Mirror surface deformation characterization method and device based on conformal orthogonal basis, equipment and medium

Abstract

This application discloses a method, apparatus, equipment, and medium for characterizing mirror deformation based on a conformal orthogonal basis, relating to the field of optical analysis technology. The method includes: selecting a rotationally symmetric optical mirror and establishing an intrinsic coordinate system that locates each point based on the mirror's intrinsic geometric parameters; constructing the eigenvalue equation of the Laplace-Beltrami operator based on this coordinate system, separating variables to obtain angular and radial equations, the solutions of which constitute a first conformal orthogonal basis; orthogonalizing the Zernike polynomial of the intrinsic coordinate system using Gram-Schmidt transformation to obtain a target radial polynomial, generating a second conformal orthogonal basis. A finite element model of the mirror is established, the nodal deformation under load conditions is solved, and the normal displacement is extracted; one of two types of bases is selected as the target conformal orthogonal basis to construct a basis function matrix, the least squares problem is solved to obtain fitting coefficients, and the combination yields the expression for the mirror deformation surface shape. This application can design an orthogonal basis fitting scheme that conforms to the inherent geometric characteristics of the optical mirror.

Description

Technical Field

[0001] This invention relates to the field of optical analysis technology, and in particular to a method, apparatus, device, and medium for characterizing mirror deformation based on a conformal orthogonal substrate. Background Technology

[0002] Optical mirrors can develop surface distortion under mechanical and thermal loads, interfering with beam transmission and degrading the imaging quality of optical systems. In optomechanical coupling simulation and active surface correction processes, the discrete nodal deformation quantities output by the finite element method need to be transformed into continuous analytical surface shapes to adapt to ray tracing calculations. Using orthogonal polynomial basis fitting of the deformation field is currently the most compatible technical approach.

[0003] Currently, the industry generally uses the classical Zernike polynomial (an orthogonal polynomial used to describe the light deviation of optical mirrors) as the basis for fitting mirror deformation. This polynomial is defined in a unit circle region of the plane, and its terms correspond to typical optical aberrations. It is the standard fitting tool for commercial optical simulation software. However, this approach has inherent geometric defects. The orthogonality of the classical Zernike polynomial is established based on planar Euclidean metrics. When fitting high-curvature rotationally symmetric aspherical surfaces such as spheres and parabolas, the surface deformation field needs to be projected onto a two-dimensional plane. The projection process introduces geometric metric distortion, directly causing the orthogonality of the basis on the surface to degenerate. Moreover, the greater the curvature of the surface, the more significant the degradation phenomenon, and the fitting error in the mirror edge region will increase sharply. At the same time, the curvature distribution of different rotational optical surfaces varies greatly, and the degree of distortion caused by planar projection changes with the surface shape. This makes the fitting effect of the Zernike polynomial unstable and inconsistent. When dealing with complex mirror deformations, a large number of basis terms are required to achieve convergence, which significantly reduces the simulation calculation efficiency and introduces additional fitting biases, making it difficult to meet the requirements of high-precision optomechanical coupling analysis.

[0004] Therefore, how to design an orthogonal basis fitting scheme that fits the geometric characteristics of the optical mirror to overcome the problem that the fitting scheme based on Zernike polynomials is insufficient in terms of fitting accuracy at the edges of large curvature surfaces due to the dependence of orthogonality on planar Euclidean metric. This is a technical problem that urgently needs to be solved. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a method, apparatus, device, and medium for characterizing mirror deformation based on a conformal orthogonal basis. This method enables the design of an orthogonal basis fitting scheme that conforms to the geometric characteristics of the optical mirror, overcoming the limitations of Zernike polynomial-based fitting schemes, which rely on planar Euclidean metrics for orthogonality and suffer from insufficient edge fitting accuracy on surfaces with large curvature. The specific solution is as follows: In a first aspect, this application provides a method for characterizing mirror deformation based on a conformal orthogonal basis, including: Determine a rotationally symmetric optical mirror and construct an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial position of each point on the optical mirror. Based on the intrinsic coordinate system, the eigenvalue equations of the Laplace-Beltramian operator are constructed. The angular equations and radial equations are determined using the variable separation method and the eigenvalue equations. The first conformal orthogonal basis is determined based on the angular solution corresponding to the angular equations and the radial solution corresponding to the radial equations. Based on the intrinsic coordinate system, the Zernike polynomial is orthogonalized using Gram-Schmidt to obtain the target radial polynomial, and the target radial polynomial is used to determine the second conformal orthogonal basis. A finite element simulation model of the optical mirror is constructed, and the finite element simulation model under a preset load condition is solved to obtain the deformation displacement of the finite element nodes in the finite element simulation model, and the normal displacement component is determined based on the deformation displacement. A basis function matrix is ​​constructed based on the target conformal orthogonal basis, and a least squares problem based on the basis function matrix and the normal displacement component is solved to obtain fitting coefficients. The fitting coefficients and the target conformal orthogonal basis are used to construct a surface shape expression for the mirror deformation of the optical mirror, so as to trigger optomechanical coupling analysis or surface shape correction operation using the surface shape expression; the target conformal orthogonal basis is either the first conformal orthogonal basis or the second conformal orthogonal basis.

[0006] Optionally, the construction of an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial positions of each point on the optical mirror includes: The polar angle and azimuth angle of the optical mirror are selected as the intrinsic geometric parameters of the optical mirror, and a corresponding spherical coordinate system is constructed based on the intrinsic geometric parameters. The spherical coordinate system is then determined as the intrinsic coordinate system that marks the spatial position of each point on the optical mirror.

[0007] Optionally, the step of constructing the eigenvalue equations of the Laplace-Beltrami operator based on the intrinsic coordinate system, determining the angular and radial equations using the variable separation method and the eigenvalue equations, and determining the first conformal orthogonal basis based on the angular solution corresponding to the angular equations and the radial solution corresponding to the radial equations includes: Construct the first expression of the Laplace-Beltramian operator on the optical mirror surface in the intrinsic coordinate system; The normalized polar angle is determined based on the polar angle and the half-aperture angle of the optical mirror surface, and the polar angle in the intrinsic coordinate system is replaced with the normalized polar angle to obtain the target intrinsic coordinate system; The first expression is transformed into a second expression based on the target intrinsic coordinate system, and an eigenvalue equation is constructed based on the second expression; The solution of the eigenvalue equation is separated into an angular function related to the azimuth angle and a radial function related to the normalized polar angle to complete the variable separation operation. Based on the variable separation operation and the separation of the eigenvalue equation, the angular equation and the radial equation are obtained. Boundary conditions are applied to the radial equation and solved to obtain the radial solution. The angular equation is then solved to obtain the angular solution. Based on the radial solution, the angular solution, and the first normalization coefficient, a first conformal orthogonal basis is obtained.

[0008] Optionally, the boundary conditions include a solution-bounded regularity condition at the optical axis of the optical mirror, and a Neumann boundary condition with a zero radial derivative at the aperture boundary of the optical mirror.

