Open Access
Add to BibSonomyAbstract
While the behavior of optically trapped dielectric spherical particles has been extensively studied, the behavior of non-spherical particles remains mainly unexplored. In this work we focus on the dynamics of oblate spheroidal particles trapped in a tightly focused elliptically-polarized vortex beam. In our experiments we used polystyrene spheroids of aspect ratio of major to minor axes equal to 2.55 and of a volume equal to a sphere of diameter 1.7μm. We demonstrate that such particles can be trapped in three dimensions, with the minor axis oriented perpendicular to both the beam polarization (linear) and the beam propagation, can spin in a circularly polarized beam and an optical vortex beam around the axis parallel with the beam propagation. We also observed that these particles can exhibit a periodic motion in the plane transversal to the beam propagation. We measured that the transfer of the orbital angular momentum from the vortex beam to the spheroid gives rise to torques one order of magnitude stronger comparing to the circularly polarized Gaussian beam. We employed a phase-only spatial light modulator to generate several vortex beam traps with one spheroid in each of them. Due to independent setting of beams parameters we controlled spheroids frequency and sense of rotation and observed hydrodynamic phase and frequency locking of rotating spheroids. These optically driven spheroids offer a simple alternative approach to the former techniques based on birefringent, absorbing or chiral microrotors.
© 2014 Optical Society of America
1. Introduction
An employment of the transfer of linear momentum from light to objects led to enormous progress in contactless manipulation with objects. Laser beams are nowadays used in hundreds of applications where microobjects, nanoobjects, or living microorganisms are trapped, delivered, sorted, deformed, or their surrounding viscosity or acting external forces are quantified [1–6]. On the top of that the transfer of spin and orbital angular momentum from light to objects has been also investigated from the theoretical and experimental point of view in the last two decades [7, 8]. Such transfer usually leads to rotation of objects around their axis or orbiting around the beam propagation axis. Transfer of spin angular momentum from an elliptically polarized beam upon a birefringent [9–14], absorbing [15], or a non-spherical [16–22] object is responsible for one way how to generate object rotation in fluid, gas [23] or vacuum [14]. Further approaches causing object rotation or orbiting have exploited the transfer of orbital angular momentum from a vortex beam to microscopic particles [24–30], screwing particles with helical structure by Gaussian beam [17, 31, 32], rotation of an asymmetrical beam profile [33–35] and illumination of a rod-like or ellipsoidal particle with optical vortex or elliptically polarized beam [18, 19, 36, 37]. Optically trapped and rotated particles have found applications in rheology as optically driven pumps [13, 21, 32, 38, 39], as a tool quantifying the properties of fluids or gases from a microscopic point of view [11, 23, 38, 40, 41] or even navigating nerve fiber grow [42]. An optically trapped ellipsoidal probe represents a new generation of photonic force microscopes [43] with the ability to detect tiny external torque in addition to tiny external force [20, 44, 45]. Ellipsoidal particles are also a subject of theoretical interest because they represent well-defined geometry to be investigated for the Brownian motion [46] and transfer of linear and angular momentum between light and particle [47–56].
In this paper we demonstrate for the first time the three dimensional (3D) spatial confinement and spinning of an oblate spheroidal particle (OSP) (i.e. an ellipsoidal particle with two identical major axes) in focused elliptically polarized vortex beam. Using spatial light modulator we generated several such optical traps with independently controlled parameters and for the first time we report phase synchronization and frequency locking of trapped and spinning OSPs.
2. Experiment
Preparation of oblate spheroidal particles
We used polystyrene spheres (Bangs Laboratories, Inc.) of Rs = 0.85μm and refractive index np = 1.59. Manufacturing of OSPs was based on published methods [57,58] which we illustrate in Fig. 1. A thick (≈ 0.5 mm) polymeric film made with polyvinyl alcohol and glycerol is used as a solid matrix to suspend a diluted sample of spherical polystyrene particles. The film is heated up to 160°C, which is higher than the glass-transition temperature Tg of the polyvinyl alcohol and polystyrene, and then pressed using thin spacers to control the final thickness of the pressed sample. Finally the OSPs were released by solving the polymeric matrix in water [58]. Figure 2(c) shows an example of OSPs observed by scanning electron microscope (SEM). OSPs are of ellipsoidal shape defined by two longer axis with equal length 2a and by a shorter axis of length 2b. Let us define the OSP aspect ratio s as s = a/b and the equivalent spherical radius Rs = (a2b)1/3 giving the same sphere volume as the OSP . All results presented in this paper were obtained using OSPs with Rs = 0.85μm and s = 2.55. The error related to the determination of the particle sizes was 5%. The released OSPs are dispersed in water and placed into the sample chamber made of two cover glasses (borosilicate 130 – 160μm thick) that are separated by spherical particles of 15μm in diameter and hermetically sealed. If not reported, the laser power in the sample cell between the cover glasses was 3.3 mW.
