Large aperture off-axis parabolic coaxial processing and fitting method

Abstract

The application discloses a large-aperture off-axis parabolic coaxial processing and fitting method, comprising the following steps: obtaining off-axis surface shape data; determining a transformation reference datum containing a reference point and a normal vector; establishing a target coaxial coordinate system with a target normal vector; calculating a coordinate transformation matrix; applying the transformation matrix and a translation transformation to transform the off-axis surface shape data to the target coaxial coordinate system to obtain coaxial surface shape data, which is presented as an equivalent coaxial aspheric surface; and finally, performing polynomial curve surface fitting on the coaxial surface shape data to obtain a coaxial mathematical model. Through strict mathematical transformation, the application converts asymmetric data into an accurate coaxial model, solves the contradiction between the symmetric design of a zero position detection system and the asymmetry of a measured part, reduces the design difficulty, cost and adjustment sensitivity, and ensures nanometer-level fitting accuracy.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of optical curved surface processing and detection, in particular to a large-aperture off-axis parabolic mirror coaxial processing and fitting method. BACKGROUND

[0002] Off-axis parabolic mirrors are core components in modern high-end optical systems, such as space telescopes, high-energy laser devices, and earth observation cameras. Compared to traditional on-axis systems, off-axis systems eliminate central obscuration, significantly improving the imaging quality and energy transmittance of the system. With the continuous development of optical technology, there is an increasing demand for large-aperture, high-precision off-axis aspherical optical elements, which play an important role in improving the performance of optical systems. In the field of advanced optical manufacturing and detection technology, the surface data processing technology of off-axis aspherical optical elements is increasingly critical, and it is of great significance to ensure the quality and performance of optical elements.

[0003] In the field of surface data processing of large-aperture, high-precision off-axis aspherical optical elements, various conventional methods have been used to solve similar problems. In the detection reference aspect, interferometers and other devices are used for non-zero position detection; in the optical modeling aspect, off-axis data is usually directly fitted; in the design of zero-position detection system, there are two traditional methods, one is to forcibly design an asymmetric compensator or computer-generated hologram to match the asymmetric off-axis mirror, and the other is to approximate the off-axis mirror as a coaxial aspherical surface for design. These methods can meet some needs to a certain extent, but each has its own characteristics and application scenarios.

[0004] However, these conventional methods of existing technology have obvious defects. When using interferometers and other devices for non-zero position detection, the optical axis of the off-axis mirror does not coincide with its mechanical reference, resulting in complex positioning and attitude errors during adjustment, which are coupled with the real surface error of the element, making it difficult to separate and making the detection result less reliable. The method of directly fitting off-axis data does not consider the specific geometric properties of the off-axis mirror as part of the "mother mirror", resulting in an unclear physical meaning of the fitting model, poor edge accuracy, and inability to provide the key initial parameter of base curve radius for optical design. The two conventional methods of zero-position detection system design also have problems, the asymmetric design increases the design, processing and adjustment difficulty of the compensation system, is costly, and introduces additional asymmetric errors; the approximate design introduces a non-negligible theoretical error, especially in the case of large aperture and large off-axis amount, which severely restricts the upper limit of detection accuracy. Therefore, there is a lack of a full-chain technical solution that can systematically solve the problem from "off-axis detection data" to "coaxial high-precision model", and then to "optical design usable parameters". SUMMARY

[0005] In order to solve the technical problems in the prior art, the application provides a large-aperture off-axis parabolic surface coaxial processing and fitting method.

