FFT计算菲涅尔衍射相位的跳变与矫正研究
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摘要
采用傅里叶变换算法计算菲涅尔衍射相位时,在相位未解包裹的情况下,接收面上提取的相位分布曲线会出现跳变,如果进行解包裹,必然会导致错误的结果。研究发现用傅里叶变换算法进行衍射计算导致接收面上相位跳变的原因,是因为快速傅里叶变换(FFT)对矩阵标注索引的方式与离散傅里叶变换(DFT)有所区别,从而导致计算结果的相位与真实相位有差异。本文提出在FFT运算前后分别进行一次倒谱的方法矫正这种相位跳变,并仿真利用单次FFT进行二维矩孔的菲涅尔衍射,用2次倒谱矫正接收面上的相位跳变,结果证明了该矫正方法的可行性。
When the phase of Fresnel diffraction is calculated by Fourier transform algorithm,the phase distribution curve of the receiving surface will be hopping when the phase is wrapped,and if it is unwrapped,it will inevitably lead to the wrong result.The reason that the phase jump on the receiving surface is caused by the Fourier transform algorithm is analyzed. It is because the FFT's way of indexing the matrix is different from that of the DFT,which leads to the difference between the phase and the real phase of the calculated result. The phase jump can be corrected by doing fftshift respectively before and after the FFT operation.The Fresnel diffraction integral of the two-dimensional moment holes is simulated by using a single FFT,and the phase jump of the receiving surface is corrected by using two times cepstrum,which proves the feasibility of the proposed method.
When the phase of Fresnel diffraction is calculated by Fourier transform algorithm,the phase distribution curve of the receiving surface will be hopping when the phase is wrapped,and if it is unwrapped,it will inevitably lead to the wrong result.The reason that the phase jump on the receiving surface is caused by the Fourier transform algorithm is analyzed. It is because the FFT's way of indexing the matrix is different from that of the DFT,which leads to the difference between the phase and the real phase of the calculated result. The phase jump can be corrected by doing fftshift respectively before and after the FFT operation.The Fresnel diffraction integral of the two-dimensional moment holes is simulated by using a single FFT,and the phase jump of the receiving surface is corrected by using two times cepstrum,which proves the feasibility of the proposed method.
引文
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[2]李俊昌.角谱衍射公式的快速傅里叶变换计算及在数字全息波面重建中的应用[J].光学学报,2009,29(5):1163-1167.
[3]Yamaguchi I,Kato J,Ohta S,et al.Image formation in phase-shifting digital holography and applications to microscopy[J].Applied Optics,2001,40(34):6177-6186.
[4]Hennelly B,Sheridan J T.Optical image encryption by random shifting in fractional Fourier domains[J].Optics Letters,2003,28(4):269-271.
[5]Singh N,Sinha A.Optical image encryption using fractional Fourier transform and chaos[J].Optics and Lasers in Engineering,2008,46(2):117-123.
[6]Liu Zhengjun,Li She,Liu Wei,et al.Image encryption algorithm by using fractional Fourier transform and pixel scrambling operation based on double random phase encoding[J].Optics and Lasers in Engineering,2013,51(1):8-14.
[7]Bruning J H,Herriott D R,Gallagher J E,et al.Digital wavefront measuring interferometer for testing optical surfaces and lenses[J].Applied Optics,1974,13(11):2693-2703.
[8]Kreis T.Digital holographic interference-phase measurement using the Fourier-transform method[J].Optical Society of America,Journal,A:Optics and Image Science,1986,3(6):847-855.
[9]王华英,于梦杰,刘飞飞.基于快速傅里叶变换的四种相位解包裹算法[J].强激光与粒子束,2013(5):1129-1133.
[10]韦春龙,陈明仪,王之江.基于一维快速傅里叶变换的相位去包裹算法[J].中国激光,1998,25(9):813-816.
[11]Roddier F,Roddier C.Wavefront reconstruction using iterative Fourier transforms[J].Applied Optics,1991,30(11):1325-1327.
[12]刘志强,宋庆和,刘超,等.基于虚拟光波场的菲涅耳衍射计算及应用研究[J].应用光学,2018,39(2):196-199.
[13]Cooley J W,Tukey J W.An algorithm for the machine calculation of complex Fourier series[J].Mathematics of Computation,1965,19(90):297-301.
[14]Gonzalez R C,Woods R E.Digital Image Processing[M].3rd ed.Beijing:Publishing House of Electronics Industry,2011.
[15]吕乃光.傅里叶光学[M].北京:机械工业出版社,2016:87-90.