Fig. 1 Manufacturing of oblate spheroidal particles. (a) Polymeric film (Polyvinyl alcohol and glycerol) with dispersed polystyrene (PS) spheres is pressed along one axis at a temperature T higher than the glass-transition temperature Tg of the polyvinyl alcohol (PVA) and polystyrene. (b) Following this method the spherical particles become oblate spheroids. (c) Image of the manufactured OSPs from scanning electron microscope-SEM (Magellan 400, FEI). Original spheres were added later to the manufactured OSP sample before observation by SEM in order to compare the sizes of spheres and OSPs. One such sphere can be seen in the lower left part of the image.
Fig. 2 Description of the experimental setup. The laser beam from Verdi V6 (wavelength λ = 532 nm, input power 6W) is expanded 8× by the telescope T1 (not shown in the figure) (f = 25 mm, Thorlabs AC127-025-A and f = 200 mm, Thorlabs AC254-200-A) is reflected from the spatial light modulator (SLM, Hamamatsu PAL-SLM, X8267–5080DB), passes through the achromatic doublet L1 (focal length f = 400 mm, Thorlabs AC254-400-A), aperture that blocks all diffraction orders except the first one, achromatic doublet L2 (f = 250 mm, Thorlabs AC254-250-A), telescope T2 (0.75X, with lenses f = 200 mm, Thorlabs AC254-200-A and f = 150 mm, Thorlabs AC254-150-A), polarizer (linear film polarizer with high extinction ratio and laser damage threshold, Thorlabs LPVISB050), quarter wave plate (QWP, multi-order quarter-wave plate,Thorlabs WPMQ05M-532), high-numerical aperture microscope objective (Olympus UPLSAPO 60× water immersion, NA = 1.2). The objective is mounted on Z piezo controller (Mad City Labs, NanoF-200) and the sample cell is mounted on XY positioning stage (Prior Scientific, Proscan II). The OSPs dispersed in the sample cell are monitored by the microscope objective (Olympus UPLSAPO, 100×, oil immersion, NA = 1.40), lens tube L3 (Achromatic doublet with f = 250 mm, Thorlabs AC254-250-A) and CCD camera (Basler acA640-100gm).
Experimental setup
The whole trapping setup is described in Fig. 2. The flexibility of the system is based on a spatial light modulator (SLM) controlling the beam parameters (i.e. beam power, beam width, intensity and phase profile) and number of beams used for trapping [59,60]. We added a blazed grating to the original phase in the SLM (giving rise to the characteristic fork dislocation grating [61]) in order to diffract the desired beam efficiently to the first order. This diffraction order is spatially filtered and focused to the sample cell with dispersed particles. Using this configuration we generated Gaussian beams and simple vortex beams with controlled beam width, intensity and topological charge, respectively, at the input aperture of the focusing objective. It determines the beam properties (diameter, intensity and phase profile) in the objective focal plane and consequently provides control of the 3D trapping and behavior of OSPs. We also employed SLM to compensate optical aberrations in the optical path in the following way: we removed both objectives and minimized the beam focus imaged at CCD camera by the method presented by Cizmar et al. [62]. The elliptical polarization of the beam is controlled with a quarter-wave plate (QWP) placed close to the entrance aperture of the objective. A linearly polarized beam before the QWP is set precisely with a high power polarizer. The dynamics of the OSPs is recorded with a video-microscopy system providing sampling time of Δτ > 2.8ms and resolution 43nm/pixel. Position and orientation of an OSP were determined from the center of mass and eigenvectors of the moment of inertia of the OSP gray-scale image intensity distribution in OSP central region, respectively [63]. The moving bright spot is caused by the illumination in combination with the tilt of the OSP. It prevents precise determination of the lateral OSP position but does not strongly influence the determination of the OSP azimuthal orientation.
3. Spheroid in a focused elliptically-polarized Gaussian beam: 3D confinement, autonomous oscillations and rotations
Similarly as spheres, OSPs were laterally trapped on the beam axis and axially behind the beam waist position. OSPs were oriented with their longer axes along the direction of the propagation of the beam. Figure 3 shows several frames of the OSP dynamics in right-hand circularly polarized (RHCP) Gaussian beam. Except anti-clockwise rotation, due to the circularly polarized beam, the lateral oscillations of the OSP are clearly visible. Analyses of CCD images gave results presented in Fig. 4 for circularly polarized (CP) and linearly polarized (LP) beams in the form of time record of the lateral position of the OSP center denoted as r⊥, the azimuthal orientation of the OSP denoted as ϕ, and diagram of OSP angular dynamics. In CP beam the particle rotates very slowly but consistently according to the sense of rotation of the polarization. Slight azimuthal nonlinearity is probably caused by the method used for determination of the OSP orientation in combination with azimuthal nonuniformity of optical intensity. In the case of LP beam, the OSP was oriented with its second longer axis along the beam polarization. The particle is randomly deviated laterally and angularly due to the Brownian motion. These observations are consistent with the previously reported observations [18,44,45] and theoretical predictions [47, 54]. Moreover, for both CP and LP beams we also observed that the particle is subjected to a lateral oscillatory motion, perpendicular to the axis of propagation of the beam. We believe this is closely related to the wobbling phenomena observed before in light-levitated prolate spheroids [37]. However, in our case the OSP is smaller and trapped in 3D.
