1. Introduction

Chirality is emerging as a tool to address innovative concepts in materials science, colloidal systems, optical and photonics devices, optomechanics, sensors, and others [1–7]. Recent results involving materials science and colloidal systems have demonstrated the relevant role of chirality in guiding coupling phenomena between particles or structures [8, 9] with external fields. In particular, because chiral particles have an intrinsic handedness their electromagnetic response depends on the wave helicity. As a consequence, chiral-type structures enable phenomena such as polarization conversion or rotation, and asymmetric transmission of electromagnetic waves [10, 11]. Based on these phenomena interesting effects have been theoretically predicted and experimentally investigated including, for example, chiral optomechanics, chiral sorting, etc. [12–17].

Recent investigations with single beam trap have shown that chiral microresonators, based on chiral polymeric particles with spherulitic configuration enable polarization-controlled trapping, spinning or orbiting through the chirality-mediated coupling of linear and angular momentum of the light. In the single beam configuration the gradient force is used for trapping and the full mechanical control is limited by the value of the particle reflectance connected to the chiral Bragg phenomenon (i.e. on the ratio of the particle size and pitch of the supramolecular helical structure, or on the position of the material stop-band with respect to the trap wavelength) [12–14, 18].

Investigations of dual beam trapping of chiral liquid crystal droplets, demonstrated the full control of the radiation pressure by photon spin, and the capability to apply it to perform optical sorting of chiral particles [15–17]. The advantage of the dual beam geometry is that the flow of linear optical momentum is neutral, along the beam axis. Particles are confined (along the optical axis), not by intensity gradients, but by balanced scattering forces. 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 [19, 20]. For instance, it provides an ideal environment in which particles can undergo self-organization mediated by optical interactions (optical binding) [21,22]. In the case of coherent and interfering counter-propagating beams with proper polarization an array of optical traps is created along the beams propagation direction (standing wave traps) [20, 23–28].

Based on the above described feature this approach exhibits large versatility and provides a useful instrument to investigate the optomechanics of chiral particles. In this paper we report an experimental investigation of chiral microparticles, exhibiting selective Bragg reflectance, in dual–interfering beams trap with different polarization configurations. We concentrate on chiral particles whose size prevents 100% Bragg reflectance of the parallel circular polarization. Based on this feature a stable equilibrium position of the particle along the beams propagation axis and the rotation can be simultaneously controlled by tuning the polarization of only one beam from circular parallel to circular antiparallel with the chirality of the particle. The reported results give interesting suggestions regarding the investigation of particle dynamics in tunable washboard potential and optical binding of Bragg onion microresonators.

2. Experimental setup

Our experimental system consisted of spherulitic microparticles obtained from photopolymerized cholesteric liquid crystal (CLC) droplets. Due to their supramolecular helical structures, they exhibit selective circular Bragg reflection for light propagating along the helical axis and wavelengths within the stop-band [12]. Light with the same handedness as the material is selectively reflected, while the light outside the stop-band or with opposite handedness is completely transmitted, see Fig. 1(b). At fixed incident wavelength circular Bragg reflection thus occurs over a range of incidence angle around the normal incidence [15].

 

Fig. 1 a) Dual-beam optical trap [29, 30] with the ability to set the polarizations of each beam by quarter wave plates (QW). Optically trapped particles are observed using the microscope from side-view. b) Red arrows show the electric field directions of rotation for the right-hand circular (RC) and left-hand circular (LC) polarization of light, respectively. The right-hand circularly polarized light going along the left-helix does not see periodicity of the helix and, therefore, does not diffract. c) optical torque and force acting on the left handed chiral particle illuminated by various combinations CP Gaussian beams are discussed. (LHCP – left handed chiral particles; RC – right-handed circularly polarized light, LC left-handed circularly polarized light, LP linearly polarized light (as a sum of LC and RC circularly polarized beams)).

