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Abstract
In this paper, a plasmonic trapping scheme including a polystyrene nanoparticle with gold cap and a metal tip tweezers was proposed. We numerically investigated the optical trapping behavior of the metal tip to this asymmetric particle. The results show that the metal tip can capture the particle at the position of the gold cap due to the strong plasmonic interaction, while other positions of the particle cannot be captured by metal tip. Furthermore, the trapping angle of the nanoparticle can be adjusted by changing the incident wavelength. Precisely controlling the trapping angle of the nanoparticles in our study has important potential applications of optical tweezers, such as in single molecule manipulation.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since Ashkin reported optical tweezers in 1987 [1], this technology has been widely applied in many fields such as physics and bio-science because of its noninvasive capture and capability to manipulate micron particles [2–4]. But conventional optical tweezers still remains some problems as limited trapping gradient force and sample damage in high power irradiation, which has blocked the nanoscale application of the technology. In order to overcome these obstacles, a new technology based on localized surface plasmon resonance (LSPR) was put forward in recent years [5]. Exploiting surface plasmon resonances supported by nanostructures, the plasmon optical tweezers can focus light beyond the diffraction limit and the gradient force of evanescent fields can be increased significantly.
So far, metal nanostructures for optical tweezers has been widely studied. These designed structures can generate a much stronger optical gradient force than conventional optical tweezers. In 1997, the stable optical trapping of dielectric particles by a metal tip was firstly studied by Lukas Novotny et al. [6]. In 2009, using a nanoaperture in a metal film, R. Quidant et al. demonstrate experimentally trapping of a single 50 nm polystyrene sphere and a self-induced-back-action (SIBA) effect was put forward [7]. In 2010, Weihua Zhang et al. reported successful trapping of 10 nm metal particles in the gap of plasmonic dipole antennas, which indicates the potential to trap nanoparticles sized only a few nanometers [8]. In 2011, K. Sasaki et al. reported the experimental trapping of 350-nm-sized nanoparticles by a gold nanoblock pair array with 200 W/cm2 laser intensities [9]. Then, using Double-Nanoholes in a Gold Film, R. Gordon et al. achieved the optical trapping of 12 nm dielectric spheres [10]. Except for the trapping of dielectric and metal particles with spherical symmetry, Aporvari et al. reported the optical trapping and angular control of a dielectric nanowire by a single nanoaperture in a low incident power [11]. Zhe Shen et al. also studied the optical trapping of two or/and more particles [12]. For many body trapping, another geometry can be optical pinning where dense area of metal coated nano-needles trap nano-objects or are pinned to them by outside light beam [13].
Polystyrene (PS) nanoparticles have been widely used for various applications, such as in nanobiotechnology [14–16] and in photonics [17]. In the study of optical tweezers, polystyrene nanoparticles are often used as targets for the optical trap. Precise and flexible manipulation of polystyrene particles by optical trap is very necessary for practical application. For example, in single molecule manipulation experiment [18], we need to manipulate biological macromolecules through trapping the polystyrene particles attached to them. In this paper, we designed a plasmonic trap scheme based on asymmetric particles, which can precisely control the trapping angle of the polystyrene particles. Janus particles, composed of two fused hemispheres of different substances, were widely investigated for its different trapping characteristics [19]. A novel kind of Janus particles proposed by ref [20], can be not only stably trapped by optical tweezers but also displaced controllably along the axis of the laser beam due to optical and thermal forces. A self-propelled cyclic round-trip motion of Janus particles was also systematically investigated [21]. A polystyrene particle with a gold cap, which is called nanocap [22], is proposed in this work. This type of asymmetric particle can be easily synthesized by chemical methods [23,24]. It is kind of Janus particles. We theoretically investigate the optical trapping of metal nanotip to this asymmetric particle. The results show that the metal tip can capture the particle at the position of the gold cap due to the strong plasmonic interaction, while other positions of the particle cannot be captured by metal tip. Furthermore, the trapping angle of the nanoparticle can be adjusted by changing the incident wavelength. Precisely controlling the trapping point of the particles in our study has important potential applications of optical tweezers, such as in single molecule manipulation.
