Mathematical modeling method and device for high-steep micro-nano structure complex surface topography

Abstract

The mathematical modeling method and apparatus for complex surface morphology of high-steep micro / nano structures are universal and can be applied to many fields such as optical element design, system design, optical processing, and morphology detection. It can screen the lens units and curved substrate of the curved microlens array at the data level, thereby analyzing and designing two different types of features, improving the flexibility of use. The method includes: (1) inputting the parameters of the curved microlens array used for modeling; (2) calculating the equation of the curved substrate; (3) calculating the normal vector of each microlens unit and the intersection point of each normal vector with the equation of the curved substrate; (4) calculating the binary mask matrix used to divide the region of the curved substrate and each microlens unit; (5) calculating the vector matrix of each microlens unit; (6) calculating the vector matrix of the curved substrate; (7) superimposing the vector matrices of the curved substrate and each microlens unit to obtain the complete vector matrix of the curved microlens array, thus completing the modeling.

Description

Technical Field

[0001] This invention relates to the field of precision optics, and more particularly to a mathematical modeling method for complex surface morphology of high-steep micro / nano structures, as well as a mathematical modeling device for complex surface morphology of high-steep micro / nano structures. Background Technology

[0002] Microlens arrays are optical elements with lens characteristics arranged periodically in an array on their surface structure. The size of their lens units is mostly from a few micrometers to hundreds of millimeters, which is much smaller than that of conventional lenses, hence the name microlens. With the advancement of micro- and nano-fabrication technology, the applications of microlenses are becoming increasingly widespread, ranging from fiber bundle coupling utilizing focusing characteristics to beam shaping and light field cameras utilizing optical signal modulation characteristics, and are involved in many fields.

[0003] In recent years, with the deepening of research on microlens arrays, curved microlens arrays, as a branch of microlens arrays, have developed rapidly. Traditional microlens arrays are mainly planar microlens arrays, in which the microlens units are distributed on a planar substrate, and the base height and optical axis direction of each unit are consistent. In contrast, the microlens units of curved microlens arrays are distributed on curved substrates, including spherical, aspherical, and even freeform surfaces, and the base height and optical axis direction of each unit are not consistent. Curved microlens arrays are mainly used in imaging systems to obtain a larger field of view and better image quality. Patents such as CN201410806265.0 and CN202011132753.X describe this technology.

[0004] Research on microlens arrays covers all aspects from component and system design to fabrication and inspection, creating a wide demand for corresponding theoretical analysis and simulation design. Therefore, mathematical modeling methods for the surface morphology of microlens arrays are crucial. Currently, mathematical modeling methods for the surface morphology of planar microlenses are relatively mature, but research on mathematical modeling methods for curved microlenses is scarce, mainly focusing on areas such as fabrication trajectory design, which cannot describe the surface morphology to guide further analysis and simulation. Therefore, a universally applicable mathematical modeling method for the surface morphology of curved microlens arrays is urgently needed. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, the technical problem to be solved by this invention is to provide a mathematical modeling method for complex surface morphology of high-steep micro-nano structures. This method is universal and can be applied to many fields such as optical element design, system design, optical processing, and morphology detection. It can screen the lens units and curved substrates of curved microlens arrays at the data level, thereby analyzing and designing two different types of features, which improves the flexibility in the process of use.

[0006] The technical solution of this invention is: a mathematical modeling method for complex surface morphology of highly steep micro / nano structures, comprising the following steps:

[0007] (1) Input the parameters of the surface microlens array used for modeling. The surface microlens array parameters include: radius of curvature R of the surface base, element diameter D, coordinates of the center of the surface base (x0, y0, z0), radius of curvature r of the microlens unit, diameter d of the microlens unit, and spacing d between the microlens units. I The number of elements in the height matrix is ​​m×n;

[0008] (2) Calculate the equation of the surface base. Based on the input radius of curvature R of the surface base, the component diameter D, and the coordinates of the center of the surface base (x0, y0, z0), calculate the equation c of the surface base. B ;

