1. Introduction

Optical manipulation of nano-particles provides a controllable, non-destructive, non-contact and highly precise technique for manipulating nano-particles for the biological, medical and chemical applications [1,2]. The pioneering observation of optical force phenomenon succeed using a single Gaussian beam to control microscopic particles in the three-dimensional space [3]. This experiment confirmed the concept of light exerting force on particles in its field and promoted the study of efficient optical translating and trapping for precisely detecting [4], sorting [5], and controllable moving [6] micro- or nanoparticles. Such optical manipulation techniques can be classified into two categories. One is the traditional free-space method, whereby free-space beams are highly focused to realize an optical trap [6,7]. The other one is the near-field method where evanescent fields surrounding photonic structures are used to interact with particles [8]. Owing to the diffraction limitation in focusing a laser beam, the conventional method needs high numerical aperture lens or/and light intensity when dealing with Rayleigh particles (which are much smaller than the wavelength). Overcoming these challenges, the approach of utilizing the optical force originating from the evanescent fields of on-chip nanophotonic structures open new opportunities to manipulate particles with low optical power.

Among various nanophotonic structures, planar optical waveguides is one of the most widely used structures in the realm of near-field optical manipulation. Unlike free-space optical trapping devices, nanophotonic waveguide can confine light well due to the small cross section and the ultra-high index contrast. As a result, different kinds of particles can be trapped by an optical waveguide over long distance. Parallel systems can be fabricated on a chip accomplishing that many trapping processes can be conducted simultaneously over a large area. In addition, waveguide-based optical manipulation devices are convenient to be integrated with microfluidic channels to preferably serve large-scale chemical and biochemical detection.

One of the earliest demonstrations of waveguide structures was presented by Kawata and Tani [9] to demonstrate particles optically driven along an optical channel waveguide. Yang and Schmidt et al also theoretically and experimentally demonstrated the optical transport of dielectric particles along nanowire waveguides in an integrated microfluidic architecture [10,11]. To improve the trapping stability, Yang and his associates continually extended to sub-wavelength slot waveguides to acquire stronger optical confinement and higher intensities to trap particles smaller than 100nm-diameter [12]. Some other emerging near-field photonic nanostructures, like plasmonic nanostructures [13] or photonic-crystal waveguides [14], have also been demonstrated for on-chip nano-particle trapping. More recently a photonic-crystal waveguide supporting multimode was also proposed to realize the optical pulling force by involving the mode conversion [17]. Basically, plasmonic nanostructures or photonic crystal waveguides usually have relatively high loss and are not suitable to manipulate particles over a long distance along the waveguide. Furthermore, plasmonic structures might cause some heat issues due to the metal absorption that will break the trapping equilibrium when increasing the input light power. In order to enhance the optical force, a slotted nanobeam-cavity with periodic integrated structures firmly trapping nano-particles is also proposed theoretically [18]. The relationship between the optical force and the quality factor, mode volume and the transmission of the cavity was analyzed in the system of waveguide-cavity-waveguide [19], which is helpful to the design and optimization of a cavity-based optical manipulation.

Recently an interesting structure waveguide based on subwavelength gratings (SWGs) is experimentally fabricated [15] and shows both bulk and surface sensitivities superior to those obtained in conventional nanophotonic waveguides [16]. This proves that the SWG waveguide offers enhanced interaction between the evanescent tail and matters, which is of great importance in optical manipulation. However, to the best of our knowledge, the SWG waveguide has not yet been proposed for optical manipulation. In this paper we given a theoretical analysis for the optical forces induced by the Bloch modes propagating along an SWG waveguide for the first time. As a periodical structure, an SWG waveguide supports periodically light field distribution along the waveguide. In contrast, the field distribution in a conventional optical waveguide is uniform along the propagation distance. As a result, it is possible to trap many nano-particles stably in period when using SWG waveguides, which is totally different from the case when using conventional optical waveguides. The separation of the trapped nano-particles can be designed easily by modifying the grating period of an SWG waveguide. This cannot be realized by using other regular photonic nanostructures, including SOI strip waveguides [10,11], nano-slot waveguides [12], plasmonic nanostructures [13] and photonic crystal waveguides [14] mentioned above. An SWG waveguide also has longer working distance in space to trap nano-particles around the waveguide than a conventional optical waveguide.

