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Abstract
The feedback control to optical tweezers is an obvious approach to improve the optical confinement. However, the electronic-based feedback controlling system in optical tweezers usually consists of complex software and hardware, and its performance is limited by the inevitable noise and time-delay from detecting and controlling devices. Here, we present and demonstrate the dual-beam intracavity optical tweezers enabling all-optical independent radial and axial self-feedback control of the trapped particle’s radial and axial motions. We have achieved the highest optical confinement per unit intensity to date, to the best of our knowledge. Moreover, both the axial and radial confinements are adjustable in real-time, through tuning the foci offset of the clockwise and counter-clockwise beams. As a result, we realized three-dimensional self-feedback control of the trapped particle’s motions with an equivalent level in the experiment. The dual-beam intracavity optical tweezers will significantly expand the range of optical manipulation in further studies of biology, physics and precise measurement, especially for the sample that is extremely sensitive to heat.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the invention of optical tweezers in 1986 by Arthur Ashkin et al. [1], optical tweezers have become an essential tool in several fields of physics [2,3], spectroscopy [4,5], biology [6,7], nanotechnology [8,9], and thermodynamics [10,11]. In these areas, higher optical confinement along the three axes has been desired to reduce the photothermal side-effect on the trapped particle [12]. We defined the optical confinement efficiency as the inverse positional variance per trapping power density on the trapped sample to characterize the confinement performance of the optical tweezers [13]. The optical confinement efficiency of most optical tweezers is usually lower than 103 mW-1 [2,14]. Many feedback control approaches have been used to improve system performance, including enhanced optical confinement efficiency and wider operational bandwidth [2,15,16]. A. E. Wallin et al. improved the optical stiffness up to ∼13 folds by using the real-time re-programmable signal processing based on field programmable gate array [17]. T. Li et al. realized the feedback bandwidth up to 75 MHz by applying the specially designed position detection system [2]. However, the feedback control system in above-mention studies usually requires complex hardware such as particle position sensing module, laser intensity modulator and electronic circuit, as well as signal processing procedures [18].
Kalantarifard et al. first presented and implemented self-feedback control of the particle’s position in the single-beam intracavity optical tweezers [19]. They achieve intracavity optical trapping inside an active laser cavity where the laser mode is directly influenced by the position of the particle. It is a novel dissipative optomechanical system with a levitated particle at the micrometer-scale [13,19,20]. The optomechanical coupling gives rise to intrinsic nonlinear feedback forces that confine microparticles efficiently at low intensity and low numerical apertures. Such systems have obtained a two orders of magnitude reduction of the average light intensity at the sample, compared with single-beam standard optical tweezers that achieve the same degree of confinement. Moreover, there is no requirement for any detection and control hardware or signal processing procedures compared with the standard optical tweezers with an extra feedback loop [19]. The intracavity optical tweezers is expected to extend its promising applications in future studies of optical manipulation, precise measurement, opto-mechanics, etc.
However, we noticed that the axial confinement efficiency is consistently about one order of magnitude lower than the radial confinement efficiency in the single-beam intracavity optical tweezers [13,19]. Our research has shown that there is only one feedback loop between the trapped particle’s three-dimensional positions and the scattering loss of the intracavity laser. That leads to the coupling effect between the particle’s radial and axial motions, and aggravates the axial confinement efficiency [13]. Besides, the trapping direction of the single-beam intracavity optical tweezers is immutable, since the unidirectional trapping beam needs to point upward to counteract the gravity. Moreover, the single-beam intracavity optical tweezers is difficult to be employed in air or vacuum, due to the weak restoring force produced by the gravity [20].
In this paper, we first present and demonstrate the dual-beam intracavity optical tweezers, enabling all-optical independent radial and axial self-feedback control of the trapped particle’s radial and axial motions. Importantly, the axial and radial nonlinear feedback optical forces are adjustable through tuning the foci offset of the clockwise and counter-clockwise beams. An experimental system is built to characterize the remarkable performance of this novel optical tweezers in air. Moreover, we fully discuss the advantages of the dual-beam intracavity optical tweezers, compared with single-beam intracavity optical tweezers, single-beam standard optical tweezers and dual-beam standard optical tweezers.