[0009] (3) Calculate the normal vector of each microlens unit and the intersection point of each normal vector with the surface base equation. According to the characteristics of the surface microlens array, each microlens unit is distributed on the surface of the surface base, and the direction of the normal vector of each microlens unit is perpendicular to the direction of the tangent at the intersection point with the surface base. Predetermine the coordinates of the intersection point of one of the microlens units with the surface base, and combine them with the surface base equation c calculated in step (2). B Based on the input microlens unit diameter d and microlens unit spacing d l Calculate the coordinates of the intersection points of all microlens units and their corresponding normal vectors;

[0010] (4) Calculate the binary mask matrix used to divide the curved substrate and each microlens unit region. Based on the normal vector of each microlens unit calculated in step (3) and its intersection with the curved substrate, combined with the input microlens unit diameter, calculate the region range of each microlens unit, and generate the binary mask matrix accordingly. Where i is the sequence number of each microlens unit; after deducting all Obtain the binarized mask matrix Mask of the curved substrate B ;

[0011] (5) Calculate the vector matrix of each microlens unit. Based on the normal vector of each microlens unit in step (3) and the coordinates of the intersection points of each normal vector with the surface basis equation in step (4), calculate the center coordinates of each microlens unit. Based on the input radius of curvature r of the microlens unit, calculate the equation of each microlens unit in combination with the center coordinates. According to the information obtained in step (4) Calculate the elevation matrix of each microlens unit:

[0012]

[0013] Where i is the serial number of each microlens unit;

[0014] (6) Calculate the surface base elevation matrix, based on the surface base equation c calculated in step (2). B The binarized mask matrix Mask of the curved substrate calculated in step (4) B Calculate the base elevation matrix of the curved surface:

[0015] s B =c B Mask B (2);

[0016] (7) Superimpose the elevation matrices of the curved substrate and each microlens unit to obtain the complete elevation matrix of the curved microlens array:

[0017]

[0018] This invention employs a surface region segmentation approach, utilizing the discrete distribution characteristics of microlens units on a curved substrate surface to divide the surface region of the curved microlens array into a curved substrate portion and individual microlens unit portions for separate analysis. At the numerical modeling level, the surface morphology of the curved microlens array is described using a vector height matrix, and the regions containing the curved substrate and each microlens unit are divided using a binary mask matrix. The surface shape of each divided region is calculated using independent surface equations, and finally, these are superimposed to form a complete mathematical model of the curved microlens array. Therefore, it possesses universality and can be applied to many fields such as optical element design, system design, optical processing, and morphology inspection. It allows for the separate screening of lens units and the curved substrate of the curved microlens array at the data level, enabling analysis and design work on two different types of features, thus improving flexibility in use.

[0019] A mathematical modeling device for complex surface morphology of highly steep micro / nano structures is also provided, comprising: an input module configured to input parameters of a curved microlens array for modeling, the parameters of which include: radius of curvature R of the curved substrate, element diameter D, coordinates of the center of the curved substrate (x0, y0, z0), radius of curvature r of the microlens unit, diameter d of the microlens unit, and spacing d between the microlens units. l The number of elements in the height matrix is ​​m×n;

[0020] The first calculation module is configured to calculate the equation of the surface base. Based on the input surface base curvature radius R, component diameter D, and surface base sphere center coordinates (x0, y0, z0), it calculates the equation c of the surface base. B ;

[0021] The second calculation module is configured to calculate the normal vector of each microlens unit and the intersection point of each normal vector with the surface base equation. Based on the characteristics of the surface microlens array, each microlens unit is distributed on the surface of the surface base, and the direction of the normal vector of each microlens unit is perpendicular to the direction of the tangent at its intersection point with the surface base. The coordinates of the intersection point of one microlens unit with the surface base are pre-determined, and combined with the surface base equation c calculated in the first calculation module... B Based on the input microlens unit diameter d and microlens unit spacing d l Calculate the coordinates of the intersection points of all microlens units and their corresponding normal vectors;

[0022] The third calculation module is configured to calculate the binary mask matrix used to divide the curved substrate and each microlens unit region. Based on the normal vectors of each microlens unit calculated in the second calculation module and their intersection points with the curved substrate, combined with the input microlens unit diameter, it calculates the region range of each microlens unit and generates the binary mask matrix accordingly. Where i is the sequence number of each microlens unit; after deducting all Obtain the binarized mask matrix Mask of the curved substrate B ;