Download Full Size | PDF

The CLC droplets were prepared from a photo-reactive mesogen doped at 10% concentration of chiral dopant left-handed ZLI-811 by Merck, with a band gap matching the trapping laser vacuum wavelength 1064 nm. The microparticles were obtained by photopolymerization of CLC droplets emulsion, with diameters ranging from 1 to 10 μm, see Figs. 2(a) and 2(b) for optical microscope image in the configuration of crossed polarizers and scanning electron microscope images, respectively. The size of the chiral pitch p was about 660 nm. Depending on the size of the particle with radius a and the chiral pitch p, the particles can reach a reflectance R up to 100%. The reflectance is related to the number of pitches with respect to the radius a, as for a planar Bragg mirror, then we consider a shape factor form (d = 2/3a as reported in previous papers [12, 13]). By the percentage of the chiral dopant we controlled that the trap wavelength is located within the stop band, then the particle circular Bragg reflectance depends only on their size radius (and, of course, on the light polarization state [18]). Based on this feature, we investigated particles with a ratio a/p in the range 1.5–6.5, to which corresponds a chiral Bragg reflectance in the range R = 7% – 50%.

 

Fig. 2 a) An optical microscope image of a small polymeric particle (in the configuration of crossed polarizers) shows the radial structure of the helix with the number of pitches to be a/p ≈ 2. b) Scanning electron microscopy proofs spherical shape of chiral microparticles, c) The asymmetric transmission of the two beams from the particle, due to the Bragg reflectance, modifies the potential in a biased cosine potential (blue curve). d) A trajectory of a left-handed chiral particles (with radius 1.3 ± 0.2 μm) initially illuminated by two counter-propagating Gaussian beams with opposite orientation of circular polarization. At 6 seconds we switched the orientation of the LB to have the same orientation of circular polarization the RB, and we observed particle’s motion in the tilted potential towards the new equilibrium position at 0 μm. (w₀ = 4.6 μm, P = 10 mW)

Download Full Size | PDF

Confining liquid crystals in micrometre-sized systems allows a wide range of thermodynamically stable supramolecular configurations exhibiting various defect topologies. In particular, CLCs droplets exhibit a variety of complex confined chiral geometries that result from the combination of chirality, elasticity, and interface properties. The self-organized internal configuration of the droplet strongly depends on the molecular orientation at the interface and on its radius, a, with respect to the nominal pitch, p, of the helical structures. Hydrophobic CLCs tend to form spherical droplets in water, characterized by planar anchoring of the molecular director $\overrightarrow{n}$ at the CLC-water interface. In the high chirality regime a/p ≫ 1, the more favourable configuration is the radial one, where the cholesteric layers are bent in concentric spherical surfaces with a single defect in the centre. When a/p > 1, the droplet exhibits either a concentric ring pattern with a radial disclination or a spiral pattern without disclination. When a/p < 1, twisted bipolar configurations are mainly expected.

We used a dual-beam optical trap which consisted of two horizontally counter-propagating (C-P) laser beams overlapping in the square glass capillary (Vitrocell, with an inner diameter of 300 μm) which was filled with the microparticles dispersed in deionized water. The trapping 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 shape and position both beams. For a detailed description of the experimental set-up see our previous papers [29, 30]. The beam waist radius w₀ of the C-P beams could be altered between 0.9 and 6 μm,vacuum wavelength of the trapping laser was 1064 nm (IPG), and the total power in the sample cell varied between 10 and 1200 mW. The distance between the beam waists was set to be less than 10 μm. The axial geometry of the set-up is illustrated in Fig. 1(b). The state of polarization of each beam was controlled by quarter-waveplates (QWs).

3. Optical radiation force

Tkachenko and Brasselet [15, 16] recently demonstrated that momentum transfer enhanced by the Bragg reflection on the chiral droplets can be used for a passive optofluidic chiral sorting in the dual-beam arrangement. However, depending on the combination of radii of the the beam waists at the droplet location, sizes of the droplets and position of the laser wavelength inside the circular photonic bandgap, the investigated phenomena refer to droplets in the Bragg regime, i.e high reflectance of circularly polarized light parallel with particles chirality (R = 100%) and to non-Bragg droplets with negligible reflectance. We explore the behaviour of solid-phase chiral particles in the partially Bragg-reflectance regime (with reflectance R = 10% – 50%), and in a dual-beam interfering trap, see Fig. 2(c). However, we studied mainly the stable trapping positions along beam propagation axis z which were not reported previously. Moreover, our solid CLC microparticles may offer some advantages (when compared to droplets) for optical manipulation experiments, since the molecular orientation is frozen-in by photo-polymerization, however do not allow to study phenomena connected for example to deformation and molecular reorientation [31].

