Calculation method for deviation angle of central axis of aspheric optical primary mirror

Abstract

The invention discloses a method for calculating a deviation angle of a central axis of an aspheric optical primary mirror, and solves the technical problem that the imaging quality is reduced due to deviation of the central axis in the assembling process of the aspheric optical primary mirror. The method can effectively detect and correct the deviation of the central axis, improves the imaging quality of an optical system, reduces the assembly difficulty, improves the production efficiency, is excellent in performance in various high-end optical application scenes due to the flexibility and adaptability, is suitable for aspheric optical primary mirrors with different sizes and shapes, and is suitable for large-scale popularization and application. No matter a small-sized optical system or a large-sized optical system, the system is suitable for wide optical equipment and instruments.

Description

Technical Field

[0001] The invention relates to a method for calculating an angle, and in particular to a method for calculating an offset angle of a central axis of an aspherical optical primary mirror. Background Art

[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 aspheric optical primary mirrors directly determines the performance of the entire optical system. However, in practical applications, due to errors in the processing and assembly process, the central axis of the aspheric optical primary mirror may be offset. This offset phenomenon often leads to a significant decrease in imaging quality, which is particularly evident in application scenarios with high spatial resolution and high spectral resolution.

[0003] At present, the detection and correction methods for the central axis offset of aspheric optical primary mirrors still face many technical challenges. Traditional detection methods often rely on complex mechanical devices and cumbersome manual adjustment processes, which are not only time-consuming and labor-intensive, but also difficult to achieve the required high-precision requirements. In addition, with the development of optical systems towards higher precision and larger apertures, the limitations of traditional methods have become increasingly obvious. Therefore, the development of an efficient and accurate method for detecting and correcting the central axis offset of aspheric optical primary mirrors has become a key technical requirement for improving the imaging quality of optical systems, reducing assembly difficulty, and improving production efficiency. Summary of the invention

[0004] The purpose of the present invention is to solve the technical problem that the imaging quality is degraded due to the central axis offset during the assembly process of the aspheric optical primary mirror, and to provide a method for calculating the central axis offset angle of the aspheric optical primary mirror.

[0005] In order to achieve the above object, the present invention adopts the following technical solution:

[0006] A method for calculating the central axis offset angle of an aspherical optical primary mirror is characterized in that it comprises the following steps:

[0007] Step 1: Establish a 3D coordinate system with the axis of the central axis of the aspherical optical primary mirror as the Z axis, and collect the first ring point set P corresponding to the M rings on the central axis. ki =(x ki ,y ki ,z ki ) ; The radius r of each ring zone k are all different, and a total of N points are collected on each ring; M≥2, N≥3; where (x ki ,y ki ,z ki ) represents the coordinates of the i-th point of the k-th ring, 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 zone k Calculate the corresponding first theoretical value of the Z axis 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 with point set P ki ' ki 'Take the average

[0017] 3.3) 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 sum of squared errors E (R,T) :

[0022]

[0023] Step 5: Minimize the error sum of squares E by nonlinear least squares method (R,T) , and get the optimized rotation matrix and translation matrix

[0024] Step 6: Repeat steps 2 to 5, and use the rotation matrix obtained in step 5 each time. and translation matrix Correspondingly replace the rotation matrix R and translation matrix T in step 2 until E (R,T) If it is less than the preset value, it converges, 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] in,<v,v'> represents the inner product of vector v and vector v', ||v|| and ||v'|| represent the modulus of vector v and vector v' respectively.

[0031] Furthermore, in step 1:

[0032] M=5.

[0033] Furthermore, in step 1:

[0034] The corresponding radii of the five ring belts are r1=150mm, r2=225mm, r3=260mm, r4=280mm, and r5=300mm.

[0035] Furthermore, in step 1:

[0036] N=12.

[0037] Furthermore, in step 1, a total of N points are collected on each ring, specifically:

[0038] N points are evenly collected on each ring.