Fig. 3 Time sequence of CCD images of an OSP trapped in 3D and rotating anti-clockwise in the right-hand circularly polarized (RHCP) Gaussian beam. Lateral oscillations of the trapped particle are visible, too. Fig. 4 shows the corresponding tracking of the centroid and orientation of the particle in the whole time record.
Fig. 4 Dynamics of the trapped OSP in the RHCP and LP Gaussian beam. Time record of the lateral position r⊥ of the OSP center, azimuthal OSP orientation ϕ and diagram of the OSP XY position and orientation in degrees (see the main text describing the tracking method).
4. Spheroid in a focused elliptically polarized vortex beam: 3D confinement and rotation
For the case of a tightly focused vortex beam with low topological charge l = ±1, ±2 we achieved confinement of the OSP in 3D, the OSP was laterally localized on the optical axis and rotated around its axis overlapping with the beam propagation axis. Figure 5 demonstrates this behavior in time series of CCD images. Image analysis gives results presented in Fig. 6. It is seen that for the same incident power, the OSP rotates about 10× faster comparing to Fig. 4 where the rotation was only driven by the CP beam. Also the lateral oscillations of r⊥ are reduced.
Fig. 5 Time sequence of CCD images of an OSP trapped in 3D and rotating clockwise in a vortex beam with topological charge l = −2 that was focused down by the objective upon the particle.
Fig. 6 Dynamics of the trapped particle in a vortex beam with l = −2 focused down upon the particle. Time record of the lateral position r⊥ of the OSP center and azimuthal OSP orientation ϕ.
The influence of the topological charge and polarization of the vortex beam on an OSP is demonstrated in Figs. 7 and 8. Figure 7(a) shows the time record of the azimuthal angle of the OSP. We also fitted polynomials of degree 3 and frames of 9 points using the Savitzky-Golay filter [64] to further smooth the measured data in Fig. 7(a) (full black lines) and to calculate the time derivatives more precisely by the analytical derivation of the polynomials to get the immediate frequency of rotation fϕ shown in Fig. 7(b) (full black lines). For comparison the blue curve denotes the immediate frequency of rotation fϕ obtained from the raw data by the following differences:
where Δt denotes the time step in the measurement.Fig. 7 Example of the determination of the immediate frequency of OSP rotation. Figure (a) presents time records of azimuthal angles ϕ, obtained from the image analysis, for different topological charges (|l| = 1, 2) and polarizations of the beam. Dashed (full) curves correspond to |l| = 1 (|l| = 2), green/red/blue colors encode RHCP/LP/LHCP polarizations. Inset in (a) shows a detail of the trajectory for a LHCP beam with l = −2. The black solid lines correspond to the filtered data. Figure (b) demonstrates the immediate frequency of rotation fϕ of the OSP as a function of time for the LHCP beam with l = −2. The black solid thin curve corresponds to the frequencies obtained from filtered ϕ dependence.
Fig. 8 Influence of vortex beam topological charge and polarization on the frequency fϕ of OSP rotation obtained from the filtered data ϕ (t). (a) Azimuthal dependence of immediate frequency of OSP rotation. Positive fϕ corresponds to anti-clockwise disk rotations in Fig. 5. Dashed (full) curves correspond to |l| = 1 (|l| = 2), green/red/blue colors encode RHCP/LP/LHCP polarizations. (b) Distribution of probability density of fϕ for above mentioned topological charges and polarizations of the vortex beam.
Figure 8(a) shows azimuthal dependence of fϕ obtained from the filtered ϕ(t) for different combinations of the topological charges |l| = 1, 2 and circular and linear polarizations of vortex beam incident on the objective and focused down upon the particle. Figure 8(b) illustrates distribution of probability density of fϕ for all the beams parameters mentioned above. These results reveal that transfer of orbital angular momentum from the beam upon a particle is the dominant mechanism in the configurations studied here and the influence of the beam polarization on the OSP rotation is about one order of magnitude weaker.
