1. Introduction

Following the discovery of the single beam gradient trap [1], the popularity of the original, counter-propagating, or dual beam trap [2] declined. However, interest has recently been revived. The main advantage of this geometry is that the flow of linear optical momentum is neutral, along the beam axis. Particles are confined, not by intensity gradients, but by balanced scattering forces, permitting trapping of a much wider range of particles. For example, it enables the trapping of nano-particles [3, 4], sub-micrometer sized particles [5–8] and even large objects [9, 10]. Furthermore, since high intensity gradients are unnecessary, the focusing optics requires only low numerical apertures, allowing very large working distances to be created and enabling the long range transport of trapped particles [11, 12]. Thus, the system is extremely versatile, and has been applied in diverse contexts. For instance, it provides an ideal environment in which particles can undergo self-organization mediated by optical interactions (optical binding) [6, 7, 13–15]. Because it is tolerant to high dielectric contrast, it lends itself well to trapping in air or vacuum, [16]. For related reasons, it is also the one configuration in which the very high forces required to deform soft matter can be obtained; single cells [17], and liquid droplet lasers [18] have been successfully stretched. Finally, the same principles operate for single beams, retro-reflected at highly reflective mirrors [9, 19].

Ellipsoids are simple, well defined non-spherical bodies. Their hydrodynamic characteristics, at low Reynolds number, were first analyzed by Perrin [20,21], who evaluated their friction coefficients and studied their diffusion properties. The latter have, only recently, been experimentally observed [22]. In addition, their geometric anisotropy allows angular momentum to be exchanged with optical fields, permitting ellipsoidal particles to be oriented or rotated [23–32]. A number of applications follow from these properties. For example, an optically trapped ellipsoidal probe can be used to detect tiny external torques as well as forces [33, 34] with precision sufficient, even, to measure the rotational stiffness of the DNA molecule [35]. Furthermore, optically trapped and rotated asymmetrical particles have found use as optically driven pumps [36], and as tools for quantifying the properties of fluids or gases in the microscopic regime [37]. Spherical birefringent particles are often trapped by optical tweezers [38] and used for similar applications. For example, Arita et al. [39] demonstrated rotation of a spherical vaterite particle with a rotational frequency up to 10 MHz.

In our recent experiments with polystyrene spheroidal microparticles we demonstrated transfer of both spin and orbital angular momentum to one or more spheroids optically trapped in several holographic tweezers, with controlled vorticity and elliptical polarization of the beams [29]. We have also observed hydrodynamic coupling of two spheroidal rotors. In this paper we have employed the advantages of dual beam geometry for optical trapping [40] and enhanced the interaction complexity by using spheroidal polystyrene microparticles. We have used coherent counter-propagating Gaussian beams with opposite circular polarization to trap and rotate single and multiple oblate spheroidal particles by transfer of spin angular momentum [41, 42] from the beams. Single particles can be continuously rotated with an angular frequency that can be theoretically predicted. Increasing the number of spheroids gives rise to optical coupling between rotational, as well as translational degrees of freedom. For pairs of spheroids, a form of angular optical binding emerges giving rise to phase locked rotation. Under the influence of optical forces, these pairs can form dimers which, for particular sizes, are seen to inhibit rotation. Similarly, spheroid triplets form trimers which execute a form of periodic motion in which circulation of the centre of mass is correlated with an oscillation in the tilt angle of the trimer axis. Finally, sets of four spheroids form optically interacting dimer pairs which hop between discrete equilibrium configurations. Understanding complex, optically induced motion of this sort could facilitate the design of optically actuated micro-robotic devices [43, 44].

2. Experimental setup

Two horizontally counter-propagating laser beams were generated by a holographic mask imprinted onto a single spatial light modulator (SLM) and transferred by relay optics. The SLM enables us to individually alter the waist of both beams. We used an infrared trapping laser (IPG, YLM-10-1064-LP-SP) with vacuum wavelength 1064 nm and, depending on the experiment, the total power in the sample cell was between 85 and 240 mW. The radius of the beam waist of both counter-propagating beams was set to 4.36 μm. A detailed description of the experimental setup can be found in our previous study [40].