[0006] The large-aperture off-axis parabolic surface coaxial processing and fitting method provided by the application adopts the following technical scheme:

[0007] The large-aperture off-axis parabolic surface coaxial processing and fitting method comprises the following steps:

[0008] S1. Obtain off-axis surface data of an off-axis aspheric surface to be measured, wherein the off-axis surface data represents a three-dimensional topography of the off-axis aspheric surface in a parent mirror coordinate system;

[0009] S2. Determine a transformation reference datum on the off-axis surface data, wherein the transformation reference datum comprises at least a geometric center point or an area center point of the off-axis aspheric surface as a reference point, and a normal vector at the reference point;

[0010] S3. Establish a target coaxial coordinate system, wherein the target coaxial coordinate system has a preset optical axis defined by a target normal vector;

[0011] S4. Calculate a coordinate transformation matrix, wherein the coordinate transformation matrix is used to rotate the normal vector at the reference point to coincide with the target normal vector;

[0012] S5. Apply the coordinate transformation matrix and a translation transformation to transform the off-axis surface data from the parent mirror coordinate system to the target coaxial coordinate system, thereby obtaining coaxial surface data, wherein the coaxial surface data represents an equivalent coaxial aspheric surface in the target coaxial coordinate system;

[0013] S6. Perform polynomial curve fitting on the coaxial surface data to obtain a coaxial mathematical model representing the equivalent coaxial aspheric surface.

[0014] In some embodiments, the step S1 of obtaining the off-axis surface data comprises:

[0015] According to a vertex curvature radius R of the parent mirror of the off-axis aspheric surface, a digital model of a parent mirror parabolic surface is established, and a mathematical expression of the digital model is:

[0016] Z(x, y) = (x 2 + y 2 ) / (2R);

[0017] According to an off-axis amount A and a projection aperture D of the off-axis aspheric surface, the off-axis surface data (x sample , y sample , z sample ) is obtained by intercepting or sampling on the parent mirror parabolic surface model.

[0018] In some embodiments, in step S2, the reference point P0 is the area center point of the off-axis aspheric surface on the mother mirror, and the coordinates of the reference point P0 are (0, A, Z(0, A));

[0019] In step S3, the target normal vector n N0 is the Z-axis direction vector of the target coaxial coordinate system.

[0020] In some embodiments, in step S4, the coordinate transformation matrix is calculated, including:

[0021] According to the normal vector n f0 and the target normal vector n N0 , the rotation axis and the rotation angle θ required for rotating n f0 to n N0 are calculated by using the Rodrigues rotation formula;

[0022] The coordinate transformation matrix R matrix is constructed based on the rotation axis and the rotation angle θ.

[0023] In some embodiments, in step S5, the coaxialization is performed, specifically including:

[0024] First, a translation operation is performed on the off-axis surface data (x sample , y sample , z sample ), so that the reference point P0 is translated to the origin of the target coaxial coordinate system;

[0025] Then, a rotation operation is performed on the translated surface data by using the coordinate transformation matrix R matrix , to obtain the coaxial surface data (x rotated , y rotated , z rotated ).

[0026] In some embodiments, in step S6, the polynomial surface fitting is performed, including:

[0027] A normalized XY polynomial matrix is constructed, wherein the normalized radius is half of the projection aperture D of the off-axis aspheric surface;

[0028] By using a singular value decomposition algorithm, the least square fitting is performed on the coaxial surface data, to solve the polynomial coefficients of the XY polynomial. When the XY polynomial matrix is constructed, the constant term is excluded, and the construction starts from the first-order term.

[0029] In some embodiments, the large-aperture off-axis parabolic surface coaxial processing and fitting method further includes:

[0030] S7. Based on the principle of differential geometry, the first fundamental form and the second fundamental form of the surface at the reference point P0 are calculated, and the Gaussian curvature K and the mean curvature H are solved;

[0031] Based on the Gaussian curvature K and the mean curvature H, the meridional radius of curvature R min and the sagittal radius of curvature R max .

[0032] In some embodiments, the large-aperture off-axis parabolic homothetic processing and fitting method further comprises:

[0033] S8. The geometric mean R min of the meridional radius of curvature R max and the sagittal radius of curvature R gauss is calculated, wherein ;

[0034] The geometric mean R gauss is taken as the base curvature radius of the equivalent coaxial aspheric surface in the optical design software.