Nevertheless due to almost perfect vortex beams with suppressed aberrations it can be clearly distinguished that the circular polarization increases or decreases speed of OSP rotation depending on the relative sign of the spin angular momentum with respect to the sign of orbital angular momentum. If the spin and orbital angular momenta have the same sign (e.g. positive l and RHCP beam or negative l and LHCP beam), the frequency of OSP rotation is higher comparing to the cases of linear polarization or CP with opposite signs. This is in qualitative agreement with the following assumption. If the photons of the incident focused vortex beam with topological charge l and polarization σz (RHCP: σz = 1, LP: σz = 0, LHCP: σz = −1) carry higher amount of axial angular momentum jz = l + σz [65–67], due to scattering at the same particle they can also transfer higher amount of angular momentum upon an OSP which results in a higher frequency of OSP rotation. As expected the frequency of OSP rotation for LP beam (σz = 0) is measured between the frequencies corresponding to LHCP and RHCP beams. Figure 9 shows linear dependence of the mean frequency of OSP rotation 〈fϕ〉 on the incident power P, proves that the trapped particle does not change its axial position and demonstrates expected transfer of angular momentum from the vortex beam to the OSP.
Fig. 9 Mean frequency of rotation of the OSP in LHCP beam with l = 1 as a function of power incident upon an OSP. Horizontal error bars correspond to the uncertainty in the power in the sample and vertical error bars correspond to the standard deviation of the estimated frequency (see Fig. 8). The solid black line represents a linear fit to the data 〈fϕ〉 = 2.28P − 0.0074.
Observed steady rotation of the OSP is the result of equilibrium between the optical torque coming from the transfer of angular momentum from the beam (Tz) and the drag torque due to the viscosity of the surrounding medium. Since the system is overdamped we can write the following Langevin equation:
where ξϕ represents the zero averaged random torque that gives rice to the rotational Brownian motion of the OSP and γ is the rotational drag coefficient of a rotating OSP along its long axis given as [68, 69]: where η is the viscosity of the medium (η ≈ 9 · 10−4 Pa s for water at room temperature) and Let us stress that in these equations we neglect both the lateral oscillations and the translational displacement. Considering that the optical torque Tz has no strong variations along ϕ, we can estimate its mean value as In order to have a quantitative idea of the amount of angular momentum transfered to the particle, let us define the dimensionless parameter [70] where ν is the frequency of light and P the power of the incident beam passing through a lateral plane in the sample cell. Using the data from Fig. 8 we summarize the mean values of in Table 1. The presented results demonstrate that using optical vortex beams with low topological charge leads to rotation of an OSP with an efficiency comparable to the most efficient birefringent rotors driven by circular polarization ( ).
Figure 10 presents results of extensive theoretical study with the goal to compare the theoretical model with experimental results and further increase values of through optimization of the axial position of the trapped OSP. We expected the same geometry as in the experiment, i.e. the incident vortex beam is focused by an objective (maximal angular aperture θ = 62°) and we used the Debye-Richards-Wolf integral formulation [71, 72] for an ideal aplanatic high NA lens to express the non-paraxial focused beam [73,74]. As the beam profile at the input aperture of the objective we considered Laguerre-Gaussian beam. We used T-matrix approach [75] and our own code to calculate the efficiencies of the transfer of linear and angular momenta from the beam upon the OSP. We looked for these quantities at various axial positions of the OSP to understand better the vortex beam interaction with OSP. Condition with negative slope indicates in Fig. 10 equilibrium position of OSP along z axis. Vertical lines in the figure indicate these positions and set the corresponding values of . Comparing these values with the experimental data in Table 1 one finds that the theoretical results of are about 3× lower comparing to the experimental results. As Fig. 10 demonstrates, the values are larger if the OSP is localized further behind the beam focus where the high-intensity vortex ring has larger radius and stronger transfer of optical angular momentum occurs. However, even if we included gravity and absorption of OSP, both pushing the particle further behind the beam focus, we have not obtained satisfactory coincidence with the experimental data. Therefore, the main reason of this discrepancy is probably the considered beam profile incident upon the input aperture of the objective. It was reported [26, 76] that the vortex beam generated by phase grating does not coincides perfectly with the Laguerre-Gaussian beam profile, considered in our theoretical model. Further it was pointed out that the radius of the high-intensity ring in focused vortex beam created using SLM (i.e. in our experiment) scales linearly with l in contrast to the ring radius of Lagguerre-Gaussian beam growing proportionally to [76]. Therefore, we conclude that our existing model, based on Laguerre-Gaussian beam, probably underestimates the transfer of angular momentum due to narrower vortex beam in the focus.