[0009] Optionally, the step of performing Gram-Schmidt orthogonalization on the Zernike polynomial based on the intrinsic coordinate system to obtain the target radial polynomial, and using the target radial polynomial to determine the second conformal orthogonal basis, includes: The radial polynomials in the Zernike polynomials are used as the initial sequence, and the initial sequence is orthogonalized by Gram-Schmidt using the weight function determined based on the intrinsic coordinate system to obtain the target radial polynomials that are orthogonal to each other under the weight function. The second conformal orthogonal basis is obtained based on the target radial polynomial, the angular mode of the azimuth angle, and the second normalization coefficient.

[0010] Optionally, the second normalization coefficient is a coefficient determined based on the target radial polynomial, the normalized polar angle, and the area element determined based on the intrinsic coordinate system.

[0011] Optionally, the step of constructing the finite element simulation model of the optical mirror, solving the finite element simulation model under a preset load condition to obtain the deformation displacement of the finite element nodes in the finite element simulation model, and determining the normal displacement component based on the deformation displacement includes: A finite element simulation model is established for the optical mirror and its supporting structure. The finite element simulation model is set with material parameters including elastic modulus, Poisson's ratio and coefficient of thermal expansion, and a preset load condition including mechanical load and thermal load is applied to the finite element simulation model to obtain the finite element equilibrium equation. Solving the finite element equilibrium equations yields the deformation displacements of the finite element nodes, including rigid body displacement components and elastic deformation components. Based on the deformation displacement, sampling points are selected on the optical mirror surface using finite element shape functions, and the displacement field at the sampling points is determined. Based on the displacement field and the normal unit vector of the optical mirror surface at the sampling points, the normal displacement component at the sampling points is obtained.

[0012] Secondly, this application provides a mirror deformation characterization device based on a conformal orthogonal substrate, comprising: A coordinate system construction module is used to determine a rotationally symmetric optical mirror and construct an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial position of each point on the optical mirror. The first basis determination module is used to construct the eigenvalue equation of the Laplace-Beltrami operator based on the intrinsic coordinate system, determine the angular equation and the radial equation using the variable separation method and the eigenvalue equation, and determine the first conformal orthogonal basis based on the angular solution corresponding to the angular equation and the radial solution corresponding to the radial equation. The second basis determination module is used to perform Gram-Schmidt orthogonalization on the Zernike polynomial based on the intrinsic coordinate system to obtain the target radial polynomial, and to determine the second conformal orthogonal basis using the target radial polynomial. The component determination module is used to construct the finite element simulation model of the optical mirror, solve the finite element simulation model under the applied preset load condition, obtain the deformation displacement of the finite element nodes in the finite element simulation model, and determine the normal displacement component based on the deformation displacement. The expression construction module is used to construct a basis function matrix based on a target conformal orthogonal basis, and to solve a least-squares problem based on the basis function matrix and the normal displacement components to obtain fitting coefficients. The fitting coefficients and the target conformal orthogonal basis are used to construct a surface shape expression for the mirror deformation of the optical mirror, so as to trigger optomechanical coupling analysis or surface shape correction operation using the surface shape expression; the target conformal orthogonal basis is either the first conformal orthogonal basis or the second conformal orthogonal basis.

[0013] Thirdly, this application provides an electronic device, comprising: Memory, used to store computer programs; A processor is used to execute the computer program to implement the aforementioned method for characterizing mirror deformation based on a conformal orthogonal basis.

[0014] Fourthly, this application provides a computer-readable storage medium for storing a computer program; wherein, when the computer program is executed by a processor, it implements the aforementioned method for characterizing mirror deformation based on a conformal orthogonal basis.

[0015] In this application, a rotationally symmetric optical mirror is determined, and an intrinsic coordinate system is constructed to mark the spatial positions of points on the optical mirror using its intrinsic geometric parameters. Based on this intrinsic coordinate system, the eigenvalue equations of the Laplace-Beltramme operator are constructed. The angular and radial equations are determined using the separation of variables and the eigenvalue equations. A first conformal orthogonal basis is determined based on the angular solutions corresponding to the angular equations and the radial solutions corresponding to the radial equations. The Zernike polynomial is orthogonalized using the intrinsic coordinate system to obtain a target radial polynomial, and a second conformal orthogonal basis is determined using the target radial polynomial. The optical mirror's... A finite element simulation model is used to solve the finite element simulation model under a preset load condition to obtain the deformation displacement of the finite element nodes in the finite element simulation model, and the normal displacement component is determined based on the deformation displacement. A basis function matrix is ​​constructed based on a target conformal orthogonal basis, and a least squares problem constructed based on the basis function matrix and the normal displacement component is solved to obtain fitting coefficients. The fitting coefficients and the target conformal orthogonal basis are used to construct a surface shape expression for the mirror deformation of the optical mirror, so as to trigger optomechanical coupling analysis or surface shape correction operation using the surface shape expression. The target conformal orthogonal basis is either the first conformal orthogonal basis or the second conformal orthogonal basis. As can be seen from the above, this application constructs an intrinsic coordinate system for a rotationally symmetric optical mirror, solves the eigenvalue equations of the Laplace-Beltramme operator based on this intrinsic coordinate system, obtains the angular and radial equations using the separation of variables method, and determines the first conformal orthogonal basis from its angular and radial solutions, respectively; simultaneously, it performs Gram-Schmidt orthogonalization on the Zernike polynomial to obtain the target radial polynomial, and then determines the second conformal orthogonal basis. A finite element simulation model of the mirror is established, and the deformation displacement under a preset load condition is solved to extract the normal displacement component. A basis function matrix is ​​constructed using either the first or second conformal orthogonal basis as the target conformal orthogonal basis, and the least squares problem formed with the normal displacement component is solved to obtain the fitting coefficients, thereby constructing the surface shape expression of the mirror deformation, which is used to trigger optomechanical coupling analysis or surface shape correction operations. In this way, this application can design an orthogonal basis fitting scheme that fits the geometric features of the optical mirror itself, so as to overcome the problem that the fitting scheme based on Zernike polynomials is insufficient in terms of fitting accuracy at the edge of large curvature surfaces due to the dependence of orthogonality on planar Euclidean metric. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0017] Figure 1 This is a flowchart of a mirror deformation characterization method based on a conformal orthogonal basis disclosed in this application; Figure 2 This is a flowchart of a specific mirror deformation characterization method based on a conformal orthogonal basis disclosed in this application; Figure 3 This is a schematic diagram of a conformal continuous base disclosed in this application; Figure 4 This is a primitive optical surface displacement distribution cloud map disclosed in this application; Figure 5 This is a fitting residual distribution cloud map obtained from a conformal cosine basis used in this invention, as disclosed in this application; Figure 6 This application discloses a fitting residual distribution cloud map obtained using a conformal Zernike basis; Figure 7 This application discloses a fitted residual distribution cloud map obtained using a classical Zernike basis; Figure 8 This is a schematic diagram of a mirror deformation characterization device based on a conformal orthogonal substrate disclosed in this application. Figure 9 This is a structural diagram of an electronic device disclosed in this application. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. 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 are within the scope of protection of the present invention.

