Square-aperture freeform optical system design method with added manufacturability constraints

Abstract

The application relates to an optical system design method, in particular to a square aperture free-form surface optical system design method with manufacturability constraints, which is used for solving the contradiction between the detection sensitivity and the system volume and weight of a space target detection system on a satellite. The square aperture free-form surface optical system design method with manufacturability constraints replaces a traditional circular aperture diaphragm with a square aperture diaphragm, generates an initial free-form surface optical system by using a free-form surface design method, represents the square region by using orthogonal ChebyShev polynomials, and adds manufacturability constraints in the design optimization process, so that higher detection sensitivity is realized in a relatively smaller volume.

Description

TECHNICAL FIELD

[0001] The present application relates to an optical system design method, in particular to a square aperture free-form optical system design method with manufacturability constraints. BACKGROUND

[0002] With the deepening of global space exploration, the number of on-orbit spacecraft has increased significantly, and the number of space debris generated by abandoned equipment and collisions has also continued to grow. These objects, collectively known as space objects, are diverse and complex in distribution, posing a serious threat to the safe operation of on-orbit spacecraft.

[0003] To address this problem, countries are actively developing space object detection systems. In this field, spaceborne detection systems have obvious advantages because they operate in space and can avoid atmospheric and weather interference. Among them, off-axis three-mirror optical systems are widely used in spaceborne detection tasks due to their optical design characteristics, such as high light collection efficiency and long focal length.

[0004] However, the detection capability of a spaceborne space object detection system is mainly determined by its detection sensitivity, which is directly related to the entrance pupil area of the optical system. The traditional solution is to increase the entrance pupil area of the optical system to improve detection sensitivity, but this method usually results in a significant increase in the volume and mass of the optical system. In space, the satellite platform has strict limitations on the volume and weight of the optical system. Therefore, increasing the entrance pupil area without increasing the system volume has become a key technical bottleneck for improving the performance of spaceborne space object detection systems.

[0005] In summary, the research of spaceborne space object detection systems needs to address the contradiction between detection sensitivity and system volume and weight, providing important technical support for the long-term sustainable development of space activities. SUMMARY

[0006] The purpose of the present application is to solve the contradiction between the detection sensitivity and the system volume and weight of the spaceborne space object detection system, and to provide a square aperture free-form optical system design method with manufacturability constraints.

[0007] To solve the above-mentioned deficiencies of the prior art, the present application provides the following technical solutions:

[0008] The square aperture free-form optical system design method with manufacturability constraints has the following steps:

[0009] Step 1, according to the design index of the off-axis three-mirror optical system, a three-mirror optical system is selected as an initial model, the initial model includes a primary mirror M1, a secondary mirror M2 and a tertiary mirror M3; the off-axis optical path layout design is performed on the initial model, and the off-axis unobstructed transmission of the light beam is realized by adjusting the relative positions of the secondary mirror M2 and the tertiary mirror M3, and an off-axis three-mirror optical system is obtained;

[0010] Step 2, Zemax and Matlab are interacted, and a point-by-point construction iteration method is applied to optimize the primary mirror M1, the secondary mirror M2 and the tertiary mirror M3 of the off-axis three-mirror optical system in step 1, and the primary mirror M'1, the secondary mirror M'2 and the tertiary mirror M'3 are obtained, and then an initial free-form optical system is obtained;

[0011] Step 3, in the initial free-form optical system obtained in step 2, ChebyShev polynomials are introduced to represent the primary mirror M'1, the secondary mirror M'2 and the tertiary mirror M'3, and optimization is performed by adding manufacturability constraints, and the primary mirror M''1, the secondary mirror M''2 and the tertiary mirror M''3 are obtained, and then a square aperture free-form optical system is obtained.

[0012] Further, the step 2 is specifically:

[0013] Step 2.1, Zemax and Matlab are interacted, and a point-by-point construction iteration method is applied to construct an initial free-form surface for the primary mirror M1, the secondary mirror M2 and the tertiary mirror M3 of the off-axis three-mirror optical system in step 1;

[0014] Step 2.1.1, a field of view is selected, and a square aperture is divided to obtain a plurality of characteristic light rays;

[0015] Step 2.1.2, the plurality of characteristic light rays are traced in Zemax to obtain the starting point coordinates and direction vectors of the characteristic light rays;

[0016] Step 2.1.3, in Matlab, the three-dimensional coordinates and normal vectors of the feature points of the characteristic light rays on the initial free-form surface to be solved are calculated point by point according to the optical object relationship and the shortest light ray algorithm;

[0017] Step 2.1.4, a fitting algorithm is used to generate an initial free-form surface expression to complete the construction of the initial free-form surface for each mirror;

[0018] Step 2.2, the initial free-form surface obtained in step 2.1 is iteratively optimized to obtain an initial free-form optical system;

[0019] Step 2.2.1, a field of view is selected, and a square aperture is divided to obtain a plurality of characteristic light rays;

[0020] Step 2.2.2, in Zemax, the plurality of characteristic light rays are traced to obtain the intersection points of the characteristic light rays and the initial free-form surface;

[0021] Step 2.2.3, calculating the normal vector of each feature point based on Fermat's principle and object-image relationship in Matlab;

[0022] Step 2.2.4, synthesizing the three-dimensional coordinates and normal vectors of the updated feature points to re-fit to generate an initial free-form surface expression;

[0023] Step 2.2.5, iteratively optimizing all initial free-form surfaces for each mirror to obtain a primary mirror M'1, a secondary mirror M'2, and a tertiary mirror M'3, and further to obtain an initial free-form optical system.

