基于瀑布型多重网格加速的复指数波前复原算法

赵旺; 董理治; 杨平; 王帅; 许冰

doi:10.7498/aps.68.20182137

摘要

激光在大气中传输时, 由于强湍流或长传输距离的影响, 畸变波前中出现由相位起点组成的不连续相位，现有波前复原算法不能有效复原不连续相位, 使得自适应光学系统校正效果下降甚至失效. 本文分析了最小二乘波前复原算法不能复原相位奇点的原因, 提出了基于瀑布型多重网格加速的复指数波前复原算法, 给出了复指数波前复原算法中迭代计算、降采样、插值计算的实现方式. 研究了该方法对不连续相位和随机连续相位的复原能力, 数值分析了采用复指数波前复原算法的自适应光学系统对大气湍流像差的校正效果. 仿真结果表明, 同等复原精度下, 相比直接迭代过程, 该方法所需浮点乘数目减少了近2个数量级, 且随着夏克-哈特曼波前传感器子孔径数目增加, 其在计算量上的优势更加明显. Rytov方差较大时, 相比直接斜率法, 自适应光学系统采用复指数波前复原算法后校正光束Strehl比提升1倍.

关键词:

Abstract

When laser beam propagates through the turbulent atmosphere, there are branch points in wavefront, which are caused by deep turbulence or long propagation distance. Conventional least-square reconstruction algorithms cannot restore the discontinuous wavefront, which severely limits correction performance of an adaptive optics system. If the incoming wavefront contains a branch cut, there is $ {\rm{2}}n{\text{π}} $ difference between the measured phase difference and the principle phase difference, which is the reason why conventional least-square reconstruction algorithms cannot reconstruct wavefront with branch points. The complex exponential reconstructor is developed to restore the discontinuous wavefront with phase difference replaced by complex exponents. However, thousands of iterations are required by the complex exponential reconstructor before converging to an acceptable solution. In order to speed up the iterative calculation, the cascadic multigrid method (CMG) is introduced in the process of wavefront reconstruction. The proposed method can be used to restore discontinuous wavefront with lower residual error similar to those reconstructed by the direct iteration. The number of float point multiplications required by the CMG method is nearly 2 orders of magnitude lower than that required by the direct iteration. The acceleration of the CMG method increases with the number of subapertures increasing. The performance of CMG method to recover continuous wavefront is also investigated and compared with conventional wavefront reconstruction algorithm based on successive over-relaxation. It is shown that the CMG method has good capability for wavefront reconstruction with high precision and low computation cost no matter whether it is applied to discontinuous or continuous wavefront. Furthermore, the CMG method is used in the adaptive optics for correcting the turbulence aberration. The direct slope wavefront reconstruction algorithm based on the assumption that the measured slope and the control voltage satisfy the linear relationship cannot restore the wavefront with branch points. As a result, the adaptive optics system with the CMG method doubles the correction quality evaluated by the Strehl ratio compared with that with the direct slope wavefront reconstruction algorithm.

Keywords:

作者及机构信息

1.
中国科学院自适应光学重点实验室, 成都　610209

2.
中国科学院光电技术研究所, 成都　610209

3.
中国科学院大学, 北京　100049

通信作者: 王帅, wangshuai@ioe.ac.cn

基金项目: 国家自然科学基金(批准号: 61875203)、国家自然科学基金青年科学基金(批准号: 11704382, 61805251)和国家自然科学基金国际(地区)合作与交流项目(批准号: 1171101412)资助的课题.

Authors and contacts

1.
Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu 610209, China

2.
Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China

3.
University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author: Wang Shuai, wangshuai@ioe.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61875203), the Young Scientists Fund of the National Natural Science Foundation of China (Grant Nos. 11704382, 61805251), and the International (Regional) Cooperation and Exchange Program of the National Natural Science Foundation of China (Grant No. 1171101412).

