1. Introduction

All-optical trapping and manipulation of small objects using far-field techniques have shown powerful application perspectives in biochemical and life sciences [1,2]. As the size of the objects to be trapped enters the nano-scale, high numerical aperture focusing and large incident power are required to create sufficient field gradient force overcoming the Brownian motion. This also renders damage in the target object, e.g., biological samples, because of the intense heat generation [2]. Recently, with the help of surface plasmons (SP), plasmonic nano-optical tweezers (PNOTs) based on metal nano-gap [3], disk [4], antenna [5,6], pillars [7,8], strips [9,10], nano-ring [11], optical lattices [12,13], and coaxial aperture [14] etc have been proposed and demonstrated to provide strong trapping force with low incident power densities due to the field localization and enhancement. Trapping of nano-objects in a non-diffraction limited regime with ultra-high accuracy has become accessible [15].

However, the main emphasis of reported efforts is on the exploration of new types of PNOTs [4,5,7–13], and strengthening of the near-field force to achieve trapping with high degree of confinement [3,6,14]. On the other hand, not much has been reported in the literature on nano-particle (NP) manipulation using the plasmonic effect. Wang et al. have developed two schemes to address this issue [8,9]. In the first approach, manipulation of the trapped NP around a gold nano-pillar was achieved by rotating the incident polarization [8]. In the second one, they controlled the particle position by tuning the relative intensity of two input beams that excited counter-propagating surface plasmon polaritons [9]. Nonetheless, all-optical transference of target objects in a predefined pathway still remains challenging. Very recently, a series of C-shaped plasmonic structures with different size or orientation has been proposed and demonstrated to behave as nano-optical conveyor belt, which can achieve the transport of target in a peristaltic fashion by controlling the resonant wavelength or rotating the incident polarization [16,17].

The key concept is to create mobile trapping potential wells in the near-field, thus the trapped particle can be manipulated along with the movement of the potential well. Previously, plasmonic structures with graded property have been proposed to localize the SP at different position for different frequencies, therefore realizing the plasmonic “Rainbow” trapping effect [18–21]. The operation window is considerably broad ranging from visible to THz. The graded plasmonic effect can be implemented by changing one of the geometry parameters.

In this article, we propose the use of plasmonic nano-disks (NDs) having graded diameters [see Fig. 1(a)] as nano-optical conveyor belt for delivering the target in a controllable manner. Simulation studies conducted using the three-dimensional (3D) finite-difference time-domain (FDTD) technique and the Maxwell stress tensor (MST) method demonstrate the feasibility of scaling such manipulation from µm to nm level. The metal is silver and target is gold NP. Our design is capable of controlling the target position continuously via rotating the polarization and switching the wavelength of incident light. Because of its simplicity, the proposed device is easy to be fabricated [22]. While manipulation of target objects in the nano-scale is clearly an important attribute for many domains within the field of bionanotechnology, we anticipate that our design holds promising potential for biochemistry and biophysics applications.

 

Fig. 1 (a) Schematic of our nano-optical conveyor belt based on a couple of graded silver NDs. (b) Normalized extinctions of ND₁ (D₁ = 150 nm, red line) and ND₂ (D₂ = 100 nm, blue line), respectively.

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2. Modeling and optical forces calculations

In our modeling study, all NDs have a thickness of 35 nm and are fabricated on silica (n = 1.5). The environment is water (n = 1.33), and the target is a gold nanoparticle (Au-NP) of diameter 30 nm, while practically the fluorescent molecule labeled target should be used for tracking the trajectory of target [8]. The choice of parameters is based on practical considerations. The light source (plane wave) illuminates the device from the top [along -Z axis, Fig. 1(a)] with X-polarized electric field. Perfect matching layers are used as absorption boundary. The relative permittivity of Ag is taken from reported data [23].

Figure 1(b) shows the normalized extinction characteristics of ND₁ (diameter D₁ = 150 nm, red line) and ND₂ (D₂ = 100 nm, blue line), respectively. Their resonant wavelengths are 775 and 622 nm, which are well separated from each other for the purpose of avoiding mutual disturbance between NDs. The respective full width at half maximum (FWHM) are 248 and 148 nm, indicating that decreasing the ND size will lead to higher Q-value and hence stronger resistance towards mutual disturbance. Moreover, it is worth pointing out that wavelengths such as 785 and 633 nm, etc., according to the practical availability of light sources, can also be chosen in the design by varying the size of NDs.

