A method for calculating the central axis deviation angle of an aspheric optical primary mirror

Abstract

The application discloses a kind of non-spherical optical primary mirror center axis offset angle calculation method, solve the technical problem that imaging quality is reduced due to center axis offset in the process of non-spherical optical primary mirror assembly, a kind of non-spherical optical primary mirror center axis offset angle calculation method is provided, center axis offset can be effectively detected and corrected, improve optical system imaging quality, reduce assembly difficulty, improve production efficiency, its flexibility and adaptability make it perform well in various high-end optical application scenarios, different sizes and shapes of non-spherical optical primary mirror have applicability, whether small or large optical system, applicable to a wide range of optical equipment and instruments.

Description

TECHNICAL FIELD

[0001] The application relates to an angle calculation method, in particular to a calculation method of an angle of a center axis offset of an aspheric optical primary mirror. BACKGROUND

[0002] In modern optical systems, aspheric optical elements are widely used due to their excellent imaging quality and compact design. As the core component of many high-end optical systems, the imaging accuracy of the aspheric optical primary mirror directly determines the performance of the entire optical system. However, in actual application, due to errors in the processing and assembly process, the center axis of the aspheric optical primary mirror may be offset. This offset phenomenon often leads to a significant decline in imaging quality, especially in high spatial resolution and high spectral resolution application scenarios.

[0003] Currently, the detection and correction methods for the center axis offset of the aspheric optical primary mirror still face many technical challenges. Traditional detection methods often rely on complex mechanical devices and tedious manual adjustment processes, which not only consume time and effort, but also are difficult to achieve the required high precision requirements. In addition, with the development of optical systems towards higher precision and larger aperture, the limitations of traditional methods become increasingly apparent. Therefore, developing an efficient and accurate detection and correction method for the center axis offset of the aspheric optical primary mirror has become a key technical requirement to improve the imaging quality of optical systems, reduce assembly difficulty, and improve production efficiency. SUMMARY

[0004] The purpose of the present application is to solve the technical problem of imaging quality decline caused by center axis offset in the assembly process of aspheric optical primary mirror, and to provide a calculation method of the center axis offset angle of aspheric optical primary mirror.

[0005] In order to achieve the above purpose, the technical scheme adopted by the present application is as follows:

[0006] A calculation method of the center axis offset angle of an aspheric optical primary mirror, characterized by comprising the following steps:

[0007] Step 1: Establish a three-dimensional coordinate system with the axis of the center axis of the aspheric optical primary mirror as the Z-axis, and collect a first ring point set P corresponding to M ring bands on the center axis ki =(x ki ,y ki ,z ki ) of the center axis of the aspheric optical primary mirror; the radius r k of each ring band is different, and N points are collected on each ring band; M≥2, N≥3; wherein (x ki ,y ki ,z ki ) represents the coordinates of the i-th point of the k-th ring band, k=1, 2, …, M, i=1, 2, …, N;

[0008] Step 2: Rotate and translate the first ring point set P ki =(x ki ,y ki ,z ki ) to the base coordinate system:

[0009] P ki ′=R·P ki +T;

[0010] Among them, the rotation matrix Translation Matrix

[0011] Get the second ring point set P ki '=(x ki ',y ki ',z ki ');

[0012] Step 3: Calculate the absolute coordinate value z ki ”;

[0013] 3.1) The radius r of each ring k Calculate the corresponding Z-axis first theoretical value of each ring in the reference coordinate system

[0014]

[0015] Where s represents the curvature radius of the aspherical optical primary mirror surface, and t represents the cone constant;

[0016] 3.2) For the same second ring point set P ki The z in ' ki 'Take the average

[0017] 3.3) Add the second ring with point set P ki 'Coordinate z in ki 'Adjust to absolute coordinate value

[0018] Step 4: Calculate the sum of squared errors;

[0019] 4.1) Take the second ring point set P ki 'The coordinates of each point (x ki ',y ki ',z ki ') in the coordinates (x ki ',y ki '), calculate the second theoretical value of the Z axis corresponding to each ring in the reference coordinate system

[0020]

[0021] 4.2) Calculate the second theoretical value and the absolute coordinate value z ki The error square sum E (R,T) :

[0022]

