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Abstract
We disclose a transporting/collecting optical sling generated by a liquid crystal geometric phase optical element with spatial variant topological charge, which shows the intriguing repelling/indrawing effect on the micro-particle along the spiral orbit. Two proof-of-concept prototypes, i.e., an optical conveyor and a particle collector, are demonstrated. Based on the distinct dynamic characteristics of the micro-particles in different sizes, we conceptually propose a design for particle sorting. Thus, our proposed method to generate a spiral optical sling with spatial variant orbital angular momentum for on-demand collecting, transporting and sorting micro-particles is substantiated, which can find extensive applications in bio-medicine, micro-biology, etc.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The marvelous facets of light are not only embodied in its amplitude, wavelength and polarization, but also manifested in phase and momentum. The light-matter interaction due to the momentum transfer of photons was first discovered by Ashkin et al. in an experiment in 1970, where the dielectric particles were accelerated and trapped under the radiation pressure generated by two counter-propagating laser beams, laying the foundation for an optical tweezer [1]. Sixteen years later, the stable optical trapping for the micro-spheres with various sizes was achieved by Ashkin et al. through the gradient force arising from a highly focused fundamental Gaussian beam, thus launching the development of optical tweezers technology [2]. As one of the world’s smallest artificial micro-tools, optical tweezer plays a significant role in the capture and manipulation of the microscopic objects, exhibiting the merits of non-contact, damage-free and high precision, and thus provides us an effective means to explore the microscopic world. Consequently, the optical manipulation has attracted extensive attention in the literature, and a variety of representative applications have sprung up, including the cooling of atoms [3], the laser cell fusion trap in biology [4], the sensing and precision measurement [5], the study of microscopic physical properties [6], etc.
With the increasing demands of application, an optical tweezer employing spatial light modulator (SLM) to generate the structured light field (i.e., the spatial variant amplitude/phase/polarization of light) via propagation phase modulation was demonstrated to simultaneously manipulate a variety of particles in real time, enriching the application scenarios of optical tweezers [7]. Moreover, some structured beams with special intensity profiles carry orbital angular momentum (OAM), showing an outstanding ability in controlling the complicated motion of particles [8]. Indeed, an OAM as the additional angular momentum in light propagation is related to a spiral phase distribution, giving rise to the spiral wave front for a vortex beam. Each photon of the beam carries the OAM of mh [9], where m and h are the topological charge and the Dirac constant, respectively. It has been reported that a typical vortex beam with a donut-like annular intensity distribution in the cross section can transfer its OAM to the particles trapped in the light field [10]. The particle manipulation under a structured beam via the joint effect of the OAM transfer and optical gradient force usually exhibits advantageous trapping performance and superior motion control characteristics over the single-beam gradient trap [11]. A great number of light beams with various exotic structured field generated by light field regulation have been extensively applied in the optical manipulation, for instance, optical-vortex-driven micro-optomechanical pumps [12], single-beam optical tweezer with large trapping range [13], optical grinder that sorts trapped particles [14], etc. Besides, Rodrigo et al. have achieved the governable motions of the particles along the gradient intensity regulated trajectories through an SLM equipped optical tweezer, which are fundamentally driven by phase gradient forces [15]. In the past three decades, SLMs have facilitated optical tweezers to manipulate the particle via various structured vortex beams and gradually become the mainstream method. However, the employment of an SLM makes the optical system complicated and unsuitable for the on-chip integration, what’s more, it substantially increases the manufacture cost of the entire optical tweezer. Metasurface with considerably compact size is also regarded as one of the possible routes to the on-chip photonic applications of optical manipulation [16,17], whereas its manufacture cost is still high, and in some circumstance, it lacks the flexibility for the applications of optical manipulation due to the absence of tunability. In this context, a cost-effective, compact and controllable functionalized photonic device for optical manipulation is highly in demand and deserves plenty of worthwhile endeavors. Liquid crystal (LC) geometric phase optical element (GPOE) as a low-cost, compact, electrically controllable device is one of the most suitable candidates for the on-chip photonic applications in optical manipulation.
