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The angular resolution of optical system is limited by the ratio of the wavelength to the aperture of the entrance pupil, indicating that the optical system with large aperture has a high spatial resolution. Sparse aperture imaging is one of the effective solutions to the problem that the telescope is bulky, heavy and difficult to manufacture. According to the self-similarity and multi-scale characteristics of fractal configuration, we propose a sparse aperture array and analyze its performance for synthetic aperture imaging system. In the array Golay-3 is used as a structural unit to expand a multi-layered fractal configuration in a self-similar manner. Given the analytical expression of the pupil function which is reduced by dimensionless parameters, we calculate the modulation transfer functions (MTFs), the practical cut-off frequencies and the middle spatial frequency characteristics of the fractal configuration under different fill factors and different outer layer rotational angles. We analyze both the MTF values and the performance parameters of the fractal structure for the cases of N = 3, 9, and 18, respectively. The results show that the decrease of fill factor does not significantly change the MTF curve nor the practical cutoff frequency in a range of fill factor between 0.0952 and 0.2246. The outer layer rotational angle has a periodicity, and the change in the angle has no large influence on the practical cutoff frequency. When the reduced aperture parameter is
$ {d_0} = 1$ and the fill factor is 22.46%, the middle spatial frequency of N = 18 array is more stable and the practical cut-off frequency is higher. Using the fractal self-similarity, the aperture of the system can be expanded effectively while maintaining the middle spatial frequency characteristics. The computing results are of scale invariance due to the adoption of the reduced aperture parameter.-
Keywords:
- fractal configuration /
- sparse aperture array /
- pupil function /
- modulation transfer function
[1] 吴泉英 2006 博士学位论文 (苏州: 苏州大学)
Wu Q Y 2006 Ph. D. Dissertation (Suzhou: Suzhou University) (in Chinese)
[2] 陈海亭, 江月松, 钟宇 2005 光学学报 25 1616
Google Scholar
Chen H T, Jiang Y S, Zhong Y 2005 Acta Opt. Sin. 25 1616
Google Scholar
[3] 赵娟, 王大勇, 张亚新, 耿则勋, 陶世荃 2009 中国激光 36 934
Zao J, Wang D Y, Zhang Y X, Geng Z X, Tao S K 2009 Chin. J. Lasers 36 934
[4] 刘丽, 江月松 2013 综合孔径成像原理与应用 (北京: 国防工业出版社) 第48−54页
Liu L, Jiang Y S 2013 Principle and Application of Synthetic Aperture Imaging (Beijing: National Defense Industry Press) pp48−54 (in Chinese)
[5] 苏显渝, 李继陶 信息光学(北京: 科学出版社)第20−26页
Su X Y, Li J T 1999 Information Opitics (Beijing: Science Press) pp20−26 (in Chinese)
[6] Meinel A B 1970 Appl. Opt. 9 2501
Google Scholar
[7] Chung S, Spie M 2004 Opt. Eng. 43 2156
Google Scholar
[8] Fiete R D 2002 Opt. Eng. 41 1957
Google Scholar
[9] Zhou C, Wang Z 2018 Opt. Eng. 26 6973
[10] 易红伟, 李英才, 樊超 2007 光子学报 36 2062
Yi H W, Li Y C, Fan C 2007 Acta. Photonica Sin. 36 2062
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Google Scholar