[0035] In some embodiments, the large-aperture off-axis parabolic homothetic processing and fitting method further comprises:

[0036] S9. The polynomial coefficients solved in step S6 are input into the optical design software as the surface type parameters of the equivalent coaxial aspheric surface;

[0037] S10. Based on the homothetic mathematical model defined by the geometric mean R gauss and the polynomial coefficients, a zero-position detection system for detecting the off-axis aspheric surface is designed; wherein the zero-position detection system is a compensator or a computer-generated hologram, and the zero-position detection system is designed as a rotationally symmetric system.

[0038] In some embodiments, the large-aperture off-axis parabolic homothetic processing and fitting method further comprises:

[0039] S11. The fitting residual of the polynomial surface fitting is calculated, and the root mean square error and the peak-to-valley error of the homothetic surface shape data are output in nanometers.

[0040] In summary, the present application includes at least one of the following beneficial technical effects:

[0041] 1. The application adopts a set of strict mathematical coordinate transformation to "right" and convert the physically asymmetric off-axis aspheric surface data to an equivalent coaxial aspheric model with the vertex at the origin and the optical axis along the Z axis, solving the contradiction between the rotational symmetry design of the detection system and the asymmetry of the measured element. Designers can use mature symmetric design scheme based on this coaxial model to convert complex asymmetric design problem into simple symmetric design problem, thereby greatly reducing the design difficulty, processing cost and adjustment sensitivity of zero position detection system;

[0042] 2. For the high-order polynomial fitting requirement after coaxialization, the application adopts singular value decomposition algorithm for least squares fitting, supplemented by normalization and constant term exclusion processing. SVD algorithm effectively overcomes the "ill-conditioned matrix" problem encountered by traditional methods in processing high-order polynomials, which helps to ensure the robustness, reliability and high precision of numerical calculation process;

[0043] 3. Based on the principle of differential geometry, the application provides a method for calculating the principal curvatures (R min ,R max ) of the off-axis mirror center point, and recommends using the geometric mean R gauss as the best base curvature radius in optical design. This provides a clear theoretical basis for setting the key reference radius parameter in Zemax and other software, solving the problem of unclear reference radius parameter selection in the prior art. The polynomial coefficients obtained by SVD fitting and R gauss can be directly imported into optical software, forming an automatic and highly practical complete closed loop from manufacturing detection to design simulation. BRIEF DESCRIPTION OF DRAWINGS

[0044] Figure 1 is a flowchart of the large-aperture off-axis parabolic surface coaxial processing and fitting method provided by an embodiment of the application;

[0045] Figure 2 is a schematic diagram of the position of the parent mirror parabolic surface and the off-axis region in the large-aperture off-axis parabolic surface coaxial processing and fitting method;

[0046] Figure 3 is a schematic diagram of the original off-axis parabolic surface model obtained in the large-aperture off-axis parabolic surface coaxial processing and fitting method;

[0047] Figure 4 is a schematic diagram of the parabolic surface after the original off-axis parabolic surface model is coaxialized;

[0048] Figure 5 is a schematic diagram of the curve surface after the data after coaxial processing is fitted by a polynomial without constant term;

[0049] Figure 6 This is a schematic diagram of the fitting error distribution of the polynomial fitting. Detailed Implementation

[0050] This application mainly employs coordinate transformation and polynomial fitting to process off-axis parabolic surface data, achieving the effect of coaxializing off-axis aspherical data and solving the problems of detection and modeling. The following is a further detailed description of this application.

[0051] Example

[0052] Please refer to Figure 1 The method for coaxializing and fitting large-diameter off-axis parabolic surfaces provided in this application includes:

[0053] S1. Obtain the off-axis surface shape data of the off-axis aspherical surface to be tested, wherein the off-axis surface shape data characterizes the three-dimensional morphology of the off-axis aspherical surface in the parent mirror coordinate system;

[0054] S2. Determine a transformation reference datum on the off-axis surface shape data. The transformation reference datum includes at least the geometric center point or area center point of the off-axis aspherical surface as a reference point, and the normal vector at the reference point.

