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Abstract
In this work, we propose a slotted photonic crystal nanobeam cavity (PCNC) to trap and sort the 120 nm and 30 nm nanoparticles. The simulation shows that the maximum optical trapping force of the 120 nm particle is 38.7 pN/mW, and that of the 30 nm particle is 10.8 pN/mW. It is calculated that the trapping threshold power of the 120 nm particle is 35.3 µW, and that of the 30 nm particle is 41.6 µW. Because the width of the slot is 100 nm, when the input power is between 35.3 µW and 41.6 µW, only the 120 nm particle can be trapped in the upper cladding of the slotted-PCNC. When the input power is greater than 41.6 µW, the 120 nm particle is still trapped in the upper cladding of the slotted-PCNC, while the 30 nm particle is trapped inside the slot of the slotted-PCNC. By properly controlling the input power and the direction of flow in the microfluidic channel, the sorting of particles can be achieved. In addition, trapping of the particles causes different redshifts of peak wavelengths. Thus, the proposed slotted-PCNC can detect particle trapping and sorting by monitoring the resonant wavelength shifts. What is the most important, compared with previous reported single particle trapping work, is that the proposed work can realize both trapping and sorting. Therefore, provided with the ultra-compact footprint and excellent performance, the proposed slotted-PCNC shows great potential for a multifunctional lab-on-a-chip system.
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1. Introduction
Optical trapping and manipulation have been widely used in physics, biochemistry and other fields since the 1980s, when Ashkin et al. experimentally focused laser beams to trap and manipulate particles in nanometer scale [1]. However, the inherent diffraction limit of these free-space systems limits the improvement of trapping strength, and the large size of the system requires high input power, resulting in serious cost consumption [2,3]. To solve these problems, many near-field optical manipulation techniques have been proposed, such as solid waveguides [4–6], slot waveguides [7,8], optical fibers [9–12], microring resonators [13,14], plasmonic tweezers [15–17] and photonic crystal cavities [18–35]. Among them, the electric field of the photonic crystal cavities is highly localized, and there are no issues of metal heating and loss of the plasma compared with plasmonic tweezers [36,37]. These advantages make the photonic crystal cavities attract widespread attention. For example, P. F Jing et al. demonstrated an approach to achieve patterned optical trapping with two-dimensional photonic crystals (2D-PC), which generated an enhanced diffraction field for optical trapping of microbeads with high efficiency [35]. However, compared with the 2D-PC, the one-dimensional (1D) photonic crystal cavities are widely used in particle trapping due to their simple structure, small size, and easy integration [36]. S. Mandal et al. proposed a photonic crystal microcavity to realize particle trapping and realized particle manipulation with the help of microfluidic channel [27]. Y. Gao et al. designed Bow-Tie-Shaped photonic crystal nanobeam cavities to achieve single particle trapping [36]. D. Q. Yang et al. proposed a slotted photonic crystal microcavity to trap particle inside the slot, which greatly enhanced the trapping force [37].
However, most previously reported works only trap and manipulate a single particle. Multi-particles sorting technology has not been studied much, and it is essential for biological testing, cell or virus performance testing, and basic drug research [38]. Up to now, a series of sorting techniques have been proposed. M. M. Wang et al. implemented a fluorescence-activated microfluidic cell sorter using all-optical switching technology and evaluated the cell performance [39]. Afterwards, some sorting techniques based on waveguides have also been studied [38,40–42]. K. Grujic et al. used the Y-branch optical waveguide to guide particles to the branch of stronger light output [40]. The use of a 3-dB splitter composed of slot waveguide and conventional waveguide achieved the sorting of 320 nm and 2 μm particles [41]. The sorting of 600 nm, 700 nm and 900 nm particles was realized by the multi-step optical waveguide distributor [42]. While most of the previous-reported sorting work structures are complicated, and they require a high input power to achieve particle sorting, which makes them uncompetitive in power-efficient applications.
To solve these limitations above, a multifunctional slotted-PCNC is proposed to trap and sort 120 nm and 30 nm particles in this work. Due to the ingenious setting of the width of the slot, the 120 nm particle can only be trapped in the upper cladding of the slotted-PCNC, while the 30 nm particle is trapped inside the slot. Since particles with different radius have different threshold powers for stable trapping, the trapping of a single particle or two different size particles can be achieved by controlling the input power of the light source. Besides, by controlling the direction of the flow in the microfluidic channel, particle sorting can also be further realized. Therefore, by flexible adjustment of the input power and the flow direction, the trapping and sorting of different size particles can be achieved in a single cavity, thus making it versatile for multifunctional applications.