[0024] Further, the step 3 is specifically:

[0025] Step 3.1, performing ray tracing in Zemax on the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3 of the initial free-form optical system in step 2 to obtain feature point data on the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3, including three-dimensional coordinates and normal vectors of the feature points;

[0026] Step 3.2, representing the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3 with 6th-order even ChebyShev polynomials, and each mirror is symmetric about the YOZ plane; using Matlab to fit the feature point data to generate a ChebyShev polynomial expression for each mirror;

[0027] Step 3.3, controlling the sum of sag differences at the edges of the square region of each mirror to be zero to eliminate the piston term and tilt term caused by the polynomials themselves;

[0028] Step 3.4, gradually improving the performance of the optical system through optimization cycles until the convergence condition is reached to obtain a primary mirror M''1, a secondary mirror M''2, and a tertiary mirror M''3, and further to obtain a square aperture free-form optical system; the convergence condition is that the change amplitude of the overall mirror sag difference in consecutive optimization cycles is less than or equal to the convergence threshold.

[0029] Further, the step 3.3 is specifically:

[0030] Step 3.3.1, controlling the sum of sag differences zsag sum at the edges of the square aperture of each mirror to be zero, and the constraint formula is:

[0031]

[0032] where a 0，2 , a 2，0 , and a 2，2 are low-order components reflecting the main curvature and asymmetry of the free-form surface in the x-y direction; a 0，4 , a 4，0is a middle order component, describing the fourth order asymmetric distribution of the free-form surface; a 0，6 6，0 is a high order component; a 2，4 4，2 is a mixed order term;

[0033] Step 3.3.2, eliminate the piston term caused by the polynomial itself:

[0034] piston = α 0，0 -α 2，0 -α 0，2 +α 4，0 +α 2，2 +α 0，4 -α 6，0 -α 4，2 -α 2，4 -α 0，6 = 0

[0035] Step 3.3.3, eliminate the tilt term caused by the polynomial itself:

[0036] tilt = α 0，1 -3α 0,3 +5α 0,5 -α 2，1 +3α 2，3 +α 4，1 = 0.

[0037] Further, the step 3.4 is specifically:

[0038] Step 3.4.1, generate the system default operation number in Zemax with the spot size as the optimization target, and optimize the mirror shape to minimize the spot size;

[0039] Step 3.4.2, modify the optimization target, take the system contrast as the optimization index, and generate the system default operation number;

[0040] Step 3.4.3, repeat steps 3.4.1-3.4.2 until the convergence condition is reached, to obtain the primary mirror M"1, the secondary mirror M"2, the three mirrors M"3, and further obtain the square aperture free-form optical system.

[0041] Compared with the prior art, the beneficial effects of the present application are:

[0042] ​​The application adds a manufacturable constraint square aperture free-form optical system design method, replaces a traditional circular aperture diaphragm with a square aperture diaphragm, generates an initial free-form optical system by using a free-form design method (point-by-point construction iteration method), represents the square region in the orthogonal ChebyShev polynomial, and adds a manufacturable constraint in the design optimization process; compared with the same order off-axis three-mirror system adding a manufacturable constraint XY polynomial representation, the system pupil area is increased by 27% under the premise of relatively smaller system volume, the detection difficulty of each mirror is reduced, and the detection system performance is excellent after defocusing. BRIEF DESCRIPTION OF DRAWINGS

[0043] Figure 1 The three-dimensional diagram of the off-axis three-mirror optical system in step 2 of the embodiment of the application adds a manufacturable constraint square aperture free-form optical system design method;

[0044] Figure 2 The point diagram of each field of view of the off-axis three-mirror optical system in step 2 of the embodiment of the application adds a manufacturable constraint square aperture free-form optical system design method;

[0045] Figure 3 The three-dimensional diagram of the initial free-form optical system in step 3 of the embodiment of the application;

[0046] Figure 4 The point diagram of each field of view of the initial free-form optical system in step 3 of the embodiment of the application;

[0047] Figure 5 The sum of sag of the square aperture edge after adding a constraint to the primary mirror M'1, the secondary mirror M'2 and the third mirror M'3 in step 4 of the embodiment of the application adds a manufacturable constraint square aperture free-form optical system design method sum The change curve diagram in the optimization cycle;

