Angular spectrum algorithm for Fresnel diffraction integral

doi:10.3969/j.issn.1000-1565.2023.03.003

Abstract

Abstract: Fresnel diffraction can be regarded as a linear space-invariant system, so the Fresnel diffraction integral can be calculated numerically in the spectral domain using the angular spectrum. First, the implementation of the angular spectrum algorithm is explained in detail, including the ideal sampling conditions, the origins and influence range of the wraparound error, and the selection of the calculation window size. Then the Fresnel diffraction field of a Gaussian beam illuminated cosine grating is numerically calculated using the angular spectrum algorithm. A comparison between the numerical solution and the analytical solution shows that the angular spectrum algorithm can achieve very high accuracy when the ideal sampling conditions are met and the calculation window size is correctly selected. Finally, a single-lens coherent imaging is numerically simulated to further demonstrate the implementation of the angular spectrum algorithm.

Key words: Fresnel diffraction, angular spectrum algorithm, ideal sampling condition, wraparound error

CLC Number:

O438

HAN Jiacheng, GUO Wenya, ZHAO Yanan, YANG Lijun, WANG Ying, ZHANG Suheng. Angular spectrum algorithm for Fresnel diffraction integral[J]. Journal of Hebei University(Natural Science Edition), 2023, 43(3): 237-247.

References

[1] SOUTHWELL W H.Validity of the Fresnel approximation in the near field[J].J Opt Soc Am, 1981, 71(1): 7-14.
[2] STEANE A M, RUTT H N.Diffraction calculations in the near field and the validity of the Fresnel approximation[J].J Opt Soc Am A, 1989, 6(12): 1809-1814.DOI: 10.1364/JOSAA.6.001809.
[3] 路彦珍,张腾远,王泽宇,等.基于衍射理论定量研究衍射调节与振幅误差[J].河北大学学报(自然科学版), 2022, 42(1): 16-21.DOI: 10.3969/j.issn.1000-1565.2022.01.003.
[4] ROSEN J, BROOKER G.Digital spatially incoherent Fresnel holography[J].Opt Lett, 2007, 32(8): 912-914.DOI: 10.1364/OL.32.000912.
[5] KUMAR S S, HONG J R.Digital Fresnel reflection holography for high-resolution 3D near-wall flow measurement[J].Opt Express, 2018, 26(10): 12779-12789.DOI: 10.1364/OE.26.012779.
[6] 汤明玉,武梦婷,臧瑞环,等.菲涅耳非相干数字全息大视场研究[J].物理学报, 2019, 68(10): 104204.DOI: 10.7498/aps.68.20182216.
[7] CHEN W, CHEN X D, SHEPPARD C J R.Optical image encryption based on diffractive imaging[J].Opt Lett, 2010, 35(22): 3817-3819.DOI: 10.1364/OL.35.003817.
[8] 王雪光,李明,于娜娜,等.基于空间角度复用和双随机相位的多图像光学加密方法[J].物理学报, 2019, 68(24): 77-86.DOI: 10.7498/aps.68.20191362.
[9] 吴军,王刚,徐刚.结合计算全息与混沌的彩色图像加密方法[J].光学学报, 2021, 41(19): 113-132.DOI: 10.3788/AOS202141.1909001.
[10] TAN X, GU B Y, YANG G Z, et al.Diffractive phase elements for beam shaping: a new design method[J].Appl Opt, 1995, 34(8): 1314-1320.DOI: 10.1364/AO.34.001314.
[11] 王磊,窦健泰,马骏,等.基于叠层衍射成像的二元光学元件检测研究[J].物理学报, 2017, 66(9): 159-167.DOI: 10.7498/aps.66.094201.
[12] 孔德键,孙小燕,董卓林,等.飞秒激光空间光束整形加工技术研究进展[J].激光与光电子学进展, 2020, 57(11): 195-214.DOI: 10.3788/LOP57.111416.
[13] 朱一帆,耿滔.谐振腔内的高质量圆对称艾里光束的产生方法[J].物理学报, 2020, 69(1): 014205.DOI: 10.7498/aps.69.20191088.
[14] 陈聪,李定国.基于快速傅里叶变换的衍射现象的数值仿真[J].大学物理, 2004, 23(9): 46-49, 62.DOI: 10.16854/j.cnki.1000-0712.2004.09.011.
[15] 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.DOI: 10.3788/AOS20092905.1163.
[16] PAPOULIS A.Pulse compression, fiber communications, and diffraction: a unified approach[J].J Opt Soc Am A, 1994, 11(1): 3-13.DOI: 10.1364/JOSAA.11.000003.
[17] BRODZIK A K.On the Fourier transform of finite chirps[J].IEEE Signal Process Lett, 2006, 13(9): 541-544.DOI: 10.1109/LSP.2006.874440.
[18] VOELZ D G, ROGGEMANN M C.Digital simulation of scalar optical diffraction: revisiting chirp function sampling criteria and consequences[J].Appl Opt, 2009, 48(32): 6132-6142.DOI: 10.1364/AO.48.006132.
[19] KELLY D P.Numerical calculation of the Fresnel transform[J].J Opt Soc Am A Opt Image Sci Vis, 2014, 31(4): 755-764.DOI: 10.1364/JOSAA.31.000755.
[20] SHEN F B, WANG A B.Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula[J].Appl Opt, 2006, 45(6): 1102-1110.DOI: 10.1364/AO.45.001102.
[21] MATSUSHIMA K, SHIMOBABA T.Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields[J].Opt Express, 2009, 17(22): 19662-19673.DOI: 10.1364/OE.17.019662.
[22] LIU J P.Controlling the aliasing by zero-padding in the digital calculation of the scalar diffraction[J].J Opt Soc Am A Opt Image Sci Vis, 2012, 29(9): 1956-1964.DOI: 10.1364/JOSAA.29.001956.
[23] MENDLOVIC D, ZALEVSKY Z, KONFORTI N.Computation considerations and fast algorithms for calculating the diffraction integral[J].J Mod Opt, 1997, 44(2): 407-414.DOI: 10.1080/09500349708241880.
[24] MAS D, GARCIA J, FERREIRA C, et al.Fast algorithms for free-space diffraction patterns calculation[J].Opt Commun, 1999, 164(4/5/6): 233-245.DOI: 10.1016/S0030-4018(99)00201-1.
[25] YU X, XIAHUI T, XIONG Q Y, et al.Wide-window angular spectrum method for diffraction propagation in far and near field[J].Opt Lett, 2012, 37(23): 4943-4945. DOI: 10.1364/OL.37.004943.
[26] 崔文乐,韩利琪,霍晓敏,等.菲涅尔衍射积分的单次傅里叶变换算法[J].大学物理, 2019, 38(2): 9-16. DOI: 10.16854/j.cnki.1000-0712.180362.
[27] 黄婉云.傅里叶光学教程[M].北京: 北京师范大学出版社, 1985.
[28] ZHANG S H.Diffraction calculation[EB/OL].(2022-04-18)[2022-07-03] https://github.com/Su-HengZhang/Diffraction-Calculation.
[29] 郎海涛,钱晓凡,戴欣冉,等.相干与非相干照明衍射受限系统成像仿真[J].激光杂志, 2014, 35(4): 17-19. DOI: 10.3969/j.issn.0253-2743.2014.04.009. (