I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. SURFACE-ENHANCED CHIR...III. LONG-DISTANCE CHIRAL...IV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESThe microfluidic chip, also known as Lab on a chip, was first proposed by Manz in the early 1990s.11. A. Manz, D. J. Harrison, E. M. J. Verpoorte, J. C. Fettinger, A. Paulus, H. Lüdi, and H. M. Widmer, J. Chromatogr. A 593, 253 (1992). https://doi.org/10.1016/0021-9673(92)80293-4 It is a micro-scale device that integrates various functional units and possesses a basic role similar to that of a real laboratory.2,32. S. Haeberle and R. Zengerle, Lab Chip 7, 1094 (2007). https://doi.org/10.1039/b706364b3. A. Wu, Y. Y. Tanaka, and T. Shimura, APL Photonics 6, 126104 (2021). https://doi.org/10.1063/5.0069371 To date, microfluidic chips have been wildly applied in many different fields, including disease diagnosis, biological analysis, and environmental monitoring.4,54. C. D. Chin, V. Linder, and S. K. Sia, Lab Chip 7, 41 (2007). https://doi.org/10.1039/b611455e5. L. Y. Yeo, H.-C. Chang, P. P. Y. Chan, and J. R. Friend, Small 7, 12 (2011). https://doi.org/10.1002/smll.201000946 In addition, it can also play an important role in reducing detection costs and improving detection efficiency and analysis accuracy. Today’s microfluidic chip technology can analyze various forms of target substances, including solid micro/nanoparticles, different solvent forms (liquid streams or droplets), gaseous states, cells, organs, and even living objects.6–96. Y. Schaerli, R. C. Wootton, T. Robinson, V. Stein, C. Dunsby, M. A. A. Neil, P. M. W. French, A. J. deMello, C. Abell, and F. Hollfelder, Anal. Chem. 81, 302 (2009). https://doi.org/10.1021/ac802038c7. V. Studer, A. Pépin, Y. Chen, and A. Ajdari, Analyst 129, 944 (2004). https://doi.org/10.1039/b408382m8. M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, Nat. Biotechnol. 23, 83 (2005). https://doi.org/10.1038/nbt10509. R. W. Applegate, Jr., J. Squier, T. Vestad, J. Oakey, D. W. M. Marr, P. Bado, M. A. Dugan, and A. A. Said, Lab Chip 6, 422 (2006). https://doi.org/10.1039/b512576fOn the other hand, the recognition of chiral molecules has gained much attention in the biochemical and pharmaceutical industries.10–1210. T. Zhang, M. R. C. Mahdy, Y. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C.-W. Qiu, ACS Nano 11, 4292 (2017). https://doi.org/10.1021/acsnano.7b0142811. S. Huang, H. Yu, and Q. Li, Adv. Sci. 8, 2002132 (2021). https://doi.org/10.1002/advs.20200213212. G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, M. A. Iatì, O. M. Maragò, and P. Biagioni, J. Phys. Chem. C 123, 28336 (2019). https://doi.org/10.1021/acs.jpcc.9b06508 Chirality describes the geometric property of a structure that doesn’t coincide with its mirror image for any shifts and turns.13,1413. V. V. Klimov, I. V. Zabkov, A. A. Pavlov, and D. V. Guzatov, Opt. Express 22, 18564 (2014). https://doi.org/10.1364/oe.22.01856414. K. Serita, E. Matsuda, K. Okada, H. Murakami, I. Kawayama, and M. Tonouchi, APL Photonics 3, 051603 (2018). https://doi.org/10.1063/1.5007681 Many biologically active molecules are chiral, and the individual enantiomer has completely different chemical behaviors.15,1615. A. Hayat, J. P. B. Mueller, and F. Capasso, Proc. Natl. Acad. Sci. U. S. A. 112, 13190 (2015). https://doi.org/10.1073/pnas.151670411216. L. Wang, A. M. Urbas, and Q. Li, Adv. Mater. 