[0019] Currently, for mirror characterization, the classical Zernike polynomial is commonly used as the basis for fitting mirror deformation. This polynomial is defined in a unit circle region of the plane, and its terms correspond to typical optical aberrations. It is the standard fitting tool in commercial optical simulation software. However, this approach has inherent geometric defects. The orthogonality of the classical Zernike polynomial relies on planar Euclidean metrics. When fitting high-curvature rotationally symmetric aspherical surfaces such as spheres and parabolas, the surface deformation field needs to be projected onto a two-dimensional plane. The projection process introduces geometric metric distortion, directly causing the orthogonality of the basis on the surface to degenerate. Moreover, the greater the curvature of the surface, the more significant the degradation phenomenon, and the fitting error in the mirror edge region will increase sharply. At the same time, the curvature distribution of different rotational optical surfaces varies greatly, and the degree of distortion caused by planar projection changes with the surface shape. This makes the fitting effect of the Zernike polynomial unstable and inconsistent. When dealing with complex mirror deformations, a large number of basis terms are required to achieve convergence, which significantly reduces the simulation efficiency and introduces additional fitting biases, making it difficult to meet the requirements of high-precision optomechanical coupling analysis. To this end, this application provides a method, apparatus, device and medium for characterizing the mirror deformation of a conformal orthogonal substrate, which can design an orthogonal substrate fitting scheme that fits the geometric characteristics of the optical mirror itself, so as to overcome the problem that the fitting scheme based on Zernike polynomials is insufficient in terms of fitting accuracy at the edge of large curvature surfaces due to the dependence of orthogonality on planar Euclidean metric.

[0020] See Figure 1 As shown, this embodiment of the invention discloses a method for characterizing mirror deformation based on a conformal orthogonal basis, including: Step S11: Determine the rotationally symmetric optical mirror and construct an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial position of each point on the optical mirror.

[0021] In this embodiment, a rotationally symmetric optical mirror refers to a curved surface formed by rotating the cross-sectional profile around the optical axis as the axis of symmetry. Such mirrors are widely used in precision optical systems such as primary mirrors of astronomical telescopes and primary and secondary mirrors of space cameras.

[0022] For the aforementioned optical mirror, an intrinsic coordinate system is used to mark each point on the mirror surface, that is, the position of the point is described directly using the inherent geometric parameters of the surface itself, without the need for an external planar coordinate system.

[0023] Specifically, the polar angle and azimuth angle of the optical mirror are selected as the intrinsic geometric parameters of the optical mirror, and a corresponding spherical coordinate system is constructed based on the intrinsic geometric parameters. This spherical coordinate system is then used as the intrinsic coordinate system to mark the spatial positions of each point on the optical mirror. The introduction of the intrinsic coordinate system allows the subsequent construction of the orthogonal basis to be directly based on the geometric measurements of the surface itself, fundamentally avoiding the measurement distortion problem caused by planar projection, and laying a geometric foundation for high-precision surface deformation characterization.

[0024] Step S12: Construct the eigenvalue equations of the Laplace-Beltrami operator based on the intrinsic coordinate system, determine the angular equations and radial equations using the variable separation method and the eigenvalue equations, and determine the first conformal orthogonal basis based on the angular solution corresponding to the angular equations and the radial solution corresponding to the radial equations.

[0025] In this embodiment, the Laplace-Beltramie operator is a generalization of the classical Laplace operator to Riemannian manifolds. Its characteristic functions form a complete orthogonal basis on the manifold, suitable for describing the function distribution on a surface. This step uses the Laplace-Beltramie operator as the core tool to construct the first conformal orthogonal basis in the surface space defined by the optical mirror.

[0026] Specifically, a first expression for the Laplace-Beltrammian operator on the optical mirror is constructed in the intrinsic coordinate system; a normalized polar angle is determined based on the polar angle and the half-aperture angle of the optical mirror surface, and the polar angle in the intrinsic coordinate system is replaced with the normalized polar angle to obtain the target intrinsic coordinate system; the first expression is transformed into a second expression based on the target intrinsic coordinate system, and an eigenvalue equation is constructed based on the second expression; the solution of the eigenvalue equation is separated into an angular function related to the azimuth angle and a radial function related to the normalized polar angle to complete the variable separation operation, and the angular equation and radial equation are obtained based on the variable separation operation and the eigenvalue equation; boundary conditions are applied to the radial equation and solved to obtain the radial solution, the angular equation is solved to obtain the angular solution, and a first conformal orthogonal basis is obtained based on the radial solution, the angular solution, and the first normalization coefficient. The first conformal orthogonal basis is based on the metric of the surface itself and has orthogonality directly on the optical mirror geometry, which can accurately and stably describe various deformation modes on the surface.

[0027] For the radial equation, appropriate boundary conditions need to be applied to the surface before solving. These boundary conditions include a bounded regularity condition at the optical axis of the optical mirror and a Neumann boundary condition where the radial derivative is zero at the aperture boundary of the optical mirror. Under these boundary condition constraints, the radial equation can be solved to obtain the radial solution.

[0028] Step S13: Perform Gram-Schmidt orthogonalization on the Zernike polynomial based on the intrinsic coordinate system to obtain the target radial polynomial, and use the target radial polynomial to determine the second conformal orthogonal basis.

[0029] In this embodiment, the Zernike polynomial is a classic tool in optics for describing wavefront aberrations, and its radial polynomial portion constitutes an orthogonal polynomial sequence on a unit circle in a plane. However, when applied to high-curvature optical surfaces, the orthogonality depends on the plane Euclidean metric, and the orthogonality condition is no longer satisfied under the intrinsic metric of the surface, leading to a decrease in fitting accuracy. Therefore, this step, while preserving the angular modal form of the Zernike polynomial, re-orthogonalizes its radial polynomial portion so that it satisfies the orthogonality condition under the intrinsic metric of the optical mirror, thereby constructing a second conformal orthogonal basis.

[0030] Specifically, the radial polynomials in the Zernike polynomials are used as the initial sequence, and the initial sequence is orthogonalized using a weight function determined based on the intrinsic coordinate system to obtain target radial polynomials that are orthogonal under the weight function. A second conformal orthogonal basis is obtained based on the target radial polynomials, the angular mode of the azimuth angle, and the second normalized coefficients. The second normalized coefficients are coefficients determined based on the target radial polynomials, the normalized polar angle, and the area element determined based on the intrinsic coordinate system.

[0031] Compared with the construction method of the first conformal orthogonal basis, the second conformal orthogonal basis makes full use of the correspondence between Zernike polynomials and optical aberrations. By reorthogonalizing it, it restores orthogonality under surface measurement, which significantly improves the fitting accuracy and stability of high curvature surface deformation while maintaining physical interpretability.

[0032] Step S14: Construct the finite element simulation model of the optical mirror, solve the finite element simulation model under the applied preset load condition, obtain the deformation displacement of the finite element nodes in the finite element simulation model, and determine the normal displacement component based on the deformation displacement.