[0023] The fourth calculation module is configured to calculate the vector matrix of each microlens unit. Based on the normal vectors of each microlens unit in the second calculation module and the coordinates of the intersection points of each normal vector with the surface basis equation in the third calculation module, it calculates the coordinates of the center of the sphere of each microlens unit. Based on the input radius of curvature r of the microlens unit and the coordinates of the center of the sphere, it calculates the equations of each microlens unit. According to the data obtained from the third calculation module Calculate the elevation matrix of each microlens unit:

[0024]

[0025] Where i is the serial number of each microlens unit;

[0026] The fifth calculation module is configured to calculate the surface basis elevation matrix, based on the surface basis equation c calculated in the first calculation module. B The binarized mask matrix Mask of the curved substrate calculated in the third calculation module B Calculate the base elevation matrix of the curved surface:

[0027] s B =c B Mask B (2);

[0028] The output module is configured to superimpose the elevation matrices of the curved substrate and each microlens unit to obtain the complete elevation matrix of the curved microlens array.

[0029]

[0030] Finally, a mathematical model of the surface morphology of the curved microlens array was completed. Attached Figure Description

[0031] Figure 1 This is a flowchart of the method described in this invention.

[0032] Figure 2 Here is the equation for the curved base in the embodiments of the present invention.

[0033] Figure 3 This refers to the binarized mask matrix of each microlens unit in the embodiments of the present invention.

[0034] Figure 4 This is the binary mask matrix for the curved substrate in the embodiments of the present invention.

[0035] Figure 5 This is the equation for the microlens unit in the embodiments of the present invention.

[0036] Figure 6 This refers to the vector height matrix of the microlens unit in the embodiments of the present invention.

[0037] Figure 7 This is the surface base elevation matrix in the embodiments of the present invention.

[0038] Figure 8 This is the complete surface microlens array elevation matrix in the embodiments of the present invention. Detailed Implementation

[0039] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0040] It should be noted that the term "comprising" and any variations thereof in the specification, claims and accompanying drawings of this invention are intended to cover non-exclusive inclusion. For example, a process, method, apparatus, product or device that includes a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products or devices.

[0041] like Figure 1 As shown, the mathematical modeling method for the complex surface morphology of this high-steep micro / nano structure includes the following steps:

[0042] (1) Input the parameters of the surface microlens array used for modeling. The surface microlens array parameters include: radius of curvature R of the surface base, element diameter D, coordinates of the center of the surface base (x0, y0, z0), radius of curvature r of the microlens unit, diameter d of the microlens unit, and spacing d between the microlens units. l The number of elements in the height matrix is ​​m×n;

[0043] (2) Calculate the equation of the surface base. Based on the input radius of curvature R of the surface base, the component diameter D, and the coordinates of the center of the surface base (x0, y0, z0), calculate the equation c of the surface base. B ;

[0044] (3) Calculate the normal vector of each microlens unit and the intersection point of each normal vector with the surface base equation. According to the characteristics of the surface microlens array, each microlens unit is distributed on the surface of the surface base, and the direction of the normal vector of each microlens unit is perpendicular to the direction of the tangent at the intersection point with the surface base. Predetermine the coordinates of the intersection point of one of the microlens units with the surface base, and combine them with the surface base equation c calculated in step (2). B Based on the input microlens unit diameter d and microlens unit spacing d l Calculate the coordinates of the intersection points of all microlens units and their corresponding normal vectors;

[0045] (4) Calculate the binary mask matrix used to divide the curved substrate and each microlens unit region. Based on the normal vector of each microlens unit calculated in step (3) and its intersection with the curved substrate, combined with the input microlens unit diameter, calculate the region range of each microlens unit, and generate the binary mask matrix accordingly. Where i is the sequence number of each microlens unit; after deducting all Obtain the binarized mask matrix Mask of the curved substrate B ;

[0046] (5) Calculate the vector matrix of each microlens unit. Based on the normal vector of each microlens unit in step (3) and the coordinates of the intersection points of each normal vector with the surface basis equation in step (4), calculate the center coordinates of each microlens unit. Based on the input radius of curvature r of the microlens unit, calculate the equation of each microlens unit in combination with the center coordinates. According to the information obtained in step (4) Calculate the elevation matrix of each microlens unit:

[0047]

[0048] Where i is the serial number of each microlens unit;

[0049] (6) Calculate the surface base elevation matrix, based on the surface base equation c calculated in step (2).B The binarized mask matrix Mask of the curved substrate calculated in step (4) B Calculate the base elevation matrix of the curved surface:

[0050] s B =c B Mask B (2);

[0051] (7) Superimpose the elevation matrices of the curved substrate and each microlens unit to obtain the complete elevation matrix of the curved microlens array:

[0052]

[0053] This invention employs a surface region segmentation approach, utilizing the discrete distribution characteristics of microlens units on a curved substrate surface to divide the surface region of the curved microlens array into a curved substrate portion and individual microlens unit portions for separate analysis. At the numerical modeling level, the surface morphology of the curved microlens array is described using a vector height matrix, and the regions containing the curved substrate and each microlens unit are divided using a binary mask matrix. The surface shape of each divided region is calculated using independent surface equations, and finally, these are superimposed to form a complete mathematical model of the curved microlens array. Therefore, it possesses universality and can be applied to many fields such as optical element design, system design, optical processing, and morphology inspection. It allows for the separate screening of lens units and the curved substrate of the curved microlens array at the data level, enabling analysis and design work on two different types of features, thus improving flexibility in use.

[0054] Preferably, in step (1), if the surface shape of the base surface is an aspherical or freeform surface, additional higher-order surface coefficients of the base surface are input. The number of terms i is determined based on the higher-order terms used, or the vector matrix of the surface basis can be entered additionally; if the surface shape of the microlens unit is aspherical or freeform, the higher-order surface coefficients of the microlens unit should be entered additionally. The number of terms i is determined based on the higher-order terms used, or by additionally inputting the vector height matrix of the microlens unit; if the curved microlens array of the target is a transmission element, the thickness t of the curved substrate is additionally input. B Microlens unit thickness t MLA The optical properties of the material, including refractive index and transmittance, as well as the R, D, r, and d parameters of the second surface.

[0055] Preferably, the elevation matrix is ​​a projection perpendicular to the principal optical axis.

[0056] Preferably, the surface base equation and microlens unit equation only give the expression form of a sphere. If these surfaces are aspherical or freeform, the calculation is performed in combination with the input high-order surface coefficients.

[0057] Preferably, the method uses tolerance analysis provided in optical design software, and the method of use is the same as that for conventional optical components.

[0058] Those skilled in the art will understand that all or part of the steps in the methods of the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium. When executed, the program includes the steps of the methods of the above embodiments. The storage medium can be ROM / RAM, magnetic disk, optical disk, memory card, etc. Therefore, corresponding to the method of the present invention, the present invention also includes a mathematical modeling device for complex surface morphologies of highly steep micro / nano structures. This device includes:

[0059] The input module is configured to input the parameters of the surface microlens array used for modeling. These parameters include: the radius of curvature R of the surface substrate, the element diameter D, the coordinates of the surface substrate sphere center (x0, y0, z0), the radius of curvature r of each microlens element, the diameter d of each microlens element, and the spacing d between microlens elements. l The number of elements in the height matrix is ​​m×n;

[0060] The first calculation module is configured to calculate the equation of the surface base. Based on the input surface base curvature radius R, component diameter D, and surface base sphere center coordinates (x0, y0, z0), it calculates the equation c of the surface base. B ;

[0061] The second calculation module is configured to calculate the normal vector of each microlens unit and the intersection point of each normal vector with the surface base equation. Based on the characteristics of the surface microlens array, each microlens unit is distributed on the surface of the surface base, and the direction of the normal vector of each microlens unit is perpendicular to the direction of the tangent at its intersection point with the surface base. The coordinates of the intersection point of one microlens unit with the surface base are pre-determined, and combined with the surface base equation c calculated in the first calculation module... B Based on the input microlens unit diameter d and microlens unit spacing d l Calculate the coordinates of the intersection points of all microlens units and their corresponding normal vectors;

[0062] The third calculation module is configured to calculate the binary mask matrix used to divide the curved substrate and each microlens unit region. Based on the normal vectors of each microlens unit calculated in the second calculation module and their intersection points with the curved substrate, combined with the input microlens unit diameter, it calculates the region range of each microlens unit and generates the binary mask matrix accordingly. Where i is the sequence number of each microlens unit; after deducting all Obtain the binarized mask matrix Mask of the curved substrate B ;