The non-spherical particles were fabricated following two different methods well described in literature [29, 35, 45, 46]. Both approaches rely on the compression of spherical particles by an externally applied force. In the simplest method, a monolayer of spherical particles is deposited between two slides and then homogeneously pressed with a vice at room temperature [35]. The deformed particles are then recovered by washing the glass surfaces with water. This method produces disc-like particles with two equal long axes, corresponding, approximately to an oblate particle. In the other method 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 (Bangs Laboratories, Inc.) of diameter 2μm and refractive index n_p = 1.59. The film is heated up to 160 °C, beyond the glass-transition temperature of the polyvinyl alcohol and polystyrene, and then pressed using thin spacers to control the final thickness of the pressed sample [29]. Finally, the resulting spheroids are released by dissolving the polymeric matrix in water [46]. Altering the stress applied during the manufacture of the spheroids provides a way of modifying their aspect ratio. For simplicity, we refer to these methods, respectively, as the cold and thermal methods.

After fabrication, the oblate spheroidal particles are dispersed in deionized water inside a glass squared capillary with an inner diameter of 100 μm (Vitrocell 8510). The capillary was inserted into an optical field of focused, counter-propagating, circularly polarized (CP) laser beams. The motion of the disc-like particles was observed by a CCD camera (Basler Pilot GigE) from a direction perpendicular to the axis of the laser beams. Note, that the absorption of water at the trapping wavelength is not negligible. However, the usage of the capillary with relatively small inner diameter sufficiently suppress any thermal fluid flows [40].

3. Experimental and theoretical results

In the following experiments we use identical, coherent, counter-propagating Gaussian beams with opposite circular polarizations. The beam propagating along the positive z axis was left-hand circularly polarized and, assuming the simplest form of the lateral Gaussian beam envelope, we can write,

 

Fig. 1 Spheroid is optically trapped on the axis of two overlapping counter-propagating beams. The motion of the particle was observed by a CCD camera from a direction perpendicular to the axis of the laser beams. The orientation of the electric field on the beam axis of both counter-propagating beams according to Eqs. (1) and (2) and the final field formed by their interference following Eq. (3) is visualized.

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3.1. Single trapped spheroid

We observe that a single spheroid ends up trapped with its center on the beam axis in one of two stable orientations. Most commonly, the symmetry axis (or the minor axis, along which the radius is minimum), is oriented perpendicularly to the beam propagation direction, so that the profile of the spheroid, in the plane perpendicular to the propagation direction, is anisotropic (an ellipse), and the particle can interact with the spin momentum in the beams. In this configuration spheroids rotate continuously about an axis perpendicular to the symmetry axis, and parallel to the beam axes (see Fig. 2(b)).

 

Fig. 2 Two stable orientations of the same oblate spheroid were observed in two counter-propagating Gaussian beam with opposite circular polarizations. a) Non-rotating spheroid, oriented with its symmetry axis parallel to the direction of the beam propagation. b) The same spheroid, oriented with its symmetry axis perpendicular to the direction of beam propagation, rotating around the horizontal beam axis (see Media 1). The original diameter of the sphere before deformation into the spheroid was 2μm and the spheroid aspect ratio is equal to 0.46 ± 0.02. The beam waists of counter-propagating beams were the same and equal to w₀ = 4.36μm, total laser power incident on the sample plane was P = 85 mW.

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Less frequently, the spheroids find their way into a second orientational equilibrium, this time with the symmetry axis parallel to the propagation direction, Fig. 2(a). In this configuration, rotation, if it occurs, cannot be detected since the profile of the spheroid is constant in the viewing direction. The preference for each of these equilibrium configurations is dependent on aspect ratio (see the Appendix section for more details). We are able to accurately predict both the existence and the stability of these equilibria with T-matrix theory [29, 47].

