curve, 4 µm, interval time 25 s) acts on the particle.
3. Numerical results and discussion
The forces exerted on the trapped particle mainly include the optical forces Fr, Fz, the gravity minus the buoyancy FG, the viscous force and the thermal fluctuations. Thus, the particle radial position r and axial position z are described by the Langevin equation [26]:
The optical gradient and scattering force depend on the particle position away from the equilibrium position and laser power, which can be written as:
Since the size of the particle was larger than the wavelength, we calculated the optical force using geometrical optics theory [26,27]. The particle was assumed to have had a step radial displacement applied (amplitude 4 µm, interval time 25 s). Figure 4 illustrates the simulation result for a polystyrene particle of 10µm-diamenter held in intracavity optical tweezers. When the particle was settled to the radial position a, the laser power increased to 47.2 mW immediately. Resultant enhanced optical scattering force broke the previous axial force equilibrium, as a result, the particle moved to position b. After approximately 1.7 s, called the recovery time, the particle was pulled back to the original radial position by enhanced optical gradient force, while the laser power also decreased down back to its previous value. The axial force equilibrium in b was broken again, and the particle dropped back to its original axial position after 13.1 seconds. We noticed that the axial offset distance was almost two-fold greater than radial offset distance, and the axial recovery time was much longer than the radial recovery time, which was in good agreement with the experimental results presented in the above section.
We further investigated how to reduce the coupling effect, in order to enhance position stability of particles trapped in intracavity optical tweezers. First, we found that both the axial offset δz and the recovery time δt cut down along with the reduction of the NA of the lens as shown in Fig. 5. Thus, it would be better to trap particles using a lower NA lens in the intracavity optical tweezers.
Fig. 5. Simulations for the axial offset δz (a) away from the equilibrium position and the recovery time δt (b) as a function of NA. The error bars stem from many simulations.
Moreover, the medium viscosity around the particle also influences the axial offset distance and the recovery time, see Fig. 6, where the NA is 0.25. The axial offset will sharply decrease with the medium viscosity’s increase until it equals to about 2×10−5 Pa.s, perhaps arising from the inertial item’s dominance in formula (3). Beyond this value, the medium viscosity has little to do with the axial offset (see Fig. 6(a)). In Fig. 6(b), the radial and axial recover times will firstly decrease and then increase with the medium viscosity’s increase.
Fig. 6. Axial offset δz (a) away from equilibrium position and the recovery time δt (b) as a function of medium viscosity. The simulation starts from the medium viscosity η=1×10−6 Pa.s because less than this the particle is not trapped any more.
To better understand this interesting phenomenon, we plotted the time evolution of the particle radial and axial positions in Fig. 7. Clearly, the lower medium viscosity permits the optical force’s pulling the particle back towards its original point more easily in the radial and axial directions (comparing Fig. 7(a) and Fig. 7(b)). However, when the medium viscosity is lower than 4×10−6 Pa•s and the particle experiences the high-Reynolds-number process, the second order differential system defined by formula (1) will become underdamped, which characterizes the radial and axial damping oscillation around the equilibrium position. This gives rise to additional recovery time (see Fig. 7(c)). It is worth noting that when the value of medium viscosity is reduced to less than η=1×10−6 Pa.s, the axial offset δz will become too large to be pulled back to the trap as shown in Fig. 7(d).
Fig. 7. Time evolution of radial and axial positions after the particle is settled at a radial offset of 3 µm for different medium viscosities: η=2×10−3 Pa•s (a), η=4×10−6 Pa•s(b), η=2×10−6 Pa•s (c) and η=1×10−6 Pa•s(d). δtr and δtz represent the radial and axial recovery times, respectively.
4. Conclusions
Since intracavity optical tweezers implement an inherent close-loop control, it characterizes a higher confinement along the three axes with lower numerical aperture lens and lower laser intensity, compared with standard optical tweezers. In addition, we pointed out another advantage of the intracavity optical tweezers that the optical confinement efficiency tends to become higher with laser power’s rising on contrast with the standard optical tweezers. A preliminary explanation has been given out and a further in-depth and careful study is needed next. It should be noted that an appropriate size of trapped particle is needed to ensure highly efficient feedback and good operation of ring cavity fiber laser.
Meanwhile we have demonstrated that the coupling effect between the particle’s radial motion and axial motion, against its axial position stability. Two methods have been proposed to depress the coupling effect through simulation. On one hand, using a lower numerical aperture lens can obtain more quick recovery after radial offset. On the other hand, we can choose appropriate medium viscosity balancing well the small viscosity force and long period damping oscillating to achieve the optimal recovery time.
In summary, we further investigated its novel characteristics about the optical confinement efficiency in the intracavity optical tweezers. We analyzed the coupling process between the particle’s radial motion and axial motion, and provided two approaches to reduce it. It is straightforward that our work will benefit the further enhancement of position stability of the trapped particle in the intracavity optical tweezers.
Funding
National Natural Science Foundation of China (61975237).
Acknowledgments
The author appreciates useful discussions with Pan Xu.
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