Chiral nanoparticle sorting method based on near-field optical chirality enhancement

By constructing a dielectric circular hole metasurface structure substrate and using near-field optical chiral enhancement method, the problem of difficulty in driving nanoscale chiral particles in the prior art is solved, and efficient detection and sorting of nanoscale chiral particles is achieved.

CN120028259APending Publication Date: 2025-05-23NORTHWESTERN POLYTECHNICAL UNIV
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Patent Information

Application Number
CN202510102977.2
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-01-22
Publication Date
2025-05-23

AI Technical Summary

Technical Problem

The prior art is difficult to effectively drive chiral particles at the nanoscale, resulting in the inability to efficiently detect and sort nanoscale chiral particles.

Method used

By constructing a metasurface structure substrate for dielectric circular holes and using near-field optical chiral enhancement methods, a single optical chiral distribution with enhanced enhancement is generated, and the detection and sorting of nanoparticles are achieved using chiral optical force.

Benefits of technology

It realizes efficient detection and sorting of chiral particles at nanoscale, can effectively control the dynamic behavior of chiral nanoparticles, and has strong scalability.

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Abstract

The invention discloses a chiral nanoparticle sorting method based on near-field optical chirality enhancement, and the method comprises the steps: constructing a medium circular hole metasurface structure substrate, etching a circular hole in a silicon substrate, enabling a beam of circularly polarized light to vertically enter the upper part of the substrate, and carrying out the separation of the circular hole metasurface structure substrate; according to the invention, enhanced single optical chiral distribution is generated around the structure, microfluid containing a large number of chiral nanoparticles is continuously injected from one side of the platform, and the nanoparticles with opposite chirality are respectively attracted and repelled by the substrate under the action of single chiral optical force, so that the detection and sorting of the chiral nanoparticles are realized. The chiral nano particle sorting method provided by the invention is high in functionality, can realize detection and sorting of nano-sized chiral particles, and can also control the dynamic behavior of the chiral particles. The device is high in expandability, structural parameters are changed, excitation of different incident wavelengths can be achieved, and the practical requirements of different biological particles can be met.
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Description

Technical Field

[0001] The present invention belongs to the technical field of chiral nanoparticle detection and sorting, and particularly relates to a method for sorting chiral nanoparticles based on near-field optical chirality enhancement. Background Art

[0002] Chirality refers to the property that a substance cannot coincide with its mirror image no matter how it is rotated or moved. Detecting and sorting substances with opposite chirality has wide applications in fields such as chemistry and biology. With the development of nanotechnology, the demand for efficient detection and sorting of chiral nanoparticles is increasing day by day.

[0003] The light field also has chirality. When chiral particles interact with the chiral light field, the optical response of the particles and the optical force exerted on them exhibit obvious chirality dependence on the light field, which is an important means for detecting and characterizing molecular chirality. However, the optical chiral force generated by the far-field excited light field currently used is limited by the diffraction limit and cannot effectively drive chiral nanoparticles at the nanoscale, thus unable to achieve the sorting of chiral nanoparticles at the nanoscale. Therefore, there are still challenges in the chiral detection and sorting of nanoparticles at the nanoscale. Summary of the Invention

[0004] In view of this, in order to overcome the above deficiencies existing in the prior art, the purpose of the present invention is to provide a platform and method for sorting chiral nanoparticles based on near-field optical chirality enhancement, which is used to solve the problems of detection and sorting of chiral nanoparticles at the nanoscale. This method constructs a substrate with a dielectric circular hole metasurface structure, and its specific structure is to etch circular holes on a silicon substrate, and the depth of the holes does not exceed the thickness of the silicon layer. A circularly polarized polarized light is vertically incident above the substrate, and an enhanced single optical chirality distribution will be generated around the structure. A microfluid containing a large number of chiral nanoparticles is continuously injected from one side of the platform. Under the action of the single chiral optical force, chiral nanoparticles with opposite chirality are respectively attracted and repelled by the substrate, thereby realizing the detection and sorting of chiral nanoparticles.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions.

[0006] On the one hand, the present invention provides a method for sorting chiral nanoparticles based on near-field optical chirality enhancement, which includes the following steps:

[0007] S1: Construction of a dielectric circular hole structure substrate based on near-field optical chirality enhancement, selecting the material of the dielectric, and determining the thickness of the dielectric, the depth of the circular hole, and the background dielectric.

