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Ray tracing plays a key role in lens design area, and it is an important tool to study the problems in physics like optics. Nowadays, ray tracing becomes ubiquitous and is widely used in optical automatic design, such as aberration analysis, optimization, and tolerance calculation. With the impulse of application requirements, optical systems like space camera develop towards large scale, high degree of accuracy and complication. The magnitude of aberrations increases exponentially with the growth of focal length and aperture, even a minor perturbation error can result in severe degeneration of image quality. As a consequence, the stringent requirements for precision, accuracy and stability of ray tracing turn higher. Reliable commercial software, for example, America’s Zemax, has high precision in ray tracing, because of commercial purpose, the process of ray tracing is a black box. It is now more important to understand what error factors are formed for ray tracing, and how these running errors can be reduced effectively. In this paper, from floating point arithmetic perspective, an error model for ray tracing is provided. This error model is suitable for not only meridional rays, but also skew rays. Starting from IEEE Standard for Binary Floating-Point Arithmetic, presentation error and rounding error are analyzed, followed by the computation process of ray’s intersection point with a quadratic surface, then rounding error expression for the intersection point is presented. In addition, error expression for distance along the ray from the reference surface to the next surface is also induced. These two error expressions are called error model, and it clearly indicates that spatial coordinates on the reference surface, direction vector and distance between the two adjacent surfaces are the main error sources. Based on the error model, some of effective measures, for instance, reprojection, spatial transformation, and direction vector’s normalization are taken to reduce the rounding error. Moreover, in the process of solving quadratic equation, conjugate number method is utilized in order to avoid increasing substantially in relative error called catastrophic cancellation. Numerical experiments and classical optical design for space camera are also given. From numerical computing view, two precision tests based on Multiple Precision Floating-Point Reliable (MPFR) library are introduced to verify our method mathematically. The experimental results show that our algorithm has the same precision (14 significant digits) as MPFR, while the existing method fails to pass tests, and has only 8 significant digits at most. Moreover, both the Cassegrain space camera and off-axis three-mirror-anastigmat space camera are used to illustrate our method’s accuracy. Experimental results indicate that our method has higher precision, more than 5 to 6 orders of magnitudes than the existing method. In addition, our algorithm has higher precision than the commercial optical design software Zemax, and residuals are 3 orders of magnitudes on average less than Zemax.
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Keywords:
- ray tracing/
- error model/
- numerical algorithm/
- optical design/
- space camera
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] -