The images acquired from the CCD camera were analyzed off-line using an in house algorithm and MATLAB software package. The position of the spheroid was identified using edge detection. Following this, inner and outer ellipses were fitted to the edges. Since the spheroids rotate around an axis perpendicular to the CCD camera, the major semiaxis of the ellipse, a_⊥, changes negligibly while the minor semiaxis variations indicate spheroid rotation. The length of the minor semiaxis of the spheroid, a_‖, is determined using the minimum value of the oscillating outer semiaxis of the ellipse. The aspect ratio of the spheroid is defined as a_⊥/a_‖. The average frequency of spheroid rotation was determined as one half of the frequency obtained using the fast Fourier transform analysis of the minor axis variations time record. The factor of one half arises due to the fact that a particular value of the semi minor axis of the ellipse corresponds to two orientations of the spheroid, separated by π rads.

Experiments were preformed with spheroids of varying aspect ratio, a_‖/a_⊥, normalizing rotation frequencies against optical power. The results are presented in Fig. 3 in the form of magenta and blue circles. The horizontal and vertical error bars correspond to the standard deviation of the aspect ratio and the normalized rotational frequency. Red/dark red and grey/black curves denote theoretical results obtained using the T-matrix calculations (we implemement the EBCM method [47, 48] in our own software) and Comsol Multiphysics (finite elements method, for details of similar calculations see [49]) respectively, to calculate optical torque. Rotational frequency was then evaluated as a ratio of the torque to the rotational drag coefficient of a perfect ellipsoid [20,21]. The thicker parts of curves indicate whether the investigated spheroid, of given aspect ratio, settled with its center on the standing wave intensity maxima (anti-node) or minima (node). The coincidence between experiment and theory is fairly good, except for the lowest aspect ratio. It should be noted that production of spheroids with smaller aspect ratios, a_‖/a_⊥ < 0.25, is very difficult and spheroids prepared by the low temperature mechanical procedure are rather deformed (see inset in Fig. 3). The shape correction to the rotational drag coefficient for such particles is not simply defined, resulting in a mismatch between model and experiment. T-matrix theoretical results are not presented for low aspect ratios (below a_‖/a_⊥ <≈ 0.25), since the conventional T-matrix approach fails to converge under these conditions. The discrepancy between T-matrix approach and Comsol Multiphysics is caused by slightly different description of incident field and by numerical errors in the Comsol calculations. The results indicate that the maximal rotational frequencies are reached for aspect ratios between 0.3 and 0.4, for particles of the fixed volume used in this study (equivalent to a sphere of diameter 2 μm).

 

Fig. 3 Comparison of experimentally and theoretically obtained frequencies (normalized to 1W) of spheroid rotation for different aspect ratios a_‖/a_⊥. Magenta and blue circles denote experimental data with standard deviations in the form of error bars. Red/dark red and grey/black lines correspond to the theoretical results obtained from the T-matrix and Comsol calculations respectively. Calculations were performed in equilibria associated with both intensity maxima and minima, the stable equilibrium being indicated by thick lines. Experimental parameters were the following: beam waists of the counter-propagating Gaussian beams w₀ = 4.36μm, depending on the experiment the total power P at the sample plane was in the range (85 – 220) mW, polystyrene spheroids were obtained from spheres of original diameter 2μm. Spheroids denoted by magenta circles were prepared by the thermal method while those in blue color were carried out by the cold one.

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3.2. Two longitudinally self-arranged rotating spheroids

The dual beam geometry provides a convenient way of observing the dynamics of multiple spheroidal particles. We begin with two similar spheroids, optically trapped and rotated in the same setup as above.