[0008] S2: Placing the dielectric circular hole metasurface structure substrate constructed in step S1 on a substrate to obtain an optical platform based on near-field optical chirality enhancement;

[0009] S3: Determine the wavelength of the incident circularly polarized plane light, and use a beam of circularly polarized plane light waves to be incident vertically to generate an optical chirality-enhanced light field around the circular hole structure;

[0010] S4: The light field force on chiral nanoparticles in the near field is calculated using the dipole approximation method to obtain the light field forces on nanoparticles with different chirality, and the motion trajectories of nanoparticles with different chirality in the light field are calculated using the Langevin equation to evaluate the sorting capability of the optical platform based on near-field optical chirality enhancement for chiral nanoparticles.

[0011] S5: continuously injecting microfluid containing a large number of chiral nanoparticles from one side of the optical platform based on near-field optical chirality enhancement to detect and sort the chiral nanoparticles.

[0012] Optionally, the method further comprises the following steps:

[0013] S6: Switching the polarization state of the incident light, such as switching left-handed circularly polarized light to right-handed circularly polarized light, can achieve a sorting effect opposite to that of S5.

[0014] Furthermore, step S1 includes the following sub-steps:

[0015] S101: According to the calculation formula of optical chirality, a dielectric material capable of generating a strong electric field and a strong magnetic field is selected, such as silicon with a refractive index of 3.57 as a material for the dielectric metasurface structure;

[0016] S102: According to the calculation of the optical chirality distribution, the depth of the circular hole is selected to be 50 nm to enhance the optical chirality around the structure;

[0017] S103: Selecting an aqueous solution as the background medium.

[0018] Further, step S2 includes placing a dielectric circular hole supersurface structure on a substrate, preferably a silicon dioxide substrate, the supersurface structure is a circular hole, the silicon layer thickness is selected to be 100 nm, and the circular hole radius is 300 nm;

[0019] Further, step S3 includes the following sub-steps:

[0020] S301: Calculate the transmittance of circularly polarized plane waves of different wavelengths under vertical incidence according to the structural parameters established in S1;

[0021] S302: Select a wavelength when the transmittance is low, calculate the optical near-field chirality distribution at this time, obtain a single chirality enhanced optical chirality distribution, and select the incident light wavelength as 1.0307 μm according to the parameters of S1;

[0022] The purpose of step S4 is to evaluate the sorting ability of the above structure for different chiral nanoparticles, including the optical field forces exerted on different chiral nanoparticles, the trapping potential wells of the structure, and the simulation of the movement trajectories of different chiral particles in the potential wells, etc., including the following sub-steps:

[0023] S401: According to the electromagnetic field distribution around the substrate of the dielectric circular hole metasurface structure calculated in S302, use the dipole approximation method to calculate the optical field force exerted on the chiral nanoparticles near the metasurface structure, and then calculate the trapping potential well;

[0024] S402: According to the optical forces exerted on the nanoparticles calculated in S401, use the Langevin equation to calculate the movement trajectories of nanoparticles with different chiralities. Determine whether the chiral nanoparticles have been effectively separated.

[0025] Furthermore, in step S5, the microfluid containing a large number of chiral nanoparticles is injected from one side of the optical platform for sorting chiral nanoparticles. Particles with opposite chiralities will be attracted and repelled by the surface of the structure respectively, thereby realizing the sorting of chiral nanoparticles.

[0026] Furthermore, optionally, in step S6, switch the polarization state of the incident light, such as switching the left-handed circularly polarized light to the right-handed circularly polarized light, to achieve a sorting effect opposite to that in step S5.

[0027] Compared with the prior art, the present invention has the following beneficial effects:

[0028] The chiral nanoparticle sorting and detection platform and method proposed by the present invention have strong functionality, can realize the detection and sorting of nanoscale chiral particles, and can also control the dynamic behavior of chiral nanoparticles.