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参考文献

[1]	Fried D L, Vaughn J L 1992 Appl. Opt. 31 2865 Google Scholar
[2]	Fried D L 1998 JOSA A 15 2759 Google Scholar
[3]	Primmerman C A, Price T R, Humphreys R A, et al. 1995 Appl. Opt. 34 2081 Google Scholar
[4]	Lukin V P, Fortes B V 2002 Appl. Opt. 41 5616 Google Scholar
[5]	Steinbock M J, Hyde M W, Schmidt J D 2014 Appl. Opt. 53 3821 Google Scholar
[6]	Le B E, Wild W J, Kibblewhite E J 1998 Opt. Lett. 23 10 Google Scholar
[7]	Fried D L 2001 Opt. Commun. 200 43 Google Scholar
[8]	Barchers J D, Fried D L, Link D J 2002 Appl. Opt. 41 1012 Google Scholar
[9]	Aubailly M, Vorontsov M A 2012 JOSA A 29 1707
[10]	Yazdani R, Fallah H 2017 Appl. Opt. 56 1358 Google Scholar
[11]	Goodman J W ( translated by Qin K C, L P S, Chen J B, Cao Q Z) 2013 Introduction to Fourier Optics (3rd Ed.) (Beijing: Publishing House of Electronics Industry) p77
[12]	Hudgin R H 1977 JOSA 67 378 Google Scholar
[13]	Hudgin R H 1977 JOSA 67 375 Google Scholar
[14]	Bornemann F A, Deuflhard P 1996 Numerische Mathematik 75 135 Google Scholar
[15]	Venema T M, Schmidt J D 2008 Opt. Express 16 6985 Google Scholar
[16]	Steinbock M J, Schmidt J D, Hyde M W 2012 Aerospace Conference Big Sky, MT, USA, 3-10 March, 2012, pp1-13
[17]	Roddier N A 1990 Opt. Eng. 29 1174 Google Scholar
[18]	Southwell W H 1980 JOSA 70 998 Google Scholar
[19]	Jr J A F, Morris J R, Feit M D 1976 Appl. Phys. 10 129
[20]	蔡冬梅, 王昆, 贾鹏, 王东, 刘建霞 2014 物理学报 63 104217 Google Scholar Cai D M, Wang K, Jia P, Wang D, Liu J X 2014 Acta Phys. Sin. 63 104217 Google Scholar
[21]	程生毅, 陈善球, 董理治, 王帅, 杨平, 敖明武, 许冰 2015 物理学报 64 094207 Google Scholar Cheng Y C, Shan Q C, Dong L Z, Wang S, Yang P, Ao M W, Xu B 2015 Acta Phys. Sin. 64 094207 Google Scholar
[22]	Fan C, Wang Y, Gong Z 2004 Appl. Opt. 43 4334 Google Scholar

施引文献

图 1 测量相位差和重建相位点的关系

Fig. 1. The relationship between phase differences and phase.

下载: 全尺寸图片幻灯片

图 2 瀑布型多重网格法示意图　(a)网格结构; (b)CMG算法计算流程

Fig. 2. Schematic of the CMG method: (a) Structure of network layers; (b) calculation process.

下载: 全尺寸图片幻灯片

图 3 CMG算法降采样过程　(a)细网格上光场; (b)粗网格上光场

Fig. 3. Downsampling process of the CMG method: (a) Data on the fine network; (b) data on the coarse network.

下载: 全尺寸图片幻灯片

图 4 CMG算法插值过程　(a)细网格光场和粗网格光场的关系; (b)待插值数据位于正方形中心; (c), (d)待插值数据位于正方形四边上

Fig. 4. Interpolation process of the CMG method: (a) The relationship between grid points on coarse network and fine network; (b) the new grid point located at the center of the unit square; (c), (d) the new grid point located on the edge of the unit square.

下载: 全尺寸图片幻灯片

图 5 (a)−(d) Phase1, Phase2, Phase3和Phase4二维分布; (e)−(h)最小二乘法波前复原结果; (i)−(l)复指数波前复原算法结果

Fig. 5. (a)−(d) Two-dimensional distribution of Phase1, Phase2, Phase3 and Phase4; (e)−(h) wavefront reconstructed by the least-squares reconstruction algorithm; (i)−(l) wavefront reconstructed by the CER algorithm.

下载: 全尺寸图片幻灯片

图 6 直接迭代和CMG算法波前复原残差　(a)子孔径数目为20 × 20; (b)子孔径数目为40 × 40; (c)子孔径数目为80 × 80

Fig. 6. Wavefront residual error of the direct iteration method and the CMG method, the number of subapertures is (a) 20 × 20; (b) 40 × 40; (c) 80 × 80.

下载: 全尺寸图片幻灯片

图 7 CMG算法和直接迭代过程所需浮点乘运算数目(a)子孔径数目为20 × 20; (b)子孔径数目为40 × 40; (c)子孔径数目为80 × 80

Fig. 7. Float point multiplications required by the CMG method and the process of the direct iteration the number of subapertures is (a) 20 × 20; (b) 40 × 40; (c) 80 × 80.