Figures 2(a)-2(b) present the distribution of the optical force F_xz in the XZ plane as experienced by the Au-NP for λ = 775 nm and 622 nm, respectively. It is clear that the optical force is significant around the resonant ND, and all resultant force vectors point towards the edge of the ND. In the non-resonant ND, we also see a trace amount of trapping force due to the cross-talk effect between the two NDs. In addition, field coupling between the target particle and the near-field of NDs is also present in the system. Such field coupling has been included in our MST calculation in order to achieve better accuracy of the final calculated optical force. Our results clearly demonstrate the wavelength-addressable character of the NDs. Consequently, if we sequentially excite the NDs by varying wavelength and polarization direction (to be demonstrated in the next section) of the incident beam, the overall effect of the optical force will always bring the target to a desired edge area of the ND where the optical field is most enhanced (i.e., hot-spot).

 

Fig. 2 Distribution in the XZ plane of the optical force F_xz exerted on an Au-NP with diameter of 30 nm for excitation wavelength of (a) 775 and (b) 622 nm, respectively. The incident light has X-polarization. The NDs are schematically shown in grey. The gap between adjacent NDs is 100 nm.

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3. Optical potentials and rotational forces

Figure 3(a) shows the electric field intensity distribution in an XY plane at 10 nm above the NDs for λ = 775 nm and 622 nm. As one can see, plasmonic resonance of ND₁ at λ = 775 nm provides strong field localization along the edge of ND₁. Meanwhile, weak field enhancement also occurs around ND₂ due to the cross-talk effect. Similar phenomenon is also present for the case of λ = 622. We take the common notation that the location of maximum intensity refers to a position where the trapping potential is strongest. Figures 3(b)-3(c) display the MST calculated force components F_x (blue ‒○‒) and F_z (red ‒□‒) experienced by the Au-NP along the X axis (Y = 0) at a distance of 10 nm above the NDs for λ = 775 nm and 622 nm, respectively. On the one hand, the horizontal force component F_x varies significantly over the resonant ND. The zero crossings of F_x indicate that the Au-NP will be dragged to the edge of resonant ND because of the intense field localization there. On the other hand, the vertical force component F_z remains negative and reaches a maximum at the edges. Therefore, the overall force will finally lead the target being trapped to the edge of ND which is in resonance.

 

Fig. 3 (a) Electric field intensity distribution at 10 nm above the NDs under illumination by 775 and 622 nm light. (b)-(c) MST calculated force components F_x (blue ‒○‒) and F_z (red ‒□‒) exerted on an Au-NP locating above the NDs with a 10 nm gap as a function of position along the X axis (Y = 0) for λ = 775 nm and 622 nm, respectively. (d) Trapping potential U_x as a function of position along the X axis (Y = 0) for λ = 775 nm (red ‒○‒) and 622 nm (blue ‒○‒), respectively. The incidences have X-polarization. The NDs are schematically shown in grey at the bottom. The gap between adjacent NDs is 100 nm.

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The trapping potential corresponds to the work needed to move a target particle from a location directly above the structure to infinity [3,16]. Figure 3(d) shows the potential results as we integrate F_x along the X axis to a position far away from the edge of NDs where F_x almost vanishes. For the case of λ = 775 nm [red ‒○‒, Fig. 3(d)], ND₁ is at resonance, and the trap depth reaches a maximum of ~230 k_BT/W/µm² around its edge. At the same time, the trapping potential is insignificant above the non-resonant ND₂. While for the case of λ = 622 nm [blue ‒○‒, Fig. 3(d)], the maximum trapping potential moves to ND₂, whose depth reaches a level of ~150 k_BT/W/µm² around the edge. However, a trace potential well due to cross-talk (i.e. spectral overlap) is also present in a region directly above ND₁. This potential has a depth of ~60 k_BT/W/µm², and is resulted from field enhancement around ND₁, despite being in a non-resonant state. The wider FWHM of ND₁ as compared to that of ND₂ means that excitation under λ = 622 nm will also give rise to a certain amount of field localization and enhancement around ND₁. This trace potential well is not sufficient to influence the delivery of the trapped target from ND₁ to ND₂. For example, if we use a laser power density of 10 mW/µm², the non-resonant field enhancement of ND₁ only results in a trap depth of 0.6 k_BT, which is not sufficient to provide any trapping effect. Consequently, one can readily see that by switching the excitation wavelength from 775 to 622 nm, the trapped target will be released from the left edge of ND₁ and then trapped by the potential well at the right edge of ND₂, where the trap depth (1.5 k_BT) and is larger than the kinetic energy of Brownian motion k_BT.