[0029] The present invention has strong scalability. By changing the structural parameters, different incident wavelengths can be excited, which can meet the practical needs of different biological particles. At the same time, the trapping and attraction of particles with opposite chiralities can be achieved by switching the left-handed and right-handed states of the incident light. Compared with the far-field optical field, the near-field optical field can break through the diffraction limit and enhance the optical chirality of the optical field. Description of the Drawings

[0030] Figure 1 is the flowchart of the method of the present invention;

[0031] Figure 2 (a) is a schematic diagram of the optical platform of the present invention, Figure 2 (b) is a schematic diagram and structural parameters of a single dielectric circular hole;

[0032] Figure 3 is the enhanced electric field intensity (a), enhanced magnetic field intensity, and distribution of enhanced optical chirality of the near field (c) around a specific structure when the left-handed circularly polarized light is incident;

[0033] Figure 4 The enhanced electric field intensity (a), enhanced magnetic field intensity (b), and the optical chirality distribution of the near-field enhancement (c) around a specific structure when right-handed circularly polarized light is incident;

[0034] Figure 5 The force distribution of the near-field light field distribution on nanoparticles with a chirality parameter κ = -1 and a radius r = 20 nm when left-handed circularly polarized light is incident; Figure 5 (b) The force distribution of the near-field light field distribution on nanoparticles with a chirality parameter κ = 1 and a radius r = 20 nm; The arrows in the figure indicate the direction of the force.

[0035] Figure 6 The trapping potential well distribution of the near-field light field distribution on nanoparticles with a chirality parameter κ = -1 and a radius r = 20 nm when left-handed circularly polarized light is incident; Figure 6 (b) The trapping potential well distribution of the near-field light field distribution on nanoparticles with a chirality parameter κ = 1 and a radius r = 20 nm;

[0036] Figure 7 The trapping potential wells experienced by nanoparticles with different radii and chirality parameters κ = ±1 at z = 120 nm when left-handed circularly polarized light is incident;

[0037] Figure 8 The trapping potential wells experienced by nanoparticles with radii r = 20 nm, chirality parameters κ = ±1, ±0.8, ±0.6, ±0.4, ±0.2, 0 at z = 120 nm when left-handed circularly polarized light is incident;

[0038] Fig. 9 When left-handed circularly polarized light is incident, (a) is the movement trajectory of nanoparticles with a chirality parameter κ = -1 and a radius r = 20 nm in the solution, Fig. 9 (b) is the movement trajectory of nanoparticles with a chirality parameter κ = 1 and a radius r = 20 nm. Detailed implementation manners

[0039] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be described in more detail below in conjunction with the accompanying drawings of the specification, but the present invention is not limited thereto.

[0040] The present invention provides a chiral nanoparticle sorting method based on near-field optical chirality enhancement, which constructs a dielectric circular hole metasurface structure substrate, and its specific structure is to etch multiple circular holes on a silicon substrate, and the depth of the holes does not exceed the thickness of the silicon layer. A beam of circularly polarized polarized light is vertically incident on the substrate, and a single optical chirality distribution with enhancement is generated around the structure. A microfluid containing a large number of chiral nanoparticles is continuously injected from one side of the platform. Under the action of a single chiral optical force, nanoparticles with opposite chirality are attracted and repelled by the substrate, respectively, thereby realizing the detection and sorting of chiral nanoparticles.

[0041] The chiral nanoparticle sorting method based on near-field optical chirality enhancement provided in an embodiment of the present invention comprises the following steps:

[0042] S1: Based on the construction of a substrate with a medium circular hole structure enhanced by near-field optical chirality, the material of the medium is selected, and the thickness of the medium, the depth of the circular hole, and the background medium are determined. Preferably, the specific implementation method of this step is as follows:

[0043] S101: According to the calculation formula of optical chirality, a dielectric material capable of generating a strong electric field and a strong magnetic field is selected, and silicon with a refractive index of 3.57 is selected as the material of the dielectric metasurface structure;

[0044] S102: According to the calculation of the optical chirality distribution, the depth of the circular hole is selected to be 50nm to enhance the optical chirality around the structure; 16 circular holes are etched on the silicon substrate. Among them, the super surface structure is a circular hole, the thickness of the silicon layer is selected to be 100nm, and the radius of the circular hole is 300nm;

[0045] S103: Selecting an aqueous solution as the background medium.

[0046] S2: Placing the dielectric circular hole metasurface structure on a silicon dioxide substrate to obtain an optical platform based on near-field optical chirality enhancement.