2. Results
2.1 Working principle of the independent self-feedback control schemes
Comparison of feedback control among standard optical tweezers with feedback control, single-beam intracavity optical tweezers and dual-beam intracavity optical tweezers is illustrated in Fig. 1. For the case of the standard optical tweezers, the trapping laser and optical tweezers operate independently as shown in Fig. 1(a). The precise position sensors, data processors, and laser modulators are required to build the external feedback control system [2,15,16]. As a result, the system performance is limited by the inevitable noise and time-delay from above devices. On the contrast, the single-beam intracavity optical tweezers realizes all-optical self-feedback control based on the intrinsic coupling effect between the scattering loss of the cavity and the particle, without requirement for any detection and control hardware or signal processing procedures [19]. However, it fails to apply independently radial and axial feedback, since the radial and axial motions of the trapped particle are fed back to the same physical quality, i.e. the scattering loss of the cavity [13], as illustrated in Fig. 1(b). In our scheme, the particle’s radial and axial motions are fed back to independent physical qualities, as shown in Fig. 1(c). The radial motion of the trapped particle will lead to identical response to the scattering losses of the clockwise (CW) beam and counter-clockwise (CCW) beam, i.e. δcw and δccw. Meanwhile, the particle’s axial motion will be coupled to the differential scattering loss, i.e. δcw - δccw. It is expected to be the optimal scheme to achieve all-optical independent axial and radial self-feedback control and therefore can obtain stronger optical confinement.
Fig. 1. Comparison of the feedback control among (a) standard optical tweezers, (b) single-beam intracavity optical tweezers and (c) dual-beam intracavity optical tweezers. P and δ represent laser power and scattering loss, respectively. The green and red arrows indicate the radial and axial tracks of the feedback loop, respectively. The yellow arrow represents the track of mixed feedback channel, where the radial and axial motions of the trapped particle are fed back to the same physical quality.
Figure 2 schematically shows the responses of the scattering loss and the intracavity laser power when a set of triangle axial displacements is applied to the particles trapped in the dual-beam intracavity optical tweezers. The cylindrical coordinate systems were established as shown in Fig. 2(a). The cylindrical coordinate system is centered at the lens’s focal point in Fig. 2(a1). The origin of the cylindrical coordinate system in Fig. 2(a2) is located at the center between two foci of the trapping lenses. Compared with the single-beam intracavity optical tweezers, we omit the optical isolator in the ring fiber cavity of the dual-beam intracavity optical tweezers as shown in Fig. 2(a). Then both the CW and CCW beams are permitted to travel and construct a dual-beam optical tweezers inside the ring cavity. The dependence of their scattering loss on the particle position is the same as that in the single-beam intracavity optical tweezers [19]. Naturally, they will produce double radial nonlinear feedback optical force and generate stronger radial optical confinement. However, outwardly tiny modification, i.e. omitting the optical isolator, will bring out a completely novel feedback mechanism for the axial position of the trapped particle.
Fig. 2. Comparison of the characteristic between (1) single-beam intracavity optical tweezers and (2) dual-beam intracavity optical tweezers. (a) The schematics of the (a1) single-beam intracavity optical tweezers and (a2) dual-beam intracavity optical tweezers. The foci of two trapping lenses in (a2) is offset to a certain value D. (b) The cavity loss as a function of the particle’s axial position. (c) The response of the intracavity optical powers and optical forces when a set of triangle axial displacement is applied on the trapped particle.
