High-density periodic two-dimensional grid structure projection method and device
By using amplitude-phase hybrid modulation of a two-dimensional grid grating, the problems of low projection density and easy disappearance of two-dimensional grid structures are solved, achieving higher density periodic projection and improving the accuracy of microscopic imaging and photolithography.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SUZHOU PHOTON BAY SCIENTIFIC INSTRUMENTS CO LTD
- Filing Date
- 2026-03-25
- Publication Date
- 2026-05-05
AI Technical Summary
In existing technologies, the projection density of two-dimensional grid structures is low and easily disappears due to aperture limitations, resulting in the loss of high-frequency components and making it impossible to achieve high-precision projection.
A two-dimensional grid grating with amplitude-phase hybrid modulation is used. By setting an opaque region and two sets of transparent regions in a single periodic unit of the grating, and making a π phase difference between the two sets of transparent regions, combined with an aperture stop that only allows (±1,±1) order diffracted light to pass through, the projection of a high-density periodic two-dimensional grid structure is achieved.
Under the same optical system and aperture stop conditions, the ability to reproduce the details of the projected pattern is improved, and the grid density is increased to 4 times that of the original, solving the problem of low projection density in traditional methods and improving the performance of equipment in fields such as microscopic imaging and photolithography.
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Figure CN121978790A_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of optics, and in particular to a method and apparatus for projecting a high-density periodic two-dimensional grid structure. Background Technology
[0002] In the field of optical projection, periodic two-dimensional grid structures are widely used in high-precision scenarios such as microscopic imaging and photolithography. The density and clarity of the projection directly determine the performance limit of the related equipment.
[0003] In existing technologies, the projection of two-dimensional grid structures often relies on pure amplitude modulation two-dimensional grid gratings. These gratings achieve light intensity modulation through the alternating arrangement of transparent and opaque regions, thereby reproducing the grid structure on the projection surface. However, in practical applications, the aperture in the optical system acts as a low-pass filter. The smaller the aperture, the stronger the blocking effect on high-frequency diffraction orders in the light field. Since the dense details of the two-dimensional grid depend on high-frequency information, the loss of high-frequency components directly leads to a decrease in the contrast of the grid structure, and in severe cases, even the complete disappearance of the grid structure. Furthermore, when using pure amplitude modulation two-dimensional grid gratings, it is necessary to ensure that all nine diffraction orders (1,1), (1,0), (1,-1), (0,1), (0,0), (0,-1), (-1,1), (-1,0), and (-1,-1) in the light field can pass through the aperture to obtain a complete two-dimensional grid projection. Summary of the Invention
[0004] This application provides a method and apparatus for projecting high-density periodic two-dimensional grid structures. The technical solution is as follows: On the one hand, a method for projecting a high-density periodic two-dimensional mesh structure is provided, the method comprising: Projection is performed using an amplitude-phase hybrid modulation two-dimensional grid grating. Each periodic cell of the two-dimensional grid grating contains an opaque region and two sets of transparent regions, and there is a π phase difference between the two sets of transparent regions. Illuminate the amplitude-phase hybrid modulation two-dimensional grid grating; The light field modulated by the amplitude-phase hybrid modulation two-dimensional grid grating is passed through the projection optical system. The aperture stop of the projection optical system allows the (±1,±1) order diffracted light in the light field to pass through, thereby realizing the projection of a high-density periodic two-dimensional grid structure.
[0005] Optionally, the amplitude-phase hybrid modulation two-dimensional grid grating uses a transparent material as a substrate.
[0006] Optionally, the surface of the opaque area of the amplitude-phase hybrid modulation two-dimensional grid grating is covered with a light-blocking layer to achieve amplitude modulation.
[0007] Optionally, there is a thickness difference between the two sets of light-transmitting regions of the amplitude-phase hybrid modulation two-dimensional grid grating. This difference causes the phase delay of the two sets of light-transmitting regions to the incident light to differ by π, resulting in a π phase difference between the two sets of light-transmitting regions and the incident light.
[0008] Optionally, the two sets of light-transmitting regions of the amplitude-phase hybrid modulation two-dimensional grid grating are arranged alternately in a chessboard pattern, and the boundary between the two sets of light-transmitting regions is an opaque region.
