Optical trapping of nanoparticles with tunable inter-distance using a multimode slot cavity

Lin Wang; Yongyin Cao; Tongtong Zhu; Rei Feng; Fangkui Sun; Weiqiang Ding

doi:10.1364/OE.25.029761

1. Introduction

Since the first demonstration of optical trapping near the focus of a single light beam [1], optical trapping with the merits of contactless and scatheless has boosted many applications in physics [2], photonics [3–6], and biologics [7], etc. In traditional optical tweezers, the trapping precision is mostly determined by the spot size of the focus, which is usually obtained by using a high numerical aperture (NA) objective, and many efforts have been devoted to decrease the spot size, such as using vector beams [8].

Although significant progresses have been made, traditional optical tweezers still encounter two main challenges when they come to nanometer scale. First, the diffraction limit of light beams prevents the further decreasing of the spot size of the focus. Second, the restoring force decreases rapidly since the gradient trapping force is proportional to 1/r³ with r radius of the nano-object [9]. In this circumstance, it is necessary to develop new schemes for optical manipulation of nano-scaled objects, and several efficient configurations have been proposed, such as the deep subwavelength nano-trapping by plasmonic antennas with enhanced light confinement [10–13], and double nanohole structure [14,15]. Dielectric structures [16–19] were also reported to trap nanoparticles, although it’s hard to obtain a nano-scaled potential well. In most of the well-established nano-tweezers, however, only the trapping of a single nanoparticle is investigated due to the tiny trapping volume [20], or the distance between the multiple trapped nanoparticles can hardly be controlled since they are randomly trapped within some limited region.

Here we propose a distinct nano-trapping scheme based on a two-mode nano-slot cavity embedded in a dielectric photonic crystal (PC) waveguide, which are usually used in the particle transporting [21–23], to generate a nano-scaled potential well that can stably trap two nanoparticles. When the first order mode is excited, the two trapped particles stick together at the center of the cavity. When the second order mode is switched on (the first order mode is switched off), however, the two particles push each other and are trapped near the two ends of the slot cavity, and the inter-distance between them is determined by the mode structure. When the two modes are excited simultaneously, the inter-distance can be tuned precisely within a rather large range by tuning the relative powers of the two modes.

2. Model and method

The schematic of the photonic crystal slot cavity is depicted in Fig. 1. The silicon waveguide is set along the x axis with the width of w = 500 nm and height of h = 220 nm, respectively. Five holes are made to both sides of the waveguide with linearly tapered radius and separations, which can be used to enhance the quality factor of the cavity [24–26]. Furthermore, a slot with the width of s = 40 nm connected the two innermost holes are introduced to form the slot cavity. In Cartesian coordinates denoted in Fig. 1, the central positions $x_{j}$ and radiuses $r_{j}$ of the 5 holes on $+ x$ axis are $x_{j} = x_{1} + \sum_{k = 1}^{j - 1} a_{k}$ , $r_{j} = r_{j - 1} + Δ r$ with $a_{j} = a_{j - 1} + Δ a$ and $r_{j} / a_{j} = 0.28$ , where $j = 2, 3, 4, 5$ is the index of the holes. Other parameters are $x_{1} = 265 nm,$ $a_{1} = 355 nm,$ $Δ r = 7 nm,$ and $Δ a = 25 nm .$ The other 5 holes on $- x$ axis are located with mirror symmetry to those on $+ x$ axis. The refractive index of the silicon is set to be n = 3.45, and the whole system is immersed in water with a refractive index of $n = 1.33$ .

Fig. 1 (a) Scheme of the slotted photonic crystal cavity. See text for the detailed structural parameters. (b) The transmission spectrum of the structure, which shows two resonances at 1401 nm and 1683 nm, respectively. (c) The electric distribution (|(E)|) of xy cross section view (z = 0 nm) for the first order mode and (d) the second order mode. The arrows indicate the in-plane electric vector with the arrow lengths proportional to the amplitudes.