[0055] S3. Establish a target coaxial coordinate system, wherein the target coaxial coordinate system has a preset optical axis, and the preset optical axis is defined by a target normal vector;

[0056] S4. Calculate a coordinate transformation matrix, which is used to rotate the normal vector at the reference point to coincide with the target normal vector;

[0057] S5. Apply the coordinate transformation matrix and a translation transformation to transform the off-axis surface data from the parent mirror coordinate system to the target coaxial coordinate system, thereby obtaining coaxial surface data. The coaxial surface data is presented as an equivalent coaxial aspherical surface in the target coaxial coordinate system.

[0058] S6. Perform polynomial surface fitting on the coaxialized surface data to obtain a coaxialized mathematical model characterizing the equivalent coaxial aspherical surface.

[0059] S7. Based on the principles of differential geometry, calculate the first and second fundamental forms of the surface at the reference point P0, and solve for the Gaussian curvature K and the mean curvature H.

[0060] S8. Calculate the meridional curvature radius R. min and the radius of curvature R of the arc max Geometric mean R gauss ,in ;

[0061] S9. Input the polynomial coefficients solved in step S6 into the optical design software as the surface parameters of the equivalent coaxial aspheric surface;

[0062] S10. Design a null detection system for detecting the off-axis aspheric surface based on the coaxial mathematical model defined by the geometric mean value R gauss and the polynomial coefficients; wherein the null detection system is a compensator or a computer-generated hologram, and the null detection system is designed as a rotationally symmetric system.

[0063] S11. Calculate the fitting residuals of the polynomial curve fitting, and output the root mean square error and the peak-to-valley error of the coaxial surface data in nanometers.

[0064] The present application converts the asymmetric off-axis surface data into coaxial surface data through a series of strict mathematical transformations and calculation steps, and performs polynomial fitting to obtain a coaxial mathematical model, while calculating key optical parameters, thereby achieving the effects of accurately characterizing the optical properties of off-axis surfaces, solving the contradictions in off-axis aspheric surface detection reference, optical modeling and null detection design, etc. The reason for the beneficial effects is that the off-axis data is converted into coaxial form through coordinate transformation, so that subsequent fitting and design can be based on a more reasonable mathematical model, avoiding the many drawbacks of traditional methods.

[0065] Specifically, step S1 includes establishing a digital model of the mother mirror parabolic surface according to the vertex curvature radius R of the off-axis aspheric surface, and the mathematical expression of the digital model is:

[0066] Z(x,y) = (x 2 + y 2 ) / (2R).

[0067] The digital model of the mother mirror parabolic surface can be constructed by computer software, such as MATLAB, Python, etc. with mathematical calculation and modeling functions. In constructing the model, the vertex curvature radius R is a key parameter, which determines the shape of the mother mirror parabolic surface. Alternatively, in some special cases, other mathematical models can also be used to approximate the mother mirror parabolic surface, such as a quadratic surface model, etc., but the parabolic surface model used in the present embodiment is more consistent with the actual situation in theory. Then, according to the off-axis amount A and the projection aperture D of the off-axis aspheric surface, the off-axis surface data (x sample , y sample , z sample ) is obtained by intercepting or sampling on the mother mirror parabolic surface model. The off-axis amount A represents the degree of offset of the off-axis mirror relative to the mother mirror, and the projection aperture D determines the size of the intercepted region. The interception or sampling method can be regular grid sampling or random sampling, and regular grid sampling can ensure the uniformity of the data for subsequent processing.

[0068] The effect of this step is as follows: Figure 2 and Figure 3 As shown. Figure 2 The positional relationship between the mother mirror parabolic surface (blue) and the off-axis region (red) is shown. Figure 3 The original off-axis parabolic model data is then displayed in the form of a 3D point cloud.