In this paper, we disclose an LC GPOE, i.e., an LC q(φ)-plate, having an elaborately designed surface anchoring with inhomogeneous angular distribution that can generate a spiral optical sling with spatial variant OAM for on-demand collecting, transporting and sorting micro-particles. By tightly focusing the emergent light on the micro-particles, the static optical sling with specific phase and intensity gradients exhibits prominent properties in optical manipulation, where the micro-particles can be captured or transported depending on the handedness of the incident circular polarization. Moreover, the facile electrical controllability of LC endows the proposed GPOE with additional potential functionality, i.e., simultaneous transporting and sorting particles in different sizes, when one or more relay optical slings are deployed. Thus, our proposed electrically controllable LC q(φ)-plate manifests significant merits, e.g., novel multi-functionality, compact structure, easy fabrication, low-cost, effective simplification of the optical system, etc. With the excellent features stated above, the said LC q(φ)-plate stands out as a promising candidate for advanced optical manipulation, and paves the way towards the integrated on-chip multi-functional photonic module for the potential applications in optical micro-manipulation, bio-medicine, micro-biology, colloid physics, chemistry, etc.
2. Principle and methods
Typically, when a circularly polarized light propagates through an LC q-plate, a structured vortex beam with the reversed circular polarization as well as the spiral wavefront of a topological charge of 2q is generated. This process occurs in accompany with the conversion of angular momentum from the spin angular momentum (SAM) to the OAM. The converged vortex beam can draw the particles towards the ring-shaped orbit with strong light intensity near the focus, and the OAM transfer from the beam to the particles gives rise to the circular motion of the particles [18].
Distinct from the conventional LC q-plate with a homogeneously varying alignment, the q-value in the alignment profile of our proposed LC GPOE as shown in Fig. 1(a) is no longer a constant, which varies with the azimuthal angle ($\varphi $) as follows:
where ${q_0} = 8$ is the initial value and $k = 6/{\pi ^2}$ is a coefficient related to the rate of angle change. The GPOE called LC q(φ)-plate is fabricated by photo-patterning an azo-dye alignment material (i.e., SD1, from DIC, Japan) with a photo-addressable device [19,20]. Thus, when a right-/left-handed circularly polarized (RHCP/LHCP) light ${E_{in}} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt 2 }}}\right.}\!\lower0.7ex\hbox{${\sqrt 2 }$}}{\left( {\begin{array}{cc} 1&{ \pm i} \end{array}} \right)^T}$ incident on the GPOE that satisfies the half-wave condition, the electric field of the LHCP/RHCP emergent light can be expressed in normalized form as the equation below:
Fig. 1. Design of the LC GPOE for generating the proposed optical sling. (a) Alignment profile of the LC q(φ)-plate. (b) The schematic of the particle manipulation via the optical sling. Diffracted beam profiles for the RHCP and LHCP emergent lights in (c) and (d) are not mutually centrosymmetric. The azimuthal angles (φ) open from different initial orientations in (c) and (d). The red scale bars denote 2 mm. The magnified images for the light intensity distributions in the rectangular areas are depicted in the corresponding insets of (c) and (d). Black arrows indicate the light-intensity gradient force in the transverse plane. (e) The light intensity distribution, (f) the normalized light-intensity gradient and (g) the normalized OAM along the spiral dashed line in the RHCP/LHCP optical sling. The positive/negative sign of the normalized gradient and OAM coincides with the increase/decrease of the corresponding physical quantities.
3. Results and discussion
3.1 Single transporting/collecting optical sling
In the experiment, we employ a self-developed optical tweezer system named LC-Photoneezer-1 with straightforward structural design, whose basic optical setup is shown in Fig. 2(a). A 532 nm laser (QS-4012A, 1W/cm2) passing through an attenuator and a polarization switch is reflected by a mirror that directs the laser beam downwards and then successively propagates through the LC q(φ)-plate, a beam splitter, an oil immersion microscope objective lens (×100, NA = 1.25, working distance at 0.17 mm) and the micro-particles sample. The handedness of the circular polarization of the 532 nm laser is controlled by the polarization switch, i.e., a passive LC quarter-wave plate (QWP) and an electrically switchable LC half-wave plate (HWP), which are placed in between the attenuator and the mirror. Specifically, after the light penetrates the LC quarter-wave plate, a RHCP light is generated. Subsequently, the RHCP light is converted to an LHCP light by the LC HWP, while it keeps its polarization state when a saturated signal voltage is applied across the HWP. The micrograph of the LC q(φ)-plate under polarizing optical microscopy (POM) is presented in the inset of Fig. 2(a). The upper CCD helps to precisely adjust the x-y-z stage for the sample holder in order to guarantee that the micro-particles in the aqueous sample are close to the focal plane of the objective lens. The lower CCD monitors the motion of the micro-particles in real time through another objective lens (×10) and an optical filter that blocks the intense 532 nm laser. The observed diffracted beam profiles for the RHCP and LHCP emergent lights are shown in Fig. 2(b) and (c), respectively, where the dashed arc arrows indicate the motion directions of the micro-particles. With time elapsing, a representative micro-particle (i.e., silicon dioxide micro-sphere, 3 µm in diameter) moves to the corresponding position as depicted in Fig. 2(d) and (e), showing the repelling and indrawing processes arising from the transporting (RHCP) and collecting (LHCP) optical slings, respectively. It is noteworthy that the counter-clockwise angular displacement of the micro-particle in Fig. 2(d) terminates after 12 s, and then the particle travels freely in the water since the magnitude of the light-intensity gradient force is insufficient to confine the micro-sphere. The transportation distance is up to several dozens of micrometers. Nevertheless, the clockwise angular displacement of the micro-particle in Fig. 2(e) ceases around 23 s, thereafter, the particle is statically captured around the central area.
