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Abstract
Waveguide optical tweezers can capture and transport nanoparticles, and have important applications in biology, physics, and materials science. However, traditional waveguide optical tweezers need to couple incident light into one end of the waveguide, which causes problems such as difficulty in alignment and low efficiency. Here, we propose a new type of waveguide optical tweezers based on spin-orbit coupling of light. Under the effect of spin-orbit coupling between the waveguide and nearby particles illuminated by a circularly polarized light, the particles experience a lateral recoil force and a strong optical gradient force, which make particles in a large area to be trapped near the waveguide and then transmitted along the waveguide, avoiding the coupling of light into one end of the waveguide. We further demonstrate that the particles can be transmitted along a curved waveguide and even rotated along a ring-shaped waveguide, and its transmission direction can be simply switched by adjusting the spin polarization of incident light. This work has significance in the research of optical on-chip nano-tweezers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well known that light carries not only energy, but also momentum [1], which enables the optical forces [2] and invention of optical tweezers [3]. Optical tweezers technology has the advantages of non-contact, non-damage, real-time detection, etc., and thus has important application value in the research fields of biology [4,5], physics [6,7] and materials [8,9]. Under the continuous exploration of researchers, a variety of optical tweezers technologies have been developed, including holographic optical tweezers [10,11], surface plasmon optical tweezers [12,13], waveguide optical tweezers [14,15], and others [16,17].
Among these optical tweezers technologies, waveguide optical tweezers is very suitable for transport and manipulation of nanoparticles in lab-on-chip systems for potential applications such as drug delivery [18,19], relying on the evanescent wave generated by the total internal reflection of the light propagating in the waveguide [20–22]. In 1992, Kawata et al. [23] experimentally verified that particles can be trapped near a waveguide and move directionally along the waveguide. Subsequently, Ng et al. [22] reported that gold nanoparticles with radius of 10 nm can be captured on the transverse channel waveguide and propelled along the waveguide with a speed of 4 nm/s. Recently, Xu et al. [24] irradiated incident light vertically on one end of a nanowire, and showed that nearby particles can be attracted by the nanowire and moved along the nanowire towards the light source. However, these waveguide optical tweezers have some limitations. For example, the incident light needs to be coupled into one end of the waveguide by tight focusing, resulting in disadvantages such as low coupling efficiency and high requirement in alignment. Besides, the transport direction of particles along the waveguide is usually fixed once the incident light is focused on one end of the waveguide. To switch the particle transport direction, it has to change the incident light to focus on the other end of the waveguide, which is difficult and time-consuming. Moreover, it is hard to achieve simultaneous transmission of particles along multiple waveguides, which requires multiple focuses on the different waveguides.
To solve these problems, we propose a new type of waveguide optical tweezers based on spin-orbit coupling of light [25–28], which can trap particles in a large area of light spot near the waveguide and then transport the particles along the waveguide, avoiding the focusing of light on the end of waveguide. In addition, the transporting direction of particles can be simply switched by adjusting the spin state of incident light. As a novel effect of light, recently the spin-orbit coupling of light has been introduced into the research of optical tweezers for various manipulation of particles, such as the lateral optical force acting on the dipole near a surface [26], and optical sorting of chiral particles [29].
Here, we demonstrate a controllable transport of nanoparticles along waveguides by the spin-orbit coupling occurs in a complex structure composed of a waveguide and nearby particles illuminated by a circularly polarized light. The spin angular momentum (SAM) of the circularly polarized light is converted into orbital angular momentum (OAM) through scattering of the particles, resulting in a unidirectional propagation mode of the waveguide, thereby generating a lateral recoil force in the opposite direction on the particles. This lateral recoil force allows the particles to move along the waveguide. At the same time, an enhanced electric field is generated in the gap region between the particle and the waveguide, due to the coupling of scattered light of particle and evanescent wave of the waveguide, producing a strong optical gradient force to attract and trap the nearby particles. Therefore, eventually particles in solution will be trapped near to the waveguide by the optical gradient force and then travel along the waveguide by the lateral recoil force. We numerically study the influence of light source and structural parameters on the optical forces, including the ellipticity of incident light, particle size, and waveguide curvature. We find that the particles can not only be transported along a straight waveguide, but also move along a curved waveguide, and even rotate clockwise or anticlockwise along a ring-shaped waveguide. This waveguide optical tweezers technology shows advantages such as no requirement of light focusing on one end of waveguide, large trap area of particles, and flexible adjustment, which further expands the application range of waveguide optical tweezers and shows great importance to the precise manipulation, detection, and sorting of nanoparticles.
