We induce spontaneous motion that is both directed and complex in micron-sized asymmetric Brownian particles in a spherically aberrated optical trap to generate microswimmers. The aberrated optical trap is prepared in a slightly modified optical tweezers configuration where we use a refractive index mismatched cover slip leading to the formation of an annular intensity distribution near the trap focal plane. Asymmetric scattering from a micro-particle trapped in this annular trap gives rise to a net tangential force on the particle causing it to revolve spontaneously in the intensity ring. The rate of revolution can be controlled from sub-Hz to a few Hz by changing the intensity of the trapping light. Theoretical simulations performed using finite-difference time-domain method verify the experimental observations. We also experimentally demonstrate simultaneous spin and revolution of a micro-swimmer which shows that complex motion can be achieved by designing a suitable shape of a micro-swimmer in the optical potential.
© 2015 Optical Society of America
Microswimmers, or artificial particles which mimic the self propelling behavior seen in the mesoscopic scale in nature have been a topic of extensive study . These take up energy from their environment and convert the energy into directed motion. The directed motion results from various controlling mechanisms including thermophoresis caused by asymmetric heating , chemical reactions resulting in gas propulsion , chemotaxis [4,5], magnetic fields , etc. It is interesting to note that in these situations, the swimmers typically exhibit ballistic motion in short time scales but diffusive or non-diffusive motion over long time scales.
Most of these schemes rely on special properties of the microswimmer that are designed to move it into a desired way (active particles). A more general approach towards controlled motion would be to tune or control the external stimulus while relaxing the limitations on the shape of the particle. In cases where light provides the external stimulus, such control could be achieved by using higher order Gaussian beams (eg. LG or Bessel beams) which have an inherent azimuthal phase gradient that results in transfer of angular momentum to particles which then exhibit motion in a circular orbit  independent of their shape. A more exotic effect has been investigated recently when a specular light pattern was used to manipulate microparticles . However, the production of higher order beams often require complex optical systems, while speckle patterns do not often have regular structures and it could be difficult to use them to generate directed motion for micro-swimmers of different shapes. Thus, an alternate approach towards design of a light field that could facilitate various shapes of microswimmers without employing higher order beams would be rather useful.
In this paper, we report the design of a light field in optical tweezers that has the capability of generating microswimmers from otherwise passive asymmetric particles, which go on to display complex rotational motion. The particles exhibit spontaneous revolution in an orbit as well as coupled revolution and spin around their axes due to interaction with the field. The light field is annular in shape so that the asymmetric particle is confined in the radial direction while having a much shallower potential in the tangential direction. The asymmetric shape of the particle leads to asymmetric scattering resulting in an azimuthal phase gradient that causes a net recoil force, and the component of the recoil force along the tangential direction of the annulus is not canceled by the trapping potential. The particle thus traverses the periphery of the annulus spontaneously. The rate of revolution can be controlled from sub-Hz to a few Hz by changing the intensity of the trapping light which can be achieved either by modifying the laser power or the annular trap diameter. For particles having sufficiently complex shape, the revolution gets coupled with rotation of the particle around its axis, so that simultaneous rotational motion around an axis and an orbit are observed in the trapping volume of the optical tweezers. Note that such asymmetric scatter under influence of a linearly polarized focused Gaussian beam has been shown to lead to spin of asymmetric particles such as specially designed rotors , and red blood cells [10, 11]. This effect, known as the ‘optical windmill effect’, can be attributed to an imbalance in scattering forces which leads to a net torque on the particle, causing it to rotate along its axis. However, while the origin of the motion we observe is also imbalanced scattering, the phenomenon is distinctly different since the unique potential in combination with the shape of the particle results in a tangential force that results in translational motion (with no spin) with a specific trajectory of the particle. As mentioned earlier, for specific shapes of particles we do observe a more complicated version of the windmill effect with spin coupled to translation as we shall discuss in detail later.
2. Experiment and observations
The formation of an annular trap using a refractive index (RI) mismatched cover-slip in optical tweezers has been described in detail by Haldar et. al.  where the trapping of 1 μm diameter polystyrene beads in the annulus was also demonstrated. The basic setup of the apparatus is shown in Fig. 1(a). The trapping light (1064 nm with around 60 mW power in the trapping plane) propagates through the microscope (Zeiss Axiovert.A1) objective of Numerical Aperture (N.A.) 1.4, after which it encounters a stratified medium consisting of 1) immersion oil of RI 1.516, 2)cover slip of thickness 250 μm and RI 1.575, 3) sample chamber where the particles to be trapped are suspended in an aqueous medium (RI 1.33), 4) top glass slide of RI 1.516. The tight focusing of the laser through the stratified medium leads to interesting diffraction effects , which results in the generation of an annular optical potential near the focal plane  as is shown in Fig. 1(b). The diameter of the annular ring can be modified by changing the z-focus of the microscope and we typically trap particles at different distances ranging from 0.5–5 μm away from the focus where the diameter of the annulus varies between 2–8 μm.