32, 1801335 (2020). https://doi.org/10.1002/adma.201801335 In this case, detecting and characterizing chiral enantiomers of these biomolecules is of considerable importance for biomedical diagnostics and pathogen analyses. A common technique for chirality discrimination is based on the all-optical separation of enantiomers,17–2317. M. H. Alizadeh and B. M. Reinhard, ACS Photonics 2, 1780 (2015). https://doi.org/10.1021/acsphotonics.5b0051618. I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 6, 36884 (2016). https://doi.org/10.1038/srep3688419. A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, New J. Phys. 15, 123037 (2013). https://doi.org/10.1088/1367-2630/15/12/12303720. L. Kang, Q. Ren, and D. H. Werner, ACS Photonics 4, 1298 (2017). https://doi.org/10.1021/acsphotonics.7b0005721. R. P. Cameron, S. M. Barnett, and A. M. Yao, New J. Phys. 16, 013020 (2014). https://doi.org/10.1088/1367-2630/16/1/01302022. G. Tkachenko and E. Brasselet, Nat. Commun. 5, 3577 (2014). https://doi.org/10.1038/ncomms457723. N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, Nano Lett. 12, 977 (2012). https://doi.org/10.1021/nl204055r where the direction of the chiral optical force acting on enantiomers is opposite. In general, the chiral optical force acting on chiral nanoparticles is very weak. The question of how to enhance the chiral optical force is an important research direction.Recent investigations have shown that the strength of the chiral optical gradient force is related to the gradient for the chirality of optical fields.24–2624. V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, Adv. Mater. 25, 2517 (2013). https://doi.org/10.1002/adma.20120517825. L. Fang and J. Wang, Phys. Rev. Lett. 127, 233902 (2021). https://doi.org/10.1103/physrevlett.127.23390226. T. Shoji and Y. Tsuboi, J. Phys. Chem. Lett. 5, 2957 (2014). https://doi.org/10.1021/jz501231h It is found that the surface-enhanced gradients of optical chirality can be achieved in various artificial nanostructures, such as photonic crystal plates,27,2827. S. S. Hou, Y. Liu, W. X. Zhang, and X. D. Zhang, Opt. Express 29, 15177 (2021). https://doi.org/10.1364/oe.42324328. Y. Liu, S. Hou, W. Zhang, and X. Zhang, J. Phys. Chem. C 126, 3127 (2022). https://doi.org/10.1021/acs.jpcc.1c09921 plasmonic elliptical nanoholes,2929. Z.-H. Lin, J. Zhang, and J.-S. Huang, Nanoscale 13, 9185 (2021). https://doi.org/10.1039/d0nr09080h split ring resonators,3030. M. H. Alizadeh and B. M. Reinhard, ACS Photonics 2, 361 (2015). https://doi.org/10.1021/ph500399k and coaxial plasmonic apertures.3131. Y. Zhao, A. A. E. Saleh, and J. A. Dionne, ACS Photonics 3, 304 (2016). https://doi.org/10.1021/acsphotonics.5b00574 These systems can enhance the gradient for the chirality of optical near-fields and make the separation of enantiomers realizable. However, these nanostructures do not allow the chirality-dependent optical manipulation over a large area, making the separation distance of enantiomers limited. In this case, how to create surface-enhanced chiral separation platforms with spatially extended properties is still very important for the complete separation of chiral particles.

In this work, we design a silicon-based microfluidic chip to achieve the separation of chiral nanoparticles over a large distance. By breaking the mirror-symmetry of a pair of lossy waveguides, two original orthogonal modes are coupled to form a vector exceptional point (EP), where the surface-enhanced superchiral gradient field can be generated in the microfluidic chip. Such a chiral gradient field can create large chiral gradient forces, pushing enantiomers toward different sides of the slot. Moreover, we construct cascade vector EPs on a single microfluidic chip to fulfill the chiral separation with a larger spatial distance. Based on particle tracking simulations, we further demonstrate the feasibility of our designed microfluidic chips. Our work suggests a useful method for the high-efficiency separation of enantiomers and shows an exciting prospect for next-generation chiral separation technologies.