[0047] S3: Determine the wavelength of the incident circularly polarized plane light, and use a beam of circularly polarized plane light waves to be incident vertically to generate an optical chirality-enhanced light field around the circular hole structure. Preferably, the specific implementation method of this step is as follows:

[0048] S301: Calculate the transmittance of circularly polarized plane waves of different wavelengths under vertical incidence according to the structural parameters established in S1;

[0049] S302: Select a wavelength when the transmittance is low, calculate the optical near-field chirality distribution at this time, obtain a single chirality enhanced optical chirality distribution, and select the incident light wavelength as 1.0307 μm according to the parameters of S1.

[0050] S4: Calculate the optical field force on the chiral nanoparticles in the near field using the dipole approximation method, obtain the optical field force on nanoparticles of different chirality, calculate the motion trajectories of nanoparticles of different chirality in the optical field using the Langevin equation, and evaluate the sorting capability of the optical platform based on near-field optical chirality enhancement for chiral nanoparticles. The specific implementation method of this step is as follows:

[0051] S401: Calculate the optical field force on the chiral nanoparticles near the structure based on the electromagnetic field distribution around the substrate of the dielectric circular hole metasurface structure calculated in S302, and then calculate the capture potential well;

[0052] S402: Based on the optical force on the nanoparticles calculated in S401, the Langevin equation is used to calculate the motion trajectories of nanoparticles of different chirality, and to determine whether the chiral nanoparticles are effectively separated.

[0053] S5: continuously injecting a microfluid containing a large number of chiral nanoparticles from one side of the optical platform based on near-field optical chirality enhancement to detect and sort the chiral nanoparticles. Nanoparticles with opposite chirality will be attracted and repelled by the surface of the structure, respectively, thereby achieving chiral nanoparticle sorting.

[0054] Optionally, the method further comprises the following steps:

[0055] S6: Switching the polarization state of the incident light, such as switching left-handed circularly polarized light to right-handed circularly polarized light, can achieve a sorting effect opposite to that of S5.

[0056] like Figure 1 As shown, the steps of the chiral nanoparticle sorting platform based on near-field optical chirality enhancement are described, and the embodiment is further explained based on this step.

[0057] Figure 2 (a) is a specific platform structure with silicon as the dielectric material, including a dielectric nanopore periodic structure on a silicon dioxide substrate, and the background is an aqueous solution of chiral nanoparticles. A circularly polarized plane light wave is vertically incident on the upper surface of the periodic structure, and then an enhanced optical chirality distribution is generated in the near-field region of the upper surface. Figure 2 (b) is a schematic diagram of the structure of a single silicon nanopore, including the silicon layer thickness D, the nanopore radius R and the depth H. In this example, D = 100nm, R = 300nm, H = 50nm, the period is 1.5um, and the incident light wavelength λ = 1.0307μm.

[0058] Figure 3 (a) and (b) show the electric and magnetic field intensity distribution in the xz plane near the silicon circular hole structure after the left-handed circularly polarized plane light wave with a wavelength of λ = 1.0307μm is incident. It can be seen that there is not only an electric field enhancement but also a magnetic field enhancement around the structure. Figure 3 (c) is the optical chirality distribution calculated according to the optical chirality density formula. It can be seen that the structure enhances the optical chirality of the incident circularly polarized light by nearly 8 times, and the optical chirality outside the Si structure is all positive, which is a single chirality. It is this single and enhanced chirality distribution that generates the chiral force capable of separating chiral particles.

[0059] Figure 4 For the enhancement of the electric field intensity (a), the enhancement of the magnetic field intensity, and the optical chirality distribution of the near-field enhancement (c) around a specific structure when right-handed circularly polarized light is incident. Compared with the case when left-handed circularly polarized light is incident, the enhancement distributions of the electric and magnetic field intensities do not change, and only the positive and negative of the near-field optical chirality distribution change. Therefore, the attraction or repulsion to different chiral particles can be changed by switching the polarization state of the incident light.