Because of their circular symmetry, the NDs have the ability of moving the target along its edge by changing the polarization angle [8]. Figure 4 shows the overall force vectors F_xy, F_xz, F_yz as a function of incident polarization direction for λ = 775 nm and 622 nm, respectively. As the incident polarization being rotated anticlockwise from a starting direction along the X axis (i.e. at zero degree) to 90°, the force in the Y direction is no longer zero. Consequently, the trapped target particle will be dragged along the edge of the ND in the XY plane in anticlockwise direction. Also the weak polarization dependence of F_xz and F_yz ensures that the potential well is always sufficient to keep the target in a trapped state. Overall, by rotating the polarization one can simply move the potential well along the edge of ND.

 

Fig. 4 The overall force vectors F_xy, F_xz, F_yz versus the incident polarization direction for (a) λ = 775 nm and (b) λ = 622 nm, respectively. The Au-NP is elevated 10 nm above the NDs, and located at X = 200 nm in (a) and −50 nm in (b). The gap between adjacent NDs is 100 nm.

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4. Discussion

The nano-optical transportation of target particle within the system can be realized through the following approach. First, by using λ = 775 nm illumination the nearby target can be trapped to the edge of ND₁. Then by slowly rotating the polarization direction the target will move around the edge until it is at a location adjacent to ND₂. Second, the target is released by ND₁ and then trapped by ND₂ by switching the wavelength from λ = 775 nm to 622 nm, and the time needed for switching the trapping force from one ND to another is on the order of 10 femtoseconds, according to the plasmon relaxation time [24]. In this process the target will readily move into the trapping region of ND₂. Consequently, manipulation of target particle arbitrarily between ND₁ and ND₂ can be controlled in the same manner, offering target confinement as well as movement at the nano-scale.

This scheme may have extensive potential applications. First, multifunctional operations are possible as the target (molecule, organism, or NP) could be manipulated to different locations for specific treatments, e.g. to perform a biochemical reaction or detection (e.g., Raman and fluorescence spectroscopy). Second, a long conveyor belt is contemplated as one duplicates the pair of graded NDs periodically in the X direction. Therefore a long working distance is realized without having to introduce any mechanical movement of optical components. Instead, one only needs to perform controlled switching of resonant wavelengths and rotation of the polarization angle.

On the other hand, in a practical implementation one should also take consideration that the photothermal effect resulted from light absorption as the heat will lead to thermal convection and thermophoretic forces which may compete with the optical trapping force [25,26]. The temperature around the plasmonic structure will increase inevitably, which results in a large temperature gradient in the surrounding. For the disk case, the resultant fluid convective velocity hardly exceeds 10 nm/s [26], which corresponds to a drag force of 2.8 × 10⁻³ fN on the target particle according to Stokes’ law [5]. This force is not significant as compared to the optical force achieved by the NDs [~70 fN at 1 mW/µm², see Fig. 3(b)]. Therefore, thermal convection is not expected to contribute significant disturbance to the optical trapping performance of our design. The thermophoresis effect is more complicated as it sometimes opposes the trapping and sometimes not [26]. Generally, low incident power density is used in order to minimize thermal effects. However, one could consider fabricating the NDs on a thermally conducting material such as chemical vapor deposited diamond or aluminum nitride for efficient heat removal from the substrate. Indeed the photothermal problem is a common issue in the field of optical trapping. In addition, it should be mentioned that the incorporation of a heat sink, e.g. Au/Cu configuration [8,17], may modify the resonant wavelength of each ND in the final design. The dimensional parameters of the system will have to be re-calculated accordingly. Another method to avoid such change is the use of transparent substrate with high thermal conductivity such as diamond.

5. Conclusion

To sum up, we have proposed a nano-optical conveyor belt design based on graded silver NDs. The position of the hot-spots, as well as the target can be changed continuously by switching the incident beam wavelength and rotating the polarization. 3D FDTD technique and MST method are used to verify the feasibility of such function. Our design provides an alternative way for transporting targets within a nano-scale region, and is promising for biochemistry and optofluidic lab-on-a-chip applications.