[0048] Figure 6 The change curve diagram of the peak to valley (PV) value of the sag of the primary mirror M'1, the secondary mirror M'2 and the third mirror M'3 after adding a constraint in step 4 of the embodiment of the application adds a manufacturable constraint square aperture free-form optical system design method;

[0049] Figure 7 The three-dimensional diagram of the square aperture free-form optical system in step 4 of the embodiment of the application adds a manufacturable constraint square aperture free-form optical system design method;

[0050] Figure 8 The ray trace of the primary mirror M''1, the secondary mirror M''2, the third mirror M''3 and the image surface of the square aperture free-form optical system in step 4 of the embodiment of the application adds a manufacturable constraint square aperture free-form optical system design method; (b), (c), (d), (e) respectively correspond to the primary mirror M''1, the secondary mirror M''2, the third mirror M''3 and the image surface;

[0051] Figure 9Three-dimensional diagram of the circular-aperture freeform optical system in step 5 of the embodiment of the present application;

[0052] Figure 10 Ray trace of the primary mirror M01, the secondary mirror M02, the tertiary mirror M03 and the image surface of the circular-aperture freeform optical system in step 5 of the embodiment of the present application; (b), (c), (d), (e) correspond to the primary mirror M01, the secondary mirror M02, the tertiary mirror M03 and the image surface respectively;

[0053] Figure 11 Point spread diagram of the square-aperture freeform optical system in step 4 of the embodiment of the present application;

[0054] Figure 12 Point spread diagram of the circular-aperture freeform optical system in step 5 of the embodiment of the present application;

[0055] Figure 13 Energy concentration curve of the square-aperture freeform optical system before defocus in step 6 of the embodiment of the present application;

[0056] Figure 14 Energy concentration curve of the circular-aperture freeform optical system before defocus in step 6 of the embodiment of the present application;

[0057] Figure 15 Relative illumination curve of the square-aperture freeform optical system before defocus in step 6 of the embodiment of the present application;

[0058] Figure 16 Relative illumination curve of the circular-aperture freeform optical system before defocus in step 6 of the embodiment of the present application;

[0059] Figure 17 Distortion grid of the square-aperture freeform optical system before defocus and the circular-aperture freeform optical system before defocus in step 6 of the embodiment of the present application; (a), (b) correspond to the square-aperture freeform optical system and the circular-aperture freeform optical system respectively;

[0060] Figure 18 Full field wave aberration of the square-aperture freeform optical system before defocus in step 6 of the embodiment of the present application;

[0061] Figure 19 Full field wave aberration of the circular-aperture freeform optical system before defocus in step 6 of the embodiment of the present application;

[0062] Figure 20 Point spread diagram of the square-aperture freeform optical system after defocus and the circular-aperture freeform optical system after defocus in step 7 of the embodiment of the present application; (a), (b) correspond to the square-aperture freeform optical system and the circular-aperture freeform optical system respectively;

[0063] Figure 21 This is the energy concentration curve of the square aperture freeform surface optical system after defocusing in step 7 of the embodiment of the present invention;

[0064] Figure 22 This is the energy concentration curve of the freeform surface optical system with a circular aperture after defocusing in step 7 of this embodiment of the invention;

[0065] Figure 23 These are the distribution diagrams of the sag difference PV values ​​of each mirror surface minus the base spherical surface in the square aperture freeform surface optical system and the circular aperture freeform surface optical system in the embodiments of the present invention; (a), (b), and (c) correspond to the primary mirror M″1, secondary mirror M″2, and tertiary mirror M″3, respectively; (d), (e), and (f) correspond to the primary mirror M01, secondary mirror M02, and tertiary mirror M03, respectively.

[0066] Figure 24 This is the result of 500 Monte Carlo simulations of the tolerance analysis of the square aperture freeform surface optical system after defocusing in step 7 of the present invention.

[0067] Figure 25 The results are from 500 Monte Carlo simulations of the tolerance analysis of the freeform surface optical system with circular aperture after defocusing in step 7 of the present invention. Detailed Implementation

[0068] The present invention will be further described below with reference to the accompanying drawings and exemplary embodiments.

[0069] A design method for a square aperture freeform surface optical system with added manufacturable constraints is proposed. In an off-axis three-mirror optical system, a square aperture stop is used to replace the traditional circular aperture stop, and manufacturable constraints are added during the optimization process to finally complete the design of the square aperture stop detection system.