[0039] Furthermore, step 2 is specifically as follows:

[0040] The following formula is used to rotate and translate the first ring point set P collected ki Transform to the base coordinate system:

[0041] P ki ′=R·P ki +T;

[0042] The rotation matrix R is the rotation angles of α, β, and γ around the x-axis, y-axis, and z-axis respectively. The formula is as follows:

[0043]

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

[0045]

[0046] Get the second ring point set P ki '=(x ki ',y ki ',z ki ').

[0047] Beneficial effects of the present invention:

[0048] 1. The present invention provides a method for calculating the central axis offset angle of an aspheric optical primary mirror, which can effectively detect and correct the central axis offset, improve the imaging quality of the optical system, reduce the difficulty of assembly, and improve production efficiency. Its flexibility and adaptability enable it to perform well in various high-end optical application scenarios, and 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 invention has good compatibility and scalability, and is easy to integrate with other optical detection systems, so as to facilitate further expansion of functions, such as combining with other automatic adjustment devices to realize a closed-loop control system.

[0050] 3. The method for calculating the central axis offset angle of the aspherical optical primary mirror provided by the present invention can adopt an automated measurement and analysis process, which greatly simplifies the detection process compared with the traditional manual adjustment method. It not only reduces the technical requirements of the operator, but also significantly reduces human errors and improves the overall work efficiency and detection accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 It is a flow chart of an embodiment of a method for calculating the central axis offset angle of an aspherical optical primary mirror of the present invention;

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

[0053] Figure 3 Schematic diagram of uniformly collecting annular point sets of different radii in an embodiment of the present invention;

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

[0055] Figure 5 It is a schematic diagram of a three-dimensional theoretical aspheric surface in an embodiment of the present invention. DETAILED DESCRIPTION

[0056] The technical solution of the present invention will be clearly and completely described below in conjunction with the accompanying drawings and embodiments. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0057] The method for calculating the central axis offset angle of the aspherical optical primary mirror of this embodiment is used to accurately measure and correct the central axis offset phenomenon of the aspherical optical primary mirror. Figure 1 As shown, the specific steps include:

[0058] Step 1: Use an aspherical optical primary mirror (such as Figure 2 The axis of the central axis is the Z axis, and a three-dimensional coordinate system is established arbitrarily. The first ring point set P corresponding to the M rings on the central axis is collected. ki =(x ki ,y ki ,z ki ) ; The radius r of each ring zone k are all different, and N points are evenly collected on each ring, such as Figure 3 As shown; M≥2, N≥3; where (x ki ,y ki ,z ki ) represents the coordinates of the i-th point of the k-th ring, k = 1, 2, ..., M, i = 1, 2, ..., N;

[0059] In this embodiment, M=5 is preferred, and the radii corresponding to the five ring zones are r1=150 mm, r2=225 mm, r3=260 mm, r4=280 mm, r5=300 mm, and N=12 mm;

[0060] Step 2: Use the following formula to rotate and translate the first ring point set P ki Transform to the base coordinate system:

[0061] P ki ′=R·P ki +T;

[0062] The rotation matrix R is the rotation angles of α, β, and γ around the x-axis, y-axis, and z-axis respectively. The formula is as follows:

[0063]

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

[0065]

[0066] Get the second ring point set P ki '=(x ki ',y ki ',z ki ');

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

[0068] In this step, the rotation matrix R 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 curvature radius of the aspherical optical primary mirror, and t represents the cone constant; in this embodiment, s = 950, t = -1.23. The formula for the primary mirror working surface describes a two-dimensional curve, such as Figure 4 As shown, a three-dimensional aspheric surface is formed by rotating along the central axis (Z axis), as shown in Figure 5 As shown, the three-dimensional expression of the aspherical surface of the optical primary mirror is as follows:

[0073]

[0074] Since the first ring has 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 aspherical surface of the optical primary mirror, we can obtain:

[0077]

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

[0079]

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

[0081] 3.4) 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 use the Levenberg-Marquardt algorithm to optimize the nonlinear least squares problem and obtain the optimized rotation matrix and translation matrix