According to the dependence of the scattering losses on the axial offset distance of the particle away from the focal spot (see Fig. 6 in Ref. [20]), for the case of the dual-beam intracavity optical tweezers with no foci offset, the scattering losses of the CW and CCW laser respond nearly identically to the particle’s axial motion and cancel out. So, the differential scattering loss of the CW and CCW lasers is independent of the particle’s axial displacement and no axial nonlinear feedback force is exerted on the trapped particle. As a result, only the weak axial scattering force gives rise to the low axial optical confinement efficiency. However, when we offset the foci of the CW and CCW beams, the scattering loss curve of the two beams will deviate from each other and show inverse response to the axial displacement of the trapped particle as shown in Figs. 2(b2), 2(c2). Then, the differential scattering loss and the resultant differential laser intensity of the two beams will generate axial nonlinear inverse feedback optical force and improve the axial trapping stability.
Obviously the higher the change rates of the differential scattering loss with respect to the displacement from the focal spot, |d(δcw-δccw)/dz| and |dδ/dr|, the larger the nonlinear axial and radial inverse feedback optical forces [20]. We calculate the |d(δcw-δccw)/dz| and |dδ/dr| according to the physical model in Ref. [20], as shown in Fig. 3. The trapped particle is treated as an equivalent spherical lensing. Then, the scattering loss is calculated by applying the ray transfer matrices of the optical devices inside the cavity [21,22]. The numerical aperture of the trapping lens is set as NA = 0.25. The wavelength of the trapped laser is 1030 nm. The silicon particle’s radii are selected as 2, 3, 6, and 8 µm respectively. The result shows that the |d(δcw-δccw)/dz| sharply increases first and then decreases when the foci offset rises. The |d(δcw-δccw)/dz| reaches its peak around D = 22 µm. Orders-of-magnitude improvement of the |d(δcw-δccw)/dz| is obtained compared with the case with no foci offset. Meanwhile, the |dδ/dr| is slightly reduced with the rise of the foci offset. Figure 3 also shows that the stronger |d(δcw-δccw)/dz| and |dδ/dr| are obtained for a larger particle. Thus, we can adjust the foci offset to optimize the axial and radial feedback optical forces.
Fig. 3. Simulation results of the scattering loss. (a) The differential scattering loss per axial displacement and (b) the scattering loss per radial displacement versus the foci offset D.
2.2 Experimental results
The experimental setup is depicted in Fig. 4(a). It comprises a continuous-wave ring-cavity fiber laser emitting along the CW and CCW directions. The Yb-doped gain medium (Yb 1200-6/125, nLIGHT, core diameter of 6 µm, cladding diameter of 125 µm) is pumped by a single-mode diode-laser at 976 nm through a wavelength division multiplexer WDM1. The residual pumping laser is coupled out of the ring-cavity by another wavelength division multiplexer WDM2, and terminates in a laser terminator. The CW and CCW lasers, centered at 1030 nm, are expanded to free space by collimators C1, C2 (ZC618FC-B, Thorlabs) and coupled into the opposite fibers through the opposite collimators, implementing a ring-cavity. We can tune the foci offset of CW and CCW beams by adjusting the focusing knob of the collimators. The lenses L1 and L2 (NA = 0.25) in the free space laser path compose a 1:1 beam expansion system. The particle is trapped by the focused CW and CCW lasers in the chamber. We use the dichroic mirror (1030 nm, R = 98 ± 0.2%; 532 nm, R < 5%) to reflect the majority of the intracavity laser. The 2% of the intracavity laser power along the collimator’s direction is sent to the power meter (PDA50B2, Thorlabs) for power monitoring. The CW and CCW powers are detected by power meters P1 and P2, respectively. The 2% of the intracavity laser power along the trapping direction is sent to three balanced photo-detectors (PDB450C, Thorlabs) for detecting the three-dimensional position of the particle [2]. The CCD is used to observe the trapped particle in air.