[0070] Specifically, the steps include the following:

[0071] Step 1: Based on optical radiation theory, establish a model for the detection sensitivity of a spaceborne space target detection system to a space target, expressed as the equivalent apparent magnitude m of the reflected light at the entrance pupil, using the following formula:

[0072]

[0073] In the formula, S is the average wavelength. d Let N be the entrance pupil area (effective light-transmitting area) of the optical system, η be the average quantum efficiency of the detector, τ0 be the transmittance of the optical system, and N be the average quantum efficiency of the detector. B N represents the number of photoelectrons generated by background noise. D T represents the number of photoelectrons produced by the dark current. snTo meet the signal-to-noise ratio threshold of a certain detection rate and false alarm rate, a is the size of a single pixel of the detector, d is the spot diameter, h is the Planck constant, and c is the speed of light;

[0074] Qualitative analysis of the detection sensitivity model, under the premise of constant volume of the optical system, the detection sensitivity m increases with the entrance pupil area S d

[0075] Step 2, according to the design index of the off-axis three-mirror optical system, select a three-mirror optical system with similar index as the initial model, the initial model includes the primary mirror M1, the secondary mirror M2 and the third mirror M3; the off-axis optical path layout design is carried out on the initial model to ensure that the light path deviates from the optical center, and the off-axis non-shielding transmission of the light beam is realized by adjusting the relative positions of the secondary mirror M2 and the third mirror M3, and the off-axis three-mirror optical system is obtained;

[0076] The design index of the off-axis three-mirror optical system includes spectral range, focal length, field of view, aperture, spot size, distortion, as shown in Table 1:

[0077] Table 1

[0078]

[0079]

[0080] Figure 1 The three-dimensional diagram of the off-axis three-mirror optical system in step 2; Figure 2 The spot diagram of each field of view of the off-axis three-mirror optical system in step 2; the average RMS size of the spot diagram is about 400μm; the curvature radius, mirror spacing and mirror deflection angle of the primary mirror M1, the secondary mirror M2 and the third mirror M3 of the off-axis three-mirror optical system are as shown in Table 2:

[0081] Table 2

[0082]

[0083]

[0084]

[0085] Step 3, interact with Zemax and Matlab, apply the point-by-point construction iteration method of free-form surface optical system construction method, the primary mirror M1, the secondary mirror M2 and the third mirror M3 of the off-axis three-mirror optical system in step 2, to obtain the primary mirror M'1, the secondary mirror M'2 and the third mirror M'3, and then obtain the initial free-form surface optical system, as the starting point of subsequent addition of manufacturable ChebyShev polynomial representation square aperture free-form surface optical system optimization;

[0086] ​Step 3.1: Using Zemax and Matlab, apply the point-by-point iterative construction method to construct the initial freeform surfaces of the primary mirror M1, secondary mirror M2, and third mirror M3 of the off-axis three-mirror optical system in Step 2;

[0087] Step 3.1.1: Select the field of view, including 11 fields of view: (0°, -2°), (0°, 0°), (0°, 2°), (2°, -2°), (2°, 0°), (2°, 2°), (1.4°, -2°), (1.4°, -1.4°), (1.4°, 0°), (1.4°, 1.4°), and (1.4°, 2°). Divide the square aperture into 31 equal parts in both the x and y directions. The number of sampling points = 31 × 31 = 961, and the number of characteristic rays = 11 × 961 = 10571.

[0088] Step 3.1.2: In Zemax, ray tracing is performed using the above field of view and aperture sampling to obtain the starting coordinates and direction vector of the feature rays;

[0089] Step 3.1.3: In Matlab, based on the optical object-image relationship and the shortest ray algorithm, calculate the three-dimensional coordinates and normal vectors of the feature points of the ray on the initial freeform surface to be determined point by point;

[0090] Step 3.1.4: Using a fitting algorithm that simultaneously considers the three-dimensional coordinates of feature points and their corresponding normal vectors, generate the initial freeform surface expression (XY polynomial representation) of the initial freeform surface to be determined;

[0091] Step 3.1.5: Complete the above steps for all initial freeform surfaces in the off-axis three-mirror optical system to complete the construction of the initial freeform surfaces;

[0092] Step 3.2: Iteratively optimize the initial freeform surface obtained in Step 3.1 to obtain the primary mirror M′1, secondary mirror M′2, and tertiary mirror M′3, thereby obtaining the initial freeform surface optical system;

[0093] Step 3.2.1: Select the 11 fields of view mentioned in step 3.1.1, and divide the square aperture into 51 equal parts in the x and y directions, resulting in a total of 28,611 feature rays;

[0094] Step 3.2.2: In Zemax, trace the feature rays to obtain the intersection points of the feature rays on the initial freeform surface and the initial freeform surface;

[0095] Step 3.2.3: Write an algorithm based on Fermat's principle and the object-image relationship to solve for the normal vector corresponding to each feature point;

[0096] Step 3.2.4: Use a fitting algorithm that simultaneously considers the updated 3D coordinates and normal vectors of the feature points to refit the initial freeform surface expression (XY polynomial representation);

[0097] Step 3.2.5, iterate all initial free-form surfaces one by one to complete the iterative optimization of the entire off-axis three-mirror optical system, to obtain the primary mirror M'1, the secondary mirror M'2, the tertiary mirror M'3, and further to obtain the initial free-form optical system as the basic model for the subsequent design of the square-aperture free-form optical system;

[0098] Figure 3 Figure 3 is a three-dimensional diagram of the initial free-form optical system in step 3; Figure 4 Figure 4 is a point diagram of each field of view of the initial free-form optical system in step 3;