[0069] In step S2, a transformation reference datum is determined on the off-axis surface shape data. Reference point P0 is the center point of the area of ​​the off-axis aspherical surface on the parent mirror, with coordinates (0, A, Z(0,A)). The center point of the area is a point with clear geometric meaning, and its position is determined by calculating the area distribution of the region enclosed by the off-axis surface shape data. The normal vector n at point P0 is calculated based on differential geometry. f0 Normal vector n f0 This indicates the direction of the normal to the surface at that point, and it plays a crucial role in subsequent coordinate transformations. Alternatively, the geometric center point of the off-axis aspherical surface can be chosen as the reference point, but the area center point better reflects the overall characteristics of the off-axis surface shape in this embodiment.

[0070] Step S3 is to establish a coaxial coordinate system for the target, with the optical axis of this coordinate system along the Z-axis and the target normal vector n. N0 The coordinates are [0, 0, 1] (or [0, 0, -1]). The establishment of the target coaxial coordinate system provides a standard reference framework for subsequent coordinate transformations, enabling off-axis surface data to be accurately transformed into this coaxial coordinate system.

[0071] In step S4, the Rodriguez rotation formula is used, based on the normal vector n f0 and target normal vector n N0 Calculate n f0 Rotate to n N0 The required rotation axis and rotation angle θ. The Rodrigues rotation formula is a classic method for calculating rotations; it can calculate the rotation axis and rotation angle using the direction information of two vectors. The coordinate transformation matrix R is constructed based on the rotation axis and rotation angle θ. matrix Coordinate transformation matrix R matrix This is used to implement coordinate rotation transformations. Alternatively, other rotation calculation methods can be used, such as quaternion rotation, but the Rodriguez rotation formula is relatively simple and intuitive to calculate in this embodiment.

[0072] Step S5 specifically includes first processing the off-axis surface shape data (x sample , y sample , z sample) a translation operation is performed to translate the reference point P0 to the origin of the target coaxial coordinate system. The translation operation is achieved by subtracting the coordinates of the reference point P0 from the coordinates of each point of the off-axis surface data. Then a rotation operation is applied to the translated surface data by the coordinate transformation matrix R_matrix to obtain the coaxial surface data (x rotated , y rotated , z rotated ). After the translation and rotation operations, the off-axis surface data is converted into the coaxial surface data, forming an "equivalent coaxial aspheric surface" with the vertex at the origin and the optical axis along the Z-axis.

[0073] The data point cloud of the "equivalent coaxial aspheric surface" is shown in Figure 4 , compared with Figure 3 , its asymmetric tilted posture has been "straightened". The implementation of this step S5 has the technical effect that, by strict rigid body transformation (translation and rotation), the asymmetric off-axis detection data is mathematically converted into a symmetric coaxial model without introducing any surface fitting error, which lays a key mathematical foundation for the subsequent high-precision fitting step (S6) and the design step (S10) of the symmetric zero position detection system.

[0074] In step S6, a normalized XY polynomial matrix is constructed, where the normalized radius is half of the projection aperture D of the off-axis aspheric surface. Normalization processing can make the data comparable within a certain range, avoiding fitting problems caused by too large data scale differences. The singular value decomposition (SVD, Singular Value Decomposition) algorithm is used to perform least squares fitting on the coaxial surface data to solve the polynomial coefficients of the XY polynomial. The singular value decomposition algorithm can effectively overcome the "ill-conditioned matrix" problem encountered by traditional Gaussian elimination method when dealing with high-order polynomials, ensuring the solution accuracy. When constructing the XY polynomial matrix, the constant term is excluded, and the construction starts from the first-order term, which helps to avoid numerical problems in the fitting process.

[0075] The fitted surface model is shown in Figure 5 . The technical effect of step S6 is to obtain a high-precision, high-numerical-stability coaxial mathematical model. The application of the SVD algorithm solves the ill-conditioned problem of high-order fitting, and the exclusion of the constant term further improves the numerical stability.