Fig. 10 Comparison of theoretical efficiencies and for linear (top) and angular (bottom) momentum transfer as a function of the axial position of the particle center. z = 0 corresponds to the position of the beam focus and λm denotes the wavelength in water or refractive index n. The OSP shorter axis is oriented perpendicular to the z and x axes. Three pairs of curves correspond to RHCP, LHCP beams with topological charges l = 0, 1, 2. Vertical lines interconnect the axial equilibrium position in both plots. Parameters used for the calculation shown in this figure are s = 2.55 and original sphere radius Rs = 0.85μm. Beam waist radius at the plane of input objective aperture was w0 = 4 mm and power at the OSP position was equal to 3.3 mW.
Figure 11 compares the frequency of the OSP rotation as a function of OSP aspect ratio (with fixed OSP volume) and beam waist radius in front of the objective. In this case we assumed the OSP is localized axially at its theoretical equilibrium position. The results predict that higher aspect radius and wider beam in front of objective lead to faster rotation of the OSP.
Fig. 11 Theoretical frequencies of OSP rotation as functions of the OSP aspect ratio s = a/b for the same parameters as in Fig. 10. The dashed curves correspond to the case of a Gaussian/vortex beam with infinite waist entering the objective. The vertical gray line and bar denote the aspect ratio of particles used in our experiments including error of their mean value.
5. Controlled rotation of many spheroids
Since we used SLM, we can generate an array of vortex beams with independent control of their properties. For example we can modify position xm, ym, zm intensity |vm| or topological charge lm in mth beam [60]. The corresponding phase configuration in the hologram is given by:
where where xj and yj denote coordinates of the jth pixel at the SLM. The azimuthal phase of each mth beam is given at the SLM by lmφj = lm tan−1(yj/xj), This configuration allows us to trap several OSPs at different positions and rotate them in different directions and angular speeds. Figure 12 presents a sequence of pictures demonstrating rotation of six particles, the left column rotates in opposite direction to the right column.Fig. 12 Sequence of pictures demonstrating the rotation of six OSPs (Rs = 0.85, μm) with aspect ratios between s = 1.7 and s = 2.2. The time interval between pictures is Δt = 0.044 s. The spheroids in the left column rotate in opposite direction to those in the right column due to opposite sign of the topological charge l = −2 and l = 2, correspondingly. The mean power per trap is approximately P = (2.6 ± 0.5) mW.
6. Hydrodynamic synchronization of two optical rotors
Optically trapped objects subjected to weak interaction of hydrodynamic origin have been used to investigate experimentally the phenomenon of hydrodynamic interaction [77–81]. An important subject of interest in all these experiments is the phase and frequency locking (synchronization) of the rotors or oscillators. It happens only under certain conditions in the case of two or more weakly-interacting autonomous rotors or oscillators when some control parameter is varied [82, 83]. The ratio between natural frequencies of rotors or oscillators without mutual interaction has to be close to a rational number in order to observe their synchronization [83].
Hydrodynamic synchronization of optical rotors represents more recent example of this effect. In 2012 di Leonardo et al. [79] demonstrated experimentally synchronization of two optically driven rotors by changing the distance between them. In their experiment the rotors were fabricated by two photon polymerization and were driven by the radiation pressure of two standard optical tweezers. Sizes of these rotors and frequencies reached by this setup were in a range 5 μm and 7 Hz, respectively, where the thermal fluctuations were week to disturb observation of synchronization. In contrast to complex microstructures we employed the OSPs and the experimental setup described above to demonstrate that frequencies and phases of two such optically trapped and rotated spheroids can be also synchronized. Figure 13 shows the process of synchronization of two interacting OSPs rotating in RHCP vortex beam with l = 1. We used OSPs of an equivalent radius Rs = 0.85μm and estimated aspect ratios s1 = 2.55 (left particle) and s2 = 2.45 (right particle) shown in the inset. Since we split the incident power equally to each trap, the difference in the aspect ratio of the particles caused slightly different frequencies of rotation of both OSPs. When we increased the total power, the mean frequencies of both rotors did not follow the linear dependence for each power level. For certain powers the difference between the mean OSPs rotational frequencies decreased (denoted by arrows in Fig. 13(a)) which led to the phase locking over certain time period of OSPs rotation (demonstrated in Fig. 13(b)). The time records of phases of both OSPs are shown in Fig. 13(c) for the time window from 8 s to 18 s of the cases denoted i, ii and iii. During this period of time, phases develop independently with respect to each other in cases i and iii or synchronously in ii. Later on (t ≳ 16 s) both particles are still synchronized but slightly out of phase in case ii.
