1. Introduction

In light of the miniaturization revolution brought by microelectronics and nanotechnologies, lab-on-a-chip or microfluidics large-scale integration (mLSI) containing hundreds or more of these function units for conducting biological tests in a microdroplet or a microchamber with the reagent of nL volume or a number of biomolecules attaching on microbeads has emerged in recent years [1–7]. In the design of those function units, it is a common requirement of manipulating and separating cells or other biological particles within a mixture in life science research or biological applications. Cell or other biological particle sorting has been widely used in microfluidic systems [8]. Many active sorting methods such as electric control [9], thermal control [10], pneumatic control [11], magnetic control [12], and acoustic control [13] have been demonstrated in microfluidics. Conventional sorting methods often involve cumbersome set-ups and is difficult to realize high-throughput on micro-scale situations. Meanwhile, the size of fluidic channels is especially difficult to diminish from micro-scale of liquid droplets to nano-scale of viruses and DNA. This problem has been partially solved by sorting and manipulating microbeads which act as carriers of biomolecules with a number of proteins or DNAs attaching on them [14]. Furthermore, near field optical trapping and manipulation are very important tools for handling object in nanometer scale [15].

To achieve sorting of the nanoparticles with smaller size in submicron scale, optical sorting, which is often used in passive sorting, is an alternative and feasible method. The non-invasive and non-contaminating nature of optical sorting is suitable for manipulation of single nanoparticle. In recent years, researchers have shown growing interest in compact and portable microfluidics platforms with high-throughput and high-efficiency feature in optical sorting [16]. With the development of bio-nanotechnology, microfluidics has gradually turn to silicon-on-insulator (SOI) platform which is an attractive platform to realize optical devices miniaturization by utilizing the mature silicon fabrication process. Many integrated optical sorting devices based on SOI platform such as inverted rib waveguide-based [17] configuration, resonator-based [18] configuration and channel waveguide-based [19,20] configuration have been reported. The inverted rib waveguide-based configuration can sort out the nanoparticles and export from different output channels regardless of the particle sizes. However, it is still difficult to realize the sorting of nanoparticles with different sizes. In addition, optical sorting based on the micro-ring resonator requires accurate excitation wavelength and could be easily affected by the temperature and fabrication tolerance. Optical sorting based on channel waveguide exist the complexity of the manipulation such as dynamically changing the power distribution by thermal tuning [19] or transferring the nanoparticles by a struck bead [20]. In brief, these methods have its own drawback and unable to simply control the sorting of different sized nanoparticles. To the best of our knowledge by far, optical sorting with multiple different sized nanoparticles to multiple ports based on the channel waveguide by single wavelength manipulation has not been appropriately considered.

In this paper, we propose a multi-level automated optofluidic sorting method based on multi-step optical waveguide splitter with single wavelength excitation. The sorting mechanism of nanoparticles originates from size dependent potential wells distribution on the parallel waveguide region. By cascading optical waveguide splitters with gradually varied coupling length or gap, we realize the multi-level sorting of nanoparticles on the multi-step optical waveguide splitter. The process of these nanoparticles entering different waveguide channels is influenced by the propulsion of the light field and the power ratio between adjacent waveguides. The threshold size of the nanoparticles which jump from one waveguide to the others and the distribution of the nanoparticles between the adjacent waveguides in a statistical approach have been estimated. Upon completion of the above work, it is found that our scheme is feasible to realize multilevel sorting of nanoparticles with a high degree of automation.

2. Optical sorting on waveguide splitter

Here, we firstly demonstrate an optical sorting method with different-sized nanoparticles, which is based on optical waveguide splitter, shown in Fig. 1(a). The structure is fabricated on top of a silica platform covered by aqueous solution. The waveguide is chosen with thickness of 340 nm and width of 300 nm. The simulated wavelength is 1550 nm with transverse magnetic (TM) mode owing to near-infrared transparency of silicon and low loss at the wavelength. In addition, optical force of TM mode has been proved to be greater than that of transverse electric (TE) mode [21]. We choose the input power of 20 mW, which is the same with that of the experiment result as reported in [20]. If the input power is too high, the laser may generate too much heat, which may affect trapping effect and induce phase shift. As long as the input power is smaller than 50 mW, the heat effect could be ignored and researchers also experimentally demonstrated nanoparticle manipulation as in ref [17]. Based on the configuration in Fig. 1(a), we have conducted simulation work on optical field by adopting the finite difference time domain (FDTD) method. As shown in Fig. 1(b), the light with TM mode was induced into the waveguide and then divided into different ports (P₂, P₁) after coupling length between adjacent waveguides. The output power ratio (P₂/P₁) between adjacent waveguide is increased along the x direction. The optical field of WG₁ along the x direction are subsided and the optical field of WG₂ along the x direction are enhanced. For the position of the red dashed line (Fig. 1(b)), by using Maxwell Stress Tensor (MST) and integral displacement, we have calculated the optical force and potential distribution of the polystyrene (PS) spheres with 500 nm, 600 nm and 700 nm. PS sphere, which has a density of around 1.05 g/m³ and about the refractive index of around 1.59, is placed on top of the waveguide. As shown in the Fig. 1(c), the trapping force of 700-nm-sized nanoparticle is bigger than that of 600-nm-sized nanoparticle and 500-nm-sized nanoparticle. This indicates that the larger the particle is, the larger the optical force will be.

