Precise switching control of liquid crystalline microgears driven by circularly polarized light

: Liquid crystalline molecules carrying photopolymerizable end groups absorb photon energy via a two-photon process, enabling the photofabrication of 3D structures. In this work, we prepared microgears with different heights and tooth lengths. These birefringent microgears can be induced to rotate by circularly polarized light. Here, we demonstrate that the use of phase plate for switching between left- and right-handed polarization reverses the optically induced rotation while maintaining the same rotational frequency. Due to the precise switching control, these birefringent microgears have advantages over previous microrotors that are fabricated from non-birefringent light-curing resins.


Introduction
Micro-objects are optically rotated by light with inherent angular momentum, including lasers with a helical phase structure (e.g., the Laguerre-Gaussian mode) and circularly polarized light [1][2][3][4][5][6][7][8][9], because micro-objects can extract the angular momentum of such light due to their absorptive or birefringent optical properties [9]. In 1936, circularly polarized light was shown to transfer both angular and linear momentum to quartz plates used as birefringent materials [1]. Laser tweezers can also be used to rotate liquid crystalline (LC) droplets, birefringent liquid particles that are surrounded by solvents, with circularly polarized light, although the shapes of such birefringent micro-objects have been limited to simple configurations of spheres or cylinders [4][5][6][7][8][9].
The incident light does not necessarily carry angular momentum for the optical rotations. Yang et al. reported that continuous rotation of chiral nematic LC droplets is driven by a linearly polarized light [10]. Another alternative strategy is to use micro-objects that can scatter light in a helical manner [11][12][13]. The photopolymerization of light-curing resins can create microscopic propellers, which have been demonstrated to rotate efficiently due to helical scattering.

Materials and methods
Nematic LC materials were prepared by mixing a 4-wt.% solution of photoinitiators (Lucirin TPO, BASF) with a reactive monomer LC (Paliocolor LC242, BASF) as the main component. The materials are transparent to near-infrared light but absorb photon energy via a two-photon process, yielding photopolymerized LC resins. The LC molecules were placed on a glass plate that had been covered in advance with a polyvinyl alcohol (PVA) film and rubbed to align the LC molecules along the rubbing direction, and the orientational order was frozen by photopolymerization [8]. Because the materials change to the nematic phase at 63 − 70 • C, they were heated to 90 • C just prior to photopolymerization to achieve the LC phase. After photopolymerization, the residual LC monomer was removed by washing with ethyl acetate. Accordingly, we manipulated the fabricated objects surrounded by the ethyl acetate solvent. Figure 1 shows a typical setup used for both fabrication and manipulation. First, we employed a photofabrication system using a mode-locked Ti:sapphire laser (Spectra Physics, Tsunami) producing near-infrared laser light with a wavelength of 730 nm, instead of UV light. The laser emits ultra-short pulses with a pulse width of 130 fs, a repetition rate of 83 MHz, and an average power of 1 W, making it possible to fabricate structures with high resolution via two-photon adsorption as follows. The focus was steered in the horizontal plane by a pair of galvano-mirrors. A focusing unit was attached to the microscope, enabling one to control the vertical position of the objective. Three-dimensional (3D) photopolymerization was conducted by repeating the above processes. Programmed control of the focusing position allowed for a variety of fabricated micro-objects. To observe the 3D microstructures using scanning electron microscope (SEM: VE-7800, Keyence), the micro-objects were Au-coated, and an acceleration voltage of 10 kV was applied. The optical manipulations were performed using a linearly polarized cw Nd:YAG laser (λ = 1064 nm). To trap the LC objects with a circularly polarized beam, a quarter-wave plate was placed in front of the objective lens (NA = 1.3). The principal axis of the phase plate had an offset angle of π/4 with respect to the incident linear polarization direction, producing circular polarization from linearly polarized light. Further rotation of the plate by π/2 switches between right-and left-handed circularly polarized light. A fast CMOS camera (PL-A741, Pixelink) was equipped with a linear polarization analyzer whose direction was perpendicular to the linear polarization plate, and images were recorded using the crossed Nicols method. Because the micro-objects are birefringent, the intensity of light transmitted through the polarizer has a period of π/2 with respect to the angle between the polarizer and the fast axis of the birefringent sample. Therefore, we determined the rotational frequencies by detecting the periodically modulated intensities of transmitted light via the aforementioned camera.