[0009] On the other hand, a high-density periodic two-dimensional grid structure projection device is provided, the device comprising: A grating module is used for projection using an amplitude-phase hybrid modulation two-dimensional grid grating. Each periodic cell of the two-dimensional grid grating has an opaque region and two sets of transparent regions, and there is a π phase difference between the two sets of transparent regions. An illumination module is used to illuminate the amplitude-phase hybrid modulation two-dimensional grid grating; A projection optical system is used to allow the light field modulated by the amplitude-phase hybrid modulation two-dimensional grid grating to pass through the projection optical system. The aperture stop of the projection optical system allows the (±1,±1) order diffracted light in the light field to pass through, thereby realizing the projection of a high-density periodic two-dimensional grid structure. Attached Figure Description
[0010] Figure 1 This is a schematic diagram of a typical periodic two-dimensional grid structure; Figure 2 A schematic diagram showing the change in the projection pattern of a pure amplitude-modulated two-dimensional grid grating as the aperture shrinks through a finite-aperture optical system; Figure 3 A schematic diagram of the critical case for projection of a two-dimensional grid grating with pure amplitude modulation; Figure 4 This is a schematic diagram of the structure of a two-dimensional grid grating with amplitude-phase hybrid modulation. Figure 5 This is a schematic diagram showing the change in the projection pattern of an amplitude-phase hybrid modulation grating as the aperture shrinks through an optical system. Figure 6 This is a schematic diagram illustrating the critical case of amplitude-phase hybrid modulation grating projection. Figure 7 This is a schematic diagram of the optical path; Figure 8 This is a schematic diagram of the focusing judgment of a microscopy system based on the grating of the present invention. Detailed Implementation
[0011] To make the objectives, technical solutions, and advantages of this application clearer, the embodiments of this application will be described in further detail below with reference to the accompanying drawings.
[0012] In this article, "multiple" refers to two or more. "And / or" describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone. The character " / " generally indicates that the preceding and following related objects have an "or" relationship.
[0013] First, we will further explain the problems with the background technology.
[0014] An aperture, acting as a low-pass filter in the system, blocks high-frequency diffraction orders. The dense arrangement of the two-dimensional grid relies on high-frequency information to reproduce details. The smaller the aperture, the greater the loss of high-frequency components, leading to a decrease in the contrast of the grid structure, or even its disappearance. In an optical system with a finite aperture, to simplify the description of the aperture's function, we assume it lies in the Fourier plane of the optical system. Simultaneously, the illumination source is a parallel source. Figure 1 This is a schematic diagram of a typical periodic two-dimensional grid structure. In the diagram, the blue area represents the light-transmitting part and the black area represents the opaque part. The overall structure presents a regular periodic arrangement, clearly showing the basic structure of a traditional pure amplitude modulation two-dimensional grid grating. The dense details of this structure rely on high-frequency information for reproduction, providing a structural reference for explaining the shortcomings of existing technologies when the aperture is reduced.
[0015] Figure 2 This is a schematic diagram showing the change in the projected pattern of a pure amplitude-modulated two-dimensional grid grating as the aperture decreases through a finite-aperture optical system. As can be observed in the figure, the originally clear periodic grid structure gradually becomes blurred as the aperture of the aperture gradually decreases, the high-frequency components are continuously lost, and it eventually degenerates into a blurry pattern without obvious structure. This directly presents the problem in the existing technology where the aperture acts as a low-pass filter, blocking high-frequency diffraction orders and causing the grid structure to disappear.
[0016] Figure 3 The diagram also shows the critical case of pure amplitude modulation two-dimensional grid grating projection. The diagram shows that when the aperture is reduced to the critical size, the four diffraction orders (±1,±1) are exactly tangent to the aperture, and all nine diffraction orders must pass through the aperture to obtain the two-dimensional grid projection.
[0017] Therefore, the maximum transmittable pattern density is determined by the aperture of the optical system. When the focal length is constant, it is mainly determined by the aperture stop. For a two-dimensional periodic grid, it can be assumed that, under ideal conditions, all nine diffraction orders (1,1), (1,0), (1,-1), (0,1), (0,0), (0,-1), (-1,1), (-1,0), (-1,-1) need to pass through the aperture stop to obtain the two-dimensional grid projection. This is illustrated in the figure below. That is, the density is maximum when the four diffraction orders (±1,±1) are tangent to the aperture stop.