 

Fig. 1 (a) Schematic of one-step optical waveguide splitter. (b) Profile of light path at L_c = 1.25 μm and Gap = 100 nm. (c) Mode profile and trapping forces distribution on adjacent waveguide corresponding to the dot line in (b). (d) Trapping potential well distribution on adjacent waveguides of PS spheres with 500 nm, 600 nm and 700 nm corresponding to the position of dashed line in (b). (e) Trapping force in z axis of PS spheres with size of 500 nm, 600 nm and 700 nm. (f) Trapping force in z axis of PS spheres with size of 500 nm at different distance d_z (10nm, 60nm, 110nm) along z direction. (g) The variation of the size threshold and the power ratio (P₂/P₁) versus coupling length between adjacent waveguides with the gap of 100 nm. (h) The variation of coupling length (L_c) versus gap with the output power ratio (P₂/P₁) of 4:1. Inset of (a) is the cross section of the one-step optical waveguide splitter.

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For 700-nm-sized nanoparticle, its radius is very close to center-to-center distance (nearly 400 nm) between two waveguide modes, and the particle should be easy to jump to the waveguide of strongly optical field from that of weakly optical field. From the viewpoint of the potential well distribution, when the nanoparticle is bigger than 600 nm, the second derivative of the potential distribution at the barrier between adjacent waveguide will small than zero, as show in Fig. 1(d). It means that the barrier between adjacent waveguide would disappears for the particle bigger than 600 nm. Therefore, the size threshold of the particle is 600 nm for the power ratio of 4:1. Figure 1(e) shows the trapping force in z axis for the different sized PS sphere. The force in z axis is always negative and it indicate that PS sphere should be trapped down to the waveguide surface. The trapping force in z axis also change with the distance (d_z) between PS sphere and the waveguide surface. As shown in Fig. 1(f), as a nanoparticle of 500 nm size approaches waveguide surface with d_z changing from 110 nm to 10 nm, the trapping force in z axis enhance obviously. It means that the PS sphere would be tightly trapped on the waveguide surface. In one word, the PS sphere will be tightly trapped on the waveguide surface in x and y axis and only can move along waveguide in x direction. In Fig. 1(g), with the increase of coupling length (L_c), the output power ratio (P₂/P₁) should be increased and the size threshold of sorting different-sized nanoparticles should be reduced. For the power ratio of 4:1, with the increase of gap between adjacent waveguide, the coupling length should be increased, as shown in Fig. 1(h). To get the output power ratio of 4:1 between the adjacent waveguides, the gaps are chosen as g₁ = 100 nm, g₂ = 150 nm and g₃ = 200 nm and the corresponding coupling lengths are predetermined as L₁ = 1.25 μm, L₂ = 2.15 μm and L₃ = 3.45 μm.

The above analysis is based on the output power ratio of 4:1 between the adjacent waveguides. To further understand the sorting ability of optical splitter, we have calculated the coupling length at different gap and the size threshold of sorting nanoparticles, which corresponding the output power ratio of 2:1 and 6:1. When the output power ratio between the adjacent waveguide was changed, adjustment of coupling length at different gap is needed, as shown in Fig. 2. Furthermore, the different output power ratio at different gap can also realize optical sorting of other different-sized PS spheres, as shown in Table 1.

 

Fig. 2 The variation of coupling length versus the gap between adjacent waveguides with the power ratio of 2:1, 4:1 and 6:1.

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Table 1. The size threshold of sorting nanoparticles at different power ratio.