[0099] Step 4, introducing ChebyShev polynomials to represent the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3 in the initial free-form optical system obtained in step 3, and optimizing by adding manufacturability constraints to obtain the primary mirror M''1, the secondary mirror M''2, and the tertiary mirror M''3, and further to obtain the square-aperture free-form optical system;

[0100] Step 4.1, ray tracing the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3 of the initial free-form optical system obtained in step 3 in Zemax to obtain the characteristic point data on the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3, including the three-dimensional coordinates and normal vectors of the characteristic points;

[0101] Step 4.2, representing the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3 with 6th-order even ChebyShev polynomials, and only keeping even terms to ensure that each mirror is symmetric about the Y0Z plane; using Matlab to fit the characteristic point data to generate the ChebyShev polynomial expression of each mirror;

[0102] The 6th-order even ChebyShev polynomial is as follows:

[0103]

[0104] In the formula, c is the curvature at the vertex of the base surface (the reciprocal of the radius of curvature), k is the conic coefficient, a i，j are the coefficients of the ChebyShev polynomial; x0 and y0 represent the normalized aperture values in the x and y directions, respectively, T i (u) and T j (v) are one-dimensional ChebyShev polynomials; i and j represent the orders in the x and y directions, respectively, i is an even number, and i+j≤6;

[0105] The ChebyShev polynomial coefficients of the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3 are shown in Table 3:

[0106] Table 3

[0107]

[0108]

[0109] Step 4.3, adding manufacturability constraints - controlling the sum of sag heights of each mirror at the edges of the square aperture to be zero, eliminating piston and tilt terms caused by the polynomial itself;

[0110] Step 4.3.1, controlling the sum of sag heights zsag of each mirror at the edges of the square aperture to be zero, with the constraint formula: sum

[0111]

[0112] where a 0，2 , a 2，0 , a 2，2 are low-order components reflecting the main curvature and asymmetry of the freeform surface in x-y directions; a 0，4 , a 4，0 are middle-order components describing the quartic asymmetry distribution of the freeform surface; a 0，6 , a 6，0 are high-order components capturing more detailed characteristics of the freeform surface; a 2，4 , a 4，2 are mixed-order terms describing complex interactive characteristics on the freeform surface;

[0113] The square aperture edges are the boundary regions defined by the maximum or minimum values of x and y;

[0114] Step 4.3.2, eliminating piston terms caused by the polynomial itself:

[0115] piston = a 0,0 - a 2,0 - a 0,2 + a 4,0 + a 2,2 + a 0,4 - a 6,0 - a 4,2 - a 2,4 - a 0,6 = 0

[0116] Step 4.3.3, eliminating tilt terms caused by the polynomial itself:

[0117] tilt = a 0,1 - 3a 0,3 + 5a 0,5 - a 2,1 + 3a 2,3 + a 4,1 = 0 ​

[0118] Step 4.4, gradually improve the performance of the optical system through an optimization cycle until a convergence condition is reached to obtain the primary mirror M"1, the secondary mirror M"2, the tertiary mirror M"3, and further obtain the square aperture freeform optical system;

[0119] Step 4.4.1, in Zemax, generate system default operation numbers with the spot size as the optimization target, and optimize the mirror shape to minimize the spot size;

[0120] Step 4.4.2, modify the optimization target, and generate system default operation numbers with the system contrast as the optimization index;

[0121] Step 4.4.3, repeat steps 4.4.1-4.4.2, each execution of steps 4.4.1-4.4.2 is one cycle, until a convergence condition is reached, a total of 9 cycles are completed, the performance of the system is gradually optimized, and finally the primary mirror M"1, the secondary mirror M"2, and the tertiary mirror M"3 are obtained, and further the square aperture freeform optical system is obtained;

[0122] The convergence condition is that the variation amplitude of the overall mirror sag difference in the continuous multiple optimization cycles is less than or equal to the convergence threshold;

[0123] The optimization results are as follows:

[0124] Figures 5-6 The sum of the sag differences of the square aperture edges after adding the constraints of the primary mirror M'1, the secondary mirror M'2, and the tertiary mirror M'3 is zsag sum , and the variation curve of the sag difference PV value in the optimization cycle;

[0125] The sag difference PV value of the primary mirror M'1 decreases from 1.38 mm to 0.06 mm, the sag difference PV value of the secondary mirror M'2 decreases from 1.62 mm to 0.28 mm, and the sag difference PV value of the tertiary mirror M'3 decreases from 4.52 mm to 0.24 mm; it can be seen that by controlling the sum of the sag differences zsag sum to be zero, the overall mirror sag difference PV value is greatly reduced, and the difficulty of freeform surface detection is greatly reduced;

[0126] After 9 cycles of optimization of the manufacturability constraints and the operation numbers with the spot size and the contrast as the optimization indexes, the square aperture freeform optical system with the preset design indexes is successfully designed; wherein, the primary mirror is a square aperture diaphragm with a size of 40 mm x 40 mm, the base spherical curvature radii of the primary mirror M"1, the secondary mirror M"2, and the tertiary mirror M"3 are -341.964 mm, -112.864 mm, and -159.690 mm respectively, the focal length is 150 mm, and the field of view angle is 4° x 4°;