2. Numerical method
The trapping configuration in our work is demonstrated in Figs. 1(a) and 1(b). We define angle rotating around Y axis as θ and around Z axis as ϕ. We firstly set the angle ϕ fixed as 0° (by default) and investigate the different optical trapping behaviors by changing the angle θ as shown in Figs. 1(a) and 1(b). In the later part of this paper, trapping behavior of this Janus particle involving both of the angles will be discussed. The cylindrical Au tip was put along the X axis, the radius of the apex was set as 10 nm, and the cone angle was 25°. The diameter of PS particles was chosen as 100 nm. The gold nanocap has a thickness of 12.5 nm and full open-angle of 90°. The nanocap is well facing the tip (θ = 0°) with a distance of 2 nm as shown in Fig. 1(a), when the PS is at the origin (x, y, z = 0) and the brim of nanocap is nearest to the tip (θ = 45°) as shown in Fig. 1(b). The modeled structure was immersed in water (refractive index of 1.33). A total field scatter field plan wave with an X direction polarization is incident from the bottom (E vector along with X direction and k vector along with Z direction) to excite longitudinal LSPR of the metal tip and its power intensity is set to be 10mw/um2.
Fig. 1. The schematic of the PS-nanocap-tip trapping system in the rectangular coordinate system with θ = 0° and ϕ = 0° (a) and θ = 45° and ϕ = 0° (b). The tip radius r = 10 nm, cone angle = 25°. The radius of PS R = 50 nm, cap thickness t = 12.5 nm, and full open-angle is 90°. The dimer distance d = 2 nm.
The finite-difference time-domain (FDTD) method [25] was employed in this work. The refractive index of PS was set to be 1.57 and the dielectric permittivity for the gold tip and nanocap was chosen from ref [26]. The mesh size for override region in the 3D FDTD simulation was 1 nm for a relatively accurate and efficient calculation. Near the 2 nm gap, the mesh grid size along the X direction was 0.2 nm for an accuracy consideration, which means the general mesh cell was non-uniform. For the optical force calculation, both Maxwell stress tensor (MST) method [27] and analytic Lorenz force definition approach [28–30] are widely applied. MST method derives results by calculating the Maxwell stress tensor (MST) and integrating it on a closed box composed of six 2D monitors, surrounding the particle. While Lorenz force definition method gets optical force by volumetrically integrating total Lorentz force per unit volume on trapped object [29]. Herein, an MST method is used. It was reported that there is an ambiguity in the placement of the monitors across the interfaces for this approach [30]. Thus in this work, we carefully ensured that calculation enclosure should only be around the object and not cross any other surface. The time-averaged force acting on the center of mass of the designed nanoparticle can be determined by:
3. Results and discussion
Figure 2(a) shows the extinction cross section calculated of the designed structure using the FDTD method. There was a resonance peak of the extinction spectrum around 717 nm for θ = 0° and an 817 nm resonance peak for θ = 45°. Figures 2(b) and 2(c) show the simulated X-Z plane spatial distribution of the electric field in the irradiation of X-polarized light of 717 nm for θ = 0° and 817 nm for θ = 45° in water respectively. It is obvious to find that there was a large electric field enhancement near the gap in both schemes because of the plasmonic interaction. It is known that different plasmon hybridization modes will be excited when two mismatched nanoparticles are in close proximity to each other [31]. For our optical force calculation, we only focus on the strongest peak of the spectra at 717 nm (θ = 0°) and 817 nm (θ = 45°), which correlate to the dipole-dipole plasmon mode. A median number incident wavelength at λ = 767 nm was also considered for comparison.
Fig. 2. (a) Extinction cross section calculated for the designed structure. Simulated X-Z plane spatial distribution of the electric field in the irradiation of x-polarized light of 717 nm, for angle θ = 0° (b) and 817 nm, θ = 45° (c).