Resolution in fabrication
We measured both the widths and depths of the photofabricated lines using SEM, from which the lateral and vertical resolutions of the present LC resin were evaluated. As the scanning speed of the focal point, v s , decreases, the occurrence probability of two-photon adsorption increases, resulting in a varying degree of photopolymerization. We thus adopted both v s and the light intensity I as photopolymerization parameters. For I ranging from 5 mW to 12.5 mW, v s was in the range of 48 µm/s to 48 × 9 µm/s. We explored a set of optimal parameters, v s and I, for the present system. Consequently, we found the optimal parameter set of v s = 48 × 9 µm/s and I = 7.5 mW, providing lines that are approximately 180 nm wide and 320 nm high. We used these parameters to photofabricate the LC resins into 3D structures with a resolution that is comparable to that obtained for conventional light-curing polymers [14].
We fabricated four kinds of microgears with different heights and/or tooth lengths. The corresponding SEM images are displayed in Fig. 2. Both of the microgears shown in Fig. 2 have a height h of h = 10 µm, with different tooth lengths. Whereas Fig. 2(a) shows a microgear with a tooth length of 1 µm and a total radius r of r = 4 µm, the microgear in Fig. 2(b) has a tooth length of 2 µm with a core identical to that of Fig. 2(a), giving r = 5 µm. When the microgears were manipulated via optical tweezers, we also used the same type of cross-section, with a different height of h = 14 µm. The aforementioned resolution enables the fabrication that produces the dimensional differences between these four types of 3D structures.  The red and blue circles in Fig. 3(a), respectively, correspond to f t 4 and f s 4 of the microgears with shorter teeth (r = 4 µm), whereas the same symbols in Fig. 3(b) respectively represent f t 5 and f s 5 of the microgears with longer teeth (r = 5 µm). We see from these data that the rotational frequencies of the microgears are proportional to the laser power, in agreement with previous results for cylindrical shapes [8]. Below, we will discuss three kinds of differences between the slopes: directional changes in rotating the same object, rotation rate differences ( f t 4 > f s 4 and f t 5 > f s 5 ) encountered while increasing the height of microgears with the same tooth length (or the same total radius ), and decreasing rotational frequencies ( f t 4 > f t 5 and f s 4 > f s 5 ) caused by extending the teeth of microgears with the same height.

Optically driven rotation
Before comparing the slopes, we present two equations relating to the optical torque Γ resulting from the exchange of momentum between the circularly polarized light and the microgear. First, regarding the incident light, Γ depends on the microgear height h as follows [6]: where E 0 is the amplitude of the electric field component of the incident light, and ∆n is the birefringence. The optical torque Γ is balanced by a viscous drag torque that is exerted on a disk having a drag coefficient of (32/3)ηr 3 in the overdamped limit [14,15]: where η is the viscosity of the surrounding solvent, r is the radius of the disk, and Ω/2π = f corresponds to the rotational frequency. It has been found that microgears can be mimicked by disks when the mean radius of r m ≡ (r + r in )/2 is used in eq. (2) for evaluating the drag torque [15], where r is the total radius including the teeth and r in is the core radius without the teeth. Equations (1) and (2) imply that the rotational frequencies (Ω/2π in eq. (2)) are proportional to the intensity of the incident light (E 2 0 in eq. (1)), which is validated from Fig. 3. In Fig. 4 (Media 1), we have demonstrated that the LC microgears allow for directional control of optical rotation with the maximum frequency of approximately 11 Hz at a laser power of 180 mW, revealing the superiority of LC micromachines over conventional ones in terms of switching efficiency. In Figs. 3(a) and 3(b), the data points are fitted by two straight lines with opposite slopes, due to switching between left-and right-handed circular polarization of the incident light. A comparison of these slopes shows that the respective differences between the absolute frequency values for the left-and right-handed rotation of the microgears are below 10% and within the error of each data point. The coincidence of rotational frequencies upon a direction reversal validates the mirror symmetry of the exerted torque due to left-and righthanded circularly polarized light. The present reversal requires only the π/2 rotation of the phase plate, which is an advantage over optically driven micromachines consisting of nonbirefringent resins [13,14].