[0127] Figure 7The three-dimensional diagram of the square-aperture freeform optical system after adding constraints in step 4; Figure 8 The ray trace of the primary mirror M"1, the secondary mirror M"2, the tertiary mirror M"3 and the image plane in step 4; (b), (c), (d), (e) correspond to the primary mirror M"1, the secondary mirror M"2, the tertiary mirror M"3 and the image plane respectively; wherein the ray distribution of the primary mirror M"1 is constrained by the aperture stop, showing a square feature; the ray distribution of the secondary mirror M"2 and the tertiary mirror M"3 is more complex, reflecting the role of freeform surface in aberration correction;

[0128] Step 5, taking the initial model in step 2 as the starting point of the design optimization of adding a circular aperture stop, sampling the sag of the data points at different polar angles at the maximum aperture, and then setting the sag difference between adjacent data points to zero as a manufacturable constraint, finally optimizing to obtain a circular-aperture freeform optical system;

[0129] Step 5.1, for the initial model in step 2, set the aperture stop to a circular aperture stop with a radius of 20mm, and other parameters remain unchanged;

[0130] Step 5.2, in the optimization process, add the spot size as a system index generation operator, and add the contrast as a system index generation operator after the first optimization;

[0131] Step 5.2.1, add the spot size as an optimization index to the system's objective function, optimize the focusing performance of the light rays, and minimize the spot size;

[0132] Set the XY polynomial low-order terms of the mirror surface as variable parameters;

[0133] Add RMS Spot Radius (Root Mean Square Spot Radius) as an optimization target using Zemax, generate system default operators, and run optimization to adjust the initial freeform shape of the mirror surface;

[0134] Step 5.2.2, add the contrast as an optimization index, and improve the imaging quality by optimizing the MTF (Modulation Transfer Function) of the system;

[0135] After the initial optimization, further increase the XY polynomial middle-order terms of the mirror surface as variables;

[0136] Add the contrast as an optimization target, generate system default operators, and gradually optimize the contrast within the target wavelength range of each field point of the system;

[0137] Step 5.3, sample the sag of the data points at different polar angles at the maximum aperture, then set the sag difference between adjacent data points to zero as a manufacturable constraint, gradually set the XY polynomial high-order terms as variables, and finally optimize to obtain a 6th-order even XY polynomial;

[0138] Sampling the sag data points on the mirror surface at different polar angle positions at the maximum mirror aperture (radius 20 mm);

[0139] Setting the sag difference of adjacent data points to zero, adding the sag difference constraint in the objective function, controlling the edge error of the mirror shape within the manufacturable range; in the later stage of optimization, gradually releasing the XY polynomial high-order terms as variables, further adjusting the mirror shape to improve system performance, obtaining a circular aperture freeform optical system, with the following parameters:

[0140] The primary mirror M01 is a circular aperture stop with a radius of 20 mm, and the base spherical curvature radii of the primary mirror M01, the secondary mirror M02, and the tertiary mirror M03 are -333.962 mm, -143.829 mm, and -162.889 mm, respectively, with a focal length of 150 mm and a field of view angle of 4°x4°;

[0141] Figure 9 A three-dimensional view of the circular aperture freeform optical system in step 5;

[0142] Figure 10 The ray traces of the primary mirror M01, the secondary mirror M02, the tertiary mirror M03, and the image plane of the circular aperture freeform optical system in step 5; (b), (c), (d), and (e) correspond to the primary mirror M01, the secondary mirror M02, the tertiary mirror M03, and the image plane, respectively;

[0143] The XY polynomial coefficients of the primary mirror M01, the secondary mirror M02, and the tertiary mirror M03 are shown in Table 4:

[0144] Table 4

[0145]

[0146]

[0147] Step 6, comparing the square aperture freeform optical system in step 4 and the circular aperture freeform optical system in step 5, it can be found that compared with the circular aperture freeform optical system, the square aperture freeform optical system has a larger entrance pupil area by 27% on the premise of a relatively smaller system volume (0.78 cubic decimeters vs 0.96 cubic decimeters), which improves the detection sensitivity of the system to space targets while reducing the detection difficulty of each mirror. After defocusing processing, the detection system has excellent performance. The specific comparison is as follows:

[0148] The spot diagrams, energy concentration, relative illumination, distortion grid, and full-field wave aberration of the square aperture freeform optical system and the circular aperture freeform optical system are shown in Figure 6 、 Figure 7 、 Figure 8 、 Figure 9 and Figure 10 , respectively;

[0149] As shown in Figures 11-12 , the full field average RMS spot diameter of the square aperture freeform optical system is 0.56μm, which is much smaller than the Airy disk diameter 5.4μm; the full field average RMS spot diameter of the circular aperture freeform optical system is 0.52μm, which is much smaller than the Airy disk diameter 6μm;