[0033] In this embodiment, finite element simulation is the core means of obtaining the structural response of an optical mirror under actual service loads, and it can accurately reproduce the deformation state of the mirror under the combined action of mechanical and thermal loads.

[0034] Specifically, a finite element simulation model is established for the optical mirror and its supporting structure. Material parameters including elastic modulus, Poisson's ratio, and coefficient of thermal expansion are set in the finite element simulation model, and preset load conditions including mechanical and thermal loads are applied to the model to obtain finite element equilibrium equations. These equations are solved to obtain the deformation displacements of the finite element nodes, including rigid displacement and elastic deformation components. Based on these deformation displacements, sampling points are selected on the optical mirror using finite element shape functions, and the displacement field at each sampling point is determined. Based on the displacement field and the normal unit vector of the optical mirror at the sampling point, the normal displacement component at that sampling point is obtained. The normal displacement component directly reflects the local convexity or concavity of the mirror along the direction of light propagation. It is a key physical quantity characterizing the impact of surface shape changes on image quality and is also direct input data for subsequent fitting analysis.

[0035] Step S15: Construct a basis function matrix based on the target conformal orthogonal basis, and solve the least squares problem based on the basis function matrix and the normal displacement component to obtain fitting coefficients. Use the fitting coefficients and the target conformal orthogonal basis to construct a surface shape expression for the mirror deformation of the optical mirror, so as to trigger optomechanical coupling analysis or surface shape correction operation using the surface shape expression; the target conformal orthogonal basis is the first conformal orthogonal basis or the second conformal orthogonal basis.

[0036] In this embodiment, after obtaining the normal displacement components and the conformal orthogonal basis, the discrete sampling point deformation data is expressed as a continuous surface shape function through basis function expansion and least squares fitting, which is then used for subsequent optomechanical coupling analysis and surface shape correction. The specific process is as follows.

[0037] The target conformal orthogonal basis is selected according to actual needs. It can be the first conformal orthogonal basis constructed in step S12 above, or the second conformal orthogonal basis constructed in step S13. Both types of bases are strictly orthogonal under the intrinsic metric of the optical mirror and can be flexibly selected according to different application scenarios.

[0038] Substituting the coordinates of each sampling point into the basis functions of each order in the target conformal orthogonal basis, the values ​​of each basis function at each sampling point are calculated to construct the basis function matrix. Then, using the normal displacement component vector at the sampling point as the fitting objective and the basis function matrix as the design matrix, a least squares problem is constructed. By minimizing the sum of squares of the fitting residuals, the least squares problem is solved to obtain the fitting coefficients corresponding to each order of basis functions. Since the target conformal orthogonal basis satisfies the orthogonality and normalization condition under surface metric, and the basis functions are independent of each other, the numerical conditions of the normal equations of the least squares problem are well-defined, the solution process is stable and efficient, and fast convergence can be achieved without adding a large number of basis terms.

[0039] After obtaining the fitting coefficients, a surface shape expression for the optical mirror deformation is constructed based on the fitting coefficients and the basis functions of each order in the target conformal orthogonal basis. This surface shape expression can provide an accurate estimate of the normal displacement at any position on the mirror surface, and fully describes the global surface shape change of the mirror surface under a preset load.

[0040] Ultimately, the constructed surface shape expression can be used to trigger two types of operations: first, optomechanical coupling analysis, which involves importing surface shape data into optical simulation software to calculate the impact of mirror deformation on the system's imaging quality, evaluate optical performance indicators such as wavefront error and Zernike aberration coefficient, and provide quantitative basis for system design optimization; second, surface shape correction, which involves feeding back the surface shape fitting results to the active optical system or precision assembly mechanism to drive the actuator to apply reverse compensation, thereby achieving closed-loop control and active correction of mirror surface shape errors and improving the on-orbit imaging performance of the optical system.

[0041] As can be seen from the above, this application, by introducing the intrinsic coordinate system of the optical mirror, eliminates the dependence of traditional Zernike polynomials on planar Euclidean metrics, fundamentally avoiding the geometric metric distortion introduced by planar projection. For the large edge corner region of high-curvature rotationally symmetric aspherical surfaces, this scheme can achieve fast convergence with a smaller number of basis terms, significantly reducing the computational scale and improving simulation efficiency. Simultaneously, this method provides two construction paths: a first conformal orthogonal basis and a second conformal orthogonal basis. The first conformal orthogonal basis is obtained by solving the eigenvalue problem of the Laplace-Beltramm operator for the surface, possessing strict mathematical orthogonality; the second conformal orthogonal basis is obtained by Gram-Schmidt orthogonalization of the radial part of the Zernike polynomial, restoring the orthogonality on the surface while preserving the physical meaning of optical aberrations. The two bases complement each other. By extracting the normal displacement components of the mirror surface through finite element simulation and combining the basis function matrix with least squares solution, the transformation from discrete nodal displacement to continuous surface shape expression is realized. The obtained surface shape results can directly drive optomechanical coupling analysis and active surface shape correction.

[0042] The following is combined with Figure 2 The schematic diagram shown illustrates the technical solution of the embodiments of this application in detail.

[0043] First, the intrinsic coordinate parameterization of the rotationally symmetric surface is performed.

[0044] An intrinsic parameterized description is established for rotationally symmetric smooth surfaces. Based on the surface's own geometric metric (first fundamental form), an intrinsic coordinate system is constructed on the surface, thereby avoiding geometric metric distortion caused by the introduction of planar projection.

[0045] Specifically, common rotationally symmetric optical surfaces include planes, spheres, parabolic surfaces, and higher-order aspherical surfaces. This invention uses the intrinsic geometric parameters of the surface to uniformly describe the above-mentioned surface types, that is, defining local geometric quantities such as length, angle, and area using the first basic form of the surface itself, without relying on a specific external coordinate system.

[0046] For rotationally symmetric optical surfaces, in spherical coordinates Below, its intrinsic metric is in the form of a line element. ;in Polar angle, The half-aperture angle of the optical surface. This is the azimuth angle. The corresponding determinant is... Area in yuan .

[0047] To facilitate numerical calculations, normalized polar coordinates are introduced. This maps the computational domain to a standard interval, achieving an intrinsic equivalent transformation from polar angles to normalized radial coordinates.

[0048] Secondly, a two-dimensional conformal orthogonal basis is constructed.

[0049] This step can include two methods for constructing conformal bases, both based on the intrinsic coordinate system. One method involves establishing a Laplace-Beltrami operator on the surface and solving the eigenvalue problem of this operator to obtain a set of mutually orthogonal basis functions in the sense of the surface's intrinsic metric, i.e., a two-dimensional conformal orthogonal basis. The other method is based on the classical Zernike polynomial and modifies it to reconstruct a polynomial basis that satisfies the orthogonality of the surface within the intrinsic geometric framework. This basis inherits the geometric structure of the surface. Its orthogonality is not affected by the surface curvature, thus overcoming the defect of orthogonality degradation of the classical Zernike polynomial on surfaces with large curvature.