Fig. 13 Synchronization of two OSPs rotating in the same direction. (inset of plot (a)). OSPs of equivalent radius Rs = 0.85μm and estimated aspect ratios s1 = 2.55 (left particle) and s2 = 2.45 (right particle) are trapped and rotated by two RHCP vortex beam of the same power and topological charge l = 1. Figure (a) shows the mean frequencies of rotation as a function of the total trapping power in the sample split equally between both traps. Figure (b) shows the phase difference Δϕ/2π = (|ϕ1| − |ϕ2|)/2π as a function of time for the data marked with arrows in plot (a). Figure (c) shows the development of the spheroids phases (ϕ1 mod 2π)/2π (blue) and (ϕ2 mod 2π)/2π (red) in time for the cases i, ii and iii.
In contrary to some other dynamic systems [83], in this experiment we show that the synchronization does not necessarily get stronger as the hydrodynamic interaction increases (the power in every trap increases). This behavior could be explained by the fact that when raising the laser power, the trap stiffness increases and thus localizes the OSP center strongly. This goes according to the conclusions [82, 84, 85] that synchronization of the rotors needs certain flexibility in the OSP rotation.
7. Conclusion
We showed experimentally many features of the dynamics of oblate spheroidal particles trapped in tightly-focused beams. Especially, we used particles with a characterized size defined by the equivalent spherical radius Rs = 0.82μm and by the aspect ratio s = 2.55. We showed that these particles can be optically trapped in 3D and constantly rotated by the angular momentum of the incident beam. Using spatial light modulator we employed focused vortex beam with topological charges l = 0, ±1, ±2 and linear, right and left hand circular polarizations. We showed that the orbital angular momentum transfer upon the spheroid is 10× higher comparing to spin angular momentum transfer and gives rise to torques efficiencies comparable to those previously reported in spin-only driven rotors using birefringent particles [11, 42]. For a simple focused Gaussian beam (l = 0) we observed a lateral oscillation effect similar to the one observed before with elongated particles [18, 37].
Owing to the high efficiency to transfer the angular momentum of light to the particles and owing to the given versatility to generate multiple traps with arbitrary intensity and phase modulation using a spatial light modulator, the experiments and techniques shown in this manuscript represent a practical solution for the generation of multiple optically-driven rotors. Using two of these rotors we successfully demonstrated synchronization of their rotation. For some power values we observed an intermittent behavior corresponding to windows in time where the rotors are locked in phase and in frequency. The experiments shown in this article pave the way for future theoretical or experimental studies concerning optical trapping and angular momentum transfer from light to well-defined non-spherical particles, synchronization of motion of several self-sustained optical rotors and practical applications in flow generation at the microscopic scale induced by simple optical rotors.
Acknowledgments
Authors acknowledge help of dr. Aleš Paták to get SEM images and financial support of our research by the Ministry of education, youth and sports of the Czech Republic, project LH12018.
References and links
1. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instr. 75, 2787–2809 (2004). [CrossRef]
2. M. J. Lang and S. M. Block, “Resource letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003). [CrossRef]
3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]
4. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles using Lasers (World Scientific, 2006).
5. A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008). [CrossRef]
6. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011). [CrossRef]
7. S. Parkin, G. Knöner, W. Singer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque on microscopic objects,” Methods Cell Biol. 82, 525–561 (2007). [CrossRef] [PubMed]
8. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011). [CrossRef]
9. R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). [CrossRef]
10. M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998). [CrossRef]
11. A. Bishop, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004). [CrossRef] [PubMed]
12. A. La Porta and M. Wang, “Optical torque wrench: Angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004). [CrossRef] [PubMed]
13. S. L. Neale, M. P. MacDonald, K. Dholakia, and T. F. Krauss, “All-optical control of microfluidic components using form birefringence,” Nat. Mater. 4, 530–533 (2005). [CrossRef] [PubMed]
14. Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013). [CrossRef] [PubMed]
15. M. Friese, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998). [CrossRef]
16. E. Higurashi, R. Sawada, and T. Ito, “Optically induced angular alignment of trapped birefringent micro-objects by linearly polarized light,” Phys. Rev. E 59, 3676–3681 (1999). [CrossRef]
17. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78, 249–251 (2001). [CrossRef]
18. Z. Cheng, P. Chaikin, and T. Mason, “Light streak tracking of optically trapped thin microdisks,” Phys. Rev. Lett. 89, 108303 (2002). [CrossRef] [PubMed]
19. P. Galajda and P. Ormos, “Orientation of flat particles in optical tweezers by linearly polarized light,” Opt. Express 11, 446–451 (2003). [CrossRef] [PubMed]
20. A. Bishop, T. Nieminen, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003). [CrossRef]
21. L. Kelemen, S. Valkai, and P. Ormos, “Integrated Optical Rotor,” Appl. Opt. 45, 2777–2779 (2006). [CrossRef] [PubMed]
22. A. Buzas, L. Kelemen, A. Mathesz, L. Oroszi, G. Vizsnyiczai, T. Vicsek, and P. Ormos, “Light sailboats: Laser driven autonomous microrobots,” Appl. Phys. Lett. 101, 041111 (2012). [CrossRef]
23. Y. Arita, A. W. McKinley, M. Mazilu, H. Rubinsztein-Dunlop, and K. Dholakia, “Picoliter rheology of gaseous media using a rotating optically trapped birefringent microparticle,” Anal. Chem. 83, 8855–8858 (2011). [CrossRef] [PubMed]
24. H. He, M. Friese, N. 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, 826–829 (1995). [CrossRef] [PubMed]
25. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997). [CrossRef] [PubMed]