2. Structure and analysis

Figure 1(a) shows the schematic configuration of an SWG waveguide on silicon. The SWG waveguide has a periodic arrangement of silicon segments covered by the upper-cladding medium. For an SWG waveguide, one should choose the grating period Λ, the duty cycle η and the segment dimensions so that the grating pitch is less than a half of the effective wavelength of the waveguide mode, i.e., Λ < λ_eff/2, where λ_eff = λ/n_eff, λ is the wavelength in vacuum, λ_eff and n_eff are respectively the effective wavelength and the effective refractive index of the waveguide mode. In this way, it avoids the formation of standing waves due to Bragg-grating effect and the opening of a band gap at the operation wavelength (e.g., λ = 1550 nm). As a consequence, the SWG waveguide supports a Bloch mode, which can propagate along the segmented waveguide losslessly in theory [15]. The Bloch mode in an SWG waveguide can be excited with very high efficiency (~99%) by the fundamental mode in an SOI strip waveguide through an adiabatic taper [20–22]. SWG waveguides can be fabricated with a regular silicon-on-insulator wafer by using a single-etch step. The SWG waveguides usually have a loss as low as 2.1 dB/cm, as demonstrated previously [15].

 

Fig. 1 Schematic configuration of an SWG waveguide.

Download Full Size | PDF

For the present case, according to the design rule of SWG waveguides, the parameters of the SWG waveguide are chosen as the grating pitch Λ = 300 nm, the waveguide height h = 220 nm, the segment width w = 450 nm and the segment length a = 150 nm. A three-dimensional finite-difference time-domain (3D-FDTD) method is used to simulate the propagation of the supported Bloch-mode along the designed SWG waveguide, as shown in Fig. 2(a)-2(d). In this example, TE polarization is considered and the refractive index the upper-cladding is chosen as n_cl = 1.33 (i.e., water) because nano-particles are usually manipulated and observed in liquid. It is can be seen that light propagates along the SWG waveguide without losses in theory. More interestingly, the electric field distribution E(x, y = 0, z) is periodical along the propagation distance z, as shown in Fig. 2(a). This can be seen very clearly from Fig. 2(b), which shows the field distribution E(x = 0, y = 0, z). In the longitudinal direction z, the electric field has peaks locating at the middle of the silicon segments. In the lateral directions (x, y), one sees that there is a peak in the middle (x = 0) of the silicon segment (e.g., z = Λ/2) while significant evanescent field enhancement is observed at the region close to the sidewalls of the silicon segment (see Fig. 2(c)). On the other hand, the electric field distribution in the slot region (e.g., z = 0) has two peaks and the amplitude is minimal at x = 0, as shown in Fig. 2(d). This special light field distribution along an SWG waveguide makes it attractive to be useful for on-chip nano-particle manipulation. For example, the particles can be trapped into the gaps between the silicon segments of the SWG waveguide. In contrast, for regular SOI strip nanowires, the particles are pulled to move along the top-surface of the waveguide. Here we give a theoretical analysis to reveal the uniqueness of the optical forces of the SWG waveguide.

 

Fig. 2 Electric field distributions of TE polarization when light propagates along the designed SWG. (a) E(x, y = 0, z); (b) E(x = 0, y = 0, z); (c) E(x, y, z = Λ/2). (d) E(x, y, z = 0). Here the parameters of the SWG waveguide are chosen as: the grating pitch Λ = 300 nm, the waveguide height h = 220 nm, the segment width w = 450 nm and the segment length a = 150 nm.