Figure 2(d) shows the trajectory of a left-handed chiral particle (with diameter about 2.6 μm) initially illuminated by two counter-propagating Gaussian beams (w₀ = 4.6 μm, P = 10 mW) with opposite handedness of circular polarization, i.e. the left-hand beam (LB) was right-handed circularly (RC) polarized (no Bragg reflection) and the right-handed beam (RB) was set to have left-handed circular (LC) polarization. Thus the optical radiation force is enhanced by the circular Bragg reflection. The overall optical force is negative according to the Fig 1(c.1) and particles moved in the tilted washboard potential and then settled in the equilibrium position at about −30 μm, see Fig. 2(d). At 6 seconds the handedness of the circular polarization of the LB was changed to have the same orientation as RB (Fig. 1(c.4)) and thus the scattering forces from both beams are balanced in the new equilibrium position at z = 0 μm.

By considering the symmetric geometries [Fig. 1(c.4) or equivalently Fig. 1(c.5)] and the beams interference, parallel orientation of circular polarization of the dual beams trap gives rise to a uniform intensity distribution along z of linearly polarized light whose polarization plane periodically rotates with the same spatial periodicity of the light wavelength (linear polarization pattern). The rotation sense of the polarization pattern is clockwise or anticlockwise depending on the LC-LC or RC-RC configuration. In such geometry, the linear polarization spatially rotates according to the cholesteric helix and with the same spatial periodicity, or in the opposite sense. However the geometry is symmetric in both cases, the scattering forces are equal and the particle has the equilibrium position at the centre, z = 0.

When the opposite handedness of the circular polarization of the dual beams trap is set [Fig. 1(c.1) or Fig. 1(c.2)], a standing wave configuration is established along z [32], however the asymmetric transmission of the two beams from the particle, due to the Bragg reflectance, modifies the potential in a biased cosine potential U(z) [33], with a biased force F₀ whose value depends on the particle reflectance, U(z) = (A/2k) cos(2kz) − F₀z. Whereas, the sign of F₀ depends on the double traps geometry and on the particle chirality. The chiral particle thus moves in the tilted standing wave potential [26, 27]. The movement of particle in the standing wave depends also on the particles size, e.g. particles of some sizes are trapped in the interference fringes, other sizes in between and some particles are insensitive and move in the smooth potential [20, 28]. The frequency of jumps of particles between adjacent potential minima is given by thermal activation via Brownian motion of the particle and can be characterized, for example, by the mean optical trap escape time [28].

4. Optical radiation torque

Due to the helicolidal supramolecular arrangement of the material molecules, such microparticles behave as omnidirectional chiral mirrors [12]. Reflection from chiral mirrors preserves the handedness of the incident light, instead of reversing it, as conventional mirrors do. Thus, by conservation of the angular momentum, the chiral particles gain mechanical angular momentum from the CP light with the same handedness and experience a torque [12].

The investigation performed with single circularly polarized laser beam [12–14] in the range 0 ≤ a/w₀ ≤ 3, where a particles radius and w₀ Gaussian beam waist, showed that spherulitic-like chiral microparticles can be, depending on the light handedness and chiral reflectance R, attracted or repelled with respect to the beam axis. In particular, it is reported that, above a certain threshold value of the chiral reflectance R_thr, the particle are attracted by antiparallel CP and repelled by parallel CP. However below this threshold, the particle is always attracted towards the beam axis either for parallel or antiparallel circular polarization. The trapping position is on the beam axis for a/w₀ > 1, while in the range 0 ≤ a/w₀ ≤ 1, for some values of R < R_thr, the particles can be also trapped in an annular region around the beam axis. Providing that the above trapping conditions are achieved, due to the angular momentum transfer upon chiral reflection of light, these particles spin around their own axis of symmetry or orbit around the beam axis and spin at the same time [12, 14]. The reflectance value can be controlled by the particle size [12] or by the material band gap position with respect to the trap wavelength [14].

Similarly, the investigation performed with the dual-beam optical trap geometry [17] showed that chiral droplets of certain sizes (with respect to the beam waist) are pushed out of the optical axis [14], nevertheless no paticle’s spin or orbiting was observed.

Our investigation of optical trapping of left-handed chiral particles in a dual-beam trap with various combinations of beam polarisation states [see Fig. 1(c)] shows that chiral particles can rotate when illuminated by laser beams with an asymmetric configuration (RC – LC or LP – LC(RC)), rotational frequency decreases, when the ellipticity angle of one beam changes from π/4 (parallel CP) to −π/4 (antiparallel CP). However they do not rotate at all for all symmetric configurations. Since the circular Bragg reflectivity of the chiral particles depends on their size (in particular on the ratio a/p), in agreement with the results reported in [12] we expect that particles (of radii a) smaller than two pitch p size exhibit negligible circular Bragg reflectivity. Indeed, no optical radiation torque was observed for particles with radius smaller than 0.8 μm.