2. Results and discussion
2.1 Origin of the lateral recoil force
Figure 1 shows the basic principle of lateral recoil force on the particles near a waveguide in water generated by the spin-orbit coupling. In Fig. 1(a), we consider a left-hand (red color) or right-hand (blue color) spin polarized plane wave source illuminating the gold nanoparticles and silica waveguides from the top side. As shown in Fig. 1(a), when the incident light beam has left-handed spin polarization, the gold particle located on the + y direction side of the waveguide receives a lateral recoil force in -x direction, while the particle on the other side of the waveguide is subjected to the force in the opposite (+x) direction. When changing the chirality of the incident light, all directions of the lateral forces on the particles are switched to the opposite direction.
Fig. 1. (a) Schematic diagram of the lateral force generated by the spin-orbit coupling. A left-hand (red color) or right-hand (blue color) spin polarized light is incident vertically along -z direction onto the silica waveguide and the nearby gold nanoparticles. The particles on different sides of the waveguide are subjected to lateral recoil forces (purple arrow) in opposite directions. (b) The normalized variation of the difference (black dotted line) and the ratio (red dotted line) between the left (the -x direction, PL) and right (the x direction, PR) energy flows in the waveguide as the incident polarization ellipticity changes. Some typical polarization states (circular/linear polarization) are labeled by blue arrows on the top. (c) Excited waveguide mode with unidirectional propagation under the illumination of a left-hand (red arrow) or right- hand (blue arrow) polarized light with polarization of Jones vector (1, ± i*0.24). With each light, the particle is located on the upper side (upper figure) or lower side (lower figure) of a single waveguide (profile denoted with two white dashed lines). (d) The relationship between the lateral force (Fx) / gradient force (Fy) and the polarization state of incident light., when the particle is located in the -y direction of the waveguide as shown in the inset.
This spin-orbit coupling induced lateral recoil force comes from the symmetry break of the structure. Although both the spin polarization of incident light and the spherical particle are rotationally symmetrical, the existence of straight waveguide breaks such symmetry, leading to the spin-to-orbit conversion effect [30]. The SAM of the incident light is converted into OAM by the particle scattering, which induces an asymmetric coupling of incident energy to the waveguide, and thus excites the unidirectional propagation mode in the waveguide. Due to the momentum conservation, the particle gains momentum in the opposite direction following the unidirectional propagation in the waveguide, resulting in the lateral recoil force. It is noted that in the plane (xy plane) perpendicular to the incident light, the particle-waveguide structure with particle in two sides of the waveguide actually has two opposite chirality; so, directions of the unidirectional propagation are opposite in such two cases, as well as the generated lateral recoil force, as shown by the purple arrows in Fig. 1(a).
To prove this spin-orbit coupling effect, in Fig. 1(b) we show results of the difference and the ratio between the energy flows in two opposite directions inside the waveguide as a function of the incident polarization state, where χ represents the ellipticity of the incident polarization [31]. All data are calculated by three-dimensional finite-difference time-domain (FDTD) simulations (Lumerical FDTD Solutions). In FDTD simulation, the minimum grid size is 1 nm near the interfaces, and perfectly matched layers are placed around the whole simulation area. The silica waveguide has a cross-section size of 300 nm×300 nm, and diameter of the particle is 90 nm. The wavelength of incident light is 532 nm, and the corresponding refractive indices of different materials are chosen as: SiO2 (n = 1.46), Au particle (n = 0.402), and water (n = 1.333). Though the comparison of the two curves in Fig. 1(b), we find that they have different positions of the positive and negative peaks, where the peaks of the energy flow difference locate exactly at the left/right circular polarization but those of the energy flow ratio appear at two typical elliptical polarization of Jones vector $({1\textrm{, } \pm \; i\ast 0.24} ).$ To observe the unidirectional propagation mode inside the waveguide, Fig. 1(c) shows the electric field distribution of the particle-waveguide structure at the two peak positions of the energy flow ratio (red curve in Fig. 1(b)). In Fig. 1(c), when the incident light is left-handed polarized and the particle is located on the upper side of the waveguide, the asymmetric coupling occurs and excites the unidirectional waveguide mode to the left. In contrast, when the particle is placed on the lower side, direction of the waveguide mode propagation is switched to the right, which would produce a left lateral recoil force on the particle, as indicated in Fig. 1(a). Besides, when the chirality of the incident light is switched, all the propagation directions of the waveguide mode change to the opposite. Thus, the chirality of the incident light determines the direction of unidirectional propagation in the waveguide, which is consistent with the previous report [32].