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As discussed in [22], the optimal power distribution for effective manipulating nanoparticles between waveguides is 2:1 or 3:1. Given the higher power splitter ratio to adjacent waveguides, the smaller nanoparticles can be sorted to weakly waveguide [19]. However, the power in the weakly waveguide of adjacent waveguides would not easy to trap and propel the particles. Therefore, the power splitter ratio of 4:1 between the two adjacent waveguides should be suitable enough to sort different nanoparticles. Therefore, we cascade optical waveguide splitter with above parameter to realize multi-level sorting on multi-step optical waveguide splitter.

3. Multi-level sorting

According to aforementioned analysis, we can go a further step to design a multi-step optical waveguide splitter for multi-level sorting nanoparticles based on the power ratio of 4:1, depicted in Fig. 3(a). The gaps between waveguides, WG₁, WG₂, WG₃ and WG₄, are labeled as g₁, g₂ and g₃ and the coupling lengths between WG₁, WG₂, WG₃ and WG₄ are respectively labeled as L₁, L₂ and L₃, depicted in Fig. 3(a). The P₁, P₂, P₃ and P₄ corresponding to the four output ports are used to guiding the different sized particles. The light was induced into the first waveguide and was then divided into different ports (P₁, P₂, P₃, P₄) after each coupling length between adjacent waveguides. The optical field profiles along the power splitter with multi-step parallel waveguides are illustrated in Fig. 3(b). It can be seen that 20% of optical power goes to P₁ and 16% of optical power goes to P₂, while 12.8% of optical power goes to P₃ and the rest of optical power goes to P₄, shown in Fig. 3(b). For example, with the input power of 20 mW and the splitter ratio of 4:1, the output power for P₁, P₂, P₃ and P₄ are 4 mW, 3.2 mW, 2.56 mW and 10.24 mW respectively. Figure 3(c) shows that the optical field of WG₁ along the x direction are subsided and the optical field of WG₂ along the x direction are enhanced, which is also applicable in Figs. 3(d) and 3(e). From simulation, we find that the depth of trapping potential for P₃ port is 7.4 k_BT, which is much larger than 1 k_BT and enough for stable trapping. For multi-step optical waveguide splitter, it takes less than 3 seconds for a particle to pass through three coupling regions and reach P₄, given the average travelling speed of particle is about 5.3 μm/s in inverted rib waveguide at the power with 50 mW in [17].

 

Fig. 3 (a) Schematic of multi-step optical waveguide splitter. (b) Corresponding profile of light path. (c) The enlarged profile represents the dotted box of I. (d) The enlarged profile represents the dotted box of II. (e) The enlarged profile represents the dotted box of III. Inset at the top of (c), (d) and (e) are the cross-sectional profile of the optical field at different positions along x direction.

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To comprehensively demonstrate the sorting of nanoparticles, we have studied the potential wells and statistical distribution of the different-sized nanoparticles with respect to nanoparticle position along y direction. By changing the location of PS sphere along y direction, the distribution of forces can be calculated between the adjacent waveguides. With the trapping force in y axis and z axis, the PS sphere can only move along waveguides in x direction. Owing to the mode coupling and the variation of the power ratio of between adjacent waveguides, the potential well distribution would change accordingly. Figure 4 shows that the potential well distributions along the y direction with the gap varying from 100 nm to 200 nm when different-sized nanoparticles were transported along the x direction. For 500-nm-sized PS sphere and gap dimension of 100 nm, as shown in Fig. 4(a), the potential wells exhibits a minimum at the center of WG₂ and a secondary minimum at the center of WG₁ during the position between −9.8 μm to −9.2 μm. The PS sphere at the minimum or secondary minimum can be trapped firmly at the center of WG₂ or the center of WG₁ and cannot jump from one to the other because the potential barrier between the adjacent waveguides isolates the movement of PS sphere. On the other hand, for 600-nm-sized PS sphere, the situation is different. The potential well distribution at −9.4 μm, shown in Fig. 4(b), has only one minimum and the PS sphere can jump from WG₁ to WG₂ at this position. We also can conclude that particles with size bigger than 600 nm also can jump to WG₂ easily, since bigger particles at secondary minimum is more likely affected by the trapping force of main minimum point, as demonstrated in our previous work [19]. The position of is the critical point for sort out particles with size smaller than 500 nm to WG₁, since these particles with size bigger than 600 nm can jump from WG₁ to WG₂. Therefore, the optical splitter I with coupling length L₁ of 1.25 μm and gap of 100 nm can sort out PS sphere smaller than 600 nm diameter to P₁ and guide bigger ones to splitter II region for further operations.