[0060] Figure 5 Is the optical force exerted by the silicon nanopores on chiral nanoparticles calculated according to the dipole approximation method. The dipole approximation method is applicable to calculating the optical force on nanoparticles with a radius much smaller than the wavelength. Figure 5 Shows the optical force distribution of chiral particles with chiral parameter κ = ±1 and radius r = 20 nm in the xz plane. The power density of the incident light is 100 mW / μm^2, and other parameters of the chiral particles are set as μ p = 1, e p = 2.1 - 0.01i. It can be seen that the forces of the structure on the nanoparticles of chiral enantiomers push the nanoparticles with chiral parameter κ = -1 Figure 5 (a)] and κ = 1 Figure 5 (b)] towards the bottom of the nanopore and push the nanoparticles away from the surface of the nanopore, respectively.

[0061] Figure 6 Shows the trapping potential well distribution of the near-field light field for nanoparticles with chiral parameter κ = ±1 and radius r = 20 nm. It can be seen that for left-handed circularly polarized light incidence, the positive and negative of the optical potential well depend on the chiral parameter κ. For κ = -1 Figure 6 (a)], the value of the optical trapping potential well is always negative, proving that it shows a downward attraction to the particles. At the same time, the value of the potential well is also large enough to overcome Brownian motion (1k B T). For κ = 1 Figure 6 (b)], the value of the optical trapping potential well is always positive, proving that it shows an upward repulsive force to the particles. At the same time, the value of the potential well is also large enough to overcome Brownian motion (1k B T), proving that the proposed scheme of the present invention can provide the optical potential well required for chiral sorting.

[0062] The present invention numerically studies the size sorting ability of the proposed scheme for chiral nanoparticles. Figure 7The capture potential well of chiral nanoparticles at z = 100 nm is plotted when κ = ±1 and the particle radius r changes from 30 nm to 5 nm. It can be seen that as the radius of the chiral particle decreases, the value of the optical potential well gradually decreases, but the positive and negative values ​​do not change. The optical potential well is greater than 1k B T can achieve weak capture. Therefore, the sorting radius limit of the chiral particles with κ=±1 in the present invention is r=10 nm.

[0063] The present invention numerically studies the working range of the chiral parameters of chiral nanoparticles in the proposed scheme. Figure 8 When the radius r = 20nm, the chiral parameters κ = ±1, ±0.8, ±0.6, ±0.4, ±0.2, and 0 nanoparticles are trapped at z = 100nm. It can be seen that when the particle has no chirality, that is, κ = 0, the optical potential well is always greater than 0 at the edge of the nanopore and equal to 0 at the center of the nanopore, which means that the optical gradient force is an upward repulsive force on the particle. As the chiral parameter becomes negative, the non-chiral gradient force is gradually suppressed by the optical chiral force, and the optical potential well begins to produce negative values. As the chiral parameter becomes -1, the range of the negative potential well gradually increases, which means that the sortable area gradually increases. On the contrary, as the chiral parameter becomes positive, the optical potential well at the center of the nanopore begins to produce positive values ​​and increases as the chiral parameter increases. The numerical results show that for chiral nanoparticles with r = 20nm, the sorting limit of the chiral parameter is about κ = ±0.2, but the sorting range at this time is smaller than that of chiral nanoparticles with κ = ±1.

[0064] Finally, the present invention uses the Langevin equation to calculate the motion trajectory of the aqueous solution with chiral parameter κ=±1 and radius r=20nm. Fig. 9 (a) shows the initial positions of the chiral nanoparticles with κ = -1 at (-200, 100), (-100, 100) (-100, 100) (0, 100), (100, 100), and (200, 100). As can be seen, the asterisks indicate the initial positions of the particles and the circles indicate the final positions of the particles. Fig. 9 (b) shows the motion trajectories of chiral nanoparticles with κ = 1 at different initial positions. It can be seen that the chiral nanoparticles with κ = 1 move in the opposite direction of the nanopore. Therefore, nanoparticles with opposite chirality are separated as expected.

[0065] The present invention proposes a method for detecting and sorting chiral nanoparticles based on near-field optical chirality enhancement. The optical platform can generate enhanced near-field optical chirality by irradiating a specific medium structure with a specific incident light field. This enhanced near-field optical chirality has a single chirality symbol, and the chiral force has opposite mechanical effects on particles of different chirality, thereby capturing nanoparticles with specific chirality. Specifically, since the near-field optical chirality generated by the interaction between the optical platform and the incident light has a single chirality, nanoparticles with the same chirality in the field will be captured and adsorbed on the platform surface, while nanoparticles with opposite chirality will be repelled by the platform, thereby enabling the detection and sorting of chiral nanoparticles.