Download Full Size | PDF

As is well known, a nano-particle can be trapped possibly due to the optical forces generated by the optical field penetrated to the liquid medium when it is in the vicinity of an optical waveguide. The optical force can be estimated by integrating the time-independent Maxwell stress tensor <T_M> on the external surface S enclosing the nano-particle [23], i.e.,

First, we consider the case when the nano-particle locates inside the gap between the silicon segments of the SWG waveguide. The position of the nano-particle is given as (x, y, z). The dependence of the optical forces (F_x, F_y, and F_z) for the nano-particle on its position (x, y, z) are calculated numerically, as shown in Fig. 3(a)-3(c). The sign of the optical force indicates the direction of the optical force. When F_x, F_y, and F_z >0, it means that the direction of the optical force is positive (i.e., + x, + y, + z), which will push the particle to move in the direction of + x, + y, or + z. Otherwise, the negative optical force will push the particle to move in the direction of −x, −y, or −z. Specially, if the slope ∂F/∂s is negative, the particle will be trapped at the position where F = 0 in the s-direction (here s = x, y, or z).

 

Fig. 3 Calculated optical forces. (a) (F)_x (x = 0, y, z = 0), (F)_y(x = 0, y, z = 0), and (F)_z(x = 0, y, z = 0); (b) (F)_x (x, y = y₀, z = 0), (F)_y(x, y = y₀, z = 0), and (F)_z(x, y = y₀, z = 0); (c) (F)_x (x = w₁/2, y = y₀, z), (F)_y(x = w₁/2, y = y₀, z), and (F)_z(x = w₁/2, y = y₀, z). (d) F_t(x, y = 0.08µm, z), where (F)_t fresents the total force of (F)_x and (F)_z, and the arrows denotes the direction of the optical force (F)_t.

Download Full Size | PDF

Figure 3(a) shows the optical forces (F_x, F_y, and F_z) for the nano-particle locating at different vertical positions y. Here the lateral position x and the longitudinal position z are fixed as x = 0 and z = 0. It can be seen that the slope ∂F_y/∂y is negative while the optical force F_y becomes zero at y = −10nm, which indicates that the particle can be trapped when it locates at y = −10nm (x = 0 and z = 0). This position (y = −10nm) for particle-trapping deviates from y = 0, which is due to the slight vertical-asymmetry of the SWG waveguide. From Fig. 3(a), it can be also seen that the optical forces F_x(x = 0, y, z = 0) and F_z(x = 0, y, z = 0) are much weaker than the optical force F_y.

Figure 3(b) shows the optical forces (F_x, F_y, and F_z) when the lateral position x of the particle varies. Here the lateral position y and the longitudinal position z are fixed as y = −10nm and z = 0. It can be seen that the maximum of the optical forces F_x(x, y = −10nm, z = 0) is about 10 pN/W around x = ± 300 and ± 150nm while the optical force F_y(x, y = −10nm, z = 0) and F_z(x, y = −10nm, z = 0) are very small (e.g., <1 pN/W). The optical force F_x(x, y = −10nm, z = 0) is distributed odd-symmetrically and one has F_x(x, y = −10nm, z = 0) = 0 when x = 0 and x = ± 210nm. When the particle moves from far away to be close to the sidewall at the left side, the optical force F_x increases from 1.5 pN/W to a positive maximum of 10 pN/W around x = −300nm, which is attributed to the enhancement of the evanescent field. When the particle further moves from the left to the right, the positive optical force F_x decreases to zero around x = −210nm and then becomes negative, which is expected according to the lateral field distribution in an SWG waveguide [see Fig. 2(a)]. One has a negative maximum of about −8 pN/W for the optical force F_x(x, y = −10nm, z = 0) around x = −150nm. When the nano-particle moves further to the middle (i.e., x = 0) of the SWG waveguide, the negative optical force F_x becomes zero. It can be seen that there are three special positions, i.e., x₀ = −210, 0 and + 210nm, where the optical force F_x(x, y = −10nm, z = 0) = 0. When x = 0, the slope ∂F_x/∂x is positive and thus the position x = 0 is not steady for the particle-trapping. In contrast, when x = x₀ = ± 210nm, the slope ∂F_x/∂x becomes negative. Consequently the particle can be trapped at x = x₀ = ± 210nm and is prevented to move in the x direction. As discussed for Fig. 3(a), the particle can be trapped at y = y₀ = −10nm and is prevented to move in the y direction. As a result, the particle is trapped at (x, y) = ( ± 210, −10) nm in the x-y plane.