2. Theory
When a particle is placed in a cavity, it will be affected by the optical force, which includes gradient force and scattering force [43]. Due to the strong optical field enhancement of the cavity, the scattering force is negligible relative to the gradient force. The optical force can be obtained by [44]:
The corresponding trapping potential is calculated by integrating the component of the optical force F along the specific path [44]:
where Ux is the trapping potential in the x-direction, and Fx represents the component of F in the x-direction. Similarly, the trapping potentials in other directions can be obtained.3. Structure design
The schematic diagram of the proposed slotted-PCNC is shown in Fig. 1(a), and its side view is shown in Fig. 1(b), which vividly illustrates two particles are trapped in different positions. As seen, it is formed by introducing a slotted structure in the middle of the suspended waveguide, and a row of elliptical holes are etched into a silicon (nsi = 3.476) waveguide with a width wnb of 700 nm and a thickness h of 220 nm. We keep the major axes (2ry) of the elliptical holes at 500 nm, the minor axes (2rx) at 220 nm, and the width of the slot (wslot) is set at 100 nm. In order to enhance the localization of the field, the structure is designed with quadratic tapering of the lattice constant from acenter = 440 nm to aend = 470 nm on both sides (a(i) = acenter + i2 (aend - acenter) / i2max) [45]. In addition, the transverse electric-like (TE-like) fundamental mode in the single mode input silicon waveguide activated by the mode source is selected for the simulation, and the green arrow in Fig. 1(a) indicates the direction of the light source. Figure 1(c) shows the TE-like band diagram with lattice constants of 440 nm and 470 nm, which is simulated by the three-dimensional finite-difference-time-domain (3D-FDTD) method. At the same time, the black point indicates the target cavity resonant frequency of 196 THz. Figure 1(d) shows the different mirror strengths with different lattice constants, according to which, aend = 470 nm is chosen to realize maximum mirror strength. The greater the mirror strength, the smaller the power leaking into the waveguide and scattering into the surroundings, which is conducive to a more localized electric field [45].
Fig. 1. (a) Top-view and (b) side-view of the slotted-PCNC, where rx = 110 nm, ry = 250 nm, wnb = 700 nm, h = 220 nm, wslot = 100 nm, acenter = 440 nm, aend = 470 nm, imax = 20 are chosen. (c) Band diagram of the TE-like mode for acenter = 440 nm (blue) and aend = 470 nm (red). The black point indicates the resonant frequency of 196 THz. (d) The mirror strength of the cavity via changing the lattice constant.
Taking into account the actual situation, the cladding layer is set to the aqueous environment, and its refractive index is 1.311. The refractive index of the polystyrene (PS) particle (nps = 1.590) is very similar to the refractive index of living cells or proteins, so we choose a PS particle with a radius of 120 nm and a PS particle with a radius of 30 nm for simulation.
We know that the optical trapping force (Fz, z component) is directly proportional to Q·T1/2∕V [23], which is related to the number of holes. Here V is the mode volume of the sotted-PCNC, and T is the cavity transmissivity. Therefore, we explore the relationship between the optical trapping force Fz and the number of holes. Here, we take a 120 nm particle as an example for simulation, and the 120 nm particle is trapped at 30 nm from the upper surface of the slotted-PCNC (x = 0, y = 0, z = 260 nm). With the number of holes on each side changing from 5 to 35, Q/V is increasing, as shown in Fig. 2(a). It can be seen from Fig. 2(b) that the cavity transmissivity T decreases gradually with the increase of the number of holes. As can be seen from Fig. 2(c), when the number of holes on each side increases from 5 to 25, the optical trapping force Fz increases, while the trapping force Fz decreases in the process of the number of holes on each side from 25 to 35. When the number of holes on each side is 25, Q·T1/2∕V reaches the maximum, and the optical trapping force Fz also reaches the maximum, which is in accordance with the previous work [23]. Figure 2(d) shows that the optical trapping force Fz is directly proportional to Q·T1/2∕V, which has guiding significance for optimizing the structure. In short, when the number of holes on each side is 25, the structure can achieve the best performance, and its trapping force can reach 70.8 pN/mW.
Fig. 2. Influence of different hole numbers on each side imax (changed from 5 to 35) on (a) Q∕V, (b) cavity transmissivity T, and (c) maximum optical trapping force Fz on a 120 nm radius PS nanoparticle. (d) Normalized optical trapping force Fz for the slotted-PCNC with different values of Q·T1/2∕V.
However, compared with the slotted-PCNC with imax = 20, the simulation time of slotted-PCNC with imax = 25 is about 1.5 times longer because of higher Q. This matters a lot especially when we do optical force calculation on small size particles with high mesh resolution, which makes the simulation time ultra-long and un-acceptable based on the available servers. Thus, under the comprehensive consideration of simulation time and performance, we choose imax = 20 as the final structure. The optical transmission spectra of the slotted-PCNC calculated by 3D-FDTD are shown in Fig. 3(a). The resonant wavelength of the slotted-PCNC is 1561.49 nm, and its Q-factor is 16149. Figures 3(b)–3(c) show the TE-like fundamental resonant mode in the x-y plane (z = 0) and x-z plane (y = 0). From the figures, we can clearly find that the field distribution is highly concentrated at the slot, which can provide a strong gradient force for particle trapping.