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The PC slot cavity defined in Fig. 1(a) supports two resonant modes for the quasi-TE mode (which is a TE-like mode with the electric field intensity mainly paralleling to the xy-plane [27]), of which the resonant spectrum is shown in Fig. 1(b). For current parameters shown above, the two resonant wavelengths for the first and second order modes are 1683 nm and 1401 nm, respectively. Using the three-dimensional finite-difference in time domain (FDTD) method, we calculate the mode pattern of these two modes, as shown in Figs. 1(c) and 1(d), respectively. In the numerical simulations, the mesh grid sizes are all set to be 2 nm along the x, y and z directions. For the 1st order mode, the bright spot of electric field is localized at the center of the slot cavity, while for the second order mode, there’re two separated bright spots located near the two ends of the slot.

To evaluate the abilities of nanoparticle trapping in this structure, we calculate the optical forces exerted on the nanoparticle by using the integral of the Maxwell stress tensor over a surface S enclosing the nanoparticle [28], which is expressed as

F = ∯_{S} 〈 {\overset{\leftrightarrow}{T}}_{M} 〉 \cdot n d s,

where

〈 {\overset{\leftrightarrow}{T}}_{M} 〉

is the time-averaged Maxwell stress tensor (MST), and n is the outward normal vector of the surface. The definition of

{\overset{\leftrightarrow}{T}}_{M}

is

{\overset{\leftrightarrow}{T}}_{M} = ε E \otimes E + μ H \otimes H + \frac{1}{2} \overset{\leftrightarrow}{I} (ε {| E |}^{2} + μ {| H |}^{2}),

where

ε

and

μ

are the permittivity and permeability of medium that the enclosed surface lies in, respectively.

\otimes

means dyadic operation, and

\overset{\leftrightarrow}{I}

is the unit tensor. We choose the polystyrene nanoparticle (with a moderate refractive index of n = 1.45), and gold nanoparticle (with the CRC model) as the typical dielectric and metallic objects.

3. Optical nano-trapping in the PC slot cavity

The incident wave in this paper propagates along $+ x$ axis direction. Figure 2(a) shows optical force acting on the polystyrene nanoparticles at different positions x when the first order mode is excited. It is noted that all the optical forces and potential wells are all calculated at the incident level of 100 mW. Results show that the dielectric particle can be trapped at the center of the slot cavity. For the particles with radius of 10, 12 and 15 nm, the trapping stiffnesses are about 0.11, 0.18, and 0.34 $pN \cdot {nm}^{-1} \cdot W^{-1}$ , respectively. This means that the objects can be tightly trapped at the center (x = 0) of the slot in x direction.

Fig. 2 Optical forces and potential well on the polystyrene particles with radius of 10 nm, 12 nm and 15 nm, respectively, along x axis. The middle row shows the structure of the slot cavity and the nanoparticles. (a) Optical forces for the 1st order mode. (b) Optical forces for the 2nd order mode. (c) Potential wells for the 1st order mode. (d) Potential wells for the 2nd order mode.

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In order to damp the Brownian motion, the depth of trapping potential well should be larger than 10 K_BT [29], where K_B is the Boltzmann constant and T is the absolute temperature (T = 300 K is selected in this work). The potential is defined as the following form

U (x) = - \int_{- \infty}^{x} F_{x} \cdot d x .

The potential for the 1st order mode trapping is shown in Fig. 2(c). The potential well is deeper than 100 K_BT for the radius of 15 nm. For the smallest particle with radius of 10 nm, the potential well is also deeper than 20 K_BT, which indicates that the nanoparticles can be trapped stably to the equilibrium position (i.e., at the center of x = 0) in the slot cavity.