We have also calculate the optical forces (F_x, F_y, and F_z) for the particle when it locates at the side of the silicon segments, as shown by the circles in Fig. 3(c). Here the position of the particle in the x-y plane is chosen as (x, y) = ( ± w₁/2, y₀), where w₁ = w + 2r + 30nm. The longitudinal position z varies from z = pΛ−Λ to z = pΛ + Λ, where p = 1. It can be seen that the optical forces (F_x, F_y, and F_z) for the particle varies periodically as the longitudinal position z varies, as shown by the circles in Fig. 3(c). This can be understood easily regarding to the periodical distribution of light field along the z direction. It can be seen that the z-components F_z of the optical force varies from a positive maximum to a negative maximum when the position of the particles moves from the front end (i.e., z = pΛ + Λ/2−a/2) to the back end (i.e., z = pΛ + Λ/2 + a/2) of any silicon segment, where a is the length of the silicon segment. When the particle moves in the liquid segment, i.e., from z = pΛ + a/2 to z = pΛ−a/2, the z-components F_z of the optical force varies from a negative maximum to a positive maximum. It can be seen that the optical force F_z becomes zero at the positions of z = pΛ(at the middle of the slot region)and z = pΛ + Λ/2(at the middle of the silicon segment region). The negative slope ∂F_z/∂z at the positions of z = pΛ + Λ/2 indicates that the particle will be trapped not to move forward or backward along the z direction. One can also see that the optical force F_x(x = w₁/2, y = y₀, z) is negative while F_x(x = −w₁/2, y = y₀, z) is positive, which indicates that the particle can be pulled to move and finally touch with the right or left sidewall of the silicon segment. In this way, the particles can be trapped at the position of ( ± w₁/2, y₀, pΛ + Λ/2). Figure 3(d) shows the optical force F_t (x, y = 0.08µm, z), which represents the total force of F_x and F_z. The direction of the arrows in this figure also points out that the particles can be trapped along the right or left sidewall of the silicon segment. In contrast, at the position of z = pΛ, the slope ∂F_z/∂z is positive, which indicates that it does not allow to trap the particles at this position. From these theoretical analysis and calculation, it is noticed that one can trap and manipulate many particles in the gap regions as well as both sidewalls of the silicon segments possibly by using an SWG waveguide. This is very different from the conventional SOI strip nanowires, in which case the particles are pulled to touch with the top-surface as well as the sidewalls of an optical waveguide and are pushed to move in the light propagation direction.

The dependence of the optical force on the particle radius is also analyzed when choosing different grating periods Λ = 300, 200, and 150nm, as shown in Figs. 4(a)-4(c), where the radius of the particle is chosen as r = 20, 50, 100nm, respectively. For this analysis, the lateral and vertical positions for the particle are fixed as x = w₁ and y = y₀, respectively. Here we focus on the analysis for the optical force F_z because it determines how the nano-particles move along the propagation distance of light. As shown in Figs. 4(a)-4(c), it is not surprising that the optical force F_z has a great dependence on the radius of the particle. It can be seen that the optical force F_z is higher when the particle has a larger radius. When the particle radius increases from 20nm to 100nm, the maximum of optical force F_z is greatly enhanced from 1 pN/W to 26 pN/W in Fig. 4(a). This is consistent with the theoretical prediction according to the formula for calculating the optical force on a Rayleigh sphere in an optical field [24]. The optical force F_x and F_y have similar dependence on the particle radius.

 

Fig. 4 Calculated optical force (F)_z (x = w₁/2, y = y₀, z) acting on different sized spheres along the SWG waveguides with different grating pitches. (a) Λ = 300nm(a); (b) Λ = 200nm; (c) Λ = 150nm.