Fig. 2. Experimental setup and particle manipulation via a single optical sling. (a) The basic experimental setup of the optical tweezer system. The inset shows the micrograph of the LC GPOE under crossed polarizers. The white scale bar represents 100 µm. Diffracted beam profiles for (b) RHCP and (c) LHCP optical slings. (d) The counter-clockwise motion of the micro-sphere (3 µm in diameter) in a transporting (RHCP) optical sling. The repelling process demonstrates the transportation of the micro-particle (see Visualization 1). The presence of the bright spot in the center is due to the non-diffracted 0th order light beam. The white dashed arrows denote the motion track of the particle. (e) The clockwise motion of the micro-particle in a collecting (LHCP) optical sling. The indrawing process evidences the capture of the micro-particle (see Visualization 2). The scale bars in (d) and (e) represent 10 µm.
In our experiment, the effective area for manipulating the micro-particles did not cover the whole optical sling largely because of the imperfect beam quality, which can be considerably improved by employing a 4f optical system with a pinhole (pupil) in the Fourier plane. Moreover, the motion of the micro-particles could be smoother when the light intensity distribution as well as the OAM distribution are elaborately tailored via the alignment pattern modification. The functionality of the LC q(φ)-plate, i.e., transportation or collection of the micro-particles, is determined by the circular polarization of the incident light which can be readily switched by the LC HWP. Therefore, the LC q(φ)-plate can serve as a functionality-switchable basic unit in an optical manipulation system. Based on this basic unit, one may develop a more sophisticated micro-particle manipulation system that includes optical conveyors for transporting the micro-particles over a long distance and particle collectors for capturing or storing the micro-particles.
3.2 Proof-of-concept prototype of an optical conveyor
To achieve a relatively long-distance transportation of the micro-particle, an additional relay transporting optical sling is employed so that the judiciously designed LC GPOE generates two end-to-end transporting optical slings. The alignment profile, the theoretical and experimental diffracted beam profiles of the GPOE are shown in Fig. 3(a), (b) and (c), respectively. Two q(φ)-plate parts are combined into one alignment pattern involving a grafting treatment [21]. The lower part of the alignment profile in Fig. 3(a) contains a superposed term of a same q(φ)-plate term and a deflection term that appropriately translates the left transporting optical sling in Fig. 3(b) and (c) (see the detailed expression for the alignment profile in Supplement 1). However, a half of the q(φ)-plate pattern in the upper part of the alignment profile is replaced by a pure polarization grating pattern to deflect the unwanted 0th order beam, and thus the left half pattern contributes to the formation of the right incomplete optical sling in Fig. 3(b) and (c). Indeed, the right-sided modified transporting optical sling in Fig. 3(c) works as a relay transporter, and a remarkable relay transportation process is clearly presented in Fig. 3(d), wherein the micro-sphere swiftly moves from the left of the image to the right due to the successive influences of the first and second transporting optical slings. It is worth noting that the direction of the particle transportation can be facilely adjusted by modifying the corresponding alignment profile of the LC GPOE. Furthermore, one may deploy multiple relay transporting optical slings to further increase the particle transportation range, and thus a practical optical conveyor that delivers the micro-particle to arbitrary position can be realized.