[0066] It should be noted that the above-described embodiments are only preferred embodiments of the present invention. For those skilled in the art, without departing from the principles of the present invention, the present invention may be modified, improved and replaced with equivalents, and these modifications, improvements and equivalent replacements are also considered to fall within the scope of protection of the claims of the present invention.

Claims

1. A chiral nanoparticle sorting method based on near-field optical chirality enhancement, characterized in that: The steps include: S1: Based on the construction of the substrate of the dielectric circular hole metasurface structure with near-field optical chirality enhancement, the material of the dielectric is selected, and the thickness of the dielectric, the depth of the circular hole and the background dielectric are determined; S2: placing the dielectric circular hole metasurface structure base constructed in step S1 on a substrate to obtain an optical platform based on near-field optical chirality enhancement; S3: Determine the wavelength of the incident circularly polarized plane light, and use a beam of circularly polarized plane light waves to be incident vertically to generate an optical chirality-enhanced light field around the circular hole structure; S4: Calculate the optical field force on chiral nanoparticles in the near field using the dipole approximation method, obtain the optical field force on nanoparticles of different chirality, calculate the motion trajectories of nanoparticles of different chirality in the optical field using the Langevin equation, and evaluate the sorting capability of the optical platform based on near-field optical chirality enhancement for chiral nanoparticles; S5: continuously injecting microfluid containing a large number of chiral nanoparticles from one side of the optical platform based on near-field optical chirality enhancement to detect and sort the chiral nanoparticles.

2. The chiral nanoparticle sorting method based on near-field optical chirality enhancement according to claim 1, characterized in that: The following steps are also included: S6: Switching the polarization state of the incident light, that is, switching from left-handed circularly polarized light to right-handed circularly polarized light or from right-handed circularly polarized light to left-handed circularly polarized light, to achieve a sorting effect opposite to step S5.

3. The chiral nanoparticle sorting method based on near-field optical chirality enhancement according to claim 1 or 2, characterized in that: The step S1 comprises the following steps: S101: According to the calculation formula of optical chirality, a dielectric material capable of generating a strong electric field and a strong magnetic field is selected; S102: according to the calculation of the optical chirality distribution, selecting the depth of the circular hole to enhance the optical chirality around the structure; S103: Selecting an aqueous solution as the background medium.

4. The chiral nanoparticle sorting method based on near-field optical chirality enhancement according to claim 1 or 2, characterized in that: The step S3 comprises the following steps: S301: Calculating the transmittance of circularly polarized plane waves of different wavelengths under vertical incidence according to the structural parameters established in step S1; S302: Select a wavelength when the transmittance is relatively low, calculate the optical near-field chirality distribution at this time, and obtain an enhanced optical chirality distribution of a single chirality.

5. The chiral nanoparticle sorting method based on near-field optical chirality enhancement according to claim 1 or 2, characterized in that: The step S4 comprises the following steps: S401: Calculate the optical field force on the chiral nanoparticles near the substrate of the dielectric circular hole metasurface structure according to the electromagnetic field distribution around the substrate of the dielectric circular hole metasurface structure calculated in step S3, and then calculate the capture potential well; S402: According to the light field force and the capture potential well of the nanoparticles calculated in step S401, the movement trajectories of the nanoparticles with different chirality are calculated using the Langevin equation to determine whether the chiral nanoparticles are effectively separated.

6. The chiral nanoparticle sorting method based on near-field optical chirality enhancement according to claim 3, characterized in that: In step S101, silicon with a refractive index of 3.57 is selected as the dielectric material of the dielectric circular hole metasurface structure substrate.

7. The chiral nanoparticle sorting method based on near-field optical chirality enhancement according to claim 6, characterized in that: The thickness of the silicon layer is selected to be 100 nm and the radius of the circular hole is selected to be 300 nm.

8. The method for sorting chiral nanoparticles based on near-field optical chirality enhancement according to claim 3, characterized in that: In step S102 , the depth of the circular hole is selected to be 50 nm.

9. The method for sorting chiral nanoparticles based on near-field optical chirality enhancement according to claim 1, characterized in that: The substrate is a silicon dioxide substrate.