Two cuts from the same experimental time records of the same two spheroids are shown in Fig. 4(a). Both spheroids rotate with frequencies close to f = 0.48 Hz, with their minor axes mutually perpendicular. The second row of Fig. 4 demonstrates this behavior showing the time dependence of the values of the minor axes obtained from ellipses fitted to the images of individual spheroids. This mutual phase delay, of π/2 rads, between the two optically bound discs was observed in all experiments including those with spheroids in contact.

 

Fig. 4 a) Several images of the behavior of two equal spheroids from the same record. Left (right) column shows rotation of two longitudinally self-arranged spheroids starting at time t₀ = 8 s (t₀ = 95.9 s) from the beginning of the experiment. Within 0.9 s the spheroids rotated by about one half of the period (see Media 2 for the whole record, recorded with the frame rate 50 Hz). The parameters of the experiment were the following: horizontally counter-propagating Gaussian beams with w₀ = 4.36μm, total power at the sample plane P = 85 mW, spheroids of equal aspect ratio 0.65 ± 0.02 were obtained by the high temperature method from polystyrene spheres of original diameter 2 μm. b) Minor axis of the rotating spheroids obtained from their pictures on the CCD camera. Blue or red curves denote the left or right spheroid.

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As shown in Fig. 1 and Eq. (3), the counterpropagating waves interfere and form a standing wave, with a period given by half the wavelength of the incident radiation, in water, i.e. λ/2 = 1.064/(2 × 1.33) = 0.4μm. Each of the intensity maxima or minima represent a possible stable equilibrium position separated from the neighboring one by a potential barrier. A slower variation, on top of the standing wave oscillation, appears due to the weak focussing of the Gaussian beams, resulting in a tilted periodic potential [50]. Providing they have sufficient thermal energy particles are able to hop over local potential maxima, and drift, in discrete steps, toward the global minima around the focal region of the beams. Such jumps are, indeed, observed in the experiment (see Figs. 5(a), 5(b) and 5(e)), with the right spheroid moving toward the left one, in discrete steps of length Δz = 0.41 ± 0.02μm, which coincides well with the standing wave period mentioned above, while the left spheroid remains in the same place. This behaviour results from an attraction that arises, in part due to the weak focusing of the Gaussian beams, and in part from optical forces of interaction. As they approach one another, the spheroids continue to rotate at approximately 0.5 Hz. Owing to the low angular resolution with which we are able to measure spheroid orientation, we were unable to resolve the dependence of rotation rate on separation. However, once in contact, rotation of spheroids with aspect ratio 0.65 was completely arrested, see Fig. 5(b). This behaviour is shown in Fig. 5(c). The periodic variations in the size of the minor ellipse axis b_s indicate rotation around z axis, which continues until the particles make contact. At this point, the spheroid-dimer shifts out of the optic axis, as shown in Fig. 5(d), which gives the x coordinates of the centres of each spheroid (labelled x_s). This surprising behaviour was observed only for this particular aspect ratio. Spheroids of greater aspect ratio continued to rotate and express periodic translational motion. The left column in Fig. 6 shows their rotation at almost one axial position from the beam waists. The right column demonstrates more complex behavior with the axial position dependent on the orientation of the spheroids.

 

Fig. 5 An example of different behavior of the same two spheroids (aspect ratio about 0.65 0.02) if they are separated (a) or in contact (b). If they are separated, the spheroids rotate with minor axes oriented perpendicularly. However, if they are in contact, the orientation of axes remains also perpendicular but they stop rotation. c) The periodic variations in the size of the minor ellipse axis b_s indicate rotation around z axis, which continues until the particles make contact. Transversal x_s and longitudinal z_s time records of particle position are plotted in part d) and e), respectively. First frame in a) and b) corresponds to time t₀ = 52.12 s and t₀ = 55.34 s, respectively in parts c)–e). See Media 3 for the whole record, recorded with the frame rate 50 Hz. The parameters of the experiment were the same as in Fig. 4.