26. L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Optics, VOL XXXIX 39, 291–372 (1999). [CrossRef]
27. M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. 41, 275–285 (2000). [CrossRef]
28. A. O’Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601:1–4 (2002).
29. K. Volke-Sepúlveda, 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 Semiclass. Opt. 4, S82–S89 (2002). [CrossRef]
30. 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, 093602 (2003). [CrossRef] [PubMed]
31. P. Galajda and P. Ormos, “Rotation of microscopic propellers in laser tweezers,” J. Opt. B: Quantum Semiclass. Opt. 4, S78–S81 (2002). [CrossRef]
32. C.-L. Lin, G. Vitrant, M. Bouriau, R. Casalegno, and P. L. Baldeck, “Optically driven Archimedes micro-screws for micropump application,” Opt. Express 19, 8267–8276 (2011). [CrossRef] [PubMed]
33. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001). [CrossRef] [PubMed]
34. A. O’Neil and M. Padgett, “Rotational control within optical tweezers by use of a rotating aperture,” Opt. Lett. 27, 743–745 (2002). [CrossRef]
35. V. Bingelyte, J. Leach, J. Courtial, and M. Padgett, “Optically controlled three-dimensional rotation of microscopic objects,” Appl. Phys. Lett. 82, 829–831 (2003). [CrossRef]
36. K. Bonin, B. Kourmanov, and T. Walker, “Light torque nanocontrol, nanomotors and nanorockers,” Opt. Express 10, 984–989 (2002). [CrossRef] [PubMed]
37. B. M. Mihiretie, P. Snabre, J. C. Loudet, and B. Pouligny, “Radiation pressure makes ellipsoidal particles tumble,” EPL (Europhysics Lett.) 100, 48005 (2012). [CrossRef]
38. J. Leach, H. Mushfique, R. di Leonardo, M. Padgett, and J. Cooper, “An optically driven pump for microfluidics,” Lab Chip 6, 735–739 (2006). [CrossRef] [PubMed]
39. K. Ladavac and D. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144–1149 (2004). [CrossRef] [PubMed]
40. Z. Cheng and T. Mason, “Rotational diffusion microrheology,” Phys. Rev. Lett. 90, 018304 (2003). [CrossRef] [PubMed]
41. R. Di Leonardo, J. Leach, H. Mushfique, J. Cooper, G. Ruocco, and M. Padgett, “Multipoint holographic optical velocimetry in microfluidic systems,” Phys. Rev. Lett. 96, 134502 (2006). [CrossRef] [PubMed]
42. T. Wu, T. A. Nieminen, S. Mohanty, J. Miotke, R. L. Meyer, H. Rubinsztein-Dunlop, and M. W. Berns, “A photon-driven micromotor can direct nerve fibre growth,” Nat. Photonics 6, 62–67 (2011). [CrossRef]
43. A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197–2210 (2004). [CrossRef]
44. L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006). [CrossRef] [PubMed]
45. B. Gutierrez-Medina, J. O. Andreasson, W. J. Greenleaf, A. La Porta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 475, 377–404 (2010). [CrossRef] [PubMed]
46. Y. Han, A. Alsayed, M. Nobili, J. Zhang, T. Lubensky, and A. Yodh, “Brownian motion of an ellipsoid,” Science 314, 626–630 (2006). [CrossRef] [PubMed]
47. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430 (2007). [CrossRef]
48. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A: Pure Appl. Opt. 9, S196–S203 (2007). [CrossRef]
49. F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007). [CrossRef]
50. F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. A 78, 013843 (2008). [CrossRef]
51. S. H. Simpson and S. Hanna, “Optical angular momentum transfer by LaguerreGaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009). [CrossRef]
52. H. Sosa-Martínez and J. C. Gutiérrez-Vega, “Optical forces on a Mie spheroidal particle arbitrarily oriented in a counterpropagating trap,” J. Opt. Soc. Am. B 26, 2109–2116 (2009). [CrossRef]
53. A. Hinojosa-Alvarado and J. C. Gutiérrez-Vega, “Geometrical optics calculation of forces and torques produced by a ringed beam on a prolate spheroid,” J. Opt. Soc. Am. B 27, 1651–1658 (2010). [CrossRef]
54. S. H. Simpson and S. Hanna, “Computational study of the optical trapping of ellipsoidal particles,” Phys. Rev. A 84, 053808 (2011). [CrossRef]
55. S. H. Simpson and S. Hanna, “Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles,” Opt. Express 19, 16526–16541 (2011). [CrossRef] [PubMed]