When the second order mode is switched on (the first order mode is switched off), the optical force and potential on each particle are shown in Figs. 2(b) and 2(d), respectively. Results show that the trapping stiffnesses are about 10 times larger than those of the first mode, which are about 1.03, 1.86, and 3.49 $pN \cdot {nm}^{-1} \cdot W^{-1}$ for the three different nanoparticles. In this case, the particles are trapped at x = ± 150 nm in the slot. Comparing to the trapping potentials shown in Figs. 2(c) and 2(d), these nanoparticles can be restricted in a smaller region using the second mode, which is only about half the spreading size of the first order mode.

It is noted that the resonant wavelength of the PC waveguide shifts obviously due to the active feedback of the object to the trapping system when it is trapped in such a strongly localized field. At equilibrium in the second mode, the resonant wavelength undergoes red shifts of 2.4, 3.2 and 3.4 nm for the nanoparticle with radius of 10, 12 and 15 nm, respectively. In a practical implementation, the post-shifted wavelengths should be used in order to resonantly trap the objects with larger optical force. This property is also used to track the successful trapping of a particle.

The enhanced manipulation of gold nanoparticles is also investigated, as shown in Fig. 3. Results show that the gold nanoparticle experiences much stronger optical force when it comes into the slot, and the optical force increases rapidly with the increasing of the particle size. For gold nanoparticles with radius of 10 nm, 12 nm and 15 nm, the optical forces of the second mode are 2.92, 2.94 and 3.25 times larger than those of the first order mode, respectively. At the incident power of 100 mW, the particle with radius of 15 nm could be trapped within the potential of 3000 $K_{B} T$ . For the 2nd order mode, the region of trapping uncertainty is less than half of that of the first mode. Since gold nanoparticles have stronger interaction with the light field, the resonant wavelengths undergo larger shift of 2.6 nm, 3.2 nm, and 4.7 nm for the radiuses of 10 nm, 12 nm and of 15 nm, respectively.

Fig. 3 Optical force on Au particles with radius of 10, 12 and 15 nm along x axis for (a) the 1st order mode and (b) the 2nd order mode. The trapping potentials on particles with radius of 10, 12 and 15 nm along x axis for the 1st and 2nd order modes are shown in (c) and (d), respectively.

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4. The trapping in the transverse directions

In the previous section, we have demonstrated that the objects can be trapped stably at different positions along the slot direction (i.e., the x axis) when different modes are switched on. One may wonder whether the trapping is stable or not along the transverse y and z directions. For this purpose, we calculated the trapping properties both in the y and z directions, and the results are shown in Fig. 4.

Fig. 4 The trapping properties in the y (left column) and z (right column) directions for the 1st order mode and the polystyrene spheres with r = 10 nm. (a) Field profile |(E)| along the y direction. It can be seen that the field intensity maximum is on the wall. (b) Optical force F_y versus the central position y of the object. (c) Potential well in the y direction. (d) Field profile |(E)| along the z direction. (e) Optical force F_z versus the central position z of the object. (f) Potential well in the z direction.

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Figure 4 show the electric field profiles, optical forces, and potential wells for the 1st order mode in the y and z directions for the dielectric sphere of r = 10 nm (For the objects of r = 12 nm and 15 nm and the second order mode, very similar results are obtained and not shown here for clarity). Figure 4(a) shows the field profile along the y direction, from which one may regard that the objects will be trapped to the slot wall since the intensity maximum is on the walls. However, due to the very small slot width (40 nm only in current case) and very small intensity gradient, a tiny F_y (see Fig. 4(b)) and a very small well depth (less than 0.08 K_BT, as shown in Fig. 4(c)) are obtained in y direction. According to Ref [29], the potential well should be larger than 10 K_BT in order to trap the object on the wall. This means that the object can move (almost) freely driven by the Brownian motion in the y direction, and can will be trapped by the physical boundary of the slot walls. Also, since the electric field intensity changes very slightly across the slot (see Fig. 4(b)), the trapping behaviors in the x direction (demonstrated in Fig. 2 and Fig. 3) do not change obviously with y. In the z direction, the results are shown in Fig. 4(d-f). Different from the case in y direction, the optical force and the potential well in z direction are large enough (30 K_BT), and can trap the objects at the center of z = 0.