Fig. 3. (a) The optical transmission spectra of the designed slotted-PCNC. Here the resonant wavelength is 1561.49 nm, and the Q-factor is 16149. (b) The electric field distribution in the x-y plane (z = 0). The electric field is mainly distributed in the slot. (c) The electric field distribution in the x-z plane (y = 0).
4. Trapping and sorting results
Figures 3(b)–3(c) show the electric field distribution of the slotted-PCNC. Due to the gradient force of the slotted-PCNC, the particles will be stably trapped. In the simulation, the 120 nm particle is trapped in the upper cladding of the slotted-PCNC because of the limitation of the width of the slot. At the same time, we assume that the 120 nm particle is trapped at a distance of 30 nm from the upper surface of the slotted-PCNC (x = 0, y = 0, z = 260 nm). However, the 30 nm particle is trapped in the center of the slotted-PCNC (x = 0, y = 0, z = 0).
The particle is moved along three coordinate axes to study the trapping characteristics of the slotted-PCNC. Figure 4 shows the trapping properties of the 120 nm particle. The optical trapping force is obtained by calculating the MST and integrating it on the outer surface of the particle, and it is normalized by input power in units of pN/mW. It can be seen from Fig. 4(a) that the zero value of the trapping force (Fx, x component) appears at x = 0 because the structure is symmetrical in the x-direction. When x > 0, the trapping force Fx received by the particle is negative, indicating that the direction of the trapping force Fx is pointing to the negative semi-axis of the x-direction, which means that when the particle is far from the center of the x-axis, the trapping force Fx will pull the particle back to the center of the x-axis. At the same time, it can be seen from Fig. 4(a) that the trapping force Fx received by the particle gradually increases, reaching the maximum at x = 100 nm, and the maximum value is 16.1 pN/mW. When x > 100 nm, the trapping force Fx on the particle gradually decreases, which is determined by the characteristics of the evanescent field. In the same way, the particle is moved along the y-direction, and the trapping force (Fy, y component) received by the simulated particle at different positions in the y-direction is shown in Fig. 4(c). The maximum trapping force Fy is 12.4 pN/mW. The trapping force (Fz, z component) in the z-direction is shown in Fig. 4(e). Since the 120 nm particle is trapped in the upper cladding of the slotted-PCNC, the trapping force Fz on the particle in the z-direction is decreasing, and the direction is negative. The maximum trapping force Fz is 38.7 pN/mW when z = 260 nm.
Fig. 4. Numerical analysis of the optical trapping force and the trapping potential for the slotted-PCNC. All calculations are done for a PS nanoparticle with a radius of 120 nm. (a) Fx, (c) Fy and (e) Fz are the trapping force along the x-axis (ranging from x = −220 nm to x = 220 nm), y-axis (ranging from y = −230 nm to y = 230 nm) and z-axis (ranging from z = 260 nm to z = 360 nm) of the device, respectively. (b), (d) and (f) show the trapping potential obtained by integrating the optical trapping force along the x-axis, y-axis and z-axis.
The corresponding trapping potential is calculated by integrating the component of the trapping force along the specific path. Figures 4(b), 4(d) and 4(f) show the corresponding trapping potentials of the 120 nm particle in x, y and z directions, respectively. The particles in the solution have Brownian motion. In order to achieve stable trapping, it is usually required that the depth of the potential energy is greater than 10 kBT, where kB represents the Boltzmann constant, and T is set to room temperature (300 K). The trapping threshold power of three directions can be calculated separately. For the x-direction, the maximum trapping potential is 530.1 kBT/mW, so the threshold in the x-direction is 18.9 µW. Similarly, according to the maximum trapping potential of 458.9 kBT/mW in the y-direction, the threshold power of 21.8 µW in the y-direction can be calculated. The maximum trapping potential in the z-direction is 283.3 kBT/mW, and the calculated threshold is 35.3 µW.
The 30 nm particle is trapped in the center of the slot (x = 0, y = 0, z = 0). Because the y-direction barrier restricts the movement of the particle, only the trapping characteristics of the particle in the x and z directions need to be studied. Figure 5(a) shows the trapping force of the particle at different positions in the x-direction. The trapping force (Fx, x component) reaches its maximum value at x = 100 nm, and the maximum force Fx is 10.8 pN/mW. The trapping potential in the x-direction is shown in Fig. 5(b). It can be seen that the maximum trapping potential is 240.2 kBT/mW, and the trapping threshold power in the x-direction can be calculated as 41.6 µW. Similarly, the trapping force (Fz, z component) and the trapping potential diagram of the particle in the z-direction can be obtained by Figs. 5(c) and 5(d). In the z-direction, the maximum trapping force Fz is 10.2 pN/mW, and the maximum trapping potential is 302.7 kBT/mW. So, the trapping threshold power in the z-direction can be calculated as 33.1 µW.