We firstly investigate the optical forces and the optical potential imposed on the asymmetric nanoparticle when the particle moves towards the metal tip along with the two representative directions: Y direction which is perpendicular to the k vector of the incident light and Z direction which is parallel to the k vector of the incident light. Figures 3(a)–3(c) illustrate the evolution of lateral optical force Fy acting on the nanoparticle when it just moving along the Y direction (0, y, 0) with different θ at incident wavelength λ = 717 nm, 767 nm, and 817 nm, respectively. One can clearly see that the lateral optical forces are acting like restoring force for angles θ = 0° and θ = 45° (black solid line and red solid line) and the value of Fy comes to the order of pN. But for other values of θ (shown as a dotted line and dashed line), the optical forces are very small, which indicates the nanoparticle could not be trapped at these angles. It is also obvious that at λ = 717 nm the optical force imposed on the nanoparticle for angles θ = 0° is larger than that for θ = 45°, at λ = 767 nm the optical force imposed on the nanoparticle for angles θ = 0° comes close to that for θ = 45°, and at λ = 817 nm the optical force for angles θ = 0° is smaller than that for θ = 45°. We also examine the near field distribution in X-Z plane at a fixed incident wavelength λ = 717 nm. The large electric field enhancement in Figs. 3(d) and 3(e) indicates the strong interaction between the nanocap and the metal tip due to the small gap for angles θ = 0° and θ = 45°. In these two cases, the optical force imposed on the nanoparticle is expected to be a larger one. While seen from Figs. 3(f)–3(h), the electric field enhancement is comparably smaller as the distance between the nanocap and metal tip increases. In these angles (θ = 90°, 135° and 180°), the optical force imposed on the nanoparticle is expected to be a smaller one.
Fig. 3. The evolution of lateral optical force Fy acting on the nanoparticle when it just moving along the Y axis (0, y, 0) with different θ at incident wavelength λ = 717 nm (a), 767 nm (b), and 817 nm(c). The simulated near field distributions in X-Z plane at λ = 717 nm with θ = 0° (d), θ = 45° (e), θ = 90° (f), θ = 135° (g) and θ = 180° (h).
We then calculate the trapping potential in these three wavelengths by integrating the optical force. The optical trapping potential Uy imposed on the designed nanoparticle with different angles θ is presented in Fig. 4. The trapping energy is normalized with the kinetic energy of Brown motion (KBT). In Fig. 4(a), the black solid line shows that at λ = 717 nm the trapping potential well depth for θ = 0° is close to 14 KBT, larger than that for θ = 45° (4 KBT, red dashed line). In Fig. 4(b), at λ = 767 nm the trapping potential well depth for θ = 0° is about 7 KBT, comes close to that for θ = 45° (12 KBT). In Fig. 4(c), at λ = 817 nm the trapping potential well depth for θ = 0° decreases to about 5 KBT. But for θ = 45° it increases to more than 14 KBT, which is larger than that for θ = 0°, on the contrary. As we know, the stable optical trapping requires a potential well depth of 10 KBT to overcome the particle's Brownian motion [5]. One can clearly see that for θ = 90°,135°,180°, the optical potential well depth is far too small to trap the nanoparticle. At the three incident wavelengths, the nanoparticle can only be trapped at the θ = 0° and θ = 45°. When the incident wavelength is at λ = 717 nm, the trapping potential well depth for θ = 0° is the deepest and the nanoparticle tends to be trapped at θ = 0°. While at λ = 817 nm, the trapping potential for θ = 45° is the deepest and the nanoparticle tends to be trapped at θ = 45°. It means that we can control the trapping angle of the nanoparticle by adjusting the incident wavelength.
Fig. 4. The optical potential Uy with different θ at incident wavelength λ = 717 nm (a), 767 nm (b), and 817 nm (c). Only when θ = 0° and θ = 45° the potential well depth can be larger than 10 KBT and for the other angles the potential well depth is too small to stably trap the nanoparticle.