Figures 3(a)–3(c) reports the measurements performed on a particle with radius a = 1.4 ± 0.2 μm. The trapped micorparticles were observed from the side-view using an in-house built optical microscope on the pixel calibrated fast camera (see Visualization 1). We acquired time series of particle positions along z and x axes, respectively [see Fig. 1(a)]. The tracking signal x can exhibit oscillatory behaviour indicating that the trapped particles orbits around axis, see Fig. 3(a). The rotation (here) seems to be almost uniform in case of RC-LC, while in the case LP-RC is clearly not uniform, however, the displayed total x displacement (radius of the orbit) is, in both cases about 200 nm. In the LP-RC trap, the particle rotates with approximately half the frequency when compared to the RC-LC trapping rotation, see violet and yellow curve in Figs. 3(a) and 3(c). In these cases, one of the beams (RC) is fully transmitted and the second one is partially reflected (depending on the particle size and polarization) and thus there is optical radiation torque, see Figs. 1(c.1) and 1(c.3). Moreover, the half value of the frequency [Fig. 3(c)] measured for LP is in good agreement with the expected one according to the optical torque Γ on a left-handed chiral mirror with negligible optical retardance [12], as a function of the ellipticity angle of the light beam,

 

Fig. 3 A chiral particle with radius a = 1.4 ± 0.2 μm optically trapped in the C-P Gaussian beams. Various combinations of the polarizations of both beams were set to study the optical torque exerted on chiral particles. a) time series of particle positions along x axes, b) x-position distribution, c) Fast Fourier transform revealed, that the observed particle rotates most intensively when illuminated by laser beams with opposite handednesses, see violet curve. When illuminated, for instance, by one linearly polarized beam (having half left-and half right-circular component) and by second one with the opposite handedness as the particle material, the particle rotates with approximately half frequency, see yellow curve (see Visualization 1).

Download Full Size | PDF

The optical trap can generally be calibrated via the so-called trapping stiffness quantity using the equipartition theorem κ = k_BT/var(x), where k_B is Boltzmann constant, T sample temperature and var(x) is variance of thermal motion of the optically trapped particle. The value of κ is higher in the case of the LC-LC configuration with respect to the one evaluated for RC-RC dual-beam trap; we note that, in the first case, the balance of both scattering forces and torques quenches the thermal fluctuation, with respect to the case where only the balance of the forces occurs. Since the above defined trapping stiffness is valid for particles optically trapped on the beam axis, the trapping stiffness determined for orbiting particles is rather disputable. One can clearly see that there are multiple maxima rather than single Gaussian distribution in the x-position distribution for rotating particle, see violet and yellow curves for orbiting particle in Fig. 3. Moreover, the particle (for various combinations of beam polarisation states) can be optically trapped at different positions along the beam axis z, consequently the local intensity profile differs and thus the comparison of the trapping stiffnesses and frequencies are not straightforward.

The optical torque can be evaluated from the balance with the rotational drag torque on a sphere rotating in a fluid at low Reynolds number Γ = 8πνa³Ω, where Ω is rotational frequency, a sphere radius and ν is viscosity of the fluid. In Fig. 4(a) are reported the values of the optical torques (normalized to the overall laser power in the sample) measured for different particles as a function of the sphere radius to beam waist ratio a/w₀. Since the optical radiation torque upon each particle size was observed only for a certain range of values of the C-P Gaussian beam waists the problem is quite complex.

 

Fig. 4 a) The optical radiation torque Γ measured for different particles as a function of the sphere radius to beam waist ratio a/w₀. b) Chiral particles with a/w₀ in the range 0 ≤ a/w₀ ≤ 1.5 orbit around optical axis. R_O is the radius of orbital motion. Both quantities are plotted for three sizes of particles a = 1.0, 1.5 and 4.3 μm.

Download Full Size | PDF

In Fig. 4(b) we report the radius of orbital motion R_O around the optical axis versus the beam waist normalized to the particle radius. Chiral particles with a/w₀ in the range 0 ≤ a/w₀ ≤ 1 orbit around optical axis which is consistent with the previous results performed with single circularly polarized beam [12–14]. The orbital motion for a/w₀ > 1 can be attributed partially to a very small asymmetry of the internal structure of the particle [31].