[0050] Firstly, for a two-dimensional conformal cosine basis (i.e., the first conformal orthogonal basis) based on the eigenmodes of the Laplace-Beltrami operator. The Laplace-Beltrami operator... It is a generalized Laplace operator defined on Riemannian manifolds (surfaces). Its characteristic modes naturally satisfy orthogonality in the sense of the intrinsic metric of the surface, and it is the theoretical basis for constructing intrinsic orthogonal basis.

[0051] In spherical coordinates The expression for the Laplace-Beltrami operator is as follows: (That is, the first expression); where u is the distribution function to be solved; converted to normalized coordinates Later (That is, the second expression).

[0052] Establish eigenvalue equations In the formula, Let be the eigenvalues. Utilizing rotational symmetry, we separate the variables in the solution, letting In the formula, It is a radial function. Let be the angular function. Substituting it into the eigenvalue equation, we can separate the angular equation and the radial equation.

[0053] The angular equation can be: .in, Let be the angular order. Its solution is a Fourier mode (a simple harmonic periodic function with a fixed angular order), taking its real form as . or .

[0054] The radial equation can be: In the formula, This is the radial order. The corresponding boundary condition is: at the optical axis... The solution must be bounded, which is a regularity condition; at the aperture boundary. Apply Neumann boundary conditions This ensures that the normal derivative of the basis functions is zero at the physical mirror boundary. The radial equation solution satisfying the above boundary conditions is denoted as... This constitutes a family of radially orthogonal functions under the corresponding intrinsic metrics.

[0055] Finally, the complete expression for a two-dimensional conformal orthogonal basis can be given by: .in It is the normalization coefficient (i.e., the first normalization coefficient), derived from... This orthogonal normalization condition is uniquely determined.

[0056] Secondly, regarding the conformal Zernike orthogonal basis (i.e., the second conformal orthogonal basis), a conformal cosine basis has already been constructed in the previous step. However, polynomial-type bases have advantages such as computational simplicity, clear physical meaning, and good compatibility with existing analysis procedures. Therefore, a polynomial-type orthogonal basis suitable for rotationally symmetric surfaces can be designed. This invention modifies the classical Zernike polynomial and reconstructs a polynomial basis that satisfies the orthogonality of the surface within the intrinsic geometric framework. This method retains the advantages of the polynomial structure while ensuring universality and uniqueness on any rotationally symmetric surface. Thus, a precise fit of the polynomial-type orthogonal basis is achieved on surfaces with large curvature.

[0057] First, we introduce the classical Zernike radial polynomial as the initial sequence, i.e. Since the polynomial is no longer orthogonal under surface metric, therefore in the weight function... By performing Gram-Schmidt orthogonalization on it, we obtain the conformal radial polynomial. .

[0058] Radial polynomial and angular mode By combining these, we can obtain the complete expression for the two-dimensional conformal Zernike basis, i.e. , of which The normalization coefficient (also known as the second normalization coefficient) is determined by the intrinsic integral, and the formula for its determination is as follows: .

[0059] Under this definition, a two-dimensional basis satisfies strict weighted orthogonality, i.e. In the formula, and For Kronek The function embodies the orthogonal property of the basis band weights. Let be the radial order of the other contrast basis in the integral. Let be the angular order of the other contrast basis in the integral.

[0060] When the surface tends to be planar, the intrinsic weight function degenerates into planar polar coordinate weights, and the conformal Zernike basis is automatically reduced to a planar Zernike polynomial, thus exhibiting strict planar compatibility.

[0061] and, Figure 3 This is a visualization of the two-dimensional conformal orthogonal basis functions constructed in this invention. The figure shows several typical low-order basis functions in the normalized parameter domain. The distribution behavior of the basis functions is shown in the form of polar coordinate pseudo-color contour plots. The value distribution over the unit circle domain, where the radial coordinate corresponds to the normalized polar angle. Angular coordinates correspond to azimuth angle In each subplot, the intensity of the color represents the magnitude of the function value. Additionally, Figure 3 In this context, n and m are related to the basis functions, where n is the radial order and m is the angular order.

[0062] Then, the mirror normal displacement is extracted.

[0063] A finite element simulation model of a rotationally symmetric optical mirror is established, and the model is solved after applying the corresponding load conditions. The structural deformation displacement is extracted at the finite element nodes of the mirror. Considering that the deflection of the light propagation direction is mainly determined by the normal displacement component of the mirror, this step extracts the normal displacement component at each node as the scalar field data to be fitted.

[0064] Specifically, a finite element model is established for the rotationally symmetric optical mirror and its supporting structure, material parameters (including elastic modulus, Poisson's ratio, and coefficient of thermal expansion) are set, and thermal loads are applied. and mechanical load The thermal load is transformed into equivalent thermal strain through thermo-mechanical coupling and then participates in the structural analysis. Solving... This is the steady-state finite element equilibrium equation. This is the overall system stiffness matrix, reflecting the mechanical properties of the optical components and supporting structure; It is the nodal displacement vector, which includes rigid body displacement components and elastic deformation components.

[0065] Based on the obtained nodal displacements By using finite element shape function interpolation, M sampling points are selected on the optical surface to obtain the displacement field at each point. In the formula, The displacement is represented by three components in three-dimensional space. Considering that the deflection of the light propagation direction is mainly determined by the normal displacement component of the mirror, the normal displacement component at each sampling point is extracted. ,in Let be the normal unit vector of the optical surface at the i-th sampling point. It is calculated at this point from the analytical expression of the optical surface shape.

[0066] Finally, surface deformation fitting based on the least squares method is performed.

[0067] The extracted mirror normal displacement field is expressed as a linear combination of the two-dimensional conformal orthogonal basis functions. The fitting coefficients corresponding to each basis function are solved by the least squares method, that is, minimizing the L2 norm of the residual between the fitted value at the node and the finite element calculation value. Finally, a continuous analytical expression of the deformable surface shape is obtained for subsequent ray tracing analysis.

[0068] Specifically, the optical surface normal displacement field It can be expressed as a linear combination of constructed two-dimensional conformal orthogonal bases. .in, It is the first basis functions, These are the corresponding substitution fitting coefficients. This represents the number of basis terms selected. In practical applications, the number of sampling points... It should be much larger than the number of base terms. (Right now This is used to construct an overdetermined system of equations, ensuring the stability of the least squares solution.

[0069] Constructing the basis function matrix Its elements are defined as: That is, the first... At the sampling point, the first The values ​​of the basis functions are determined. The displacement sampling vector is then... The fitting coefficient vector is obtained by solving... This least squares problem is solved.

[0070] Since the constructed conformal basis satisfies orthogonality in the sense of the intrinsic metric of the surface, the matrix is ​​uniform when the sampling points are evenly distributed. Approaching a diagonal matrix provides favorable numerical conditions, which can further improve the computational efficiency and numerical stability of the fit.