[0023] Step 5, minimize the error square sum E (R,T) by the nonlinear least square method, to obtain the optimized rotation matrix and translation matrix

[0024] Step 6, repeat steps 2-5, and replace the rotation matrix R and translation matrix T in step 2 with the rotation matrix and translation matrix obtained in step 5 in each repetition until E (R,T) is less than a preset value, i.e. convergence, and the final rotation matrix and translation matrix are the optimal rotation matrix R' and translation matrix T';

[0025] Step 7, calculate the offset vector v':

[0026] v' = R' Transpose · (v - T');

[0027] wherein v is an arbitrary vector along the central axis of the optical primary mirror in the reference coordinate system;

[0028] Step 8, calculate the offset angle θ of the central axis of the optical primary mirror:

[0029]

[0030] wherein <v, v'> represents the inner product of vector v and vector v', and ||v|| and ||v'|| represent the lengths of vector v and vector v', respectively.

[0031] Further, in step 1:

[0032] M = 5.

[0033] Further, in step 1:

[0034] The radii corresponding to the 5 annular zones are r1 = 150 mm, r2 = 225 mm, r3 = 260 mm, r4 = 280 mm, and r5 = 300 mm, respectively.

[0035] Further, in step 1:

[0036] N = 12.

[0037] Further, in step 1, N points are collected on each ring belt, specifically:

[0038] N points are collected uniformly on each ring belt.

[0039] Further, step 2 is specifically:

[0040] The collected first ring belt point set P ki is converted to the reference coordinate system by rotation and translation using the following formula:

[0041] P ki '=R·P ki +T;

[0042] Wherein, the rotation matrix R is respectively rotated by α, β, γ angles around x axis, y axis and z axis, and the formula is as follows:

[0043]

[0044] The translation matrix T is respectively translated by a, b, c distances along x axis, y axis and z axis, and the formula is as follows:

[0045]

[0046] The second ring belt point set P ki '=(x ki ',y ki ',z ki ') is obtained.

[0047] The beneficial effects of the present application are:

[0048] 1. The method for calculating the center axis offset angle of the aspheric optical primary mirror provided by the present application can effectively detect and correct the center axis offset, improve the imaging quality of the optical system, reduce the assembly difficulty, and improve the production efficiency. Its flexibility and adaptability make it perform well in various high-end optical application scenarios. It is applicable to aspheric optical primary mirrors of different sizes and shapes, whether it is a small or large optical system, and is suitable for a wide range of optical equipment and instruments.

[0049] 2. The method of the present application has good compatibility and expandability, is easy to integrate with other optical detection systems, and is convenient for further expanding functions, such as combining with other automatic adjustment devices to realize a closed-loop control system.

[0050] 3. The method for calculating the center axis offset angle of the aspheric optical primary mirror provided by the present application can use an automatic measurement and analysis process, which greatly simplifies the detection process compared to the traditional manual adjustment method. Not only does it reduce the technical requirements of the operator, but it also significantly reduces human error, improves overall work efficiency and detection accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0051] Figure 1 is a flowchart of an embodiment of a method for calculating the center axis offset angle of an aspherical optical primary mirror of the application;

[0052] Figure 2 is a cross-sectional view of a mechanical device of an aspherical optical primary mirror;

[0053] Figure 3 is a schematic diagram of uniformly collecting ring band point sets of different radii in an embodiment of the application;

[0054] Figure 4 is a two-dimensional curve graph of the working surface of the primary mirror in an embodiment of the application;

[0055] Figure 5 is a schematic diagram of a three-dimensional theoretical asphere in an embodiment of the application. DETAILED DESCRIPTION

[0056] The technical solutions of the application will be described in detail below with reference to the drawings and embodiments. Obviously, the described embodiments are only a part of the embodiments of the application, rather than all the embodiments. Based on the embodiments of the application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the application.