II. SURFACE-ENHANCED CHIRAL GRADIENT OPTICAL FORCES IN MICROFLUIDIC CHIPS WITH VECTOR EXCEPTIONAL POINTS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. SURFACE-ENHANCED CHIR... <<III. LONG-DISTANCE CHIRAL...IV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESIn silicon-based micro/nano-fluidics, various functional units are generally created by modifying or combining some simple waveguide structures.32–3432. L. Yin, H. Zhang, H. Zhang, S.-M. Yang, and W. Zhang, Int. J. Biosci., Biochem. Bioinf. 9, 237 (2019), The School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai, PR China. https://doi.org/10.17706/ijbbb.2019.9.4.237-24733. W. Jiao, G. Wang, Z. Ying, Y. Zou, H.-p. Ho, T. Sun, Y. Huang, and X. Zhang, Opt. Lett. 41, 2652 (2016). https://doi.org/10.1364/ol.41.00265234. Y. Yan, J. Wang, S. Chang, Y. Geng, L. Chen, and Y. Gan, RSC Adv. 9, 38814 (2019). https://doi.org/10.1039/c9ra06428a In this case, we first consider a simple microfluidic chip, which is composed of a silicon (Si) slot waveguide, a silicon dioxide (SiO2) substrate, and an overlying cladding layer, as shown in Fig. 1(a). The structure is immersed in water to simulate the liquid environment for chiral separations. Here, w1 and w2 represent the width of the two arms (arm1 and arm2), ws is the slot width between the two arms, and h marks the thickness of the Si layer. The refractive indices of the silicon strips, SiO2 substrate, and background are set to nsi = 3.609 + i0.0021, ns = 1.45, and nb = 1.33, respectively. The intrinsic loss of silicon can introduce non-Hermitian properties into the structure, triggering the appearance of vector EPs. The vector EP is formed by coupling a pair of optical modes with orthogonal polarizations.3535. T. Wu, W. Zhang, H. Zhang, S. Hou, G. Chen, R. Liu, C. Lu, J. Li, R. Wang, P. Duan, J. Li, B. Wang, L. Shi, J. Zi, and X. Zhang, Phys. Rev. Lett. 124, 083901 (2020). https://doi.org/10.1103/physrevlett.124.083901 It is different from conventional EPs induced by the interplay between two optical modes with the same polarization. The eigenstate of the vector EP can induce strong and uniform superchiral fields, which is useful for chiral separation.Eigenmodes of the mirror-symmetric waveguide slot (the widths w1 and w2 are equal) can be classified into two types: transverse electric (TE)-like mode and transverse magnetic (TM)-like mode. The left and right charts in Fig. 1(b) show the intensity distributions (the colormap) and polarization directions (the arrows) of the TM-like and TE-like modes in the system with w1 = w2 = 256 nm. Here, the refractive index of the covering dielectric layer (nc) is chosen to be 1.679. Other structural parameters are set as ws = 81.4 nm and h = 245 nm. For the TE-like (TM-like) mode, the electric field is mostly parallel (perpendicular) to the mirror-symmetric plane, and the magnetic field is mostly perpendicular (parallel) to the mirror-symmetric plane.3636. X. Zhi-Gang, L. Zhi-Yuan, and Z. Dao-Zhong, Chin. Phys. Lett. 25, 2089 (2008). https://doi.org/10.1088/0256-307x/25/6/045 It can be seen that the distributions of E and H fields in the left and right waveguides are identical. It is important to note that the strength of the magnetic (electric) field in the slot is relatively weak for the TM-like (TE-like) mode. Therefore, it is expected that the structure would not give rise to the large optical chirality [C = −0.5ɛ0μ0ω Im(E* · H)], which requires enhanced electric and magnetic fields with parallel polarizations, by uniquely exciting the TM-like or TE-like mode.To solve this issue, we tune the width of the right waveguide (arm2) to break the mirror symmetry so that the original orthogonal TE-like and TM-like modes can couple with each other. The effective refractive index (neff) corresponds to the eigenvalue of two-coupled waveguides. In this case, the effective Hamiltonian of the system can be written as36–4036. X. Zhi-Gang, L. Zhi-Yuan, and Z. Dao-Zhong, Chin. Phys. Lett. 25, 2089 (2008). https://doi.org/10.1088/0256-307x/25/6/04537. Y. Huang, Y. Shen, and G. Veronis, Opt. Express 27, 37494 (2019). https://doi.org/10.1364/oe.27.03749438. W. D. Heiss, J. Phys. A: Math. Theor. 45, 444016 (2012). https://doi.org/10.1088/1751-8113/45/44/44401639. S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, J. Lightwave Technol. 34, 5258 (2016). https://doi.org/10.1109/jlt.2016.260989940. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Nat. Mater. 