Design of space optical systems and analysis of their thermal stability
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摘要:
离轴反射式光学系统可用于对地遥感与测绘，因此要求光学系统兼具大视场、高像质与稳定内方位元素。针对传统PW法所得初始结构轴外像质量差、相机成像过程中存在指向热漂移等问题。本文推导了孔径光阑在次镜处、主三镜沿轴同距条件下三反光学系统结构像差系数，并引入主、三镜四次非球面项像差，增加优化变量，构建像质评价函数，结合远心约束条件，利用GA-SQP算法得到视场离轴下准远心的三反初始结构。进一步优化得到一款焦距260 mm、F数为10，视场为7×30°准远心离轴三反光学系统，其MTF在77 lp/mm处大于0.25，最大畸变为2%，最大主光线倾角为2.3°。针对该设计，采用微晶玻璃为基底，钛合金为结构材料，对系统进行有限元热分析。基于TRIAD算法，定量分析系统光轴在6.8 °C温差条件下绕相机坐标系X/Y/Z三轴旋转变化：绕X轴−0.728″、绕Y轴
1.0816 ″、绕Z轴11.045″，表明光学系统具有较高热稳定性的同时也对无控条件下在轨测绘数据误差修正具有重要意义。Abstract:Off-axis reflective optical systems are widely employed in Earth observation and mapping owing to their advantages of wide field of view (FOV), high image quality, and stable interior orientation elements. However, conventional designs based on the point-by-point (PW) method often suffer from degraded off-axis imaging performance and thermally induced pointing drift. In this study, the third-order aberration coefficients of a three-mirror optical system are analytically derived under the condition that the aperture stop is located at the secondary mirror and that the primary and tertiary mirrors are equally spaced along the optical axis. To further enhance imaging performance, fourth-order aspheric terms are introduced on both the primary and tertiary mirrors, thereby increasing the degrees of freedom for optimization. A comprehensive image-quality evaluation function incorporating quasi-telecentric constraints is constructed, and a hybrid genetic algorithm-sequential quadratic programming (GA-SQP) approach is employed to obtain an optimized initial configuration. The resulting system achieves a focal length of 260 mm, an F-number of 10, and a 7° × 30° FOV, with a modulation transfer function (MTF) above 0.25 at 77 lp/mm, a maximum distortion of 2%, and a maximum chief-ray angle of 2.3°. Microcrystalline glass and titanium alloy are adopted as the mirror substrate and structural materials, respectively. Finite-element thermal analysis is performed under a 6.8 °C temperature gradient, and the optical axis rotation, evaluated using the TRIAD algorithm, is −0.728″ about the X-axis,
1.0816 ″ about the Y-axis, and 11.045″ about the Z-axis. These results confirm the excellent thermal stability of the proposed design and underscore its potential for reducing in-orbit mapping errors under uncontrolled thermal environments. -
图 5 基于GA算法的目标函数迭代曲线
Figure 5. Objective Function Convergence Curve Based on the GA Algorithm
图 14 基于 TRIAD 算法的三轴计算精度蒙特卡洛分析（质心计算精度0.1像素）
Figure 14. Monte Carlo analysis of TRIAD-based three-axis accuracy with 0.1-pixel centroiding precision
图 15 基于 TRIAD 算法的三轴计算精度蒙特卡洛分析（质心计算精度0.05像素）
Figure 15. Monte Carlo analysis of TRIAD-based three-axis accuracy with 0.05-pixel centroiding precision
图 16 基于 TRIAD 算法的三轴计算精度蒙特卡洛分析（质心计算精度0.01像素）
Figure 16. Monte Carlo analysis of TRIAD-based three-axis accuracy with 0.01-pixel centroiding precision
图 18 光机结构有限元分析
Figure 18. Finite Element Analysis (FEA) of the Opto-mechanical Structure
表 1 光学系统设计指标
Table 1. Design Specifications of the Optical System
Parameter Technical target Focal length/mm 260 Entrance pupil diameter/mm 26 Waveband/nm 380-780 Field of view/° 7×30 Pixel size/μm 6.45 Distortion <2.5% 
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表 2 变量取值范围
Table 2. Value Ranges of the Variables
Variables Value Ranges $ {\alpha _2} $ [0.1,10] $ {\beta _1} $ [0.1,10] $ {\beta _2} $ [0.1,10] $ {k_1} $ [−30,30] $ {k_2} $ [−30,30] $ {k_3} $ [−30,30] $ {A_{41}} $ [−1E-9, 1E-9] $ {A_{43}} $ [−1E-9, 1E-9]
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表 3 评价函数权重取值
Table 3. Weight Values of the Evaluation Function
Wight Value Ranges W1 40 W2 80 W3 60 W4 30 W5 2
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表 4 基于GA-SQP算法优化结果
Table 4. Optimization Results Based on the GA-SQP Algorithm
Variable Value $ {\alpha _2} $ 1.3629 $ {\beta _1} $ 3.2459 $ {\beta _2} $ 0.3629 $ {k_1} $ − 1.8096 $ {k_2} $ − 0.7225 $ {k_3} $ 0.0952 $ {A_{41}} $ − 8.71565 e-11$ {A_{43}} $ 8.92229 e-11
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表 5 基于GA-SQP算法得到的初始结构参数
Table 5. Initial Optical System Parameters from GA-SQP Optimization Algorithm
名称 半径/mm 厚度/mm 圆锥常数 四次项 PM − 441.4465 − 119.3785 − 1.8096 − 8.71565 e-11SM − 154.9514 119.3785 − 0.7225 0 TM − 238.7571 − 162.7017 0.0952 8.92229 e-11
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表 6 局部优化后系统参数
Table 6. Locally Optimized System Parameters
名称 半径/mm 厚度/mm 圆锥常数 4次项 6次项 8次项 PM −429.35 −111.9 − 1.8748 1.824E-10 −2.832E-14 7.367E-19 SM −160 111.9 − 1.4195 0 0 0 TM −242.5 −163.2 − 0.7177 −6.55E-9 −8.30E-14 1.447E-18
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表 7 有限元分析结果
Table 7. FEA Results
名称 Tx/μm Ty/μm Tz/μm Rx/″ Ry/″ Rz/″ PM 7.76 −6.73 1.75 3.93 1.64 4.26 SM 4.85 −6.72 10.15 3.75 − 0.0343 3.74 TM 9.44 −5.02 1.77 0.469 −2.994 3.031
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