[0063] The fourth calculation module is configured to calculate the vector matrix of each microlens unit. Based on the normal vectors of each microlens unit in the second calculation module and the coordinates of the intersection points of each normal vector with the surface basis equation in the third calculation module, it calculates the coordinates of the center of the sphere of each microlens unit. Based on the input radius of curvature r of the microlens unit and the coordinates of the center of the sphere, it calculates the equations of each microlens unit. According to the data obtained from the third calculation module Calculate the elevation matrix of each microlens unit:

[0064]

[0065] Where i is the serial number of each microlens unit;

[0066] The fifth calculation module is configured to calculate the surface basis elevation matrix, based on the surface basis equation c calculated in the first calculation module. B The binarized mask matrix Mask of the curved substrate calculated in the third calculation module B Calculate the base elevation matrix of the curved surface:

[0067] s B =c B Mask B (2);

[0068] The output module is configured to superimpose the elevation matrices of the curved substrate and each microlens unit to obtain the complete elevation matrix of the curved microlens array.

[0069]

[0070] Finally, a mathematical model of the surface morphology of the curved microlens array was completed.

[0071] Preferably, in the input module, if the surface shape of the base surface is an aspherical or freeform surface, the higher-order surface coefficients of the base surface are additionally input. The number of terms i is determined based on the higher-order terms used, or the vector matrix of the surface basis can be entered additionally; if the surface shape of the microlens unit is aspherical or freeform, the higher-order surface coefficients of the microlens unit should be entered additionally. The number of terms i is determined based on the higher-order terms used, or by additionally inputting the vector height matrix of the microlens unit; if the curved microlens array of the target is a transmission element, the thickness t of the curved substrate is additionally input. B Microlens unit thickness t MLA The optical properties of the material, including refractive index and transmittance, as well as the R, D, r, and d parameters of the second surface.

[0072] Preferably, the elevation matrix is ​​a projection perpendicular to the principal optical axis.

[0073] Preferably, the surface base equation and microlens unit equation only give the expression form of a sphere. If these surfaces are aspherical or freeform, the calculation is performed in combination with the input high-order surface coefficients.

[0074] Preferably, the device uses tolerance analysis provided in optical design software, and the method of use is the same as that of conventional optical components.

[0075] The following details specific embodiments of the present invention.

[0076] The target element described in this embodiment is a reflective curved surface microlens array. Both the curved substrate of the element and the microlens unit are spherical. The surface morphology is modeled using the method proposed in this invention.

[0077] According to step one, input the parameters of the curved microlens array used for modeling. The input parameters described in this embodiment are as follows: curved substrate radius of curvature R = 25mm, element diameter D = 5mm, curved substrate sphere center coordinates (0, 0, 0), microlens unit radius of curvature r = 5mm, microlens unit diameter d = 1mm, and microlens unit spacing d. l =1mm, number of elements in the height matrix (1024×1024). Since the curved substrate of the target element and the microlens unit are both spherical and the surfaces are reflective, no additional parameters need to be input.

[0078] Based on step two, calculate the equation c of the curved surface base. B The calculation results are as follows Figure 2 As shown.

[0079] According to step three, the normal vector of each microlens unit and the intersection point of each normal vector with the surface substrate are calculated. Based on the input parameters described in the embodiment and the calculation results of this step, the total number of microlens units is 21.

[0080] According to step four, the binary mask matrix used to divide the curved substrate and each microlens unit region is calculated. Specifically, the binary mask matrix for each microlens unit described in the embodiment... Arrange in order, such as Figure 3 As shown; the curved substrate binarization mask matrix described in the embodiment. B like Figure 4 As shown.

[0081] According to step five, calculate the elevation matrix of each microlens, and in the process, calculate the equations of each microlens unit sequentially. and the vector matrix of each microlens unit Taking the microlens number (i=17) as an example, the microlens unit equation described in the embodiment... and the vector matrix of the microlens unit Each as Figure 5 and Figure 6 As shown.

[0082] According to step six, calculate the surface base elevation matrix s. B The surface base elevation matrix s described in the embodiment B like Figure 7 As shown.