[12] Golay M J E 1971 J. Opt. Soc. Am. 61 272
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Google Scholar
[14] Cassaing F, Mugnier L M 2018 Opt. Lett. 43 4555
[15] Tcherniavski I, Kahrizi M 2005 Opt. Lett. 44 103201
[16] 钱霖, 吴泉英, 吴峰, 沈为民 2005 光学学报 25 1030
Google Scholar
Qian L, Wu Q Y, Wu F, Shen W N 2005 Acta Opt. Sin. 25 1030
Google Scholar
[17] 韩骥, 王大勇, 刘汉承, 伏西洋, 郭红锋, 陶世荃 2007 光电子·激光 18 649
Han J, Wang D Y, Liu H C, Fu X Y, Guo H F, Tao S K 2007 Optronics Lasers 18 649
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Google Scholar
[19] 刘丽, 江月松, 王长伟 2009 光学学报 29 2774
Liu L, Jiang Y S, Wang C W 2009 Acta Opt. Sin. 29 2774
[20] 刘政, 王胜千, 饶长辉 2012 61 039501
Google Scholar
Liu Z, Wang S Q, Rao C H 2012 Acta Phys. Sin. 61 039501
Google Scholar
[21] 龙伟军, 王治乐, 周彦平 2004 光学学报 24 1009
Google Scholar
Long H W, Wang L Z, Zhou Y P 2004 Acta Opt. Sin. 24 1009
Google Scholar
[22] 李兰芳 2004 博士学位论文 (南京: 南京理工大学)
Liu L F 2004 Ph. D. Dissertatio (Nanjing: Nanjing University of Science and Technology) (in Chinese)
[23] 刘肖尧, 梁忠诚, 郝未倩, 赵瑞, 孔梅梅, 陈陶, 张月 2019 光学学报 39 0811003
Liu X Y, Liang Z C, Hao W Q, Zhao R, Kong M M, Chen T, Zhang Y 2019 Acta Opt. Sin. 39 0811003
[24] 周程灏, 王治乐, 朱峰 2017 中国光学 10 25
Zhou C H, Wang Z L, Zhu F 2017 Chin. Opt. 10 25
[25] 朱华, 姬翠翠 2011分形理论及其应用 (北京: 科学出版社) 第10−16页
Zhu H, Ji C C 2011 Fractal Theory and Application (Beijing: Science Press) pp10−16 (in Chinese)
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图 1 分形阵列结构
Figure 1. Fractal array configuration.
图 2 结构特征 (a)子孔径直径与填充因子曲线图; (b)结构层数与包围圆半径关系
Figure 2. Configuration characteristics: (a) Sub-aperture diameter and fill factor curve; (b) the relationship of the number of fractal configuration and the radius of aperture.
图 3 分形阵列MTF(F = 22.46%) (a)三维MTF; (b) MTF俯视图
Figure 3. MTF of fractal array (F = 22.46%): (a) There-dimensional MTF; (b) top-view MTF.
图 4 分形结构随填充因子变化MTF曲线 (a)沿fx归一化频率方向; (b)沿fy归一化频率方向
Figure 4. MTF curves of fractal array with different fill factor: (a) Normalized frequency along fx - axis; (b) normalized frequency along fy - axis.
图 5 分形阵列随外环旋转角度变化MTF曲线 (a)沿fx归一化频率方向; (b)沿fy归一化频率方向
Figure 5. MTF curves of fractal array with different outer layer rotational angles: (a) Normalized frequency along fx - axis; (b) normalized frequency along fy - axis.
图 6 实际截止频率随外层旋转角的变化曲线
Figure 6. The curve of the practical frequency with outer layer rotational angles.
图 7 N = 3, N = 9阵列结构
Figure 7. N = 3, N = 9 array configuration.
图 8 3种阵列的MTF曲线(F = 22.46%) (a)沿fx归一化频率方向; (b)沿fy归一化频率方向
Figure 8. MTF curves of three kinds of array configuration (F = 22.46%): (a) Normalized frequency along fx - axis; (b) normalized frequency along fy - axis.
表 1 分形阵列在不同填充因子下的特性指标
Table 1. Characteristics of fractal array with different fill factors.
${d_0}$ 0.5 0.6 0.7 0.8 0.9 1.0 填充因子 0.0952 0.1219 0.1485 0.1747 0.2001 0.2246 实际截止频率 0.4518 0.6284 0.6350 0.6382 0.6382 0.6382 中频特性 0.0560 0.0632 0.0507 0.0726 0.0683 0.0632 
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表 2 3种阵列的特性指数
Table 2. Characteristics of three kinds of array configuration.