 

Fig. 4 The potential depths along y direction of the different-sized PS spheres at different gaps. (a) The potential depths at four different positions along x direction from −9.8 μm to −9.2 μm with a 500 nm PS sphere at g₁ = 100 nm. (b) The potential depths at four different positions along x direction from −9.8 μm to −9.2 μm with a 600-nm-sized PS sphere at g₁ = 100 nm. (c) The potential depths at four different positions along x direction from −5.8 μm to −5.2 μm with a 600–nm-sized PS sphere at g₂ = 150nm. (d) The potential depths at four different positions along x direction from −5.8 μm to −5.2 μm with a 700–nm-sized PS sphere at g₂ = 150nm. (e) The potential depths at four different positions along x direction from −0.4 μm to 0.2 μm with a 700–nm-sized PS sphere at g₃ = 200nm. (f) The potential depths at four different positions along x direction from −0.4 μm to 0.2 μm with a 800-nm-sized PS sphere at g₃ = 200nm.

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Figures 4(c) and 4(d) show the potential well distribution for 600-nm-sized PS sphere and 700nm-sized PS sphere at splitter II region during the position of −5.8 μm to −5.2 μm. Similarly, for 700-nm-sized PS sphere at position of −5.4 μm, there is no potential barrier and only one minimum point at WG₃. Therefore, at splitter II with coupling length L₂ of 2.15 μm and gap of 150 nm, these PS spheres with size smaller than 600 nm diameter go to P₂, while those PS sphere with size bigger than 700 nm diameter would jump to the WG₃ and go to splitter III region for further operations. Similarly, at splitter III with coupling length L₃ of 3.45 μm and gap of 200 nm, as shown in Figs. 4(e) and 4(f), it implies that PS spheres with size bigger than 800 nm can jump to WG₄ and PS spheres smaller than 700 nm diameter will be sorted out from P₃. After the above three steps, we can sort the different-sized PS spheres to different ports at the output power ratio of 4:1 between the adjacent waveguides.

To further demonstrate the above results in a more realistic way, we also investigate the sorting effect statistically by introducing Brownian motion to a large number of nanoparticles [23]. Finally, we can obtain the statistical distribution of the particles between two adjacent waveguides. With the input power of 20 mW, we use the sample of 1000 nanoparticles in our simulation to demonstrate their distribution at different position. Figure 5 shows one-dimensional histograms of 1000 nanoparticles distribution at different position. The dotted line in Fig. 5 represents the position of the center of adjacent waveguides in y direction. As can be seen in Fig. 5(b), for the distribution at the position of −9.4 μm, 99% of the 600-nm-sized particles transfer to WG₂ and only 1% of the particles stay on WG₁. This is consistent with the conclusion from Fig. 4. However, there are more particles distributed on the WG₁ at other position along x direction and cannot jump to WG₂, as shown in Figs. 5(a) and 5(b). Similarly, particles distribution for splitter II and splitter III are demonstrated in Figs. 5(c)-5(f). At the position of −5.4 μm, 98% of 700- nm-sized particles appear on the WG₃ and only 2% of them go to WG₂, as shown in Fig. 5(d). On the other hand, at the origin of x axis, 97% of the 800-nm-sized particles go to WG₄ and only 3% of them stay on WG₃, as shown in Fig. 5(f). The statistical simulation results agree well with the potential well distribution analysis depicted in Fig. 4. It is again confirmed that different-sized PS sphere being automatically categorized into different ports can be realized.

 

Fig. 5 One dimensional histograms along y direction for PS spheres with different sizes at different gaps. (a) The nanoparticles with diameters of 500 nm at different position along x direction with g₁ = 100 nm. (b) The nanoparticles with diameters of 600 nm at different position along x direction with g₁ = 100 nm. (c) The nanoparticles with diameters of 600 nm at different position along x direction with g₂ = 150 nm. (d) The nanoparticles with diameters of 700 nm at different position along x direction with g₁ = 150nm. (e) The nanoparticles with diameters of 700 nm at different position along x direction with g₃ = 200 nm. (f) The nanoparticles with diameters of 800 nm at different position along x direction with g₃ = 200 nm.

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4. Conclusion

In summary, we have demonstrated a multi-level sorting of dielectric nanoparticles based on optical splitter with multi-step waveguide by single wavelength excitation in nanofluidic system. Numerical simulations show that the nanoparticles with different size can be categorized and exported from separated ports by the propulsion of the optical force. Furthermore, the distribution of the different-sized nanoparticles at different position which includes Brownian movement in statistics agree well with the potential distribution of the different-sized particles without considering the Brownian movement. The optical splitter with multi-step waveguides has a high degree of auto-sorting and can potentially be used to sort various different-sized particles simultaneously in a lab-on–a-chip system.