[0057] The method for calculating the center axis offset angle of an aspherical optical primary mirror in this embodiment is used to accurately measure and correct the center axis offset phenomenon of an aspherical optical primary mirror. As shown in Figure 1 , the method specifically comprises the following steps:

[0058] Step 1: Establish a three-dimensional coordinate system with the axis of the center axis of the aspherical optical primary mirror (as shown in Figure 2 ) as the Z-axis, and collect a first ring band point set P ki =(x ki ,y ki ,z ki ) corresponding to M ring bands on the center axis; the radii r k of each ring band are different, and N points are uniformly collected on each ring band, as shown in Figure 3 ; M≥2, N≥3; wherein (x ki ,y ki ,z ki ) represents the coordinates of the i-th point of the k-th ring band, k=1, 2, …, M, i=1, 2, …, N;

[0059] In this embodiment, it is preferred that M=5, the radii of the five ring bands are r1=150 mm, r2=225 mm, r3=260 mm, r4=280 mm, and r5=300 mm, and N=12 mm.

[0060] Step 2, the collected first annulus point set P ki is converted to the reference coordinate system by rotation and translation using the following formula:

[0061] P ki ' = R·P ki +T;

[0062] wherein the rotation matrix R is a rotation of angles a, b, g around the x-axis, y-axis, and z-axis respectively, and the formula is as follows:

[0063]

[0064] The translation matrix T is a translation of distances a, b, c along the x-axis, y-axis, and z-axis respectively, and the formula is as follows:

[0065]

[0066] The second annulus point set P ki ' = (x ki ',y ki ',z ki ');

[0067] It should be noted that the reference coordinate system is a three-dimensional coordinate system that has not been offset in an ideal case;

[0068] The rotation matrix R in this step takes the initial value The translation matrix T takes the initial value

[0069] Step 3, calculate the absolute coordinate value z ki ";

[0070] 3.1) The working surface of the optical primary mirror follows a specific formula:

[0071]

[0072] Wherein s represents the radius of curvature of the aspheric optical primary mirror, and t represents the conical constant; in this embodiment, s = 950, t = -1.23. The primary mirror working surface formula describes a two-dimensional curve, as shown in Figure 4 , which is rotated along the central axis (Z-axis) to form a three-dimensional asphere, as shown in Figure 5 , and the three-dimensional expression of the asphere of the optical primary mirror is as follows:

[0073]

[0074] Since the first annulus point set P ki is measured at different radii r kTheoretically, for any annulus, each point on it satisfies the relationship between the radius r of the circle and the coordinates (x, y):

[0075] x 2 +y 2 =r 2 ;

[0076] Substituting the above relationship into the three-dimensional expression of the aspheric surface of the optical primary mirror, we can obtain:

[0077]

[0078] 3.2) The radius r of each ring k Calculate the corresponding Z-axis first theoretical value of each ring in the reference coordinate system

[0079]

[0080] 3.3) For the same second ring point set P ki The z in ' ki 'Take the average

[0081] 3.4) Add the second ring with point set P ki 'Coordinate z in ki 'Adjust to absolute coordinate value

[0082] Step 4: Calculate the sum of squared errors;

[0083] 4.1) Take the second ring point set P ki 'The coordinates of each point (x ki ',y ki ',z ki ') in the coordinates (x ki ',y ki '), calculate the second theoretical value of the Z axis corresponding to each ring in the reference coordinate system

[0084]

[0085] 4.2) Calculate the second theoretical value and the absolute coordinate value z ki The sum of squared errors E (R,T) :

[0086]

[0087] Step 5: Minimize the error sum of squares E by nonlinear least squares method (R,T)and the Levenberg-Marquardt algorithm is used to optimize the non-linear least square problem to obtain the optimized rotation matrix and translation matrix

[0088] Step 6, repeat steps 2-5, and replace the rotation matrix R and translation matrix T in step 2 with the rotation matrix R and translation matrix T obtained in step 5 each time until E and translation matrix Step 7, calculate the offset vector v' as follows: (R,T) Step 8, calculate the offset angle θ of the optical primary mirror central axis as follows:

[0089] Step 7, calculate the offset vector v' as follows:

[0090] v' = R' -1 · (v - T').

[0091] wherein v is an arbitrary vector along the direction of the optical primary mirror central axis in the reference coordinate system. Since the rotation matrix R' is an orthogonal matrix, its inverse matrix R' -1 is equal to its transpose matrix R' Transpose . Therefore, the coordinate inverse transformation can be expressed as:

[0092] v' = R' Transpose · (v - T').

[0093] Step 7, calculate the offset vector v' as follows:

[0094]

[0095] wherein <v, v'> represents the inner product of the vector v and the vector v', and ||v|| and ||v'|| represent the lengths of the vector v and the vector v', respectively.