[0071] Finally, the obtained continuous analytical form of the deformable surface shape expression is... It can be directly used as input for subsequent ray tracing models, enabling high-precision information transfer between discrete finite element deformation results and continuous optical analysis.

[0072] The conformal orthogonal basis constructed in this invention is based on the intrinsic geometric metric (first fundamental form) of rotationally symmetric surfaces. The orthogonality of the basis functions is strictly valid under the metric meaning of the surface itself, independent of the assumptions of a planar Euclidean coordinate system and a planar circular domain. Therefore, even for rotationally symmetric optical surfaces with large curvature, the basis used in this invention will not suffer orthogonality degradation due to planar projection, thus maintaining uniform fitting accuracy across the entire aperture range, effectively suppressing the increase of fitting error in edge regions, and overcoming the inherent defect of severe edge distortion in Zernike polynomials on large curvature surfaces.

[0073] Furthermore, since the constructed conformal orthogonal basis satisfies the orthogonal normalization condition under the intrinsic metric of the surface, when the sampling points are relatively uniformly distributed on the optical surface, the basis function matrix constructed in the least squares fitting is... It approximates a diagonal matrix, has a good matrix condition number, and the solution process is numerically stable. It can reduce the number of basis terms required while ensuring fitting accuracy and improving computational efficiency.

[0074] The conformal orthogonal basis function system overcomes the dependence of classical Zernike polynomials on planar circular domains in terms of fitting accuracy on rotationally symmetric surfaces. This invention outputs a continuous analytical form of deformable surface representation, which can be directly used as input for ray tracing models. It exhibits good compatibility with mainstream optical design and ray tracing software, requiring no special modifications to geometric optics algorithms. Furthermore, the method framework of this invention is applicable to various surface types, including planes, spheres, parabolas, and various high-order rotationally symmetric aspherical surfaces, demonstrating strong versatility. It can be widely applied to optomechanical coupling simulation analysis and active surface correction in high numerical aperture (NA) optical systems.

[0075] In addition, to test the substrate's ability to characterize mid-to-high frequency errors in a two-dimensional scene, the deformation signal simulating residual errors in optical element processing was constructed as shown below, including three components: periodic ripples, random roughness, and local defects: ; in, For the intrinsic normalized radial coordinates of the surface ( , (where half the aperture angle is the curved surface). It is the azimuth angle. This represents the maximum value of the normalized radial coordinates. The periodic ripple amplitude is set to... The number of cycles is Angular order (Simulated axisymmetric machining ripples); random noise amplitude Used to characterize the random roughness of optical surfaces; local defect amplitude Pulse width central position This simulation was used to model localized machining defects. The simulation employed a 2D surface with a radius of curvature of 100 mm, a conic coefficient K=0, a half-diameter of 100 mm, and included aspherical coefficients of orders 4 to 12. A 150×150 sampling grid (polar angle × azimuth angle) was used, with the half-diameter corresponding to a radial diameter of 100 mm. The first 40 terms were used for fitting the data for all three substrates. The fitting residuals are shown in Table 1 below. Table 1. Fitting residuals of the basis

[0076] The simulation results above demonstrate that the rotationally symmetric mirror deformation fitting method based on two conformal substrates proposed in this invention has significant advantages in high-precision characterization of rotationally symmetric optical surfaces with large curvature, thus verifying the effectiveness and feasibility of the technical solution of this invention.

[0077] Reference Figure 4 , Figure 5 , Figure 6 as well as Figure 7 That is, the two substrate fitting methods of this invention and the traditional Zernike polynomial method for fitting the normal displacement residuals on a rotationally symmetric optical mirror. Figure 4 This is the original optical surface displacement distribution cloud map. Figure 5 The image shows the fitted residual distribution cloud map obtained using the conformal cosine basis of this invention. Figure 6 This is a contour map of the fitted residuals obtained using a conformal Zernike basis. Figure 7 This is a contour plot of the fitted residuals obtained using the classic Zernike basis. All plots are presented in pseudo-color contour plot format. Figure 5 and Figure 6 The residuals in the pores are relatively uniformly distributed across the entire aperture range; Figure 5 The corresponding residual RMS (Root Mean Square) = 2.471e-04 mm, PV (Peak-to-Valley) = 4.536e-04 mm. Figure 6The corresponding residuals are RMS = 5.262e-04 mm and PV = 1.221e-03 mm. Figure 7 A significant increase in residuals is observed in the central region, with residual RMS = 1.145e-03 mm and PV = 2.827e-03 mm. The above comparison verifies that the two conformal substrates provided by this invention exhibit higher fitting accuracy and a more uniform residual distribution on large curvature rotationally symmetric optical surfaces.

[0078] Furthermore, it should be noted that the fitting object of this invention is the normal displacement field of the optical mirror, regardless of the type of load that causes the deformation. Therefore, this method is not only applicable to mirror deformation under the coupled action of thermal and mechanical loads, but also to mirror normal deformation caused by other types of external factors such as clamping force, gravity, vibration load, and irradiation deformation. It can also be used for high-precision description of target surface shape in active surface correction systems, and has a wide range of applications.

[0079] The technical framework of this invention provides two methods for constructing deformation based on intrinsic conformal bases. This framework applies to general Riemannian manifolds (surfaces) and is not limited to rotationally symmetric cases. For rotationally symmetric surfaces, this invention utilizes their symmetry to separate variables, obtaining analytical angular basis functions and numerical radial basis functions. For general optical surfaces that are not rotationally symmetric, variable separation is no longer applicable, but the Laplace-Beltrami eigenvalue problem can be directly solved numerically on the surface mesh using the finite element method or finite difference method to obtain the intrinsic orthogonal basis of the corresponding surface. Thus, the core method framework of this invention can be extended to the deformation fitting of non-rotationally symmetric mirrors.

[0080] In the fitting process, the number of basis terms N can be adaptively determined according to the convergence of the actual fitting residuals: the number of basis terms is gradually increased, and the process stops when the root mean square (RMS) of the full aperture fitting residuals is lower than the preset accuracy threshold, thereby achieving the optimal balance between fitting accuracy and computational efficiency and adapting to engineering application scenarios with different accuracy requirements.

[0081] Accordingly, see Figure 8 As shown, this application provides a mirror deformation characterization device based on a conformal orthogonal substrate, comprising: The coordinate system construction module 11 is used to determine the rotationally symmetric optical mirror and construct an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial position of each point on the optical mirror. The first basis determination module 12 is used to construct the eigenvalue equation of the Laplace-Beltrami operator based on the intrinsic coordinate system, determine the angular equation and the radial equation using the variable separation method and the eigenvalue equation, and determine the first conformal orthogonal basis based on the angular solution corresponding to the angular equation and the radial solution corresponding to the radial equation. The second basis determination module 13 is used to perform Gram-Schmidt orthogonalization on the Zernike polynomial based on the intrinsic coordinate system to obtain the target radial polynomial, and to determine the second conformal orthogonal basis using the target radial polynomial. The component determination module 14 is used to construct the finite element simulation model of the optical mirror, solve the finite element simulation model under the applied preset load condition, obtain the deformation displacement of the finite element nodes in the finite element simulation model, and determine the normal displacement component based on the deformation displacement. The expression construction module 15 is used to construct a basis function matrix based on the target conformal orthogonal basis, and to solve the least squares problem constructed based on the basis function matrix and the normal displacement component to obtain fitting coefficients. The fitting coefficients and the target conformal orthogonal basis are used to construct a surface shape expression of the mirror deformation of the optical mirror, so as to trigger an optomechanical coupling analysis operation or a surface shape correction operation using the surface shape expression; the target conformal orthogonal basis is the first conformal orthogonal basis or the second conformal orthogonal basis.