We then discuss the longitudinal optical force Fz when the nanoparticle approaching metal tip along the Z direction. Figures 5(a)–5(c) show the evolution of lateral optical force Fz acting on the nanoparticle when it just moving along the Z direction (0, 0, z) with different θ at incident wavelength λ = 717 nm, 767 nm, and 817 nm, respectively. We can see that the overall behavior of Fz is similar to that of Fy. The optical forces for θ = 0° and θ = 45° have a form of restoring force and their values are comparably larger than other angles. It is seen that at λ = 717 nm the optical force imposed on the nanoparticle for angles θ = 0° is larger than that for θ = 45°, and on the contrary, at λ = 817 nm the optical force for angles θ = 45° is larger than that for θ = 0°. However, there are some differences comparing with the trapping behavior of Fy. We can see that at larger angles (θ = 90°, 135° and 180°) Fz increases and becomes optical pushing force. In addition, the two edges of force curves of Fy for θ = 45° also become optical pushing force. The total optical force of the nanoparticle is the sum of gradient force and scattering force. The gradient force, which leads to the optical pulling force of asymmetric particle, is mainly contributed by the strong field enhancement caused by plasmon interaction. The scattering force causes the optical pushing force of the asymmetric particle. As the propagation direction of incident light is along the Z direction, the gold nanocap will scatter strongly which leads to the increase of the optical pushing force in the Z direction. For larger angles (θ = 90°, 135° and 180°), the pulling gradient forces are comparably small and the optical pushing forces are totally larger than pulling forces, which leads to a total pushing force. For θ = 45°, the optical pushing force is smaller than pulling gradient force only when the nanoparticle is close to metal tip. As long as the distance between the nanoparticle and metal tip increases and pulling gradient force decreases, the pushing force becomes larger and a total pushing force appears.
Fig. 5. The evolution of longitudinal optical force Fz acting on the nanoparticle when it just moving along Z axis (0, 0, z) with different θ at incident wavelength λ = 717 nm (a), 767 nm (b), and 817 nm (c). And the corresponding optical trapping potential for λ = 717 nm (d), 767 nm (e), and 817 nm (f). The other angles were not shown as they have no potential well.
In the corresponding optical potential Uz imposed on the nanoparticle, as shown in Figs. 5(d)–5(f), the effects of the pulling force can be easier to see. For larger angles, there are no potential wells, the relative optical potential only for θ = 0°, 45° at three different incident wavelengths is illustrated. One can clearly see from Figs. 5(d)–5(f) that for θ = 45° (red solid line) the nanoparticle will be pushed quickly to the center when it approaches metal tip from minus z-direction, if the potential well depth is not large enough, it is possible that the nanoparticle is pushed out from the trapping center. Now we examine the optical potential well depth in the Z direction. At λ = 717 nm the trapping potential well depth for θ = 0° (as shown in Fig. 5(d), black solid line) is about 12 KBT, evidently larger than that for θ = 45° (about 1 KBT, red dashed line). At λ = 767 nm the trapping potential well depth for θ = 0° (as shown in Fig. 4(e), black solid line) is about 6 KBT, almost equals to that for θ = 45° (red solid line). At λ = 817 nm the trapping potential well depth for θ = 0° decreases to less than 5 KBT (as shown in Fig. 5(e), black solid line). But for θ = 45° (red solid line) it increases to more than 10 KBT, which is larger than that for θ = 0°. It is seen that for θ = 0°, 45° the nanoparticle can be stably trapped. When the incident wavelength is at λ = 717 nm (λ = 817 nm) the trapping potential well depth for θ = 0° (θ = 45°) is the deepest and the nanoparticle tends to be trapped at θ = 0° (θ = 45°). When the incident wavelength is at λ = 767 nm, θ = 0° and θ = 45° schemes share the same trapping potential well depth and the nanoparticle tends to oscillate between θ = 0° and θ = 45°. The corresponding optical trapping potential for x direction is also calculated, as shown in Fig. 6. The optical potential imposed on the nanoparticle behaves similarly among the three translational directions, and this can lead to precise control of trapping angel θ through changing the incident wavelength λ. Comparing with the traditional optical trapping of the Janus particle, our design allows metal tip to trap the particles at more precise locations (at the metal cap), however, the optical trapping of particles by metal tips is localized and limited to 50-100 nanometers.
Fig. 6. The calculated optical potential Ux exerting on PS nanoparticle when moving along X axis at incident wavelength λ = 717 nm (a), 767 nm (b), and 817 nm (c).
Now we examine the optical force and optical potential imposed on the PS particle (with no gold cap). Figure 7 shows the lateral optical force Fy (a), longitudinal optical force Fz (c), and the lateral optical force Fx (e) acting on PS at incident wavelength λ = 717 nm, 767 nm, and 817 nm and the corresponding optical potential Uy (b), Uz (d), and Ux (f). It is clear to see that the values of the optical force and optical potential imposed on the PS particle decrease by 1∼2 order of magnitude, compared with that on designed nanoparticle (with gold cap) for θ = 0° and θ = 45°. It means the PS particles with diameter 100 nm cannot be stably trapped by the metal tip. The introduction of the gold cap increases the interaction between target nanoparticle and metal tip, which makes it capable to be trapped.