Download Full Size | PDF

When the grating period Λ varies, the optical force F_z has similar dependence on the longitudinal position z. In particular, the optical force becomes pretty different for a large nano-particle in an SWG waveguide with small period, as shown by the results for r = 100nm and Λ = 150nm given in Fig. 4(c). This is attributed to the pretty large scattering force on the large particles in the SWG waveguide with Λ = 150nm. One should notice that the optical force F_z becomes smaller for the SWG waveguide with a smaller grating period Λ. When the duty cycle is fixed, a larger grating period is preferred when a stronger optical force is desired for the particle manipulation. On the other hand, one should notice that the grating period is required to be small enough to support Bloch-modes and avoid the grating reflection. For example, the grating period Λ should be smaller than 330 nm for the present case when considering the operation wavelength λ = 1550nm. In this case, the optical force can be further enhanced slightly in comparison with the case of Λ = 300nm shown in Fig. 4(a). It is also possible to chirp the separation of the particles trapped by modifying the period Λ adiabatically along the SWG waveguide.

Finally, we give a comparison between the optical forces for a particle with a radius of r = 50nm around an SWG waveguide and an SOI strip nanowire by assuming they have the same core widths, i.e., w = 450nm. Figure 5 shows the dependence of the optical forces (F_x, F_y, F_z) on the lateral position x. Here, the vertical position y and the longitudinal position z of the particle are fixed as y = 0 and z = 0. It can be seen that the SWG waveguide provides stronger optical force than the SOI strip nanowire due to the significant evanescent field enhancement in the SWG waveguide. This also enables that the SWG waveguide to have a longer work distance for trapping particles than the SOI strip nanowire, which is pretty helpful for optical manipulation. One can also see that the optical force decreases quickly for both types of optical waveguides when the lateral position of the nano-particle becomes far away from the waveguide core. This is due to the exponential decay of the evanescent field. It can be seen that the optical forces F_x are negative for both waveguides, which indicates that the particle is pulled toward the sidewall of the silicon core. The trap stiffness along y-axis is 0.25 pN nm⁻¹ W⁻¹ at the equilibrium point by calculating the slope of the force-distance curve with the formula given in [12]. In contrast, the trap stiffness in a regular SOI nanowire and a nano-slot waveguide is ~0.1 pN nm⁻¹ W⁻¹ and ~0.2 pN nm⁻¹ W⁻¹, respectively [12]. It can be seen that there is significant improvement when using the present SWG waveguide structure. By using the formula given in [25], we also theoretically calculate the work required for the 50 nm nano-particle to escape from the trap of the SWG waveguide. The theoretical result shows that the trapping potential is more than 400 K_bT/W for the nano-particle locating at the position of x = 300 nm. It means that one can achieve 10K_bT for realizing a stable trap as long as the optical power is greater than 25 mW.

 

Fig. 5 Calculated optical force (F)_x (x, y = y₀, z = Λ/2) acting on a 50nm radius sphere for a nanowire waveguide and a SWG waveguide with a core with w = 450nm.

Download Full Size | PDF

3. Conclusions

In this paper, we have given a detailed analysis for optical force on nanoparticles in the evanescent field of a SWG waveguide. It is demonstrated that particles can be trapped in the gaps of the SWG waveguide by analyzing the F_x force profile along the lateral direction. Meanwhile, the unique propagating mode makes the SWG waveguide able to form optical traps at the sidewalls in the evanescent field, realizing numbers of three-dimensional optical traps at the ends of silicon segments at a time. The comparison of the forces for different particle radius indicates that the magnitudes of forces acting on particles are strongly dependent on the particle radius. The analysis of varying the grating pitch of the SWG waveguide shows the change of separation distance holding particles. We also conclude that the SWG waveguide can attract particles in further district away from it, compared with a normal nanowire waveguide. This theoretical analysis is very important for the following experimental work to be done in the future. It is also well known that strong field enhancement can be achieved by introducing a high-Q optical cavity [26]. Therefore, it is possible to further enhance the optical force for stronger particles trapping by using optical micro-cavities based on SWG waveguides in the future.