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Fig. 6 Snapshots of every second frame from Media 4 for two spheroids of larger aspect ratio 0.3 in contact. The left (right) record starts at time t₀ = 0 (t₀ = 22.76 s) from the beginning of the experiment. The parameters of the experiment were the following: horizontally counter-propagating Gaussian beams with w₀ = 4.36μm, and the total power at the sample plane P = 240 mW, polystyrene spheroids were obtained by the low temperature method, described above, from spheres of original diameter 2μm.

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The main features of the observed behaviour can be reproduced theoretically, using superposition T-matrix theory [51]. Although this approach is most commonly applied to spherical scatterers, it can be generalized to arbitrary, low symmetry structures, in a straight forward manner. This method is applied to a pair of representative spheroids immersed in counter-propagating Gaussian beams, as described above. Although the particles used in the calculations have aspect ratio 0.5, the qualitative behaviour they show is completely generic. In the model, the centres of the spheroids lie on the propagation axis, equidistant from the focal point, with their symmetry axes perpendicular to this axis. As mentioned above, the force along the beam axis oscillates as the separation between the spheroids increases (see Fig. 7(a)), revealing a sequence of stable equilibrium configurations for the spheroids. The two curves in the figure correspond to two different rotational states; config 1 refers to the case where the symmetry axes of the spheroids are parallel, whilst for config 2 they are perpendicular (in both cases they remain perpendicular to the beam axis). Although the general form of this force curve derives from the intensity distribution of the incident field, the strong attraction, at low separations is substantially due to the optical interaction between the particles. It is for this reason that the curves for config 1 and config 2 differ at low separation. The mutual phase delay, between the spheroids, as they rotate is also captured by the model. Owing to the geometric anisotropy, each spheroid interacts with the angular momentum of the incident light, modifying the torque on its partner. The difference between the optical torque experienced by one spheroid as a function of the phase advance relative to its partner varies, approximately, as ≈ sin(2ϕ₁), where ϕ₁ is the angle between the symmetry axis of the first and second spheroid. The stable relative orientation occurs when ϕ₁ = π/2, i.e. when the minor axes of the spheroids are mutually perpendicular, as observed experimentally. Figure 7(b) shows similar curves for a series of equilibrium separations along the beam axis showing that the orientational interaction decreases with separation.

 

Fig. 7 a) Axial force on a single spheroid, as a function of the distance from a second, symmetrically displaced particle. b) Calculations of relative torque of two axially trapped spheroids as a function of the angle between their symmetry axes.

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3.3. Two self-arranged spheroids in unusual configurations

Figure 8 shows two examples of atypical spheroid rotation. The left panel shows optically self-arranged spheroids orientationed differently with respect to the z axis; whilst the particle on the right rotates about the z axis, the left does not appear to spin and only moves vertically depending on the rotation of the right one. The right panel shows a different situation that occurs when two spheroids are in contact but arranged perpendicularly to the z axis. Again their rotation around the z axis is clearly demonstrated.

 

Fig. 8 Another stable configuration for two spheroids. Left: The right particle spins horizontally while the left one does not rotate, but oscillates vertically depending on the orientation of the one on the right (see attached Media 5 starting at t = 8.12 s). The parameters of the experiment were the following: horizontally counter-propagating Gaussian beams with w₀ = 4.36μm, and total power at the sample plane P = 240 mW, polystyrene spheroids of aspect ratio 0.25 were obtained by the low temperature method described above from spheres of original diameter 2μm. Right: Two spheroids of aspect ratio close to 0.65 arranged vertically and rotating horizontally around z axis (see attached Media 6 starting at t = 1.66 s). The parameters of the experiment were the following: horizontally counter-propagating Gaussian beams with w₀ = 4.36μm, and the total power at the sample plane P = 240 mW, polystyrene spheroids were obtained by the low temperature method described above from spheres of original diameter 2μm.