According to the analysis presented above, one can see that the stable trapping along the y and z directions are achieved by the slot wall and the potential well, respectively. In the x direction, however, the trapping can be tuned by the selective excitation of the resonant modes. When only the first order mode is switched on, all the nanoparticles will be trapped at the center of the slot of x = 0. When only the second order mode is switched on, the objects will be separately trapped at two different positions.

5. The inter-distance tuning by the two modes

In this section, we introduce a powerful method of tuning the inter-distance between the two trapped particles. When the second mode is switched on, two nanoparticles are trapped at a distance 150 nm (see Fig. 2(d) and Fig. 3(d)) from the center of the slot. Then the first order mode is launched into the system (the second order mode is still on). Using the combined incident source, we can tune the equilibrium position by controlling the incident power ratio $α = P_{1 st} / P_{2 nd}$ . Here $P_{1 st}$ and $P_{2 nd} = 100 mW$ are the incident powers of the 1st and 2nd order mode, respectively. The total optical force is the sum of the contributions from both the modes,

F = α F_{1} + F_{2},

where F₁ and F₂ are the optical forces of the 1st and 2nd order modes, respectively.

Figure 5 shows the trapping forces acting on the polystyrene particle with radius of 12 nm tuned with different power ratio $α$ . When the power ratio used in the trapping varies between 0 and 1.5, the equilibrium position of the nanoparticles can be tuned continuously from 150 nm to 100 nm away from the slot center of x = 0. When α keeps increasing, such as to 2.5, the equilibrium position jumps to the center of slot. Between the value of 1.5~2.5, the trapping force is small and the stiffness is also weak. This shows that the inter-distance of the two nanoparticles can be switched between the two values of 0 nm (1st order mode only) and 200 nm (2nd order mode only). Also, precise tune of the inter-distance between 200~300 nm can be achieved by the superposition of the two modes when $α$ changes from 0 to 1.5.

Fig. 5 Optical forces as a function of power ratio α (see Eq. (4)) on the polystyrene particle with a radius of 12 nm. The inter-distance between the trapped particles can be tuned from 200 nm to about 300 nm when α changes from 0 the 1.5.

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Before conclusion, it is noted that the incident power of 100 mW is used in the optical force and potential calculations. In this case, even the polystyrene nanoparticles with a radius as small as 10 nm can be trapped. However, when the incident power is decreased due to a very low coupling efficiency, such as to 1 mW, none of the three polystyrene spheres could be trapped inside the slot. In this case, the minimum metallic sphere can be trapped is r~11 nm. This means that the incident power should be larger than some threshold value in order to stably manipulate the target objects in practice.

6. Conclusion

In summary, we propose a nano-trapping mechanism for two nanoparticles using a two-mode slot photonic crystal cavity, where the inter-distance between the two nanoparticles can be tuned precisely. When the first order mode is excited, both the nanoparticles are trapped at the center of the cavity. When the second mode is excited, the two nanoparticles will be trapped at two separated equilibrium positions with a distance ~150 nm away from the slot center. When both the modes are excited simultaneously, the slot cavity cannot only be used to trap the nanoparticles, the inter-distance between them can also be tuned between 200 nm and 300 nm by using different relative power ratio of the two modes. The scheme proposed here can be extend straightforwardly to any other nano-tweezers that supports multiple modes, and the inter-distance may be tuned in a more precise scale. The methods and results reported here provides a powerful tool to control the interaction of two nano-objects via tuning the separation between them, and it may have potential applications in various related disciplinary.

Funding

National Natural Science Foundation of China (NSFC) (Grant no. 11474077, 11404083, and 11704088).

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Optical trapping of nanoparticles with tunable inter-distance using a multimode slot cavity

Abstract

1. Introduction

2. Model and method

3. Optical nano-trapping in the PC slot cavity

4. The trapping in the transverse directions

5. The inter-distance tuning by the two modes

6. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (4)

Optics Express