[0076] Step S7 is based on the principle of differential geometry. At the reference point P0, the Gaussian curvature K and the mean curvature H of the point are obtained by solving the first and second fundamental forms of the surface. Then the meridional curvature radius R min and the sagittal curvature radius R max(i.e. two principal radii of curvature). Since the off-axis parabolic mirror is an astigmatism point at this point (not the vertex), R min and R max are not equal.

[0077] The output parameter step (S8) calculates the geometric mean of the meridional radius of curvature R min and the sagittal radius of curvature R max , R gauss , and takes R gauss as the base curvature radius of the equivalent on-axis aspheric surface in the optical design software.

[0078] The implementation of these two steps (S7, S8) has the technical effect that they provide the optical designer with a clear and optimized reference curvature R gauss . This solves the blank in the prior art that the designer lacks a theoretical basis when choosing the meridional, sagittal or average value as the base curvature radius in the optical software (such as Zemax). The use of R gauss better balances the "fitting error" and "design error", which is another innovation of the present application.

[0079] The model application and output steps (S9, S10) include inputting the polynomial coefficients solved in step S6 into the optical design software (such as Zemax) as the surface parameters of the equivalent on-axis aspheric surface, and can be directly applied to the "extended polynomial" surface. The designer can design a zero-position detection system for detecting off-axis aspheric surfaces based on the on-axis mathematical model defined by the base curvature radius R gauss and the polynomial coefficients, and the zero-position detection system is a compensator (Null Lens) or a computer generated hologram (CGH, Computer Generated Hologram), and is designed as a rotationally symmetric system. This converts the complex asymmetric design problem into a more mature, simpler and lower cost symmetric design problem.

[0080] Step S11 calculates the fitting residual of the polynomial surface fitting, and outputs the root mean square (RMS) error and the peak-to-valley (PV) error of the on-axis surface shape data in nanometers (nm). For example, in a specific test of the present application (as shown in Figure 6 , the fitting RMS error can reach 0.6804 nm, and the fitting PV error is 5.6551 nm, showing high fitting accuracy. The root mean square error and the peak-to-valley error can comprehensively evaluate the accuracy of the fitting, and provide a quantitative index for the reliability of the model.

[0081] The implementation principle of the embodiment is: through a series of strict mathematical transformations and calculation steps, the asymmetric off-axis surface data is converted into coaxial surface data, and polynomial fitting is performed to obtain a coaxial mathematical model, and key optical parameters are calculated. This method solves the contradiction in off-axis aspheric surface detection reference, optical modeling and zero position detection design, improves the reliability of the detection result, provides accurate initial parameters for optical design, and reduces the design, processing and adjustment difficulty of the zero position detection system, has strong practicability and innovation, and has significant improvement and contribution to the prior art.

[0082] The specific embodiments of the application described above do not constitute a limitation on the protection scope of the application. Any various other corresponding changes and modifications made according to the technical concept of the application should be included in the protection scope of the application.