[0018] ,in, It's the wavelength. is the focal length, and D is the diameter of the aperture stop. It is the minimum period of the projected light intensity.
[0019] Therefore, since a high-density two-dimensional periodic light intensity distribution needs to be projected, this application designs a method to reduce the period of the light field through phase modulation without reducing the period of the light intensity distribution. This invention reduces the period of the light field by superimposing phase modulation on a two-dimensional grating. Specifically, it employs a two-dimensional grid grating with phase-amplitude hybrid modulation, as shown in the figure. Figure 4 The diagram shows a schematic of an amplitude-phase hybrid modulation two-dimensional grid grating. The black area in the diagram is the opaque part, and the blue and red areas are the transparent parts. It is clear that there is a π phase difference between the blue transparent area and the red transparent area. Example 1
[0020] To address the problems of low projection density and easy disappearance of structures due to aperture limitations in the prior art for two-dimensional grids, this application provides a high-density periodic two-dimensional grid structure projection method, specifically implemented as follows: The use of amplitude-phase hybrid modulation two-dimensional grid grating.
[0021] A single periodic unit of a two-dimensional grid grating (typically sized as follows) The grating contains an opaque region and two sets of transparent regions. A fixed π-phase difference exists between the two sets of transparent regions. This phase difference is used to modulate the light field distribution, concentrating the light energy at specific diffraction orders while suppressing redundant diffraction orders. This grating differs from traditional pure amplitude modulation gratings; it reconstructs the diffraction characteristics of the light field through a hybrid approach of "amplitude modulation (transparent / opaque) + phase modulation (π-phase difference)".
[0022] The illumination method of the grating.
[0023] A parallel light source is used to uniformly illuminate the amplitude-phase hybrid modulation two-dimensional grid grating, ensuring the stability and consistency of the incident light field and providing a basis for subsequent light field modulation and diffraction order control.
[0024] Projection transmission of light field.
[0025] The light field modulated by the grating is incident on the projection optical system, whose aperture stop is set to allow only the (±1,±1) order diffracted light in the light field to pass through. Because this invention suppresses redundant diffraction orders such as (±1,0), (0,±1), and (0,0) through phase modulation, there is no need to reserve light transmission space for these orders. Therefore, with the same aperture stop size, a higher density two-dimensional grid structure can be transmitted. The minimum period of its projected light intensity is given by the formula... The decision was made that, compared to traditional methods, under the same optical system parameters, the grid period can be reduced by half, and the grid density can be increased to four times the original.
[0026] This method achieves selective preservation of diffraction orders through amplitude-phase hybrid modulation, solving the problems of low projection density and easy structure disappearance caused by the loss of high-frequency components in traditional pure amplitude-modulated gratings. With a fixed optical system and aperture stop, it can project higher-density periodic two-dimensional grid structures, improving the detail reproduction capability of the projected pattern and providing a higher-precision projection solution for fields such as microscopy and photolithography, thereby enhancing the performance of related equipment. Under the same optical projection system and aperture stop conditions, it effectively overcomes the limitation of the aperture stop on projection density, achieving stable projection of higher-density periodic two-dimensional grid structures. Compared to traditional amplitude-modulated gratings, the grid period can be reduced by half, and the grid density increased to four times.
[0027] Furthermore, the amplitude-phase hybrid modulation two-dimensional grid grating uses a transparent material as the substrate. The transparent material must meet requirements such as good optical uniformity, stable refractive index, high transmittance (transmittance ≥90% in the working wavelength range), and strong chemical stability to ensure that the incident light can stably penetrate the substrate without introducing additional non-uniform phase shift.
[0028] Fused silica is preferably used as the substrate material. Fused silica has excellent light transmittance in the visible to near-infrared band, high refractive index uniformity, and strong thermal and chemical stability, making it suitable for etching, coating, and other processes in grating fabrication while meeting the optical characteristics required for phase modulation. For example, in Embodiment 1 of this invention, a stepped amplitude-phase hybrid grating is fabricated using a fused silica substrate. The stepped structure is formed by selective etching of the substrate, providing a physical basis for subsequent phase difference modulation.