Fig. 5. Numerical analysis of the optical trapping force and the trapping potential for the device. All calculations are done for a PS nanoparticle with a radius of 30 nm. (a) Fx and (c) Fz are the trapping force along the x-axis (ranging from x = −180 nm to x = 180 nm) and z-axis (ranging from z = −220 nm to z = 220 nm) of the device, respectively. (b) and (d) show the trapping potential obtained by integrating the optical trapping force along the x-axis and z-axis.
Table 1 clearly shows the trapping performance of the two different size particles in different directions. As can be seen from Table 1, the threshold power of stable trapping for the 120 nm particle is 35.3 µW, while that for the 30 nm particle is 41.6 µW. Therefore, by adjusting the input power of the light source, single particle or two size particles can be trapped flexibly.
Table 1. Trapping performance comparison of the different size particles
Figure 6(a) shows the resonant wavelength shifts with the input power p. Considering the threshold power of the 120 nm and the 30 nm particles, if the input power is controlled between 35.3 µW and 41.6 µW, only the 120 nm particle can be trapped stably. So the resonant wavelength changes from 1561.49 nm to 1564.36 nm. When the input power is greater than 41.6 µW, both particles can be trapped at the same time. The resonant wavelength changes from 1564.36 nm to 1564.48 nm. It can be seen from Fig. 6(b) that the transmission is obviously reduced after the particles are trapped. This is because the trapped particles break the effective coupling between the input/output waveguide mode and the cavity resonant mode.
Fig. 6. Analysis of the resonant wavelength shifts with the trapping process. (a) The resonant wavelength shifts with the input power. (b) The influence of the trapping process on the transmission spectra.
According to the previous analysis in this article, the trapping positions of the two different size particles are different, and the trapping state of particles can be detected by monitoring the shifts of the resonant wavelength. Table 1 shows that the threshold power of stable trapping for the 120 nm particle is 35.3 µW, while that for the 30 nm particle is 41.6 µW. If we cleverly control the input power and the flow direction of the microfluidic channel, the sorting of the two particles can be achieved. Figure 7 clearly shows the principle of the particle sorting of the proposed slotted-PCNC. The black arrow represents the input direction of the light source, and the green arrow represents the flow direction of the microfluidic channel.
Fig. 7. Schematic diagram of the sorting principle. The black arrow represents the input direction of the light source, and the green arrow represents the flow direction of the microfluidic channel.
First, the input power of the light source is adjusted to between 35.3 µW and 41.6 µW, and the flow direction in the microfluidic channel is consistent with the direction of light transmission. The aqueous solution is injected through the microfluidic channel until the resonant wavelength of the slotted-PCNC doesn't shift. Under this power, only the 120 nm particle is being trapped, so the separated solution only contains the 30 nm particle. Then, the input power is reduced to below 35.3 µW, and the 120 nm particle can't be stably trapped. So the 120 nm particle is also separated from the solution. Finally, the sorting of the different size particles is realized by a simple way of controlling the input power.
5. Conclusion
In this article, we propose a slotted-PCNC to achieve the trapping and sorting of the 120 nm particle and the 30 nm particle. The slotted-PCNC is composed of a row of elliptical holes etched into the slotted waveguide. In order to achieve better electric field enhancement, the slotted-PCNC is designed with quadratic tapering of the lattice constant, keeping the hole size and slot width. Because the slot width is only 100 nm, the 120 nm particle can only be trapped in the upper cladding of the slotted-PCNC, while the 30 nm particle is trapped in the center of the slot. According to the 3D-FDTD simulation results, the maximum optical trapping force of the 120 nm particle is 38.7 pN/mW, and that of the 30 nm particle is 10.8 pN/mW. It is calculated that the trapping threshold power of 120 nm particle is 35.3 µW, and that of 30 nm particle is 41.6 µW. In addition, when the input power is appropriately controlled, adjusting the flow direction of the microfluidic channel can realize particle sorting. Therefore, such device with characteristics of ultra-compact footprint and excellent efficiency will be facilitated to the development of multifunctional on-chip applications.
Funding
National Natural Science Foundation of China (61431003, 61634006); Key Technologies Research and Development Program (2016YFB0402405, 2017YFA0205903).
Acknowledgments
The authors would like to thank Zixing Gou and Tongyu Nie for their discussion.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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