阵列结构 N = 3 N = 9 N = 18 实际截止频率 0.2778 0.2778 0.6382 中频特性 0.1515 0.0571 0.0632
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[1] 吴泉英 2006 博士学位论文 (苏州: 苏州大学)
Wu Q Y 2006 Ph. D. Dissertation (Suzhou: Suzhou University) (in Chinese)
[2] 陈海亭, 江月松, 钟宇 2005 光学学报 25 1616
Google Scholar
Chen H T, Jiang Y S, Zhong Y 2005 Acta Opt. Sin. 25 1616
Google Scholar
[3] 赵娟, 王大勇, 张亚新, 耿则勋, 陶世荃 2009 中国激光 36 934
Zao J, Wang D Y, Zhang Y X, Geng Z X, Tao S K 2009 Chin. J. Lasers 36 934
[4] 刘丽, 江月松 2013 综合孔径成像原理与应用 (北京: 国防工业出版社) 第48−54页
Liu L, Jiang Y S 2013 Principle and Application of Synthetic Aperture Imaging (Beijing: National Defense Industry Press) pp48−54 (in Chinese)
[5] 苏显渝, 李继陶 信息光学(北京: 科学出版社)第20−26页
Su X Y, Li J T 1999 Information Opitics (Beijing: Science Press) pp20−26 (in Chinese)
[6] Meinel A B 1970 Appl. Opt. 9 2501
Google Scholar
[7] Chung S, Spie M 2004 Opt. Eng. 43 2156
Google Scholar
[8] Fiete R D 2002 Opt. Eng. 41 1957
Google Scholar
[9] Zhou C, Wang Z 2018 Opt. Eng. 26 6973
[10] 易红伟, 李英才, 樊超 2007 光子学报 36 2062
Yi H W, Li Y C, Fan C 2007 Acta. Photonica Sin. 36 2062
[11] Miller N J, Dierking M P, Duncan B D 2007 Appl. Opt. 46 5933
Google Scholar
[12] Golay M J E 1971 J. Opt. Soc. Am. 61 272
Google Scholar
[13] Cornwell T J 1988 IEEE Trans. Antennas Propag. 36 1165
Google Scholar
[14] Cassaing F, Mugnier L M 2018 Opt. Lett. 43 4555
[15] Tcherniavski I, Kahrizi M 2005 Opt. Lett. 44 103201
[16] 钱霖, 吴泉英, 吴峰, 沈为民 2005 光学学报 25 1030
Google Scholar
Qian L, Wu Q Y, Wu F, Shen W N 2005 Acta Opt. Sin. 25 1030
Google Scholar
[17] 韩骥, 王大勇, 刘汉承, 伏西洋, 郭红锋, 陶世荃 2007 光电子·激光 18 649
Han J, Wang D Y, Liu H C, Fu X Y, Guo H F, Tao S K 2007 Optronics Lasers 18 649
[18] Liu L, Jiang Y S, Wang H Y, He Y T 2011 Opt. Eng. 50 53202
Google Scholar
[19] 刘丽, 江月松, 王长伟 2009 光学学报 29 2774
Liu L, Jiang Y S, Wang C W 2009 Acta Opt. Sin. 29 2774
[20] 刘政, 王胜千, 饶长辉 2012 61 039501
Google Scholar
Liu Z, Wang S Q, Rao C H 2012 Acta Phys. Sin. 61 039501
Google Scholar
[21] 龙伟军, 王治乐, 周彦平 2004 光学学报 24 1009
Google Scholar
Long H W, Wang L Z, Zhou Y P 2004 Acta Opt. Sin. 24 1009
Google Scholar
[22] 李兰芳 2004 博士学位论文 (南京: 南京理工大学)
Liu L F 2004 Ph. D. Dissertatio (Nanjing: Nanjing University of Science and Technology) (in Chinese)
[23] 刘肖尧, 梁忠诚, 郝未倩, 赵瑞, 孔梅梅, 陈陶, 张月 2019 光学学报 39 0811003
Liu X Y, Liang Z C, Hao W Q, Zhao R, Kong M M, Chen T, Zhang Y 2019 Acta Opt. Sin. 39 0811003
[24] 周程灏, 王治乐, 朱峰 2017 中国光学 10 25
Zhou C H, Wang Z L, Zhu F 2017 Chin. Opt. 10 25
[25] 朱华, 姬翠翠 2011分形理论及其应用 (北京: 科学出版社) 第10−16页
Zhu H, Ji C C 2011 Fractal Theory and Application (Beijing: Science Press) pp10−16 (in Chinese)
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