To prove that the spin-controlled unidirectional waveguide mode can produce the lateral recoil force on the particle, influence of the incident polarization state on the optical force exerted on the particle are calculated, as shown in Fig. 1(d) (the particle is located in the -y direction of the waveguide in the inset). The optical forces on the particles are calculated using the Maxwell stress tensor method [33],
2.2 Effect of particle size
Figure 2 shows the effect of particle size on the lateral force. The lateral force exerted on the particle depends on the unidirectional waveguide mode excited by scattering of the particle. Since the size of particles has a greater impact on scattering, therefore, the particle size plays an important role in scattering as well as the optical force. Considering that the incident light is a left-handed circularly polarized light, and the gap distance between the particle and the waveguide is 10 nm (particle is on the -y direction side of the waveguide), effects of particle size on the lateral force are achieveable by gradually changing its diameter. As shown in Fig. 2(a), with size of the particle getting larger, the absolute values of both the gradient force Fy and lateral force Fx increase, but the gradient force is an order of magnitude larger than the lateral force. This indicates that when the particle-waveguide structure is illuminated by a chiral light, the particle will first be attracted towards the waveguide surface by the gradient force, and then transport along the waveguide under the effect of lateral force. Figure 2(b) shows the influence of particle size on the energy flow difference between both sides of the waveguide, which also presents the opposite trend as that of lateral force, verifying that the change of lateral force is always determined by the recoil of the difference of the two directional energy flows of the waveguide mode. In order to reveal the near-field effect of particle size, we compare the near-field properties of two cases with particle diameter of 50 nm and 100 nm in Figs. 2(c)–2(d). In Fig. 2(c), the electric field distribution near the gap region shows that the near-field intensity in the gap increases with the particle size, due to the fact that larger nanoparticle could scatter more incident light, and thus enhance the coupling efficiency from the scattered light to the waveguide mode, resulting in an enhancement of the excited waveguide mode [Fig. 2(d)] and the corresponding lateral recoil force and gradient force. Actually, we find that nanoparticles with diameter of 20 nm - 150 nm have same changing trend in both lateral force and energy flow difference as that in Fig. 2(a) and Fig. 2(b), thus nanoparticles with a wide range of diameter can be transported unidirectionally along the waveguide due to the spin-orbit coupling of light.
Fig. 2. Influence of particle size on optical forces. (a) The optical lateral (black dotted line, Fx) and gradient (red dotted line, Fy) forces as function of particle diameter (from 50 nm to 100 nm). (b) The difference of energy flow of both sides of the waveguide with the change of particle diameter. (c) For particle diameters of 50 nm and 100 nm, the electric field distribution |Ey| in the gap region. The white dotted line represents the boundary of the waveguide. (d) Real part of electric field |Re(Ey)| in the xy plane for excited waveguide mode at the particle diameter of 50 nm and 100 nm. The two white dotted lines indicate the both boundaries of the waveguide. White scale bar in (c) and (d) are 30 nm and 400 nm, respectively.
2.3 Effect of gap size
To study the effect of the gap size, the lateral forces with gradually changed gap sizes are calculated, with a left-handed circularly polarized beam incident onto a 90 nm particle, as shown in Fig. 3(a). With increased gap size, the absolute values of the lateral force and the gradient force both show a downward trend. As shown in Fig. 3(b), the energy flow difference of the waveguide also presents similar downward trend, which means that large gap size decreases the energy flow and corresponding lateral force. To further study the near-field effect of the gap size, two cases with gap size of 10 nm and 80 nm are compared in Figs. 3(c)–3(d). The results show that the near-field intensity in the gap region decreases as the gap increases [Fig. 3(c)], due to the fact that larger gap distance leads to lower coupling efficiency between the scattered light of particle and waveguide mode. Thus, a weaker near-field intensity in the gap generates a smaller gradient force, and the lower coupling efficiency results in decreased intensity of the excited waveguide mode [Fig. 3(d)] and smaller lateral recoil force. It should be noted that, for the incident light power of 100 mW and particle diameter of 90 nm, the depth of trapping potential well reaches -4.16KBT (KB is the Boltzmann constant, and T=300 K is the temperature) at the gap size of 10 nm calculated by previous method [34], which is sufficient to overcome the Brownian motion of particles.