56. J. Trojek, L. Chvátal, and P. Zemánek, “Optical alignment and confinement of an ellipsoidal nanorod in optical tweezers: a theoretical study,” J. Opt. Soc. Am. A 29, 1224–1236 (2012). [CrossRef]
57. C. C. Ho, A. Keller, J. A. Odell, and R. H. Ottewill, “Preparation of monodisperse ellipsoidal polystyrene particles,” Colloid. Polym. Sci. 271, 469–479 (1993). [CrossRef]
58. J. A. Champion, Y. K. Katare, and S. Mitragotri, “Making polymeric micro- and nanoparticles of complex shapes,” Proc. Natl. Acad. Sci. USA 104, 11901–11904 (2007). [CrossRef] [PubMed]
59. E. Dufresne, G. Spalding, M. Dearing, S. Sheets, and D. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72, 1810–1816 (2001). [CrossRef]
60. R. D. Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007). [CrossRef] [PubMed]
61. H. He, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995). [CrossRef]
62. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010). [CrossRef]
63. D. B. Phillips, D. M. Carberry, S. H. Simpson, H. Schaefer, M. Steinhart, R. Bowman, G. M. Gibson, M. J. Padgett, S. Hanna, and M. J. Miles, “Optimizing the optical trapping stiffness of holographically trapped microrods using high-speed video tracking,” J. Opt. 13, 044023:1–8 (2011). [CrossRef]
64. A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964). [CrossRef]
65. S. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670–678 (1994). [CrossRef]
66. K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011). [CrossRef]
67. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825:1–7 (2010).
68. F. Perrin, “Mouvement brownien d’un ellipsoide - i. dispersion dielectrique pour des molecules ellipsoidales,” J. Phys. Radium 5, 497–511 (1934). [CrossRef]
69. F. Perrin, “Mouvement brownien d’un ellipsoide (II). rotation libre et d’polarisation des fluorescences. translation et diffusion de molecules ellipsoidales,” J. Phys. Radium 7, 1–11 (1936). [CrossRef]
70. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405–413 (2001). [CrossRef]
71. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959). [CrossRef]
72. L. Novotny and B. Hecht, Principles of nano-Optics (Cambridge University Press, Cambridge, 2006). [CrossRef]
73. P. Török and P. R. T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605 (2004). [CrossRef] [PubMed]
74. A. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13360–13374 (2007). [CrossRef] [PubMed]
75. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011). [CrossRef]
76. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003). [CrossRef] [PubMed]
77. M. Vilfan, A. Potočnik, B. Kavčič, N. Osterman, I. Poberaj, A. Vilfan, and D. Babič, “Self-assembled artificial cilia,” Proc. Natl. Acad. Sci. USA 107, 1844–1847 (2009). [PubMed]
78. N. Bruot, L. Damet, J. Kotar, P. Cicuta, and M. C. Lagomarsino, “Noise and synchronization of a single active colloid,” Phys. Rev. Lett. 107, 094101 (2011). [CrossRef] [PubMed]
79. R. Di Leonardo, A. Búzás, L. Kelemen, G. Vizsnyiczai, L. Oroszi, and P. Ormos, “Hydrodynamic synchronization of light driven microrotors,” Phys. Rev. Lett. 109, 034104 (2012). [CrossRef] [PubMed]
80. A. Curran, M. P. Lee, M. Padgett, J. M. Cooper, and R. Di Leonardo, “Partial synchronization of stochastic oscillators through hydrodynamic coupling,” Phys. Rev. Lett. 108, 240601:1–4 (2012). [CrossRef]
81. J. Kotar, L. Debono, N. Bruot, S. Box, D. Phillips, S. Simpson, S. Hanna, and P. Cicuta, “Optimal hydrodynamic synchronization of colloidal rotors,” Phys. Rev. Lett. 111, 228103 (2013). [CrossRef] [PubMed]
82. N. Uchida and R. Golestanian, “Generic conditions for hydrodynamic synchronization,” Phys. Rev. Lett. 106, 058104 (2011). [CrossRef] [PubMed]
83. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, 2001). [CrossRef]
84. M. Reichert and H. Stark, “Synchronization of rotating helices by hydrodynamic interactions,” Eur. Phys. J. E 17, 493–500 (2005). [CrossRef] [PubMed]
85. T. Niedermayer, B. Eckhardt, and P. Lenz, “Synchronization, phase locking, and metachronal wave formation in ciliary chains,” Chaos 18, 037128 (2008). [CrossRef] [PubMed]