[0082] In some specific embodiments, the coordinate system construction module 11 specifically includes: The first coordinate system determination unit is used to select the polar angle and azimuth angle of the optical mirror as the intrinsic geometric parameters of the optical mirror, and construct a corresponding spherical coordinate system based on the intrinsic geometric parameters, and determine the spherical coordinate system as the intrinsic coordinate system that marks the spatial position of each point on the optical mirror.

[0083] In some specific embodiments, the first substrate determination module 12 specifically includes: An expression construction unit is used to construct the first expression of the Laplace-Beltramm operator on the optical mirror surface in the intrinsic coordinate system. The second coordinate system determination unit is used to determine the normalized polar angle based on the polar angle and the half-aperture angle of the optical mirror surface, and replace the polar angle in the intrinsic coordinate system with the normalized polar angle to obtain the target intrinsic coordinate system. An equation construction unit is used to transform the first expression into a second expression based on the target intrinsic coordinate system, and to construct an eigenvalue equation based on the second expression. The first equation determination unit is used to separate the solution of the eigenvalue equation into an angular function related to the azimuth angle and a radial function related to the normalized polar angle, so as to complete the variable separation operation, and obtain the angular equation and the radial equation based on the variable separation operation and the separation of the eigenvalue equation. The first basis determination unit is used to apply boundary conditions to the radial equation and solve it to obtain the radial solution, solve the angular equation to obtain the angular solution, and obtain the first conformal orthogonal basis based on the radial solution, the angular solution and the first normalization coefficient.

[0084] In some specific embodiments, the boundary conditions include a solution-bounded regularity condition at the optical axis of the optical mirror, and a Neumann boundary condition with a zero radial derivative at the aperture boundary of the optical mirror.

[0085] In some specific embodiments, the second substrate determination module 13 specifically includes: A polynomial determination unit is used to take the radial polynomials in the Zernike polynomials as an initial sequence and use the weight function determined based on the intrinsic coordinate system to perform Gram-Schmidt orthogonalization on the initial sequence to obtain target radial polynomials that are mutually orthogonal under the weight function. The second basis determination unit is used to obtain the second conformal orthogonal basis based on the target radial polynomial, the angular mode of the azimuth angle, and the second normalization coefficient.

[0086] In some specific embodiments, the second normalization coefficient is a coefficient determined based on the target radial polynomial, the normalized polar angle, and the area element determined based on the intrinsic coordinate system.

[0087] In some specific embodiments, the component determination module 14 specifically includes: The model building unit is used to establish a finite element simulation model of the optical mirror and the supporting structure of the optical mirror; The second equation determination unit is used to set material parameters including elastic modulus, Poisson's ratio and thermal expansion coefficient in the finite element simulation model, and apply preset load conditions including mechanical load and thermal load to the finite element simulation model to obtain the finite element equilibrium equation. The displacement determination unit is used to solve the finite element equilibrium equation to obtain the deformation displacement of the finite element nodes, including rigid body displacement components and elastic deformation components. The component determination unit is used to select sampling points on the optical mirror surface based on the deformation displacement using finite element shape functions, determine the displacement field at the sampling points, and obtain the normal displacement component at the sampling points based on the displacement field and the normal unit vector of the optical mirror surface at the sampling points.

[0088] Furthermore, embodiments of this application also disclose an electronic device, Figure 9This is a structural diagram of an electronic device 20 according to an exemplary embodiment. The content of the diagram should not be construed as limiting the scope of this application. The electronic device 20 may specifically include: at least one processor 21, at least one memory 22, a power supply 23, a communication interface 24, an input / output interface 25, and a communication bus 26. The memory 22 stores a computer program, which is loaded and executed by the processor 21 to implement the relevant steps in the conformal orthogonal basis-based mirror deformation characterization method disclosed in any of the foregoing embodiments. Furthermore, the electronic device 20 in this embodiment may specifically be a computer.

[0089] In this embodiment, the power supply 23 is used to provide operating voltage for each hardware device on the electronic device 20; the communication interface 24 can create a data transmission channel between the electronic device 20 and external devices, and the communication protocol it follows can be any communication protocol applicable to the technical solution of this application, and is not specifically limited here; the input / output interface 25 is used to acquire external input data or output data to the outside world, and its specific interface type can be selected according to specific application needs, and is not specifically limited here.

[0090] In addition, the memory 22, as a carrier for resource storage, can be a read-only memory, random access memory, disk or optical disk, etc. The resources stored thereon can include operating system 221, computer program 222, etc., and the storage method can be temporary storage or permanent storage.

[0091] The operating system 221 is used to manage and control the various hardware devices on the electronic device 20 and the computer program 222, which may be Windows Server, Netware, Unix, Linux, etc. In addition to including a computer program capable of performing the conformal orthogonal basis-based mirror deformation characterization method disclosed in any of the foregoing embodiments, the computer program 222 may further include computer programs capable of performing other specific tasks.

[0092] Furthermore, this application also discloses a computer-readable storage medium for storing a computer program; wherein, when the computer program is executed by a processor, it implements the aforementioned disclosed method for characterizing mirror deformation based on a conformal orthogonal basis. Specific steps of this method can be found in the corresponding content disclosed in the foregoing embodiments, and will not be repeated here.

[0093] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since it corresponds to the method disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to in the method section.

[0094] Those skilled in the art will further recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, computer software, or a combination of both. To clearly illustrate the interchangeability of hardware and software, the components and steps of the various examples have been generally described in terms of functionality in the foregoing description. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0095] The steps of the methods or algorithms described in conjunction with the embodiments disclosed herein can be implemented directly by hardware, a software module executed by a processor, or a combination of both. The software module can be located in random access memory (RAM), main memory, read-only memory (ROM), electrically programmable ROM, electrically erasable programmable ROM, registers, hard disk, removable disk, CD-ROM, or any other form of storage medium known in the art.