[0029] The selection of a transparent substrate is a prerequisite for achieving amplitude-phase hybrid modulation: on the one hand, the transparent substrate allows incident light to penetrate into the light-transmitting area, ensuring that the light field can be effectively modulated; on the other hand, the stable refractive index and uniformity ensure that the phase difference between the two sets of light-transmitting areas can be precisely controlled, avoiding phase shift errors caused by the characteristics of the substrate itself, thereby ensuring the accuracy of diffraction order control and ensuring the realization of high-density projection.
[0030] Furthermore, the opaque area of the amplitude-phase hybrid modulation two-dimensional grid grating is covered with a light-blocking layer. The light-blocking layer must have characteristics such as high light-blocking rate (≥99% within the working wavelength range), strong adhesion to the substrate and the transparent area, and easy patterning. The amplitude modulation of the incident light (i.e., the "opaque" characteristic) is achieved through the physical blocking of the light-blocking layer.
[0031] For example, a patterned chromium film can be deposited on the substrate surface. Chromium (Cr) is preferably used as the light-blocking layer material. The chromium layer has excellent light-blocking performance and strong adhesion to transparent substrates such as fused silica. It can be prepared by sputtering. As in Embodiment 1 of the present invention, a 120 nm thick chromium layer is deposited on the surface of a quartz substrate (including a stepped structure) by a global sputtering process. Subsequently, the light-blocking layer is patterned by a process of spin coating, overlay exposure / development, and wet etching (or Ar ion milling). Only the chromium layer in the opaque area within the periodic unit is retained, while the chromium layer in the transparent area is etched away, thereby forming an amplitude modulation structure of "transparent area (no light-blocking layer) + opaque area (with light-blocking layer)".
[0032] The light-blocking layer enables precise amplitude modulation. By completely blocking incident light from the opaque areas, the light-blocking layer clearly distinguishes the transparent and opaque regions of the grating, defining an effective area for subsequent phase modulation. Simultaneously, the patterning precision of the light-blocking layer directly determines the periodic accuracy of the two-dimensional grid structure, thus affecting the clarity and density of the projected pattern. The combination of the light-blocking layer and the phase modulation structure constitutes an amplitude-phase hybrid modulation system, providing a means to control the amplitude dimension for suppressing redundant diffraction orders.
[0033] Furthermore, there is a thickness difference between the two sets of light-transmitting regions of the amplitude-phase hybrid modulation two-dimensional grid grating, and the thickness difference Δd satisfies (where λ is the working wavelength and n is the refractive index of the substrate material at the working wavelength), this difference causes the two sets of transparent regions to have a phase delay of π for the incident light (i.e., a phase difference of π).
[0034] Taking a fused silica substrate (refractive index n≈1.46, operating wavelength λ) as an example, if the substrate corresponding to a set of transparent areas is etched using ICP etching, then the corresponding etching depth is... This makes the physical thickness of the substrate in this group of transparent areas Δd less than that in another group of transparent areas.
[0035] The π phase difference between the two sets of transparent regions is the core of achieving diffraction order manipulation. Through this phase difference, redundant diffraction orders such as (0,±1), (±1,0), and (0,0) are suppressed or eliminated due to phase interference, and the light field energy is concentrated in the (±1,±1) order diffracted light. This phase manipulation results in the light field mainly having four diffraction orders (±1,±1), eliminating the need to reserve light transmission space for redundant orders. Therefore, a higher density grid structure can be transmitted under the same aperture stop, solving the problem of low projection density caused by high-frequency component loss in traditional technologies.
[0036] Furthermore, the two sets of transparent regions in the amplitude-phase hybrid modulation two-dimensional grid grating are arranged alternately in a chessboard pattern. That is, adjacent transparent regions (horizontally and vertically adjacent) belong to different groups (with a π phase difference), and an opaque region is set at the boundary between the two sets of transparent regions, forming a periodic arrangement structure of "one set of transparent regions - opaque regions - another set of transparent regions". The arrangement period is consistent with the periodic unit of the two-dimensional grid (typical value). ).