Fig. 3. Influence of gap size on optical forces. (a) The optical lateral (black dotted line, Fx) and gradient (red dotted line, Fy) forces as function of gap size (from 10 nm to 80 nm). (b) The difference of energy flow of both sides of the waveguide with the change of gap size. (c) For gap size of 10 nm and 80 nm, the electric field distribution |Ey| in the gap region. The white dotted line represents the boundary of the waveguide. (d) Real part of electric field |Re(Ey)| in the xy plane for excited waveguide mode at the gap size of 10 nm and 80 nm. The two white dotted lines indicate the both boundaries of the waveguide. White scale bar in (c) and (d) are 50 nm and 400 nm, respectively.
2.4 Effect of waveguide curvature
Compared with light that can only travel in a straight line in free space, the optical signal can be transmitted to any different direction through the curved waveguide structure, which is very useful in on-chip optical signal transmission [35]. Consequently, it is important to investigate whether the particle still can be transported along the curved waveguide, when curvature of the waveguide is changed. Figure 4(a) shows schematic diagram of a bent waveguide composed of two straight waveguides in two sides and a quarter of circular waveguide in the center, and the gold nanoparticle is located near the upper or lower side of the curved part. Then, effects of the radius of the curved waveguide on the optical gradient and lateral forces are analyzed. Figures 4(c) and 4(d) show the relationship between the lateral/gradient force on the particle and the radius of the curved waveguide, for the particle near the lower or upper side of the curved waveguide, respectively. As the radius increases, it can be found that the absolute value of lateral force becomes larger for both sides of the waveguide, while the gradient force shows a slight decrease, and both forces tend to a stable value at very large radius just like the case of a straight waveguide. To further study the effect of the radius in near field, we consider the near-field electric field distribution of two cases with waveguide radius of 5 μm and 50 μm in Fig. 4(b). The comparison of the two cases shows that when radius of the waveguide is smaller, the electric field of the excited waveguide mode is weaker inside the curved waveguide. This is because in the case of smaller radius most part of the scattered light directly passes through the curved waveguide, leading to a lower coupling efficiency to the waveguide mode, just like the well-known bending loss of curved waveguides or fibers [36]. As a result, the lateral recoil force becomes larger with a curved waveguide. In Figs. 4(c) and 4(d), the change of lateral force is much less than an order of magnitude in the radius range of 1-150 μm and the gradient force is almost constant, indicating that the influence of the waveguide curvature is not strong to the optical forces. Thus, in such bend waveguide structures, the nanoparticle in both sides still can be trapped near the waveguide by the gradient force, and then transported along the curved waveguide by the lateral force. These results prove that our method can transport nanoparticles to arbitrary direction through the bend waveguides, and thus has significance in the research of optical on-chip nano-tweezers.
Fig. 4. Influence of the curvature of waveguide on optical forces. (a) Schematic diagram of a bent waveguide composed of two straight waveguides and a quarter of circular waveguide. The silica waveguide with a width of 300 nm, R is the radius of curvature of the inner ring, and the diameter of the gold particles is 90 nm. The incident light illuminates inward perpendicular to the structure. (b) The near-field electric field |Re(Ey)| distribution of two cases with waveguide radius R = 5 μm and 50 μm, respectively. The white dotted line represents the boundary of the waveguide. The upper right inset shows the position of the displayed near-field electric field include the right part of the waveguide. (c) The relationship between the lateral/gradient force on the particle and the radius of the curved waveguide, for the particle located in the lower side of the curved waveguide. (d) Same as (c) except the particle located in the upper side of the curved waveguide.