[0083] Based on step seven, the surface equations of the curved substrate and each microlens unit are superimposed to obtain the complete surface microlens array elevation matrix s. Fin The complete curved surface microlens array vector matrix s described in the embodiment. Fin like Figure 8 As shown.

[0084] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A mathematical modeling method for complex surface morphology of high-steep micro / nano structures, characterized by: It includes the following steps: (1) inputting parameters of a curved microlens array for modeling, the parameters of the curved microlens array comprising: a curved base radius R, an element diameter D, a curved base spherical center coordinate (x0, y0, z0), a microlens unit radius r, a microlens unit diameter d, and a microlens unit pitch d l , the number of elements m x n of the vector height matrix; (2) Calculate the curved base equation, according to the input curved base curvature radius R, element diameter D, curved base spherical center coordinates (x0, y0, z0), calculate the equation c of the curved base B ; (3) Calculate the normal vector of each microlens unit, and the intersection point of each normal vector and the curved surface base equation. According to the characteristics of the curved surface microlens array, each microlens unit is distributed on the surface of the curved surface base, and the direction of the normal vector of each microlens unit is perpendicular to the tangent direction of the intersection point of the curved surface base. The coordinates of the intersection point of one of the microlens units and the curved surface base are determined in advance, and the curved surface base equation c B According to the input microlens unit diameter d, the microlens unit spacing d l , the intersection point coordinates and the corresponding normal vector of all microlens units are calculated; (4) Calculate the binary mask matrix used to divide the curved substrate and each microlens unit region. Based on the normal vector of each microlens unit calculated in step (3) and its intersection with the curved substrate, combined with the input microlens unit diameter, calculate the region range of each microlens unit, and generate the binary mask matrix accordingly. Where i is the sequence number of each microlens unit; after deducting all Obtain the binarized mask matrix Mask of the curved substrate B ; (5) Calculate the vector matrix of each microlens unit. Based on the normal vector of each microlens unit in step (3) and the coordinates of the intersection points of each normal vector with the surface basis equation in step (4), calculate the center coordinates of each microlens unit. Based on the input radius of curvature r of the microlens unit, calculate the equation of each microlens unit in combination with the center coordinates. According to the information obtained in step (4) Calculate the elevation matrix of each microlens unit: Where i is the serial number of each microlens unit; (6) Calculate the matrix of the height of the curved base vector, according to the curved base equation c B calculated in step (2) and the binary mask matrix Mask of the curved base calculated in step (4) B , Calculate the matrix of the height of the curved base vector: s B = c B • Mask B (2); (7) Superimpose the elevation matrices of the curved substrate and each microlens unit to obtain the complete elevation matrix of the curved microlens array: Finally, a mathematical model of the surface morphology of the curved microlens array was completed.

2. The mathematical modeling method for complex surface morphology of high-steep micro / nano structures according to claim 1, characterized in that: In step (1), if the surface shape of the base surface is a non-spherical or free-form surface, additional input of the higher-order surface coefficients of the base surface is required. The number of terms i is determined based on the higher-order terms used, or the vector matrix of the surface basis can be entered additionally; if the surface shape of the microlens unit is aspherical or freeform, the higher-order surface coefficients of the microlens unit should be entered additionally. The number of terms i is determined based on the higher-order terms used, or by additionally inputting the vector height matrix of the microlens unit; if the curved microlens array of the target is a transmission element, the thickness t of the curved substrate is additionally input. B Microlens unit thickness t MLA The optical properties of the material, including refractive index and transmittance, as well as the R, D, r, and d parameters of the second surface.

3. The mathematical modeling method for complex surface morphology of high-steep micro / nano structures according to claim 2, characterized in that: The elevation matrix is ​​a projection perpendicular to the principal optical axis.

4. The mathematical modeling method for complex surface morphology of high-steep micro / nano structures according to claim 3, characterized in that: The surface basis equation and microlens unit equation only give the expression form of a sphere. If these surfaces are aspherical or freeform, the calculation is performed in combination with the input high-order surface coefficients.

5. The mathematical modeling method for complex surface morphology of high-steep micro / nano structures according to claim 4, characterized in that: This method uses tolerance analysis provided in optical design software, and the usage is the same as for conventional optical components.