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3.4. Three and four longitudinally self-arranged spheroids

Increasingly complex behavior can be expected for more than two spheroids. The left panel of Fig. 9 illustrates the dynamics of three self-arranged spheroids. The particles have spontaneously aggregated to form a quasi-linear trimer. The movie Media 7 reveals periodic, circulatory motion of the centre of mass in a plane parallel to the axial direction. Motion of the centre is correlated with oscillation of the trimer axis, suggesting a coupling between rotational and translational degrees of freedom. The right panel of Fig. 9 shows the behaviour produced by sets of four spheroids. This time a pair of dimers are formed. Similarly to the case of single dimers discussed above, neither of these dimers undergo axial rotation. Instead they are seen to move into equilibrium configurations in which their centers are displaced from the beam axis in opposite directions. The dimers simultaneously jump between opposing equilibria. These hopping events could be stochastic in origin, but also appear to correlate with minor adjustments to the inclination of the dimer axis (see attached movie Media 8). Unlike individual spheroids, dimers appear to be repelled from one another and, as time progresses, they stochastically hop over the potential maxima associated with the standing wave, increasing their separation as they do so. When the axial separation exceeds about 15 μm their equilibrium configurations appear to stabilize, and no further hopping is observed.

 

Fig. 9 Complex behavior of three and four optically bound spheroids. Left: Depending on the orientation of the spheroids, their axial and lateral positions are changed (see Media 7). The parameters of the experiment were the following: horizontally counter-propagating Gaussian beams with w₀ = 4.36μm, and the total power at the sample plane P = 240 mW, polystyrene spheroids of aspect ratio 0.5 were obtained by the high temperature method from spheres of original diameter 2 μm. Right: Optically self-arranged spheroid dimers ( Media 8 from t = 5.2 s) changing their vertical position depending on angle of rotation of each couple. Parameters are the same as in the right record in Fig. 8.

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4. Conclusion

To start with, we concentrated on a single spheroid trapped in two counter-propagation Gaussian beams with opposing circular polarization states. We observed two stable orientations with the spheroid minor axis either parallel, or perpendicular, to the beam axis. In the former (parallel) case, the spheroid rotation, if present, was not visible in contrast to the perpendicular orientation, in which rotation is easily detected. We have measured rotational frequencies of spheroids of various aspect ratios and found that they are in reasonable coincidence with numerical simulations based on finite element method (COMSOL) and with calculations performed with the T-matrix approach.

Secondly, we have observed that two optically self-arranged spheroids orient themselves with their minor axes perpendicular both to each other, and to the beam axis. This configuration is preserved as they rotate, constituting a form of optically induced phase locking of the particle orientations. This behaviour is captured by superposition T-matrix theory, which exposes the underlying structure to the coupling mechanism which is seen to rely on a delicate interplay of the spin momentum carried by the incident field, and the orbital angular momentum generated by scattering from non-spherical bodies [23]. A future article will discuss this issue in greater depth.

Weak gradients associated with the envelope of the incident intensity, in combination with optical binding forces, cause the trapped spheroids to gradually and stochastically approach one another. This process takes place in sequences of discrete jumps, corresponding to one half of the incident wavelength, as particles hop over the potential maxima that result from the interference of the beams. Once in contact, the rotation of spheroids of lower aspect ratio was seen to halt, whilst for larger aspect ratios it was preserved.

A number of atypical configurations of pairs of spheroids were also observed. In one, the minor axis of one spheroid is aligned parallel to the propagation direction whilst that of the other is perpendicular. This second spheroid rotates continuously, whilst rotation of the first is not apparent. In another example, the spheroids arrange themselves laterally and rotate about the beam axis.

The dynamical behaviour of three and four spheroids becomes progressively more complex. Sets of three spheroids are observed to aggregate into quasi-linear trimers. The subsequent motion involves a periodic circulation of the centre of mass of the trimer in a plane parallel to the propagation direction, coupled with small oscillations of the trimer axis. Groups of four spheroids form pairs of non-rotating dimers, which we observe to hop between equilibria. Unlike individual spheroids, these dimers repel and, over time, they increase their separation by hopping over potential maxima. When sufficiently removed from one another, a completely stable configuration is acquired.