The asymmetric particles used in this experiment were pea-pod shaped soft oxometalates (SOMs)  (see Fig. 1(c), the average size is 1.5(5)×0.50(3) μm ), which in some cases were loaded with a catalyst to render further asymmetry in their shapes  (see Fig. 1(d), average size is now 2.0(4) × 0.65(3) μm). We observe spontaneous revolution of single pea-pod particles along the periphery of this annular potential. Snapshots of the motion of a typical revolving single pea-pod are shown in Fig. 2(a) – 2(d) ( Media 1). The orbit diameter is around 2 μm, which can be altered by changing the z-focus of the microscope. A trapped particle follows a new orbit with a different revolution velocity. This is demonstrated in Figs. 2(e) – 2(j), where the sub-figures 2e–2(g) show the motion of a catalyst loaded pea-pod in an orbit of diameter around 8 μm, while 2(h) – 2(j) show motion of the same pea-pod in a smaller orbit of diameter around 6.5 μm. The orbiting velocity is enhanced in the second case, where we estimate the velocity by tracking the position of the pea-pod using video analysis. This is done for different laser powers (Fig. 2(k)), as well as different orbit diameters at the same input trapping beam power (Fig. 2(l)) for the same pea-pod. Fig. 2(k) shows that the revolution velocity increases linearly with increasing laser power, while Fig. 2(l) shows that the velocity on the y axis fits well to a law (fit exponent ‘b’ of r = 1.96±0.08), where r is the orbit diameter - both of which are expected since an increase in intensity of the trapping light (caused both by increase in power where the intensity changes linearly, and reduction in radius where the intensity has inverse quadratic dependence) leads to larger scatter from the particle. The highest velocity we attain is around 50 μm/sec in the case when the circumference was 18.8 μm ( Media 2 and Media 3). This corresponds to a rotation rate of around 2.6 Hz, while on the lower scales, we also obtain rotation of around 0.4 Hz. It is also interesting to note that we achieve both clockwise and anticlockwise revolution of pea-pods - the sense of revolution being dependent on the particular shape asymmetry of the pea-pod and its orientation in the trap (as we show later), and also the initial direction of velocity as the pea-pod enters the annular trap.
3. Numerical simulations
To understand the physics of the motion of the particles, we performed a numerical simulation of our physical system. The problem is to essentially propagate a highly focused Gaussian beam through the stratified medium described earlier so as to create the annular intensity distribution shown in Fig. 1(b), and then model the scattered intensity off a pea-pod shaped particle situated in the annulus. We mimic a pea-pod by a truncated cone like structure as is shown in Fig. 3(a). Our goal is to calculate the scattering force acting on the particle and compare it with the drag force due to the viscous medium (water) surrounding the particle.
The time averaged force acting on a body due to certain field distribution can be calculated from the Maxwell Stress Tensor (MST). Integrating the MST over a closed surface around the body thus provides the net force acting on it. Lumerical, a commercial-grade simulator based on the finite-difference time-domain (FDTD) method, was used to perform the calculations. The propagation of the tightly focused beam in the stratified medium generates the annular shaped harmonic potential which is radially symmetric. We consider a diameter of around 6 μm that is obtained about 4 μm away from the focal plane. The cone shaped particle was placed inside the potential and we assumed a rectangular pillbox around the particle for the MST calculations. Figure 3(b) shows that the introduction of the particle breaks the radial symmetry in the intensity distribution of the field, which would lead to the generation of a varying azimuthal phase component that causes angular momentum transfer to the particle. This imbalance also results in a non-zero force component in the tangential direction which is ultimately responsible for the torque exerted on the trapped particle. We calculate the net force acting on the particle by integrating the MST over the surface of the pillbox. To confirm whether the force is position independent and tangential to the ring as long as the particle resides inside the trap, we placed the particle at 40 points equidistant from each other inside the annular potential. From the scattered electromagnetic field as shown in Fig. 3(b), we calculated the MST using the following equation,
Since the average body force is the total force in the particle isFig. 3(c). It can be observed that both the x and y components of the force acting on particle vary sinusoidally with a phase difference of 90deg between them. Thus the force vector remains tangential to the ring in the xy plane. The magnitude of the force for the particular ring diameter used in the simulation (around 6 μm) is 6×10−13 N. Figure 3(d) shows a quiver plot of the force vector tangentially varying on the xy plane of the annulus. The magnitude of the force remains constant throughout the ring, and is directed tangentially along the annulus so that the test particle would keep moving in a circular trajectory in the clockwise direction (note that the direction can be changed depending on the direction of scatter imbalance). This explains the spontaneous revolution of the pea-pod particle that we observe in our experiments. Note also that the effective drag coefficient for an equivalent prolate ellipsoid of similar dimensions is about 1 ×10−13 Ns/μm, which implies that the velocity of motion of the particle is of the order of 10 μm/sec thus conforming to our observation of revolution speeds of the order of few 10s’ of μm/sec (the actual value would depend entirely on the absolute value of the intensity of the annulus which could be modified experimentally by changing the z-focus of the microscope).