18, 783 (2019). https://doi.org/10.1038/s41563-019-0304-9with neffTE=nr,effTE+jni,effTE(neffTM=nr,effTM+jni,effTM) being the complex refractive index of the original decoupled TE-like (TM-like) mode. The parameter δ quantifies the coupling strength. The eigenvalues of the above Hamiltonian are derived asneff±=nr,effTE+nr,effTM2+jni,effTE+ni,effTM2±δ2+nr,effTE−nr,effTM+jni,effTE−ni,effTM42.(2)We note that the vector EP arises at the square-root branch point when two eigenvalues coalesce. This condition can be achieved with nr,effTE = nr,effTM and |δ| = ni,effTE−ni,effTM/2. Therefore, the TE-like and TM-like modes should possess the same real part but different imaginary parts of refractive indices to create an EP. Then, by tuning the coupling strength and intrinsic losses of two modes, the eigenvectors of the two modes can coalesce, giving rise to the vector EP. It is noted that the relationships between structural parameters (e.g., width, loss) with the complex refractive index of the TE-like (TM-like) waveguide mode neffTE (neffTM) and the effective coupling coefficient δ can be numerically obtained (see S1 in the supplementary material for details). Figures 1(c) and 1(d) show the real and imaginary parts of the effective refractive index in the two-parameter space (the width w2 and the dielectric loss of Si strips γ). It is noted that the EP corresponds to the branch point with two eigenmodes splitting. To further illustrate the formation of vector EPs, we plot variations of real and imaginary parts of the effective refractive index as a function of the loss, when the width w2 is fixed to 138.4 nm, as shown in Figs. 1(e) and 1(f). It is clearly shown that when the loss γ is less than 0.0021, the real part of the effective refractive index is separated and the imaginary part intersects. When the loss γ is greater than 0.0021, the imaginary part of the effective refractive index is separated and the real part intersects. These phenomena are consistent with the special topology of the eigenvalue Riemann surface in the vicinity of the EP. In Figs. 1(g) and 1(h), we further plot the real and imaginary parts of the effective refractive index for the system as a function of the width w2 when the loss of Si strips is set as γ = γep = 0.0021 (being consistent with the real loss4141. C. Schinke, P. Christian Peest, J. Schmidt, R. Brendel, K. Bothe, M. R. Vogt, I. Kröger, S. Winter, A. Schirmacher, S. Lim, H. T. Nguyen, and D. MacDonald, AIP Adv. 5, 067168 (2015). https://doi.org/10.1063/1.4923379). It proves the existence of EP when the width w2 of arm2 is set to138.4 nm (both real and imaginary parts of the effective refractive index being identical).In the following, to illustrate the near-field characteristics of the vector EP in our proposed microfluidic chip system, we numerically calculate the enhancement of near-field chirality by exciting the vector EP with the effective refractive index being neff = 2.90252 + i0.00232. Here, the incident power is 100 mW and the wavelength is set as λ = 900 nm. Spatial distributions of the electric field, magnetic field, and chiral optical field around two waveguides are presented in S2 of the supplementary material, where these fields are mainly localized in the arm1. Then, we calculate the spatially averaged enhancement of the chiral field at different regions [the arm1 (C1/C10), the slot between two arms (Cs/Cs0), and the arm2 (C2/C20) as a function of the width w2], as shown in Fig. 2(a). Here, Ci0 (with i = 1, s, and 2) represents the spatial average value of the chiral optical field in different regions of the mirror-symmetric structure. It can be seen that significant enhancements of all types of near fields appear in different regions. Especially, the spatial average of the chiral optical field in the slot (Cs) is enhanced by six orders of magnitude. Figure 2(b) presents the spatially averaged enhancement of chiral gradient fields, where a three-order enhancement is also obtained. Such an extremely enhanced chiral gradient field has potential implications for the separation of chiral nanoparticles.Next, we analyze the effect of optical forces on the chiral particle. Generally speaking, when the radius of a chiral particle is much smaller than the wavelength, the influence of the nanoparticle on the property of optical forces can be ignored, indicating that nanoparticles can be treated as effective diploes. In addition, we note that the dominant component of the optical force acting on chiral particles is the optical gradient force (see S3 in the supplementary material), which can be expressed as Fg = Fe + Fh + Fk with Fe, Fh, and Fk being the electric field gradient force, magnetic field gradient force, and chiral gradient force, respectively. In particular, these different gradient forces can be described by42–4442. S. B. Wang and C. T. Chan, Nat. Commun. 