6. A mathematical modeling device for complex surface morphology of high-steep micro / nano structures, characterized in that: It includes: The input module is configured to input the parameters of the surface microlens array used for modeling. These parameters include: the radius of curvature R of the surface substrate, the element diameter D, the coordinates of the surface substrate sphere center (x0, y0, z0), the radius of curvature r of each microlens element, the diameter d of each microlens element, and the spacing d between microlens elements. l The number of elements in the height matrix is ​​m×n; The first calculation module is configured to calculate the equation of the surface base. Based on the input surface base curvature radius R, component diameter D, and surface base sphere center coordinates (x0, y0, z0), it calculates the equation c of the surface base. B ; The second calculation module is configured to calculate the normal vector of each microlens unit and the intersection point of each normal vector with the surface base equation. Based on the characteristics of the surface microlens array, each microlens unit is distributed on the surface of the surface base, and the direction of the normal vector of each microlens unit is perpendicular to the direction of the tangent at its intersection point with the surface base. The coordinates of the intersection point of one microlens unit with the surface base are pre-determined, and combined with the surface base equation c calculated in the first calculation module... B Based on the input microlens unit diameter d and microlens unit spacing d l Calculate the coordinates of the intersection points of all microlens units and their corresponding normal vectors; The third calculation module is configured to calculate the binary mask matrix used to divide the curved substrate and each microlens unit region. Based on the normal vectors of each microlens unit calculated in the second calculation module and their intersection points with the curved substrate, combined with the input microlens unit diameter, it calculates the region range of each microlens unit and generates the binary mask matrix accordingly. Where i is the sequence number of each microlens unit; after deducting all Obtain the binarized mask matrix Mask of the curved substrate B ; The fourth calculation module is configured to calculate the vector matrix of each microlens unit. Based on the normal vectors of each microlens unit in the second calculation module and the coordinates of the intersection points of each normal vector with the surface basis equation in the third calculation module, it calculates the coordinates of the center of the sphere of each microlens unit. Based on the input radius of curvature r of the microlens unit and the coordinates of the center of the sphere, it calculates the equations of each microlens unit. According to the data obtained from the third calculation module Calculate the elevation matrix of each microlens unit: Where i is the serial number of each microlens unit; The fifth calculation module is configured to calculate the surface basis elevation matrix, based on the surface basis equation c calculated in the first calculation module. B The binarized mask matrix Mask of the curved substrate calculated in the third calculation module B Calculate the base elevation matrix of the curved surface: s B =c B ·Mask B (2); The output module is configured to superimpose the elevation matrices of the curved substrate and each microlens unit to obtain the complete elevation matrix of the curved microlens array. Finally, a mathematical model of the surface morphology of the curved microlens array was completed.

7. The mathematical modeling device for complex surface morphology of high-steep micro / nano structures according to claim 6, characterized in that: In the input module, if the surface shape of the base surface is aspherical or freeform, additional inputs are made of the higher-order surface coefficients of the base surface. The number of terms i is determined based on the higher-order terms used, or the vector matrix of the surface basis can be entered additionally; if the surface shape of the microlens unit is aspherical or freeform, the higher-order surface coefficients of the microlens unit should be entered additionally. The number of terms i is determined based on the higher-order terms used, or by additionally inputting the vector height matrix of the microlens unit; if the curved microlens array of the target is a transmission element, the thickness t of the curved substrate is additionally input. B Microlens unit thickness t MLA The optical properties of the material, including refractive index and transmittance, as well as the R, D, r, and d parameters of the second surface.

8. The mathematical modeling device for complex surface morphology of high-steep micro / nano structures according to claim 7, characterized in that: The elevation matrix is ​​a projection perpendicular to the principal optical axis.

9. The mathematical modeling device for complex surface morphology of high-steep micro / nano structures according to claim 8, characterized in that: The surface basis equation and microlens unit equation only give the expression form of spheres. If these surfaces are aspherical or freeform, the calculation is performed in combination with the input high-order surface coefficients.

10. The mathematical modeling device for complex surface morphology of high-steep micro / nano structures according to claim 9, characterized in that: The device uses tolerance analysis provided in optical design software, and its usage is the same as that of conventional optical components.

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