5. Radiation optical torque enhanced by optical binding

The D-B interfering trap offers a useful way to investigate the dynamics of optically bound particles. Figure 5 shows the trajectory of two optically bound rotating chiral particles a₁ = 1.0 ± 0.1 μm and a₂ = 1.2 ± 0.1 μm) optically trapped in the C-P Gaussian beams (beam waist w₀ = 4.6 μm, power in the sample P = 140 mW) with opposite handedness. A multi-stability can be observed, i.e. the optically bound spheres jump between two stable positions in the direction x. These configurations are a consequence of the optical interaction. Hydrodynamic forces are proportional to velocity; they affect the time constants of the system and the frequency with which spheres jump between stable positions, but they are not responsible for the equilibrium positions themselves. Such structure can rotate – similarly to the single sphere – close to the optical axis (with radius about 500 nm) or oscillate with amplitude about 3 μm (continuing spinning).

 

Fig. 5 Trajectories (transversal coordinate x) of single chiral particle (blue line, a = 1.1 ± 0.2 μm) and two optically bound chiral particles with very similar sizes (orange line, a₁ = 1.0 ± 0.1 μm and a₂ = 1.2 ± 0.1 μm) optically trapped in the C-P Gaussian beams (beam waist w₀ = 4.6 μm, power in the sample P = 140 mW) with opposite handedness. A multi-stability was observed for the optically bound particles (see Visualization 2).

Download Full Size | PDF

In order to find an explanation of such complex dynamics we make use of some features of the Bragg onion microresonators [34]. Their modes can be classified into core modes that are confined by Bragg reflection and cladding modes that are confined by total internal reflection. The first ones – Fabry-Perot type – possess relatively small angular quantum number and their field distribution is mostly confined within the central core of the spherical microresonator. In contrast, the cladding modes have large angular quantum number, are similar to the whispering-gallery modes in disk resonators, and the electromagnetic fields of these modes are concentrated within the cladding layer.

In the experiment with chiral spherical particles, trapped microresonators spinning due to the transfer of spin angular momentum from the LCP beam, due to a small asymmetry of the particle or of the trapping system, the rotation of both particles is eccentric and the equatorial planes of the two spheres (parallel to z) do not match always. Despite the orbiting of the particles the transversal distance of the sphere centres is enough to sustain the mode coupling responsible of the optical binding of the particles. When, during the rotational dynamic of the two particles, the equatorial planes of the two spheres coincides, the coupling strengths of the cladding modes strongly increases due to the maximal spatial overlap of the modes, reducing also the spatial gap between the particles. When strong coupling occurs, the bright spots between the particles (see Visualization 2) disappear. In such a condition, the two spheres in contact can exhibit a stronger Bragg reflectance for rays propagating within the involved equatorial region, as a result of a coupling of the core modes (like a Bragg particle with a larger size). The reflectance of the bi-sphere like-particle in that region increases at least from 0.1 (one sphere, 2 pitch) to 0.25 (bi-sphere, 4 pitch for rays propagating almost along z). According to the expected increase of the optical radiation torque and an unchanged value of w₀/a lower than 1, the bi-sphere moves off axis to orbit on a larger distance. Due to the continuous rotation of the two spheres this strong binding condition is easily missed; as the coupling force reduces, the particles separate enough reducing their reflectance, and then recover the previous orbit. However the “jump” (asymmetric displacement) of the particles off axis can be accounted for a small asymmetry of the system (of the double-traps geometry or of the particle internal structure).

6. Conclusion

We have investigated the dynamics of chiral spherulitic microparticles in the dual-beam optical trap. Due to their supramolecular helical structures, they exhibit selective circular Bragg reflection for light propagating along the helical axis and wavelengths within the stop-band. The light with the same handedness of the material is selectively reflected, while the light outside the stop-band or with opposite handedness is completely transmitted. Due to the strong asymmetric response of the particles to light with a specific helicity and wavelength, their trapping position and rotational frequency can be controlled by properly combining the polarization state of the two light beams. Here symmetric and asymmetric polarization configurations of dual- interfering beams traps have been investigated. Based on the polarization controlled asymmetric transmission of the chiral particles, a tunable wash-board potential was created enabling to control the trapping position along the beams axis. Asymmetric configurations display polarization controlled rotation of the trapped particles. Moreover, the two optically bound particles exhibited a complex dynamics which is probably given by dynamic matching of internal modes of both particles.