光线始点 算法类别 交点$ P $坐标 残差$ |F(P_x, P_y, P_z)| $ $ P_x $ $ P_y $ $ P_z $ $ O(0, 0, -4\sqrt{2}(\sqrt{3}+1) $ MPFR 0 5.65685424949238 –5.65685424949238 1.42108547152020$ \times 10^{-14} $ 本文方法 0 5.65685424949238 –5.65685424949237 1.27897692436818$ \times 10^{-13} $ 原有方法 0 5.65685425535717 –5.65685423933425 3.04772385106844$ \times 10^{-9} $ $ O_1(0, -10^5, -10^5\sqrt{3}-4\sqrt{2}(\sqrt{3}+1) $ MPFR 0 5.65685424949238 –5.65685424949238 1.42108547152020$ \times 10^{-14} $ 本文方法 0 5.65685424949238 –5.65685424949237 1.27897692436818$ \times 10^{-15} $ 原有方法 0 5.65685252461116 –5.65685719583416 1.38192605021459$ \times 10^{-5} $ $ O_2(0, -10^6, -10^6\sqrt{3}-4\sqrt{2}(\sqrt{3}+1) $ MPFR 0 5.65685424949238 –5.65685424949238 1.42108547152020$ \times 10^{-14} $ 本文方法 0 5.65685424949238 –5.65685424949237 1.27897692436818$ \times 10^{-13} $ 原有方法 0 5.65709940693341 –5.65642958320677 2.03067029689663$ \times 10^{-3} $ $ O_3(0, -10^7, -10^76\sqrt{3}-4\sqrt{2}(\sqrt{3}+1) $ MPFR 0 5.65685424949238 –5.65685424949238 1.42108547152020$ \times 10^{-14} $ 本文方法 0 5.65685424949238 –5.65685424949237 1.27897692436818$ \times 10^{-13} $ 原有方法 0 5.66135398671031 –5.64906043559312 3.71872321756541$ \times 10^{-2} $ $ O_4(0, -10^8, -10^8\sqrt{3}-4\sqrt{2}(\sqrt{3}+1) $ MPFR 0 5.65685424949238 –5.65685424949238 1.42108547152020$ \times 10^{-14} $ 本文方法 0 5.65685424949238 –5.65685424949237 1.27897692436818$ \times 10^{-13} $ 原有方法 0 6.69213041663169 –3.86370331048965 4.28718721530413$ \times 10^{0} $ $ O_5(0, -10^9, -10^9\sqrt{3}-4\sqrt{2}(\sqrt{3}+1) $ MPFR 0 5.65685424949238 –5.65685424949238 1.42108547152020$ \times 10^{-14} $ 本文方法 0 5.65685424949238 –5.65685424949237 1.27897692436818$ \times 10^{-13} $ 原有方法 0 –15.9352867603302 –43.0555386543273 2.04371277294824$ \times 10^{3} $ 
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光线编号 算法类别 交点$ P $坐标 残差$ |F(P_x, P_y, P_z)| $ $ P_x $ $ P_y $ $ P_z $ 1 MPFR 0.379693092393262 0.541344968359402 –0.750185830858461 3.33066907387547$ \times 10^{-16} $ 本文方法 0.379693092393262 0.541344968359402 –0.750185830858461 3.33066907387547$ \times 10^{-16} $ 原有方法 0.379693074143900 0.541344942340492 –0.750185794801938 9.61269086552363$ \times 10^{-8} $ 2 MPFR –0.278818336478534 0.915513779528807 0.289991128719866 0 本文方法 –0.278818336478534 0.915513779528807 0.289991128719866 0 原有方法 –0.278818323077561 0.915513735526051 0.289991114781891 9.61269088772809$ \times 10^{-8} $ 3 MPFR –0.831816250860921 –0.330299029958951 0.446076535598833 2.22044604925031$ \times 10^{-16} $ 本文方法 –0.831816250860921 –0.330299029958951 0.446076535598833 2.22044604925031$ \times 10^{-16} $ 原有方法 –0.831816210880958 0.330299014083639 0.446076514158853 9.61269082111471$ \times 10^{-8} $ DownLoad:
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序号 表面类型 曲率半径 厚度 材料 膜层 圆锥系数 0 物面 标准面 Inf 500000000.000000 ———— 0 0 1 (孔径) 标准面 Inf 4500.000000 ———— 0 0 2 光阑(孔径) 标准面 –10, 478.500000 –3686.000000 MIRROR 0 –1.05000 3 标准面 –3, 846.00000 3689.000000 MIRROR 0 –2.50000 4 标准面 Inf 807.00000 ———— 0 0 5 标准面 Inf 2, 700.000000 ———— 0 0 6 标准面 1560.200000 150.000000 SILICA 0 0 7 标准面 5398.400000 210.000000 ———— 0 0 8 标准面 –2746.700000 110.000000 SILICA 0 0 9 标准面 4516.200000 185.000000 ———— 0 0 10 标准面 –2790.900000 100.000000 SILICA 0 0 11 标准面 1837.600000 150.000000 ———— 0 0 12 标准面 Inf 0.700000 BK7 0 0 13 标准面 Inf 0.750000 ———— 0 0 14 像面 标准面 Inf ———— ———— 0 0 DownLoad:
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面序号 算法类别 交点$ P $坐标 残差$ |F(P_x, P_y, P_z)| $ $ P_x $ $ P_y $ $ P_z $ OBJ 本文方法 0 4.00000000000000$ \times 10^{6} $ 0 美国商业软件Zemax 0 4.00000000000000$ \times 10^{6} $ 0 1 本文方法 0 1.53512783172994$ \times 10^{3} $ 0 6.86463863530662$ \times 10^{-9} $ 美国商业软件Zemax 0 1.53512783170000$ \times 10^{3} $ 0 2.30754721997073$ \times 10^{-8} $ 2 本文方法 0 1.49999999999948$ \times 10^{3} $ –1.07335208171627$ \times 10^{2} $ 4.68958205601666$ \times 10^{-13} $ 美国商业软件Zemax 0 1.50000000000000$ \times 10^{3} $ –1.07335208170000$ \times 10^{2} $ 1.70095404428139$ \times 10^{-9} $ 3 本文方法 0 4.16656071439095$ \times 10^{2} $ –2.24707333697088$ \times 10^{1} $ 1.70530256582424$ \times 10^{-13} $ 美国商业软件Zemax 0 4.16656071440000$ \times 10^{2} $ –2.24707333700000$ \times 10^{1} $ 1.93832505601677$ \times 10^{-10} $ 4 本文方法 0 1.25622942628122$ \times 10^{2} $ 0 1.14219744773436$ \times 10^{-9} $ 美国商业软件Zemax 0 1.25622942630000$ \times 10^{2} $ 0 3.02019032005774$ \times 10^{-9} $ 5 本文方法 0 6.23424396084419$ \times 10^{1} $ 0 9.31940746795590$ \times 