Fig. 7. (a) Lateral optical force Fy acting on PS nanoparticle when it moves along Y axis at incident wavelength λ = 717 nm, 767 nm, and 817 nm. (b) The corresponding optical potential Uy. (c) Longitudinal optical force Fz acting on PS nanoparticle when it moves along Z axis at the three incident wavelengths. (d) The corresponding optical potential Uz. (e) Lateral optical force Fx acting on PS nanoparticle and (f) the corresponding optical potential Ux.
Actually, the system under investigation is noncentrosymetric and has only one rotation symmetry axis—x axis, which means it has five degrees of freedom, three translational ones and two rotational (about y and z axes). Now we would like to add an another rotational degrees of freedom, the angle ϕ around z axes. One can carefully notice that in our configuration, only small angle schemes (θ = 0°, 45°; ϕ = 0°) can be stably trapped. Considering the confined trapping angle and the rotation symmetry, only one case remains interesting—designed nanoparticle with θ = 45° and ϕ = 45°. Its trapping behavior was shown in Fig. 8. It can be seen that the largest optical pulling force was comparably less than that of schemes θ = 0°, ϕ = 0°, and θ = 45°, ϕ = 0°. From Figs. 8(b) and 8(f), it is shown that the optical potential well depth in Y direction and X direction is less than 10 KBT. For Fz, as shown in Fig. 8(c), appealing forces can be observed only when incident wavelength λ = 767 nm and 817 nm and the corresponding potential Uz shown in Fig. 8(d) is no more than 2 KBT. Thus, the scheme with θ = 45° and ϕ = 45° cannot be stably captured. However, in the following discussion, we find that the optical torque and force acting on the nanoparticle in this situation will make it rotate and come into the stable potential well, attaining the stable trapping.
Fig. 8. (a) Lateral optical force Fy acting on designed asymmetric nanoparticle with angle θ = 45° and ϕ = 45° when it moves along Y axis at incident wavelength λ = 717 nm, 767 nm, and 817 nm. (b) The corresponding optical potential Uy. (c) Longitudinal optical force Fz acting on the designed nanoparticle when it moves along Z axis at the three incident wavelengths. (d) The corresponding optical potential Uz. (e) Lateral optical force Fx acting on the designed nanoparticle and (f) the corresponding optical potential Ux.
We calculate the optical forces acting on the asymmetric nanoparticle with angle θ = 45° and ϕ = 45° as shown in Fig. 9. The particle location is at the origin. The red cube represents MST analysis group. The optical forces in only two planes was illustrated for easier observation and other planes seeing very small forces. The blue arrows show the optical force around the particle in X-Z plane when the incident wavelength is at λ = 817 nm and angle θ = 45° and ϕ = 45°. The main torque can be easily seen by guide of the white arrows. In the trapping progress, when nanoparticle approaching trapping area, it will experience an optical torque. The optical torque and force acting on the nanoparticle will make it rotate and come close to the metal tip, achieving the stable trapping.
Fig. 9. The optical force distribution on the asymmetric nanoparticle with angle θ = 45° and ϕ = 45°. The red cube represents MST analysis group and the white arrows indicate main in-plane forces.
4. Conclusions
In this study, we have proposed a spherical-symmetry-breaking particle and numerically investigated the optical trapping behavior of the metal tip tweezers to this asymmetric particle. The metal tip can capture the particle at the position of the gold cap due to the strong plasmonic interaction, while other positions of the particle cannot be captured by metal tip. Furthermore, the trapping angle of the nanoparticle can be adjusted by changing the incident wavelength. Our study provides a method of capturing nanoparticles with precise angular selectivity.
Funding
National Natural Science Foundation of China (11304164, 11504185, 61178004); Fundamental Research Funds for the Central Universities; Natural Science Foundation of Tianjin City (06TXTJJC13500); Science and Technology Commission of Tianjin Binhai New Area (BHXQKJXM-PT-ZJSHJ-2017003); National Science Fund for Talent Training in Basic Sciences (J1210027).
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