Fig. 4. Experimental setup and results. (a) The schematic of the experimental setup. Two red arrows represent the CW and CCW directions. WDM represents the wavelength division multiplexer. (b) The response of the intracavity optical powers when the particle moves along the axial direction. The periodic motion of the particle is executed by modulating the optical power of the 532 nm laser. (c) The radial (□, blue) and axial (◊, red) optical confinement efficiency as a function of the foci offset D.
The pumping power is set as about 300 mW in the experiment. We use a collimated single-mode laser with the wavelength of 532 nm to push the particle along the axial direction, implementing a set of periodic motion. Result shows that the light powers of the CW and CCW laser vary with the axial motion of the trapped particle, as illustrated in Fig. 4(b). When the particle moves along the CW direction (-z direction), the CCW laser power increases and the CW laser power decreases. When the particle moves along the CCW direction (+z direction), the CCW laser power decreases and the CW laser power increases. The differential power of the CW and CCW laser (Pcw-Pccw) shows the same trend as the particle’s axial displacement, which is in great agreement with the prediction as illustrated in Fig. 2(c2).
We also measured the optical confinement property of the dual-beam intracavity optical tweezers for different foci offsets [13], as shown in Fig. 4(c). The optical confinement efficiency is defined as the inverse positional variance per trapping intensity It, written as
3. Discussion
Figure 5 illustrates the comparison of the experimental confinement efficiency between dual-beam intracavity optical tweezers, single-beam intracavity optical tweezers, single-beam standard optical tweezers and dual-beam standard optical tweezers for particles of various sizes. We set the foci offset of two beams as about 24 µm in the dual-beam intracavity optical tweezers when the SiO2 sphere is trapped. Others included in this figure are compiled from published experiments [2,13,19]. It shows that, as a whole, the intracavity optical tweezers realizes stronger optical confinement than the standard optical tweezers due to the intrinsic self-feedback control scheme. Moreover, the axial confinement efficiency achieved in the dual-beam intracavity optical tweezers is the highest among the above-mentioned optical tweezers, which is one order of magnitude higher than that for the dual-beam standard optical tweezers, and two orders of magnitude higher than that for the single-beam intracavity optical tweezers. The improvement of the optical confinement efficiency is remarkable for the particle with larger diameter, even though the dual-beam intracavity optical tweezers is configured with lower-numerical-aperture lenses than single-beam standard optical tweezers and dual-beam standard optical tweezers.
Fig. 5. Experimental results for the radial (□) and axial (◊) optical confinement efficiency of the dual-beam intracavity optical tweezers (red, NA = 0.25), compared with single-beam intracavity optical tweezers (blue, (a) NA = 0.12, (b) NA = 0.25), single-beam standard optical tweezers (purple, NA = 1.25) and dual-beam standard optical tweezers (brown, NA = 0.68). The data except for the dual-beam intracavity optical tweezers is compiled from published experiments for published experiments: (a) Ref. [19], (b) Ref. [13], (c) Ref. [2].
It is instructive to compare dual-beam intracavity optical tweezers with the dual-beam standard optical tweezers and single-beam intracavity optical tweezers. For the case of dual-beam standard optical tweezers, the axial direction is mainly balanced by the scattering force, while the exerted gradient force is relatively weak [23,24]. However, the stronger radial laser gradient creates the larger radial optical force. Naturally, the radial confinement efficiency is inherently higher than the axial confinement efficiency. We remark that regulating the foci offset of the dual-beam standard optical tweezers can effectively improve the axial trapping stiffness and the axial confinement efficiency [25,26]. However, the foci offset will also lead to negative optical stiffness and multi-stability [27]. Moreover, this approach only alters the static force distribution, and the optical confinement efficiency is still limited by the trapping stiffness per unit intensity at the sample [13]. On contrast, we can optimize the dynamical self-feedback control in the dual-beam intracavity optical tweezers, through tuning the foci offset of the CW and CCW beams. Then, it will generate stronger feedback optical force to counteract the dynamical motion of the trapped particle, due to optomechanical coupling. The dual-beam intracavity optical tweezers exhibits one order of magnitude higher axial confinement efficiency than that for the dual-beam standard optical tweezers, surpassing the limit from the weak scattering force.