5, 3307 (2014). https://doi.org/10.1038/ncomms430743. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, Opt. Express 18, 11428 (2010). https://doi.org/10.1364/oe.18.01142844. A. Y. Bekshaev, J. Opt. 15, 044004 (2013). https://doi.org/10.1088/2040-8978/15/4/044004Fe=14Reαee∇E2,Fh=14Reαmm∇H2,Fk=−12Reαem∇ImH⋅E*.(3)Here, αee, αmm, and αem represent the electrical polarizability, magnetic polarizability, and electromagnetic polarizability of the nanoparticle (detailed expressions are given in S3 of the supplementary material). It is noted that the electric field gradient force and the magnetic field gradient force are related to gradients for the square of electric and magnetic fields. The chiral gradient force is related to the gradient of optical chirality.Then, we put a chiral nanoparticle with a radius of rp = 4 nm into the slot 1 μm away from the entrance port, as shown in the inset of Fig. 2(c). The total length of the microfluidic chip is chosen to be 2 μm. In addition, the refractive index of the chiral particle is taken as nr = 1.43, and the particle chirality is taken as κ = ±0.5 for enantiomers. We calculate each component of optical gradient forces along the x axis (the electric field gradient force Fex, the magnetic field gradient force Fhx, and the chiral gradient force Fkx), as presented in Fig. 2(c). It can be seen that the non-chiral optical forces Fe and Fh are identical for enantiomers, and only the chiral gradient force Fk depends on the chirality of enantiomers. Consequently, the chiral gradient force Fk, which can be used to separate enantiomers, plays a dominant role in our designed structure with vector EPs.Furthermore, spatial distributions of the intensity and direction for the total optical force acting on enantiomers (κ = ±0.5) are shown in Figs. 3(a) and 3(b). The color represents the magnitude of the total gradient force Fg, and the arrow illustrates the direction of the gradient force. In addition, enlarged views in the x–y plane are given in the bottom insets. It is clearly shown that the direction of the optical force acting on a nanoparticle with κ = 0.5 is opposite to that with κ = −0.5. Therefore, the optical force can push chiral nanoparticles with κ = 0.5 and κ = −0.5 to the left and right sides in the waveguide slot.

III. LONG-DISTANCE CHIRAL SEPARATIONS BY MICROFLUIDIC CHIPS WITH CASCADE VECTOR EXCEPTIONAL POINTS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. SURFACE-ENHANCED CHIR...III. LONG-DISTANCE CHIRAL... <<IV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESTo further demonstrate the feasibility and efficiency of our designed microfluidic chips, we perform particle tracking simulations to illustrate the whole separation process of chiral nanoparticles based on finite element methods. It is worth noting that there are two types of forces determining the movement of chiral nanoparticles. Besides the optical force, the Brownian force FB, which can be expressed as FB=ς12πrpηkBT/Δt,45,4645. E. E. Michaelides, Int. J. Heat Mass Transfer 81, 179 (2015). https://doi.org/10.1016/j.ijheatmasstransfer.2014.10.01946. M. Kim, J. Colloid Interface Sci. 269, 425 (2004). https://doi.org/10.1016/j.jcis.2003.08.004 also plays an important role. Here, kB is the Boltzmann constant, T represents the absolute temperature, η marks the dynamic viscosity, and Δt is the time interval of the random force. In addition, the parameter ς defines a normally distributed random number with a zero mean and a unit variance in space. In the following simulation, we set the dynamic viscosity and temperature as η = 0.9 mPa⋅s and T = 293 K.To improve the separation distance of chiral particles, the designed microfluidic chip is composed of two parts, including the straight-waveguide region and the output region with two curved arms, as shown in Fig. 4(a). Here, the lengths of the straight waveguide and the output region are equal to l1 = 2 μm and l2 = 1 μm. In addition, the radius R of the bending arm is taken as 1.155 μm with the radian angle θ being 60°. The relatively small length of the output region can maintain the direction of chiral optical forces. In simulations, it is noted that the nanoparticles with opposite chiral parameters are randomly released at the input port of the microfluidic chip at t = 0 s. In addition, these two types of chiral nanoparticles are randomly released into the microfluidic chip with a period of 5 × 10−9 s. Moreover, the initial velocities of both types of nanoparticles are