Finally, we note that complex systems, such as the one studied here, are defined by a very large set of parameters and it is not possible to investigate the role played by each of them in a single article. Nevertheless, many of the observed phenomena may be considered generic. The effect of other parameters, such as optical power and particle size may be qualitatively inferred. For example, for a fixed configuration of particles, all of the forces and torques, including binding effects, are proportional to the incident optical power. Furthermore, all velocities and angular velocities are proportional to the applied forces and torques [52]. For this reason, the deterministic evolution of the system from a particular initial state is independent of input power, when a scaled time coordinate is used. The same cannot be said of stochastic processes, such as stochastic hopping. In this case, as power is increased the hopping frequency decreases exponentially [50]. Thus, increasing optical power increases the rate of optically driven processes whilst inhibiting thermally driven ones. It is also possible to imagine the effect of changing particle size. As described in [50], the size of a sphere determines the strength of the interaction with the optical standing wave. For spheroids, the behavior is more complex, but the diameter of the particle parallel to the beam axis (normal to the interference fringes), has a similar influence.

In summary, we have shown that the system comprising a circularly polarized dual beam trap, and non-spherical colloidal particles, can exhibit extremely rich dynamical behaviour. The phenomena described here depend critically on the way in which optical spin and orbital angular momentum interacts with multiple, anisotropic bodies. Understanding of the various forms of motion displayed, could assist the design of optically driven micro-robotic systems [43].

A. Orientation stability of a spheroid in a standing wave

The position and orientation stability of the spheroid can be derived from analysis of stiffness matrix elements [48]. The key matrix elements are $K_{zz}^{tt}=\frac{\omega }{P}\partial F_z/\partial (kz)$ and $K_{yy}^{rr}=\frac{\omega }{P}\partial T_y/\partial {\varphi }_y$ that describe stability with respect to spheroid translation across fringes and rotation along y axis. The normalization pre-factors include the light angular frequency ω and beam power P. Only if both elements are positive we may observe stable trapping in a given orientation as can be seen in Figs. 2 and 8. Figure 10 shows the stiffnesses calculated for a spheroid oriented with is symmetry axis parallel to the beams axes (left) and perpendicular to it (right). The thicker parts of curves denote the stable orientations and positions with respect to the standing wave node and anti-node. Only spheroids with an aspect ratio from sub-intervals around 0.4 and 0.6 are able to adopt the rotation-free orientation (spheroid symmetry axis aligned along z-axis). On the other hand, the rotating configuration is possible with every assumed aspect ratio (symmetry axis perpendicular to z-axis). Also note, if the equilibrium of the rotation-free orientation occurs in the bright fringe, the rotating orientation is stable in a dark fringe (and vice versa).

 

Fig. 10 Elements $K_{yy}^{rr}$ and $K_{zz}^{tt}$ of the stiffness matrix. The left column corresponds to the non-rotating orientation when the spheroid minor axis (SMA) points along the z-axis. The right column corresponds to the rotating one (the stiffness is evaluated when the SMA points along the x - axis). The blue and orange curves are drawn thick in intervals of aspect ratio, that allow for the stable equilibria. The gray curves show the incoherent parts of $K_{yy}^{rr}$, in graphs of $K_{zz}^{tt}$ is zero.

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Acknowledgments

The research was supported by the Ministry of Education, Youth and Sports of the Czech Republic (MEYS) (projects LD14069, LO1212), COST action (project MP1205), CONACYT-ASCR Mexico-Czech Republic bilateral cooperation project (project 171478) and the Czech Science Foundation (project GA14-16195S). The research infrastructure was supported by MEYS, ASCR, and EC (projects LO1212, RVO:68081731, and CZ.1.05/2.1.00/01.0017).

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