[0150] As shown in Figures 13-14 , the square aperture freeform optical system and the circular aperture freeform optical system both have 80% of the energy concentrated within a diameter of 4.6μm, and 90% of the energy concentrated within a diameter of 9.2μm;

[0151] As shown in Figures 15-16 , on the image plane, the illumination uniformity of the square aperture freeform optical system and the circular aperture freeform optical system on the image plane is 98% and 97% respectively;

[0152] As shown in Figure 17 , the maximum grid distortion of the square aperture freeform optical system and the circular aperture freeform optical system is 0.39% and 0.34% respectively, which meets the design index;

[0153] As shown in Figures 18-19 , the full field RMS wave aberration of the square aperture freeform optical system and the circular aperture freeform optical system is 0.039λ and 0.017λ respectively;

[0154] Since the spot on the image plane of the system is much smaller than the size of a single pixel of the detector, in order to improve the position interpolation accuracy, the system is defocused to make the spot distributed in 2x2 pixels; after the two systems are defocused by 0.07mm, the geometric spot diameter of the two systems is close to 2 times the pixel size, which meets the application requirements of the spaceborne detection system, as shown in Figure 20 ;

[0155] At this time, the energy concentration curve of the spot is as shown in Figures 21-22 , both systems have 85% of the energy concentrated within a diameter of 18.4μm;

[0156] The sag difference PV values of the primary mirror M''1, the secondary mirror M''2 and the tertiary mirror M''3 after subtracting the base sphere are 0.06mm, 0.28mm and 0.24mm respectively, as shown in Figure 23 (a)(b)(c), the sag difference PV values of the primary mirror M01, the secondary mirror M02 and the tertiary mirror M03 after subtracting the base sphere are 0.24mm, 0.43mm and 0.38mm respectively, as shown in Figure 23 (d)(e)(f), by adding the manufacturability constraints, the sag difference PV values of the corresponding mirror after subtracting the base sphere of the curve are reduced by 75%, 35% and 37% respectively, and the difficulty of curve surface detection is greatly reduced;

[0157] In summary, the square aperture freeform surface optical system, compared to the circular aperture freeform surface optical system, offers advantages in terms of system size (0.78 dm). 3 vs 0.96dm 3 The entrance pupil area increased by 27% (1600mm). 2 vs1257mm 2 At the same time, the PV value of the elevation difference is reduced by at least 35%, while other system indicators, such as field of view, focal length, image size, and energy concentration, are similar. According to the detection sensitivity model obtained in step 1, the system can detect the weakest spatial target with an equivalent magnitude increase of 0.24 at the system entrance pupil, which means that it can detect the light reflected by even weaker spatial targets, thus improving the system's ability to detect weak spatial targets.

[0158] Step 7: Perform tolerance analysis on the square aperture freeform surface optical system in step 4 after defocusing and the circular aperture freeform surface optical system in step 5 after defocusing;

[0159] The energy concentration of the system within an image plane diameter of 18.4 μm (i.e., twice the pixel size) after defocusing was used as the index for tolerance analysis. Under conventional processing and assembly levels, the tolerance allocation results of each lens are shown in Table 5. This tolerance allocation table is used for both systems.

[0160] Table 5

[0161]

[0162]

[0163] Under the aforementioned tolerance constraints, 500 Monte Carlo simulations were performed to assess the feasibility of system energy concentration at different wavelengths and fields of view. Figures 24-25 As shown, for a square aperture freeform surface optical system, the energy concentration value within 18.4 μm of the image plane has a 90% probability greater than 87.0%; for a circular aperture freeform surface optical system, the energy concentration value within 18.4 μm of the image plane has a 90% probability greater than 85.2%. Tolerance analysis results show that both systems meet the requirements for visible light space target detection.

[0164] Therefore, the initial structure is obtained by using the freeform surface design method - the construction iteration method. Then, the mirror is characterized by the orthogonal Cheby-Shev polynomial freeform surface in the square region. During the optimization process, manufacturable constraints are added to optimize the off-axis three-mirror system with square aperture stop. This system can increase the entrance pupil area while keeping the volume relatively smaller, thereby improving the optical system's ability to detect space targets.