For the case of single-beam intracavity optical tweezers, the axial motion of the trapped particle is fed back to the same physical quality, i.e. the scattering loss of the intracavity laser, as its radial motion [13]. That leads to the coupling effect between the particle’s radial and axial motions, and reduces the axial confinement efficiency. On contrast, dual-beam intracavity optical tweezers has fundamental differences in that it creates a separated self-feedback loop between the axial displacement and their differential scattering loss through offsetting the foci of the CW and CCW beams. Though it seems as if we only omit the optical isolator compared with the single-beam intracavity optical tweezers, we indeed achieve independent self-feedback control of the trapped particle’s radial and axial motions in the dual-beam intracavity optical tweezers. Obviously, the breakthrough of dual-beam intracavity optical tweezers is not just using “dual-beam” instead of “single-beam”, compared with single-beam intracavity optical tweezers.
On the other hand, the trapped particle in the single-beam intracavity optical tweezers is located at a fixed position relative to the focal spot, which is decided by the axial optical force and the particle’s gravity force [20]. According to Fig. 3(a), the change rates of the scattering losses, with respect to the displacement from the focal spot, are fixed. Thus, we are unable to regulate the axial and radial nonlinear inverse feedback optical forces, unless altering the particle’s property or optical configurations. On contrast, we can achieve an adjustable axial and radial nonlinear inverse feedback optical forces in real-time, through tuning the foci offset of the CW and CCW beams in the dual-beam intracavity optical tweezers. This scheme allows one to regulate both the axial and radial confinement efficiencies according to different application requirements. The dual-beam intracavity optical tweezers shown two orders of magnitude higher confinement efficiency than that for the single-beam intracavity optical tweezers in the experiment. Also we achieved three-dimensional self-feedback control of the trapped particle’s motions with equivalent level. By the way, the direction of the trapping beam in the dual-beam intracavity optical tweezers is arbitrary, without the restriction due to gravity in the single-beam intracavity optical tweezers [19].
4. Conclusion and perspective
In conclusion, we have presented and realized the dual-beam intracavity optical tweezers, enabling independent self-feedback control of the trapped particle’s radial and axial positions. In the experiment, the highest axial and radial confinements per unit intensity to date have been obtained in the dual-beam intracavity optical tweezers, compared to all other state-of-the-art optical tweezers. Moreover, we can adjust the radial and axial optical confinements to different requirements in real-time. Add to it that both the intracavity optical tweezers can operate with very low-NA lenses, and at very low average power, thus we expect that the dual-beam intracavity optical tweezers will open up a broader application field for optical manipulation, especially in biology where the sample is extremely sensitive to light intensity or needs equivalent confinement along three-dimensions [12]. In addition, it can help to pre-cool the center-of-mass motion of the micrometer-scale particle before further quantum ground cooling in higher frequency domain than electronic-based feedback controlling system, thanks to the intracavity self-feedback [28].
By the way, the dual-beam intracavity optical tweezers may be a very novel micrometer-scale levitated cavity optomechanical system based on dissipative coupling. That is intrinsically different from the cavity optomechanical system with levitated nano-particle based on the dispersive coupling [29,30]. We anticipate it might be intriguing to extend the concepts and experiments observed in micrometer-scale levitated cavity optomechanical system towards a new territory when it operates in air or vacuum.
Funding
National Natural Science Foundation of China (61975237, 11904405); National University of Defense Technology (ZZKY-YX-07-02, ZK20-14).
Acknowledgments
The authors gratefully acknowledge the support from Independent Scientific Research Project of National University of Defense Technology (ZZKY-YX-07-02) and Scientific Research Project of National University of Defense Technology (ZK20-14). The authors want to thank the valuable assistance from Bin Luo at the Beijing University of Posts and Telecommunications.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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