Fig. 3. A proof-of-concept prototype of an optical conveyor with an additional relay transporting optical sling. (a) Alignment profile of the LC GPOE. The lower part is photo-patterned according to the alignment profile expression with a superposed term of a same spatial variant topological charge term and a deflection term, while the upper part contains a half pattern for the said spatial variant topological charge and the other for a pure polarization grating. (b) Theoretical diffracted beam profile. (c) Experimental diffracted beam profile. The dashed arc arrows indicate the motion directions of the micro-particle. (d) Successive frames of the video image showing the motion of the micro-particle in these two end-to-end transporting optical slings (see Visualization 3). The dashed arrows denote the motion track of the micro-particle. The scale bar represents 10 µm.
3.3 Proof-of-concept prototype of a particle collector
Generally, a particle collector is an essential unit in a complete micro-particle transportation system. As a result, we ought to deploy a collecting optical sling, which is connected to a transporting optical sling to collect the particles. To achieve this goal, the alignment profile of the LC GPOE is modified as depicted in Fig. 4(a). Note that the alignment patterns in Fig. 3(a) and Fig. 4(a) are both designed for the LHCP incident light. The lower part of the alignment profile contains an additional deflection term (see the detailed expression for the alignment profile in Supplement 1) in order to ensure a good connection between two optical slings. The theoretical and experimental diffracted beam profiles of the LC GPOE are illustrated in Fig. 4(b) and (c), respectively. As we can see in Fig. 4(d), the micro-particle is transported by the transporting optical sling as time goes from 0 to 0.6 s, and after a curved motion it gets captured by the collecting optical sling at 4.1 s. Consequently, a fabulous optical collector that is connected to a particle transportation port is demonstrated. Ideally, one may change the input circular polarization to switch the functionality of the GPOE so that the micro-particle captured in the collector will be released and move back along the optical sling from the right to the left.
Fig. 4. A proof-of-concept prototype of a particle collector connected to a transporting optical sling. (a) Alignment profile of the LC GPOE. The lower part pattern is based on a superposed term of a same spatial variant topological charge term and a deflection term, in order to properly connect the tail of the left-sided transporting optical sling to the right-sided collecting optical sling. (b) Theoretical diffracted beam profile. (c) Experimental diffracted beam profile. The dashed arc arrows indicate the motion directions of the micro-particle. (d) Successive frames of the video image showing the motion of the micro-particle in the connected collecting and transporting optical slings (see Visualization 4). The dashed arrows denote the motion track of the micro-particle. The scale bar represents 10 µm.
3.4 Discussion on a design for particle sorting
Based on the theoretical calculation as well as the experimental observation, it turns out that the light-intensity gradient force confines the micro-particle in the spiral orbit and the OAM transfer drives it to move along the orbit. As the motion speed of the micro-particle is determined by various factors, e.g., the size and mass of particle, the viscous force from water, etc., the dynamic characteristics of the micro-particles in different sizes in the same transporting optical sling is dissimilar, and thus it provides the possibility to discriminate the particles in a specific size. In Fig. 5(a), it shows the plot of the angular displacement (Φ) versus time for the micro-sphere in each size (i.e., 2 µm, 2.5 µm, 3 µm and 4 µm in diameter) manipulated by the transporting optical sling. The micro-sphere of 3 µm in diameter presents the fastest motion speed as well as the largest angular displacement among these four kinds of particles. Moreover, the time-dependent first derivative of the angular displacement as illustrated in Fig. 5(b) manifests that all these kinds of silicon dioxide micro-spheres tend to get pulled into the spiral orbit, obtain the relatively fast initial velocity, and then the rates of the angle change go all the way down to 0. In particular, the micro-sphere of 3 µm in diameter is quite distinctive from the rest, indicating that sorting out the particle in a specific size from a blend of particles is possible.
Fig. 5. Dynamic characteristics of micro-particles in different sizes and the design for particle sorting. (a) Angular displacements versus time for the micro-spheres in different sizes (i.e., 2, 2.5, 3 and 4 µm in diameter). (b) The time-dependent first derivative of the angular displacement for the particle in each size. (c) Design for the particle sorting method. Through the ITO etching process, the LC cell is divided into two separate electrically controllable parts. The two adjacent LC GPOEs in Part 1 and Part 2 have the alignment profiles given in (d). (e) Particle transported by a single transporting optical sling when the LC GPOE in Part 1 is switched on and that in Part 2 is switched off. (f) Particle successively transported by two end-to-end transporting optical slings when LC GPOEs in Part 1 and Part 2 are both switched on.