[0037] like Figure 4 As shown in the grating structure of the present invention, the black area is an opaque area, and the red and blue areas are two sets of translucent areas: the red and blue translucent areas are arranged alternately in a chessboard pattern, and any two horizontally or vertically adjacent translucent areas must be different colors (i.e., different sets), and the black opaque area serves as the boundary between the red and blue translucent areas. For example, within a periodic unit, the upper left corner is a red translucent area, with black opaque areas immediately adjacent to its right and below, and blue translucent areas to the right and below the opaque areas, and so on, forming a uniformly alternating arrangement structure.
[0038] The advantage of this arrangement is that it ensures a uniform distribution of the π phase difference between adjacent transparent regions, making the diffraction interference effect of the light field periodic and symmetrical. This precisely suppresses the (0,±1), (±1,0), and (0,0) order diffracted light, ensuring that the energy is concentrated in the (±1,±1) order. The opaque region serves as the boundary between the two sets of transparent regions, avoiding direct contact between transparent regions with different phases, reducing edge interference noise, and improving the contrast of the projected pattern. The chessboard arrangement makes the periodicity of the two-dimensional grid structure more stable and the uniformity of the projected pattern better. Under the same aperture conditions, it can achieve a higher density grid projection, and the structure is not easy to disappear due to the reduction of the aperture, which is significantly better than the arrangement of traditional pure amplitude modulation gratings.
[0039] like Figure 5 This paper demonstrates the distribution of diffraction orders and corresponding apertures, as well as the corresponding projected light intensity structure, when switching from a pure amplitude-modulated two-dimensional grid grating to the amplitude-phase hybrid modulation grating proposed in this invention, under the critical condition of projection. It can be seen that at this point, there are only four diffraction orders (±1, ±1), and the diffraction orders have sufficient distance from the edge of the aperture. The projection density of the projected light intensity at this time is... Figure 3 Similarly, both are 8×8.
[0040] like Figure 6 A schematic diagram of the critical case of amplitude-phase hybrid modulation grating projection.
[0041] In the upper sub-image, the original pattern of the hybrid modulation grating is shown as a checkerboard-like periodic structure with alternating red, black, and light blue colors. This corresponds to the physical structure of the grating surface consisting of "opaque chromium layer (black area) + two sets of π phase difference transparent areas (red and light blue areas)". This clearly demonstrates the arrangement of the basic units of the hybrid modulation grating, which is the structural basis for subsequent diffraction and projection effects.
[0042] In the lower left sub-figure, the schematic diagram of the Fourier plane diffraction order distribution shows that the aperture stop has been reduced to a critical size, allowing only four diffraction orders (±1,±1) to pass just through the edge of the aperture stop (the remaining redundant diffraction orders are completely blocked). The effective diffraction orders and the blocked orders are distinguished by color.
[0043] The diagram in the lower right corner shows the projection result under critical conditions: the grid still maintains complete periodicity, the unit edges are sharp, the density is uniform, and there is no distortion or blurring. This directly proves the advantage of the hybrid modulation technology in being able to stably project a high-density grid under critical aperture conditions.
[0044] In one possible implementation, such as Figure 7 The schematic diagram illustrates a high-density periodic two-dimensional grid projection device, including the core optical path principle. From left to right along the light propagation direction, the left side is the illumination module (a parallel light source, such as a helium-neon laser with a wavelength of λ=632.8nm), used to provide a uniform and stable incident light field. The middle section is the grating module (an amplitude-phase hybrid modulation two-dimensional grid grating, with a fused silica substrate and a checkerboard-patterned opaque chromium layer and two sets of π-phase difference transparent regions on its surface). The incident light is modulated by this grating to form a light field containing specific diffraction orders. The right side is the projection optical system, including an aperture stop (located in the system's Fourier plane, allowing only (±1,±1) order diffracted light to pass through) and an imaging lens. The modulated light field, after being filtered by the aperture stop and imaged by the lens, ultimately forms a high-density periodic two-dimensional grid on the right projection surface. Arrows in the diagram indicate the light propagation path, clearly defining the functional connections between each module and presenting the complete technical process of "illumination-modulation-projection."