2.5 Particle rotation with ring-shaped waveguide
The spin-orbit coupling of light has been proved to have the capability of rotating nanoparticles in some optical tweezers systems [37]. Here we also study the circular motion of nanoparticles driven by the lateral recoil force in an on-chip waveguide system. Figure 5(a) shows a three-dimensional schematic diagram of the structure, including a ring-shaped silicon waveguide with a cross-section size of 300 nm × 300 nm and an inner radius of 3 μm on a silica substrate. Two cases of particle located at the inner and outer side of the waveguide are considered as shown in Fig. 5(d), where θ is the angle of particle position in polar coordinates. Figures 5(b) and 5(e) show the optical forces in x and y directions as function of angle θ for the two cases of inner and outer particles, respectively. When a right-circularly polarized light is vertically incident onto the entire structure, we find that the particles in both sides of the ring-shaped waveguide are subjected to varying forces in x and y directions depending on the angle θ. As θ changes from 0 to 360o, the particle position rotates once around the circular waveguide, and the forces Fx and Fy tend to change in a sine or cosine shape. For θ=0, Fx and Fy correspond to the gradient force and the lateral force, respectively, but for θ=90o Fx and Fy denote the lateral force and the gradient force, respectively. Furthermore, we transform the Cartesian coordinates to polar coordinates, and thus the forces Fx and Fy are converted to the forces in the radial direction (Fρ) and angular direction (FΦ), as shown in Figs. 5(c) and 5(f) for inner and outer particles, respectively. It can be found that, in the polar coordinates, the particle is always subjected to stable forces in the radial and angular directions, which represent the gradient force and lateral force, respectively. Obviously, the radial gradient force is always greater than the angular force, which means the particle will first be attracted towards the waveguide by the stronger gradient force, and then rotated along the ring-shaped waveguide clockwise (inner particle) or counterclockwise (outer particle) by the lateral force. It is worth noting that the rotation of particle due to the spin-to-orbit conversion usually keeps a fixed direction (clockwise or anticlockwise) depending on the charity of incident light [38,39]. Here, the particles can rotate both clockwise and counterclockwise due to the opposite chiral symmetries formed by the particle in two sides of the ring-shaped waveguide, which would deepen the understanding of the spin-orbit coupling of light. In addition, when the radius of the ring waveguide changes, the changing trend of both the lateral force and the gradient force for the inner and outer particles is same as that in Fig. 4(c) and Fig. 4(d), since the change of radius actually modifies the curvature of waveguide near the particle.
Fig. 5. Force analysis of an on-chip circular waveguide-particle coupling structure. (a) The three-dimensional schematic diagram of the structure, including a ring-shaped silicon waveguide (red color) with an inner radius of 3μm on a silica substrate. A right-hand (blue color) spin polarized light is incident vertically along -z direction to the silicon waveguide and the nearby gold nanoparticles in water. The particles on different sides of the waveguide are subjected to lateral recoil forces (purple arrow) in opposite directions. (b) The relationship between the optical forces in x and y directions and the angle θ for the case of inner particle. (c) The forces in the radial direction (ρ) and angular direction (Φ) for the case of inner particle in polar coordinates. (d) Schematic diagram of the ring-shaped waveguide (red color) in xy plane, indicating the angle θ and the lateral force direction on inner and outer particles. (e) Same as (b) except for the case of outer particle. (f) Same as (c) except for the case of outer particle.
3. Conclusion
In this work, we propose and numerically study a new type of waveguide optical tweezers based on spin-orbit coupling of light, with great advantages including no requirement of light focusing on one end of waveguide, large trap area of particles, and flexible adjustment. The spin-orbit coupling occurs in the waveguide-particle structure, resulting in a unidirectional propagation mode of the waveguide and generating a lateral recoil force in the opposite direction on the particles. A strong electric field of gap mode is formed between the waveguide and nearby particles, which exerts a strong optical gradient force on the particles to ensure that the particles are trapped near the surface of waveguide. Even for a curing or circular waveguide, the particles still can be transported and rotated along the waveguide under the effect of gradient and lateral recoil forces. This work strengthens the manipulation capbility of waveguide optical tweezers technology, and shows great significance for the precise manipulation, detection and sorting of nanoparticles. In addition, the waveguide optical tweezers has potential applications in many fields, such as particle manipulation in microfluidic chips [40], particle detection and sorting [41], or drug delivery [42] to designated locations in nanomedical researches.
Funding
National Natural Science Foundation of China (NSFC) (91750205, U1701661, 61935013, 61975128); Leading Talents of Guangdong Province Program (00201505); Natural Science Foundation of Guangdong Province (2016A030312010, 2019TQ05X750, 2018A030310553); Science and Technology Innovation Commission of Shenzhen (JCYJ20180507182035270, KQTD2017033011044403, ZDSYS201703031605029, JCYJ20180305125418079, JCYJ20190808140609410).
Disclosures
The authors declare no conflicts of interest.
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