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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.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:
- adaptive optics /
- wavefront reconstruction /
- cascadic multigrid method /
- branch point
[1] Fried D L, Vaughn J L 1992 Appl. Opt. 31 2865Google Scholar
[2] Fried D L 1998 JOSA A 15 2759Google Scholar
[3] Primmerman C A, Price T R, Humphreys R A, et al. 1995 Appl. Opt. 34 2081Google Scholar
[4] Lukin V P, Fortes B V 2002 Appl. Opt. 41 5616Google Scholar
[5] Steinbock M J, Hyde M W, Schmidt J D 2014 Appl. Opt. 53 3821Google Scholar
[6] Le B E, Wild W J, Kibblewhite E J 1998 Opt. Lett. 23 10Google Scholar
[7] Fried D L 2001 Opt. Commun. 200 43Google Scholar
[8] Barchers J D, Fried D L, Link D J 2002 Appl. Opt. 41 1012Google Scholar
[9] Aubailly M, Vorontsov M A 2012 JOSA A 29 1707
[10] Yazdani R, Fallah H 2017 Appl. Opt. 56 1358Google 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 378Google Scholar
[13] Hudgin R H 1977 JOSA 67 375Google Scholar
[14] Bornemann F A, Deuflhard P 1996 Numerische Mathematik 75 135Google Scholar
[15] Venema T M, Schmidt J D 2008 Opt. Express 16 6985Google 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 1174Google Scholar
[18] Southwell W H 1980 JOSA 70 998Google 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 104217Google Scholar
Cai D M, Wang K, Jia P, Wang D, Liu J X 2014 Acta Phys. Sin. 63 104217Google 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 094207Google Scholar
Cheng Y C, Shan Q C, Dong L Z, Wang S, Yang P, Ao M W, Xu B 2015 Acta Phys. Sin. 64 094207Google Scholar
[22] Fan C, Wang Y, Gong Z 2004 Appl. Opt. 43 4334Google Scholar
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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.
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.
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 -
[1] Fried D L, Vaughn J L 1992 Appl. Opt. 31 2865Google Scholar
[2] Fried D L 1998 JOSA A 15 2759Google Scholar
[3] Primmerman C A, Price T R, Humphreys R A, et al. 1995 Appl. Opt. 34 2081Google Scholar
[4] Lukin V P, Fortes B V 2002 Appl. Opt. 41 5616Google Scholar
[5] Steinbock M J, Hyde M W, Schmidt J D 2014 Appl. Opt. 53 3821Google Scholar
[6] Le B E, Wild W J, Kibblewhite E J 1998 Opt. Lett. 23 10Google Scholar
[7] Fried D L 2001 Opt. Commun. 200 43Google Scholar
[8] Barchers J D, Fried D L, Link D J 2002 Appl. Opt. 41 1012Google Scholar
[9] Aubailly M, Vorontsov M A 2012 JOSA A 29 1707
[10] Yazdani R, Fallah H 2017 Appl. Opt. 56 1358Google 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 378Google Scholar
[13] Hudgin R H 1977 JOSA 67 375Google Scholar
[14] Bornemann F A, Deuflhard P 1996 Numerische Mathematik 75 135Google Scholar
[15] Venema T M, Schmidt J D 2008 Opt. Express 16 6985Google 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 1174Google Scholar
[18] Southwell W H 1980 JOSA 70 998Google 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 104217Google Scholar
Cai D M, Wang K, Jia P, Wang D, Liu J X 2014 Acta Phys. Sin. 63 104217Google 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 094207Google Scholar
Cheng Y C, Shan Q C, Dong L Z, Wang S, Yang P, Ao M W, Xu B 2015 Acta Phys. Sin. 64 094207Google Scholar
[22] Fan C, Wang Y, Gong Z 2004 Appl. Opt. 43 4334Google Scholar
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