[0096] The above is only a specific embodiment of the present application, but the protection scope of the present application is not limited thereto, any change or replacement within the technical scope disclosed in the present application should be covered in the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for calculating the angle of the center axis offset of an aspheric optical primary mirror, characterized in that, Comprising the following steps: Step 1: Use the axis of the central axis of the aspheric optical primary mirror as the Z axis to arbitrarily establish a three-dimensional coordinate system, and collect the first ring point set P corresponding to the M rings on its central axis. ki =(x ki ,y ki ,z ki ); The radius r of each ring k are all different, and N points are collected on each ring; M≥2, N≥3; among them, (x ki ,y ki ,z ki ) represents the coordinates of the i-th point in the k-th ring, k = 1, 2, ..., M, i = 1, 2, ..., N; Step 2, converting the collected first annulus point set P by rotation and translation ki = (x ki ,y ki ,z ki ) to the reference coordinate system: P ki ’=R·P ki +T; where the rotation matrix translation matrix Obtaining a second ringed point set P ki ' = (x ki ', y ki ', z ki ) ; Step 3, calculating absolute coordinate values z ki "; 3.1) The radius r of each ring k Calculate the corresponding Z-axis first theoretical value of each ring in the reference coordinate system Wherein s represents the radius of curvature of the aspheric optical primary mirror, t represents the conic constant; 3.2) on the same second set of points P ki z in ki average 3.3) Adjust the coordinates z in the second ring-banded point set P ki ' to absolute coordinate values ki ​ Step 4, calculate the error sum of squares; 4.1) Take the second ring point set P ki 'The coordinates of each point (x ki ',y ki ',z ki ') in the coordinates (x ki ',y ki '), calculate the second theoretical value of the Z axis corresponding to each ring in the reference coordinate system 4.2) Calculate the second theoretical value and the absolute coordinate value z ki the error square sum E (R,T) : Step 5, minimize the sum of squared errors E by nonlinear least squares (R,T) , resulting in an optimized rotation matrix and translation matrix Step 6, repeat steps 2-5 and use the rotation matrix obtained in step 5 for each repetition and translation matrix corresponding to replace the rotation matrix R and translation matrix T in step 2 until E (R,T) is less than a preset value, i.e. convergence, and the final rotation matrix and translation matrix are the optimal rotation matrix R' and translation matrix T'. Step 7, calculate the offset vector v': v' = R' Transpose • (v - T') ; Wherein v is an arbitrary vector along the central axis of the optical primary mirror in the reference coordinate system; Step 8, calculate the offset angle θ of the central axis of the optical primary mirror: Wherein <v, v'> represents the inner product of vector v and vector v', ||v|| and ||v'|| represent the length of vector v and vector v' respectively.

2. The method for calculating the central axis offset angle of the aspheric optical primary mirror according to claim 1, wherein: In step 1: M＝5。 3. The method of claim 2, wherein the center axis offset angle is calculated by: ###0001### where: a is the center axis offset angle; and d is the distance between the center of the aspheric optical primary mirror and the center of the aspheric optical secondary mirror. In step 1: The radii corresponding to the 5 annular zones are r1=150mm, r2=225mm, r3=260mm, r4=280mm, and r5=300mm.

4. The method of claim 1 or 2 or 3, wherein, In step 1: N＝12。 5. The method of claim 4, wherein the angle of the center axis of the aspheric optical primary mirror is calculated by: In step 1, N points are collected on each annular zone, specifically: N points are uniformly collected on each annular zone.

6. The method of claim 5, wherein the center axis offset angle of the aspheric optical primary mirror is calculated by: ###0001### where, θ is the center axis offset angle of the aspheric optical primary mirror, R is the radius of the aspheric optical primary mirror, and h is the height of the aspheric optical primary mirror. Step 2 is specifically: The collected first annulus point set P is converted into the reference coordinate system by rotation and translation using the following formula ki converted into the reference coordinate system: P ki ′= R · P ki + T; Wherein the rotation matrix R is the rotation of α, β, γ angles around the x, y, z axes respectively, and the formula is as follows: The translation matrix T is the translation of a, b, c distances along the x, y, z axes respectively, and the formula is as follows: obtaining a second ringed point set P ki ' = (x ki ', y ki ', z ki ).

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