identical. The spatial distribution of chiral particles at different times is presented in Fig. 4(b). The red dots represent chiral particles with κ = −0.5, and the blue dots correspond to chiral particles with κ = 0.5. We can see that chiral nanoparticles are moving under the action of the optical force and Brownian force, and enantiomers are in a mixed state at t = 8 × 10−8 s. With time being increased to t = 6.2 × 10−7 s, it is clearly shown that enantiomers are gradually separated from each other, exhibiting a tendency for chiral separation. When t = 9.8 × 10−7 s, a part of the chiral nanoparticles has arrived at the output region, and the enantiomers of opposite chirality in this region are completely separated. In particular, we find that the separation distance of enantiomers at the output port is about |Δx| = 404 nm.To achieve a more complete chiral separation in the microfluidic chip, we integrate two waveguide pairs with an intermediate transition area and an output region in a single microfluidic chip, as shown in Fig. 5(a). It is worthy to note that the vector EP can be created by adjusting the width of arm1 and the material loss γ when the width of the slot channel ws is changed. In this case, two waveguide pairs can both sustain the vector EPs. Such a design is able to guarantee the same direction of chiral optical forces in these two regions, amplifying chiral separations. The slot width and length of the first (second) waveguide pairs are ws = 81.4 nm and L1 = 2 μm (ws2 = 109.9 nm and L3 = 1 μm). In addition, the width w3 of arm1 and the intrinsic loss γ2 are taken as 254.69 nm and 0.001 for the second waveguide pairs, respectively (see S4 in the supplementary material for details). Structural parameters and losses of the intermediate transition area are gradually varied from the first waveguide pairs to the second waveguide pairs, and the associated length equals 0.5 μm. It is noted that the best performance on the chiral separation requires a suitable length of the intermediate transition area (not too long or too short). The short intermediate transition region cannot adiabatically maintain the light states of two straight-waveguide regions, which degrades the effect of chiral separations. In addition, the performance of the microfluidic chip also gets worse with the length of the intermediate region becoming too long. This is due to the fact that the intermediate transition region does not support the vector EP, making the chiral gradient force smaller than the non-chiral optical forces and the Brownian motion force in this region. In this case, the chiral particles separated by the first straight-waveguide section will be mixed again in the long-length intermediate transition region, making the final chiral separation only originate from the second straight-waveguide section (see S5 in the supplementary material for numerical results on chiral separations with different lengths of the intermediate transition region). The fourth part is a bent waveguide that is exactly the same as that used in Fig. 4(a).In simulations, the same initial condition is used with respect to Fig. 4(b). Figure 5(b) displays the distribution of enantiomers at t = 1.5 × 10−6 s. It is clearly shown that the enantiomers in the optimized microfluidic chip can be completely separated at a larger distance. In particular, chiral nanoparticles with κ = 0.5 can cling to the bending arm1 in the output region. In contrast, chiral nanoparticles with opposite chirality are attached to the bending arm2. The separation distance between enantiomers at the output port is |Δx2| = 837 nm, which is more than twice that in the original design. In this case, we can see that the complete separation of chiral nanoparticles can be realized on the microfluidic chip. It is important to note that our design can be further extended to microfluidic chips with more waveguide pairs sustaining vector EPs, where a much longer separation distance can be achieved.It is worth noting that, in real experiments, our designed microfluidic chips are hard to work exactly at the exceptional point but can only work in the vicinity of it. In this case, to illustrate the feasibility of experimental chiral separations, we numerically investigate the motion of chiral nanoparticles when the designed microfluidic chips are slightly deviated from the ideal vector EP induced by the fabrication tolerance and imprecision of the operating wavelength (see S6 of the supplementary material for details). We find that our designed silicon-based microfluidic chips can still work in the vicinity of vector EPs.