Claims

1. A method for designing an optical system with a square aperture freeform surface that incorporates manufacturable constraints, characterized in that: Includes the following steps: Step 1: Based on the design specifications of the off-axis three-mirror optical system, select the three-mirror optical system as the initial model. The initial model includes the primary mirror M1, the secondary mirror M2, and the third mirror M3. Perform off-axis optical path layout design on the initial model, and achieve off-axis unobstructed transmission of the beam by adjusting the relative positions of the secondary mirror M2 and the third mirror M3 to obtain the off-axis three-mirror optical system. Step 2: Using Zemax and Matlab, apply the point-by-point iterative construction method to optimize the primary mirror M1, secondary mirror M2, and third mirror M3 of the off-axis three-mirror optical system in Step 1, to obtain the primary mirror M′1, secondary mirror M′2, and third mirror M′3, and thus obtain the initial freeform surface optical system. Step 2.1: Using Zemax and Matlab, apply the point-by-point iterative construction method to construct the initial freeform surfaces of the primary mirror M1, secondary mirror M2, and third mirror M3 of the off-axis three-mirror optical system in Step 1; Step 2.1.1: Select the field of view, divide the square aperture, and obtain multiple feature rays; Step 2.1.2: Traverse multiple feature rays in Zemax to obtain the starting coordinates and direction vectors of the feature rays; Step 2.1.3: In Matlab, based on the optical object-image relationship and the shortest ray algorithm, calculate the three-dimensional coordinates and normal vectors of the feature points on the initial freeform surface to be determined point by point. Step 2.1.4: Use a fitting algorithm to generate the initial freeform surface expression, and complete the initial freeform surface construction frame by frame; Step 2.2: Iteratively optimize the initial freeform surface obtained in Step 2.1 to obtain the initial freeform surface optical system; Step 2.2.1: Select the field of view, divide the square aperture, and obtain multiple feature rays; Step 2.2.2: In Zemax, trace multiple feature rays to obtain the intersection points of the feature rays with the initial freeform surface; Step 2.2.3: Calculate the normal vector of each feature point in Matlab based on Fermat's principle and the object-image relationship; Step 2.2.4: Combine the updated 3D coordinates and normal vectors of the feature points to refit and generate the initial freeform surface expression; Step 2.2.5: Iteratively optimize all initial freeform surfaces mirror by mirror to obtain primary mirror M′1, secondary mirror M′2, and tertiary mirror M′3, thereby obtaining the initial freeform surface optical system; Step 3: In the initial freeform surface optical system obtained in Step 2, Cheby-Shev polynomials are introduced to characterize the primary mirror M′1, secondary mirror M′2, and tertiary mirror M′3. The system is then optimized by adding manufacturability constraints to obtain the primary mirror M"1, secondary mirror M"2, and tertiary mirror M"3, thereby obtaining a square aperture freeform surface optical system. Step 3.1: Perform ray tracing on the primary mirror M′1, secondary mirror M′2, and tertiary mirror M′3 of the initial freeform surface optical system in Step 2 in Zemax to obtain feature point data on the primary mirror M′1, secondary mirror M′2, and tertiary mirror M′3. The feature point data includes the three-dimensional coordinates and normal vectors of the feature points. Step 3.2: Represent the primary mirror M′1, secondary mirror M′2, and tertiary mirror M′3 using 6th-order even-degree Cheby-Shev polynomials, with each mirror surface being symmetric about the YOZ plane; use Matlab to fit the feature point data to generate the Cheby-Shev polynomial expression for each mirror surface; Step 3.3: Control the sum of the sag differences of each mirror at the edge of the square region to be zero, thus eliminating the piston and tilt terms caused by the polynomial itself; Step 3.3.1: Control the sum of the sag differences (zsag) of the square aperture edges of each mirror. sum If the value is zero, the constraint formula is: In the formula, a 0，2 a 2，0 a 2，2 These are low-order components, reflecting the dominant curvature and asymmetry of the freeform surface in the xy directions; a 0，4 a 4，0 It is an intermediate-order component, describing the fourth-order asymmetric distribution of the freeform surface; a 0，6 a 6，0 It is a higher-order component; a 2，4 a 4，2 It is a mixed-order term; Step 3.3.2: Eliminate the Piston term caused by the polynomial itself: piston = a 0，0 -a 2，0 -a 0，2 +a 4，0 +a 2，2 +a 0，4 -a 6，0 -a 4，2 -a 2，4 -a 0，6 =0 Step 3.3.3: Eliminate the tilt term caused by the polynomial itself: tilt = a 0，1 -3a 0，3 +5a 0，5 -a 2，1 +3a 2，3 +a 4，1 =0; Step 3.4: By optimizing the loop, the performance of the optical system is gradually improved until the convergence condition is reached, and the primary mirror M"1, secondary mirror M"2, and tertiary mirror M"3 are obtained, thus obtaining a square aperture freeform surface optical system; the convergence condition is that the change amplitude of the overall mirror sagittal difference is less than or equal to the convergence threshold in multiple consecutive optimization loops.

2. The design method for a freeform surface optical system with added manufacturable constraints on a square aperture according to claim 1, characterized in that, Step 3.4 specifically involves: Step 3.4.1: In Zemax, with the image spot size as the optimization target, generate the system default operands and optimize the mirror shape to minimize the image spot size; Step 3.4.2: Modify the optimization objective, using system contrast as the optimization metric, and generate the system default operands; Step 3.4.3: Repeat steps 3.4.1 to 3.4.2 until the convergence condition is reached, and obtain the primary mirror M"1, secondary mirror M"2, and tertiary mirror M"3, and then obtain the square aperture freeform surface optical system.

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