It is clear that the azimuthal phase gradient is obtained due to the asymmetry in scattering, and would therefore not work for symmetric particles such as spheres or ellipsoids. However, even for a symmetric object, asymmetric scattering can be obtained by choosing the material appropriately - for eg. by designing the symmetric particle as a Janus particle , where one side of the particle may be coated by a high reflectivity material such as gold. We use such a design as shown in Fig. 3(e), with a polystyrene sphere coated on one side by gold. The scattering off the gold surface is much stronger than from the polystyrene, as a result of which, when such a particle is placed in the annulus, there is large imbalance in the scattered intensity as is seen in Fig. 3(e). This leads to a non-zero tangential force similar to that seen for the pea-pod, and the equal magnitude and direction of the force along the annulus as shown in Fig. 3(f) (we calculate the force at 4 points now due to simulation constraints) makes it clear that the Janus particle would also revolve spontaneously in the ring. In the simulation, we fixed the orientation with respect to the axis of the particle (only a single arbitrary orientation was chosen). It is important to note that the scattering force due to the radial component of the scattering from the particle is balanced by the gradient force due to the annular trap, while the tangential component is always directed in a way to move the particle along the ring. The orientation angle would change the magnitude of the radial and tangential components, which would essentially determine the velocity of rotation. Note also that our configuration is more efficient in trapping Janus particles which are notoriously difficult to trap with Gaussian beams due to their high scattering. In our configuration, the intensity in the annulus is much lower than that at the center of focused Gaussian beams typically used in optical tweezers (by more than an order of magnitude, see Fig. 5 of Ref. ), so that the net scattering force in the radial direction is around 3×10−12 N for the orientation of the particle used in the simulation, which is actually smaller compared to the gradient force (calculated from the gradient of the intensity profile) which comes to be 8×10−12 N. However, a more comprehensive exercise on the efficacy of our trap towards trapping Janus particles is currently ongoing in our laboratory.
4. Experimental demonstration of coupled rotational motion
We observe more complex circular motion for a particle with a shape shown in Fig. 4. The micro-rotor is opportunistically obtained by pea-pods that coalesce together in the aqueous solution. In these cases, rotational and translational motion are coupled as is demonstrated in the motion of the particle of the shape shown in Fig. 4. Here, we see a particle spinning along its axis, as well as getting transported in a circular orbit. Thus, the axial spin gets coupled with the revolution along the annulus to exhibit motion as shown in Fig. 4 ( Media 4). Note that, for such as eventuality, the rotational motion needs to be phase-matched with the revolution, such that the orientation of the particle is always adjusted to subject it to a net tangential force as demonstrated in the force profiles shown in Fig. 3. If the motion has a component in the radially outward direction, the particle would encounter the gradient force due to the intensity ring and get confined. Only the tangential motion manifests itself, and for this to occur, the particle must spin around its axis by 360 degrees during the course of one single revolution. This is exactly seen in Fig. 4(a) and 4(e), where the orientations of the particle are the same, signifying that the rotor has indeed undergone rotational motion of 360 degrees around its axis as it completes one entire orbit of revolution.
In conclusion, we demonstrate a technique for facilitating microswimmers demonstrating motion in a circular orbit by designing a annular potential in an optical trap. The potential is obtained by using a simple linearly polarized fundamental Gaussian input beam having no intrinsic angular momentum. The potential is designed by introducing spherical aberration into our system by using a stratified medium with the help of an RI mismatched cover slip. The annular potential results in an imbalance in scattering for trapped asymmetric particles that leads to the generation of a tangential force along the annulus, which causes the particles to revolve spontaneously along it. We demonstrate such motion experimentally on pea-pod shaped SOM particles achieving linear velocities of the range of 50 μm/s over orbit diameters varying between 2–8 μm. The velocity of a single particle may be modified by increasing the light intensity in the annulus by changing the z-focus of the trapping microscope. We validate our observations with a thorough simulation of the system using Lumerical, where the force calculated using the MST turns out to be equal in magnitude and tangential in direction along the entire annulus for a conical particle resembling a pea-pod (Fig. 3(d), as well as for a spherical Janus particle (Fig. 3(f)). We also generate complex motion consisting of rotation coupled with revolution in micro-rotors formed out of peapods coalescing with each other in the aqueous solution. The technique can be used to generate microswimmers out of a large range of asymmetric particles as confirmed in our simulations, lead to coupled spin and revolution in a controlled manner by designing micro-rotors using lithographic techniques that are rather well-developed presently , and also design other complex potentials (say, by using a multi-annular trap where particles can shift between traps spontaneously) where further intricate motion of microswimmers may be facilitated.
AM and BR have equal contribution in the work. The authors acknowledge Dr. Soumyajit Roy of Dept. of Chemical Sciences, IISER Kolkata for providing the pea-pod samples and IISER Kolkata for funding and laboratory facilities.
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