10^{-10} $ 美国商业软件Zemax 0 6.23424396090000$ \times 10^{1} $ 0 1.49004364402572$ \times 10^{-9} $ 6 本文方法 0 –1.49943013287087$ \times 10^{2} $ 7.22185051869110$ \times 10^{0} $ 2.16459739021957$ \times 10^{-9} $ 美国商业软件Zemax 0 –1.49943013290000$ \times 10^{2} $ 7.22185051870000$ \times 10^{0} $ 8.48085619509220$ \times 10^{-7} $ 7 本文方法 0 –1.53288434563366$ \times 10^{2} $ 2.17676371435573$ \times 10^{0} $ 4.69299266114831$ \times 10^{-10} $ 美国商业软件Zemax 0 –1.53288434560000$ \times 10^{2} $ 2.17676371440000$ \times 10^{0} $ 1.51018321048468$ \times 10^{-6} $ 8 本文方法 0 –1.62777534965079$ \times 10^{2} $ –4.82758062839639$ \times 10^{0} $ 1.53522705659270$ \times 10^{-9} $ 美国商业软件Zemax 0 –1.62777534960000$ \times 10^{2} $ –4.82758062840000$ \times 10^{0} $ 1.67483449331485$ \times 10^{-6} $ 9 本文方法 0 –1.68760549772527$ \times 10^{2} $ 3.15420842636740$ \times 10^{0} $ 1.62981450557709$ \times 10^{-9} $ 美国商业软件Zemax 0 –1.68760549770000$ \times 10^{2} $ 3.15420842640000$ \times 10^{0} $ 1.14553768071346$ \times 10^{-6} $ 10 本文方法 0 –1.84834455036683$ \times 10^{2} $ –6.12729217781634$ \times 10^{0} $ 4.67116478830576$ \times 10^{-9} $ 美国商业软件Zemax 0 –1.84834455040000$ \times 10^{2} $ –6.12729217780000$ \times 10^{0} $ 1.32187415147200$ \times 10^{-6} $ 11 本文方法 0 –1.94567071101658$ \times 10^{2} $ 1.03295178756579$ \times 10^{1} $ 3.26690496876836$ \times 10^{-9} $ 美国商业软件Zemax 0 –1.94567071100000$ \times 10^{2} $ 1.03295178760000$ \times 10^{1} $ 1.89214915735647$ \times 10^{-6} $ 12 本文方法 0 –2.18853927906069$ \times 10^{2} $ 0 5.68434188608080$ \times 10^{-13} $ 美国商业软件Zemax 0 –2.18853927910000$ \times 10^{2} $ 0 3.93043819713057$ \times 10^{-9} $ 13 本文方法 0 –2.18933499087856$ \times 10^{2} $ 0 1.08002495835535$ \times 10^{-12} $ 美国商业软件Zemax 0 –2.18933499090000$ \times 10^{2} $ 0 2.14291162592417$ \times 10^{-9} $ 14 本文方法 0 –2.19063914207259$ \times 10^{2} $ 0 7.95807864051312$ \times 10^{-13} $ 美国商业软件Zemax 0 –2.19063914210000$ \times 10^{2} $ 0 2.74022227131354$ \times 10^{-9} $ DownLoad:
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序号 表面类型 曲率半径 厚度 材料 膜层 净口径 机械半径 圆锥系数 0 物面-标准面 Inf Inf ———— 0 Inf Inf 0 1 标准面 Inf 633.705045750000 ———— 0 367.133805130855 367.133805130855 0 2 偶次非球面 –2692.02764249986 –633.705045750000 MIRROR 0 271.014896814696 271.014896814696 –2.60071000000000 3 光阑-偶次非球面 –963.158099999982 633.705045750000 MIRROR 0 45.500175946250 45.500175946250 –0.76689900000000 4 偶次非球面 –1476.14695499995 –1132.48912577100 MIRROR 0 251.605408250266 251.605408250266 0.01160100000000 5 像面-标准面 Inf ———————— ———— 0 231.196769640532 231.196769640532 0 DownLoad:
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面序号 算法类别 交点$ P $坐标 残差$ |F(P_x, P_y, P_z)| $ $ P_x $ $ P_y $ $ P_z $ OBJ 本文方法 0 Inf Inf 0 美国商业软件Zemax 0 Inf Inf 0 1 本文方法 0 –2.01651556349517$ \times 10^{2} $ 0 4.62705429526977$ \times 10^{-11} $ 美国商业软件Zemax 0 –2.01651556350000$ \times 10^{2} $ 0 4.36727987107588$ \times 10^{-10} $ 2 本文方法 0 –1.03337392022989$ \times 10^{2} $ –1.98221886405031$ \times 10^{0} $ 6.78487441703801$ \times 10^{-13} $ 美国商业软件Zemax 0 –1.03337392020000$ \times 10^{2} $ –1.98221886410000$ \times 10^{0} $ 1.63615205928275$ \times 10^{-10} $ 3 本文方法 0 4.55199992296308$ \times 10^{1} $ –1.07580474621568$ \times 10^{0} $ 3.79696274421804$ \times 10^{-14} $ 美国商业软件Zemax 0 4.55199992300000$ \times 10^{1} $ –1.07580474620000$ \times 10^{0} $ 3.30953042748661$ \times 10^{-11} $ 4 本文方法 0 2.52430579518327$ \times 10^{2} $ –2.17467853091218$ \times 10^{1} $ 1.98810369209101$ \times 10^{-13} $ 美国商业软件Zemax 0 2.52430579520000$ \times 10^{2} $ –2.17467853090000$ \times 10^{1} $ 4.12472341438513$ \times 10^{-10} $ 5 本文方法 0 2.32101258958268$ \times 10^{2} $ 0 6.82121026329696$ \times 10^{-13} $ 美国商业软件Zemax 0 2.32101258960000$ \times 10^{2} $ 0 1.73130842995306$ \times 10^{-9} $ DownLoad:
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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