波面重构中非圆域Zernike正交基底构造方法
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摘要
针对非圆域波面拟合中Zernike多项式失去正交特性、拟合系数交叉耦合的问题,提出非圆域Zernike正交基底函数构造方法。以圆Zernike为基底,采用Gram-Schimdt正交组构造方法,线性表出单位正交基底。通过构造不同遮光比环形光阑下的正交基底与环Zernike多项式进行比较,验证了此方法的正确性。然后采用圆Zernike多项式和构造的新基底对矩形光阑下的波面进行了拟合,从拟合残余误差、各项基底系数的稳定性、传递矩阵的条件数等分析,结果表明针对特定的非圆域构造的新基底可靠性和抗扰动能力优于圆Zernike多项式。此方法不需要具体求出基底的解析表达式,不同非圆域仅是正交化系数矩阵发生改变,为非圆域正交基底构造提供了一种新途径。
To solve the problem of Zernike circle polynomials lost it's orthogonality and fitting coefficients cross coupling when reconstruct wavefront in non-circular domain.A non-circular orthogonal Zernike basis construction method is proposed.In the method,the circular Zernike is used as basis and the Gram-Schimdt orthogonal group construction method is adopted.The correctness of the method is verified by comparing Zernike annular polynomials with new basis which construct for different obscuration ratio.For a wavefront data in square aperture,the results fitted with Zernike circle polynomials and new basis are compared in terms of fitting accuracy,stability and anti-perturbation capacity.The experimental results show that,in the wavefront fitting of an interferogram with non-circular aperture,new basis demonstrate better fitting stability and anti-perturbation capacity.This method doesn't need to find out the analytical expression and only changes the orthogonal coefficient matrix in different non circular domains,provides a new way for the construction of the orthogonal basis of non-circular domains.
To solve the problem of Zernike circle polynomials lost it's orthogonality and fitting coefficients cross coupling when reconstruct wavefront in non-circular domain.A non-circular orthogonal Zernike basis construction method is proposed.In the method,the circular Zernike is used as basis and the Gram-Schimdt orthogonal group construction method is adopted.The correctness of the method is verified by comparing Zernike annular polynomials with new basis which construct for different obscuration ratio.For a wavefront data in square aperture,the results fitted with Zernike circle polynomials and new basis are compared in terms of fitting accuracy,stability and anti-perturbation capacity.The experimental results show that,in the wavefront fitting of an interferogram with non-circular aperture,new basis demonstrate better fitting stability and anti-perturbation capacity.This method doesn't need to find out the analytical expression and only changes the orthogonal coefficient matrix in different non circular domains,provides a new way for the construction of the orthogonal basis of non-circular domains.
引文
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[6]侯溪,伍凡,杨力.基于Zernike环多项式的环孔径波面拟合方法[J].红外与激光工程,2006,35(5):23—526.HOU Xi,WU Fan,YANG Li.Wavefront fitting with Zernike annular polynomials for circular and annular pupils[J].Infrared and Laser Engineering,2006,35(5):23—526.
[2]刘克,李艳秋,刘景峰.带有分割遮拦环形干涉图的波面拟合[J].红外与激光工程,2008,37:778—784.LIU Ke,LI Yanqiu,LIU Jingfeng.Wavefront fitting method for annular interferogram with obscurations[J].Infrared and Laser Engineering,2008,37:778—784.
[3]TATIAN B.Aberration balancing in rotationally symmetric lenses[J].Journal of the Optical Society of America,1974,64:1083—1091.
[4]MAHAJAN V N.Zernike annular polynomials for imaging systems with annular pupils[J].Journal of the Optical Society of America,1981,71:75—85.
[5]李萌阳,李大海,王琼华.用方形区域内的标准正交多项式重构波前[J].中国激光,2012,39:153—160.LI Mengyang,LI Dahai,WANG Qionghua.Wavefront reconstruction with orthonormal polynomials in a square area[J].Chinese Journal of Lasers,2012,39:153—160.
[6]侯溪,伍凡,杨力.基于Zernike环多项式的环孔径波面拟合方法[J].红外与激光工程,2006,35(5):23—526.HOU Xi,WU Fan,YANG Li.Wavefront fitting with Zernike annular polynomials for circular and annular pupils[J].Infrared and Laser Engineering,2006,35(5):23—526.