[0096] Finally, it should be noted that in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0097] The technical solutions provided in this application have been described in detail above. Specific examples have been used to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this application. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A method for specular deformation characterization based on conformal orthogonal basis, characterized in that, include: Determine a rotationally symmetric optical mirror and construct an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial position of each point on the optical mirror. Based on the intrinsic coordinate system, the eigenvalue equations of the Laplace-Beltramian operator are constructed. The angular equations and radial equations are determined using the variable separation method and the eigenvalue equations. The first conformal orthogonal basis is determined based on the angular solution corresponding to the angular equations and the radial solution corresponding to the radial equations. The Zernike polynomial is orthogonalized using the intrinsic coordinate system to obtain the target radial polynomial, and the second conformal orthogonal basis is determined using the target radial polynomial. A finite element simulation model of the optical mirror is constructed, and the finite element simulation model under a preset load condition is solved to obtain the deformation displacement of the finite element nodes in the finite element simulation model, and the normal displacement component is determined based on the deformation displacement. A basis function matrix is ​​constructed based on the target conformal orthogonal basis, and a least-squares problem based on the basis function matrix and the normal displacement components is solved to obtain fitting coefficients. The fitting coefficients and the target conformal orthogonal basis are used to construct a surface shape expression for the mirror deformation of the optical mirror, so as to trigger optomechanical coupling analysis or surface shape correction operation using the surface shape expression; the target conformal orthogonal basis is either the first conformal orthogonal basis or the second conformal orthogonal basis. The step of performing Gram-Schmidt orthogonalization on the Zernike polynomial based on the intrinsic coordinate system to obtain the target radial polynomial, and using the target radial polynomial to determine the second conformal orthogonal basis, includes: The radial polynomials in the Zernike polynomials are used as the initial sequence, and the initial sequence is orthogonalized by Gram-Schmidt using the weight function determined based on the intrinsic coordinate system to obtain the target radial polynomials that are orthogonal to each other under the weight function. The second conformal orthogonal basis is obtained based on the target radial polynomial, the angular mode of the azimuth angle of the optical mirror, and the second normalization coefficient.

2. The method of claim 1, wherein, The construction of an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial positions of each point on the optical mirror includes: The polar angle and azimuth angle of the optical mirror are selected as the intrinsic geometric parameters of the optical mirror, and a corresponding spherical coordinate system is constructed based on the intrinsic geometric parameters. The spherical coordinate system is then determined as the intrinsic coordinate system that marks the spatial position of each point on the optical mirror.

3. The method of claim 2, wherein, The process of constructing the eigenvalue equations of the Laplace-Beltrammian operator based on the intrinsic coordinate system, determining the angular and radial equations using the separation of variables and the eigenvalue equations, and determining the first conformal orthogonal basis based on the angular solutions corresponding to the angular equations and the radial solutions corresponding to the radial equations includes: Construct the first expression of the Laplace-Beltramian operator on the optical mirror surface in the intrinsic coordinate system; The normalized polar angle is determined based on the polar angle and the half-aperture angle of the optical mirror surface, and the polar angle in the intrinsic coordinate system is replaced with the normalized polar angle to obtain the target intrinsic coordinate system; The first expression is transformed into a second expression based on the target intrinsic coordinate system, and an eigenvalue equation is constructed based on the second expression; The solution of the eigenvalue equation is separated into an angular function related to the azimuth angle and a radial function related to the normalized polar angle to complete the variable separation operation. Based on the variable separation operation and the separation of the eigenvalue equation, the angular equation and the radial equation are obtained. Boundary conditions are applied to the radial equation and solved to obtain the radial solution. The angular equation is then solved to obtain the angular solution. Based on the radial solution, the angular solution, and the first normalization coefficient, a first conformal orthogonal basis is obtained.

4. The method of claim 3, wherein, The boundary conditions include a bounded regularity condition at the optical axis of the optical mirror and a von Neumann boundary condition with zero radial derivative at the aperture boundary of the optical mirror.

5. The method of claim 1, wherein, The second normalization coefficient is a coefficient determined based on the target radial polynomial, the normalized polar angle, and the area element determined based on the intrinsic coordinate system.

6. The method according to any one of claims 1 to 5, wherein, The process of constructing a finite element simulation model of the optical mirror, solving the finite element simulation model under a preset load condition, obtaining the deformation displacement of the finite element nodes in the finite element simulation model, and determining the normal displacement component based on the deformation displacement includes: A finite element simulation model is established for the optical mirror and its supporting structure. The finite element simulation model is set with material parameters including elastic modulus, Poisson's ratio and coefficient of thermal expansion, and a preset load condition including mechanical load and thermal load is applied to the finite element simulation model to obtain the finite element equilibrium equation. Solving the finite element equilibrium equations yields the deformation displacements of the finite element nodes, including rigid body displacement components and elastic deformation components. Based on the deformation displacement, sampling points are selected on the optical mirror surface using finite element shape functions, and the displacement field at the sampling points is determined. Based on the displacement field and the normal unit vector of the optical mirror surface at the sampling points, the normal displacement component at the sampling points is obtained.

7. A specular deformation characterization apparatus based on conformal orthogonal basis, characterized in that, include: A coordinate system construction module is used to determine a rotationally symmetric optical mirror and construct an intrinsic coordinate system that uses the intrinsic geometric parameters of the optical mirror to mark the spatial position of each point on the optical mirror. The first basis determination module is used to construct the eigenvalue equation of the Laplace-Beltrami operator based on the intrinsic coordinate system, determine the angular equation and the radial equation using the variable separation method and the eigenvalue equation, and determine the first conformal orthogonal basis based on the angular solution corresponding to the angular equation and the radial solution corresponding to the radial equation. The second basis determination module is used to perform Gram-Schmidt orthogonalization on the Zernike polynomial based on the intrinsic coordinate system to obtain the target radial polynomial, and to determine the second conformal orthogonal basis using the target radial polynomial. The component determination module is used to construct the finite element simulation model of the optical mirror, solve the finite element simulation model under the applied preset load condition, obtain the deformation displacement of the finite element nodes in the finite element simulation model, and determine the normal displacement component based on the deformation displacement. The expression construction module is used to construct a basis function matrix based on the target conformal orthogonal basis, and solve the least squares problem based on the basis function matrix and the normal displacement component to obtain the fitting coefficients. The fitting coefficients and the target conformal orthogonal basis are used to construct the surface shape expression of the mirror deformation of the optical mirror, so as to trigger the optomechanical coupling analysis operation or the surface shape correction operation using the surface shape expression. The target conformal orthogonal basis is either the first conformal orthogonal basis or the second conformal orthogonal basis; The second basis determination module includes: A polynomial determination unit is used to take the radial polynomials in the Zernike polynomials as an initial sequence and use the weight function determined based on the intrinsic coordinate system to perform Gram-Schmidt orthogonalization on the initial sequence to obtain target radial polynomials that are mutually orthogonal under the weight function. The second basis determination unit is used to obtain the second conformal orthogonal basis based on the target radial polynomial, the angular mode of the azimuth angle of the optical mirror, and the second normalization coefficient.

8. An electronic device, characterized in that, include: Memory, used to store computer programs; A processor for executing the computer program to implement the mirror deformation characterization method based on a conformal orthogonal basis as described in any one of claims 1 to 6.

9. A computer-readable storage medium, characterized in that, Used to store computer programs; wherein, when the computer programs are executed by a processor, they implement the mirror deformation characterization method based on a conformal orthogonal basis as described in any one of claims 1 to 6.

Images