To achieve the task of sorting particles, we conceptually propose a facile and feasible design as presented in Fig. 5(c). The LC cell has two separate electrically controllable parts, i.e., Part 1 and Part 2, whose substrate is prepared through indium-tin oxide (ITO) etching process. The alignment profiles of the LC GPOEs in Part 1 and Part 2 are shown in Fig. 5(d), where the straight lines correspond to the areas of polarization gratings that deflect the unwanted 0th order beam. Both the alignment profile expressions for the curved patterns in Part 1 (upper part) and Part 2 (lower part) contain a superposed term of a spatial variant topological charge term and a deflection term. Each of the LC GPOEs is independently switched between high efficiency under half-wave condition (i.e., ON state under low voltage) and none efficiency (i.e., OFF state under saturated high voltage) by an electric controller. When the LC GPOE in Part 1 is switched on and that in Part 2 is switched off, a single transporting optical sling is generated as depicted in Fig. 5(e). When both LC GPOEs in Part 1 and Part 2 are switched on, two end-to-end transporting optical slings emerge as illustrated in Fig. 5(f). In the sorting process, the particles enter the sorting area one by one, and each of the particles is transported by the first transporting optical sling, whereas the relay transporting optical sling is switched on/off according to the dynamic characteristics of the particle in each size. Only the particles in target size that timely arrive at the joint spot during the operating window of the electric signal for the second optical sling can be successively transported and directed to a certain particle collector. The sorting criterion of the proposed method is based on the dynamic characteristics of particles in an optical sling, which provides the particle sorting with high precision and can be used to avoid the human error in size discrimination through conventional visual method. Additionally, when this sorting method is adopted in a more advanced dynamic scanning technology, a more powerful particle manipulation tool can be expected, and thus larger scale automated particle sorting can be achieved.
4. Conclusions
In conclusion, we disclose a spiral transporting/collecting optical sling generated by an LC q(φ)-plate under the LHCP/RHCP incidence, which shows the repelling/indrawing effect on the micro-particle along the spiral orbit. Based on this property, a proof-of-concept prototype of an optical conveyor has been demonstrated via a relay transportation of a micro-sphere by using two end-to-end transporting optical slings, and a proof-of-concept prototype of a particle collector has also been verified through the collection (capture) of a micro-sphere by adopting two connected collecting and transporting optical slings. Moreover, the study of the dynamic characteristics of the micro-particles in the same transporting optical sling unveils that the angular displacement as well as the motion speed of the micro-spheres in different sizes are dissimilar, particularly, the micro-sphere of 3 µm in diameter is quite distinctive from the rest, indicating the feasibility of sorting out the particle in a specific size from a blend of particles. In this context, we conceptually propose a facile design that includes two independently controlled LC GPOEs to generate two end-to-end electrically switchable transporting optical slings for the particle sorting. Consequently, we have substantiated an LC q(φ)-plate that can generate a spiral optical sling with spatial variant OAM for on-demand collecting, transporting and sorting micro-particles. With the prominent features of the electrically controllable LC q(φ)-plate, our proposed method is promising in advanced optical manipulation, e.g., lab-on-a-chip system for particle manipulation [16,17], the micro-manipulation of the particles with exotic shapes [22], the exploration of fundamentals of the optical forces [18,23,24] and so on so forth. We believe that the LC q(φ)-plate can find extensive applications in optical micro-manipulation, bio-medicine, micro-biology, colloid physics, chemistry, etc.
Funding
National Key Research and Development Program of China (SQ2022YFA1200117); National Natural Science Foundation of China (62275081, 62035008, 61822504, 51873060, 11874026) ; Shanghai Municipal Education Commission (2021-01-07-00-02-E00107); Shanghai Education Development Foundation (21SG29); The State Key Laboratory of applied optics (SKLAO2020001A07).
Acknowledgment
This work is supported by National Key Research and Development Program of China (SQ2022YFA1200117); National Natural Science Foundation of China (NSFC) (Nos. 62275081, 62035008, 61822504, 51873060, and 11874026), the Fundamental Research Funds for the Central Universities, Innovation Program of Shanghai Municipal Education Commission, Scientific Committee of Shanghai (2021-01-07-00-02-E00107), “Shuguang Program” of Shanghai Education Development Foundation and Shanghai Municipal Education Commission (21SG29), and the State Key Laboratory of applied optics (No. SKLAO2020001A07). Xiaoqian Wang, Mian Wu and Bo Ni contribute equally to this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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