下载: 全尺寸图片幻灯片

图 8 CMG算法和SOR算法波前复原残差统计结果

Fig. 8. Wavefront residual statistics of the CMG method and SOR method.

下载: 全尺寸图片幻灯片

图 9 变形镜驱动器和哈特曼波前传感器子孔径匹配关系

Fig. 9. Matching relation between actuators of deformable mirror and subapertures of Shack-Hartmann sensor.

下载: 全尺寸图片幻灯片

图 10 不同Rytov方差时, 自适应光学系统校正前后远场光强分布及其峰值Strehl比

Fig. 10. Far field intensity and Strehl ratio of laser beam before and after corrected by the adaptive optics system.

下载: 全尺寸图片幻灯片

图 11 不同Rytov方差时, 自适应光学系统校正光束Strehl比

Fig. 11. Strehl ratio of laser beam after corrected by the adaptive optics system in different Rytov number.

下载: 全尺寸图片幻灯片

表 1 直接迭代和CMG算法波前复原时间(单位: s)

Table 1. Time required by the direct iteration and CMG method (in s).

	子孔径数目20 × 20		子孔径数目40 × 40		子孔径数目80 × 80
	直接迭代	CMG算法	直接迭代	CMG算法	直接迭代	CMG算法
Phase1	4.261	0.081	67.39	0.271	1400	0.920
Phase2	5.112	0.119	77.61	0.319	2134	0.852
Phase3	4.184	0.103	54.56	0.519	1424	1.339
Phase4	1.891	0.097	18.43	0.494	370.8	1.308

下载: 导出CSV

[1]	Fried D L, Vaughn J L 1992 Appl. Opt. 31 2865 Google Scholar
[2]	Fried D L 1998 JOSA A 15 2759 Google Scholar
[3]	Primmerman C A, Price T R, Humphreys R A, et al. 1995 Appl. Opt. 34 2081 Google Scholar
[4]	Lukin V P, Fortes B V 2002 Appl. Opt. 41 5616 Google Scholar
[5]	Steinbock M J, Hyde M W, Schmidt J D 2014 Appl. Opt. 53 3821 Google Scholar
[6]	Le B E, Wild W J, Kibblewhite E J 1998 Opt. Lett. 23 10 Google Scholar
[7]	Fried D L 2001 Opt. Commun. 200 43 Google Scholar
[8]	Barchers J D, Fried D L, Link D J 2002 Appl. Opt. 41 1012 Google Scholar
[9]	Aubailly M, Vorontsov M A 2012 JOSA A 29 1707
[10]	Yazdani R, Fallah H 2017 Appl. Opt. 56 1358 Google Scholar
[11]	Goodman J W ( translated by Qin K C, L P S, Chen J B, Cao Q Z) 2013 Introduction to Fourier Optics (3rd Ed.) (Beijing: Publishing House of Electronics Industry) p77
[12]	Hudgin R H 1977 JOSA 67 378 Google Scholar
[13]	Hudgin R H 1977 JOSA 67 375 Google Scholar
[14]	Bornemann F A, Deuflhard P 1996 Numerische Mathematik 75 135 Google Scholar
[15]	Venema T M, Schmidt J D 2008 Opt. Express 16 6985 Google Scholar
[16]	Steinbock M J, Schmidt J D, Hyde M W 2012 Aerospace Conference Big Sky, MT, USA, 3-10 March, 2012, pp1-13
[17]	Roddier N A 1990 Opt. Eng. 29 1174 Google Scholar
[18]	Southwell W H 1980 JOSA 70 998 Google Scholar
[19]	Jr J A F, Morris J R, Feit M D 1976 Appl. Phys. 10 129
[20]	蔡冬梅, 王昆, 贾鹏, 王东, 刘建霞 2014 物理学报 63 104217 Google Scholar Cai D M, Wang K, Jia P, Wang D, Liu J X 2014 Acta Phys. Sin. 63 104217 Google Scholar
[21]	程生毅, 陈善球, 董理治, 王帅, 杨平, 敖明武, 许冰 2015 物理学报 64 094207 Google Scholar Cheng Y C, Shan Q C, Dong L Z, Wang S, Yang P, Ao M W, Xu B 2015 Acta Phys. Sin. 64 094207 Google Scholar
[22]	Fan C, Wang Y, Gong Z 2004 Appl. Opt. 43 4334 Google Scholar

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