[0088] Step 6: Repeat steps 2 to 5, and use the rotation matrix obtained in step 5 each time. and translation matrix Correspondingly replace the rotation matrix R and translation matrix T in step 2 until E (R,T) If it is less than the preset value, it converges, and the final rotation matrix and translation matrix are the optimal rotation matrix R' and translation matrix T';

[0089] Step 7: Calculate the offset vector v':

[0090] v'=R' -1 ·(v-T');

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

[0092] v'=R' Transpose ·(v-T');

[0093] Step 8: Calculate the offset angle θ of the central axis of the optical primary mirror:

[0094]

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

[0096] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions within the technical scope disclosed by the present invention should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A method for calculating the central axis offset angle of an aspherical optical primary mirror, characterized in that: The following steps are involved: Step 1: Establish a 3D coordinate system with the axis of the central axis of the aspherical optical primary mirror as the Z axis, and collect the first ring point set P corresponding to the M rings on the central axis. ki =(x ki ,y ki ,z ki ) ; The radius r of each ring zone k are all different, and a total of N points are collected on each ring; M≥2, N≥3; where (x ki ,y ki ,z ki ) represents the coordinates of the i-th point of the k-th ring, k = 1, 2, ..., M, i = 1, 2, ..., N; Step 2: Rotate and translate the first ring point set P ki =(x ki ,y ki ,z ki ) to the base coordinate system: P ki ’=R·P ki +T; Among them, the rotation matrix Translation Matrix Get the second ring point set P ki '=(x ki ',y ki ',z ki '); Step 3: Calculate the absolute coordinate value z ki ”; 3.1) The radius r of each ring zone k Calculate the corresponding first theoretical value of the Z axis of each ring in the reference coordinate system Where, s represents the curvature radius of the aspherical optical primary mirror surface, and t represents the cone constant; 3.2) For the same second ring with point set P ki ' ki 'Take the average 3.3) The second ring with point set P ki 'Coordinate z in ki 'Adjust to absolute coordinate value Step 4: Calculate the sum of squared errors; 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 sum of squared errors E (R,T) : Step 5: Minimize the error sum of squares E by nonlinear least squares method (R,T) , and get the optimized rotation matrix and translation matrix Step 6: Repeat steps 2 to 5, and use the rotation matrix obtained in step 5 each time. and translation matrix Correspondingly replace the rotation matrix R and translation matrix T in step 2 until E (R,T) If it is less than the preset value, it converges, 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: in,<v,v'> represents the inner product of vector v and vector v', ||v|| and ||v'|| represent the modulus of vector v and vector v' respectively.

2. The method for calculating the central axis offset angle of the aspherical optical primary mirror according to claim 1, characterized in that: In step 1: M＝5。 3. The method for calculating the central axis offset angle of the aspherical optical primary mirror according to claim 2, characterized in that: In step 1: The corresponding radii of the five ring belts are r1=150mm, r2=225mm, r3=260mm, r4=280mm, and r5=300mm.

4. The method for calculating the central axis offset angle of the aspherical optical primary mirror according to claim 1, 2 or 3, characterized in that: In step 1: N＝12。 5. The method for calculating the central axis offset angle of the aspherical optical primary mirror according to claim 4, characterized in that: In step 1, a total of N points are collected on each ring, specifically: N points are evenly collected on each ring.

6. The method for calculating the central axis offset angle of the aspherical optical primary mirror according to claim 5, characterized in that: Step 2 is as follows: The following formula is used to rotate and translate the first ring point set P collected ki Transform to the base coordinate system: P ki ′=R·P ki +T; The rotation matrix R is the rotation angles of α, β, and γ around the x-axis, y-axis, and z-axis respectively. The formula is as follows: The translation matrix T is the translation distance a, b, c along the x-axis, y-axis, and z-axis respectively. The formula is as follows: Get the second ring point set P ki '=(x ki ',y ki ',z ki ').

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