Claims

1. A large aperture off-axis parabolic coaxialization process and fitting method, characterized in that, Comprising: S1. Obtain off-axis surface data of an off-axis aspherical surface to be measured, the off-axis surface data representing a three-dimensional topography of the off-axis aspherical surface in a primary mirror coordinate system; S2. Determine a transformation reference datum on the off-axis surface data, the transformation reference datum comprising at least a reference point P0 and a normal vector at the reference point P0, wherein the reference point P0 is a geometric center point or an area center point of the off-axis aspherical surface; S3. Establishing a target coaxial coordinate system, the target coaxial coordinate system has a preset optical axis, the preset optical axis is determined by a target normal vector n N0 Definitions; S4. Computing a coordinate transformation matrix for rotating the normal vector at the reference point P0 to the target normal vector n N0 coincide; S5. Apply the coordinate transformation matrix and a translation transformation to transform the off-axis surface data from the primary mirror coordinate system to the target on-axis coordinate system, thereby obtaining on-axis surface data, which presents an equivalent on-axis aspherical surface in the target on-axis coordinate system; S6. Perform polynomial curve fitting on the on-axis surface data to obtain an on-axis mathematical model representing the equivalent on-axis aspherical surface; specifically comprising: constructing a normalized XY polynomial matrix, wherein the normalized radius is half of the projection aperture D of the off-axis aspherical surface; excluding the constant term when constructing the XY polynomial matrix, and starting from the first-order term; using a singular value decomposition algorithm to perform least squares fitting on the on-axis surface data to obtain the on-axis mathematical model representing the equivalent on-axis aspherical surface; S7. Based on the principles of differential geometry, calculate the first fundamental form and the second fundamental form of the surface at the reference point P0, and solve to obtain the Gaussian curvature K and the mean curvature H; Based on the Gaussian curvature K and the mean curvature H, a meridional radius of curvature R at the reference point P0 is calculated min and a sagittal radius of curvature R max ; S8. Computing the geometric mean of the meridional radius of curvature R min and the sagittal radius of curvature R max where the geometric mean of R and R is used as the base curve radius of curvature in the optical design software for characterizing the equivalent coaxial aspheric surface. the geometric mean of R and R is used as the base curve radius of curvature in the optical design software for characterizing the equivalent coaxial aspheric surface.​ 2. The large aperture off-axis parabolic homothetic processing and fitting method of claim 1, wherein, Step S1 obtains the off-axis surface data, comprising: According to the vertex curvature radius R of the off-axis aspherical surface, a digital model of a primary mirror paraboloid is established, and the mathematical expression of the digital model is: Z(x, y) = (x 2 + y 2 ) / (2R); According to the off-axis amount A and the projection aperture D of the off-axis aspherical surface, the off-axis surface shape data (x sample , y sample , z sample ) is obtained by intercepting or sampling on the primary mirror parabolic model.

3. The large aperture off-axis parabolic homothetic processing and fitting method of claim 1, wherein, In step S2, the reference point P0 is the area center point of the off-axis aspherical surface on the primary mirror, and its coordinates are (0, A, Z(0, A)); In step S3, the target normal vector n N0 is the Z-axis direction vector of the target coaxial coordinate system.

4. The large aperture off-axis parabolic homothetic processing and fitting method of claim 1, wherein, Step S4 calculates the coordinate transformation matrix, comprising: Using the Rodrigues' rotation formula, the rotation axis and rotation angle needed to rotate n f0 to n N0 are calculated from the normal vector n f0 and the target normal vector n N0 ;​ based on the rotation axis and the rotation angle constructing the coordinate transformation matrix R matrix .

5. The large aperture off-axis parabolic homothetic processing and fitting method of claim 1, wherein, Step S5 performs on-axis, specifically comprising: First, a translation operation is performed on the off-axis surface data (x sample , y sample , z sample ) so that the reference point P0 is translated to the origin of the target on-axis coordinate system; Then, the coordinate transformation matrix R is applied to the translated surface data matrix The rotation operation is performed on the translated surface data to obtain the coaxial surface data (x rotated , y rotated , z rotated ).

6. The large aperture off-axis parabolic homothetic processing and fitting method of claim 1, wherein, Further comprising: S9. Input the polynomial coefficients solved in step S6 into an optical design software as the surface type parameters of the equivalent on-axis aspherical surface; S10. based on the geometric mean value and the coaxial mathematical model defined by the polynomial coefficients, a null detection system for detecting the off-axis aspheric surface is designed; wherein the null detection system is a compensator or a computer generated hologram, and the null detection system is designed as a rotationally symmetric system.

7. The large aperture off-axis parabolic homothetic processing and fitting method of claim 6, wherein, Further comprising: S11. Calculate the fitting residual of the polynomial curve fitting, and output the root mean square error and the peak-to-valley error of the on-axis surface data in nanometers.

Images