[0045] Therefore, this application discloses a high-density periodic two-dimensional grid structure projection method, which employs an amplitude-phase hybrid modulation two-dimensional grid grating. Each periodic cell of the grating contains an opaque region and two sets of transparent regions, with a π-phase difference between the two sets of transparent regions. The grating is then illuminated, and two-dimensional grid projection is performed using a projection optics system. This method, through phase and amplitude hybrid modulation, effectively overcomes the limitation of the aperture on projection density under the same projection optics system and aperture stop conditions, achieving stable projection of a higher-density periodic two-dimensional grid structure. Compared to traditional amplitude-modulated grids, the grid period is significantly reduced, and the grid density is significantly improved. Example 2
[0046] Furthermore, this embodiment applies the hybrid modulation two-dimensional grid grating of Example 1 to the microscopic focusing calibration scenario of biological samples (such as tissue sections).
[0047] The system includes a parallel light source, a hybrid modulation grating, a 40x microscope objective, and a CMOS acquisition module. During operation, the high-density grid projected by the grating is superimposed on the sample microscopic image.
[0048] For details, please refer to the following: Figure 8 The images on the left and right are two microscopic images of the same metallographic sample. Even with similar resolution, the details of the superimposed high-density grid differ. In the left image, the grid cell edges are sharper and the periodic arrangement is more regular; in the right image, the grid exhibits slight edge blurring. Utilizing the high-density grid characteristics of this invention's grating, even with similar overall sample resolution, the "optimal focus position" (the left image corresponds to the optimal focus) can be distinguished by the regularity of the grid details. This solves the problem of "difficulty in judging similar resolution" when relying solely on sample details in traditional methods, achieving a high focusing accuracy. (50X, 0.8NA objective lens).
[0049] The sequence numbers of the embodiments in this application are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.
[0050] Those skilled in the art will understand that all or part of the steps of the above embodiments can be implemented by hardware, or by a program instructing related hardware. The program can be stored in a computer-readable storage medium, such as a read-only memory, a disk, or an optical disk. The above descriptions are merely optional embodiments of this application and are not intended to limit the application. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of this application should be included within the scope of protection of this application.
Claims
1. A projection method for a high-density periodic two-dimensional mesh structure, characterized in that, The method includes: Projection is performed using an amplitude-phase hybrid modulation two-dimensional grid grating. Each periodic cell of the two-dimensional grid grating contains an opaque region and two sets of transparent regions, and there is a π phase difference between the two sets of transparent regions. Illuminate the amplitude-phase hybrid modulation two-dimensional grid grating; The light field modulated by the amplitude-phase hybrid modulation two-dimensional grid grating is passed through the projection optical system. The aperture stop of the projection optical system allows the (±1,±1) order diffracted light in the light field to pass through, thereby realizing the projection of a high-density periodic two-dimensional grid structure.
2. The method as described in claim 1, characterized in that, The amplitude-phase hybrid modulation two-dimensional grid grating uses a transparent material as its substrate.
3. The method as described in claim 1, characterized in that, The opaque area of the amplitude-phase hybrid modulation two-dimensional grid grating is covered with a light-blocking layer to achieve amplitude modulation.
4. The method as described in claim 1, characterized in that, The amplitude-phase hybrid modulation two-dimensional grid grating has a thickness difference between the two sets of light-transmitting regions. This difference causes the two sets of light-transmitting regions to have a phase delay difference of π with respect to the incident light, resulting in a π phase difference between the two sets of light-transmitting regions with respect to the incident light.
5. The method as described in claim 1, characterized in that, The two sets of light-transmitting regions of the amplitude-phase hybrid modulation two-dimensional grid grating are arranged alternately in a chessboard pattern, and the boundary between the two sets of light-transmitting regions is an opaque region.
6. A high-density periodic two-dimensional grid structure projection device, characterized in that, The device includes: A grating module is used for projection using an amplitude-phase hybrid modulation two-dimensional grid grating. Each periodic cell of the two-dimensional grid grating has an opaque region and two sets of transparent regions, and there is a π phase difference between the two sets of transparent regions. An illumination module is used to illuminate the amplitude-phase hybrid modulation two-dimensional grid grating; A projection optical system is used to allow the light field modulated by the amplitude-phase hybrid modulation two-dimensional grid grating to pass through the projection optical system. The aperture stop of the projection optical system allows the (±1,±1) order diffracted light in the light field to pass through, thereby realizing the projection of a high-density periodic two-dimensional grid structure.