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基于格林函数法的奇型Mathieu-Gaussian光束

吴琼 任志君 杜林岳 胡海华 顾颖 杨朝凤

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基于格林函数法的奇型Mathieu-Gaussian光束

吴琼, 任志君, 杜林岳, 胡海华, 顾颖, 杨朝凤

Odd version Mathieu-Gaussian beam based on Green function

Wu Qiong, Ren Zhi-Jun, Du Lin-Yue, Hu Hai-Hua, Gu Ying, Yang Zhao-Feng
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  • 根据光束传播的独立性和叠加性原理,引入了一组能够产生第一类(2n+2阶)奇型Mathieu-Gaussian光束的虚光源点.利用虚源点技术和格林函数法,计算得到第一类奇型Mathieu-Gaussian光束的严格解析积分表达式.利用该表达式得到了轴上光场分布的积分解析解.以三阶非旁轴修正为例,得到了第一类奇型Mathieu-Gaussian光束保留到三阶非旁轴修正项的轴上光场分布精确解.
    Like the theoretical pattern of non-diffracting Bessel beams, ideal non-diffracting Mathieu beams also carry infinite energy, but cannot be generated as a physically realizable entity. Mathieu-Gaussian beams can be experimentally generated by modulating ideal Mathieu beams with a Gaussian function, and thus they are a kind of pseudo-non-diffracting beams with finite energy and finite transverse extent. The research of Mathieu-Gaussian beam propagating characteristics in free space is of great significance. In order to analytically study the propagation of Mathieu-Gaussian beams, the Mathieu function is expanded into the superposition of a series of Bessel functions in polar coordinates based on the superposition principle of light waves. It means that the Mathieu-Gaussian beam can be converted into accumulation of the infinite terms of the Bessel beams with different orders. According to the properties of the Bessel function, the free-space propagation properties of Mathieu-Gaussian beams can be studied in the circular cylindrical coordinates. Thus, a group of virtual optical sources are introduced to generate the odd Mathieu-Gaussian beams of the first kind, i.e., (2n+2)th-order, which is a family of Mathieu-Gaussian beams. Using the virtual source technique and the Green function, we derive the rigorous integral formula for the odd Mathieu-Gaussian beams of the first kind. Taking for example the first three orders with non-paraxial corrections, the analytical solution of the on-axis field of odd Mathieu-Gaussian beams of the first kind is further obtained from the integral formula. The axial intensity distribution of the odd Mathieu-Gaussian beams of the first kind is numerically calculated by the integral formula. The simulation results show that the calculation results obtained with the paraxial theory and the rigorous integral expressions of non-paraxial Mathieu-Gaussian beams are obviously different when the propagation distance of the odd Mathieu-Gaussian beams of the first kind is small. The calculation results of the two methods are coming closer and closer with the increasing propagation distance. The results indicate that the correct results can be obtained with the paraxial theory when we study the propagation of Mathieu-Gaussian beams in the far-field, but the non-paraxial theory must be used to obtain correct results when we study the propagation of Mathieu-Gaussian beams in the near-field. Owing to the complexity of the non-paraxial theory, it is difficult to obtain the exact analytic solutions of Mathieu-Gaussian beams in the near-field with the classical diffraction theory. Based on the superposition principle of light waves, by introducing the virtual source technique and the Green function, the complex Mathieu-Gaussian function can be expanded into the superposition of a series of simple Bessel functions, and the axial intensity distributions of Mathieu-Gaussian beams in the far-field and the near-field can be studied well. It will also provide a feasible method to study other complex beams propagating in free space.
      通信作者: 吴琼, wuqiong@zjnu.cn
    • 基金项目: 国家自然科学基金(批准号:11674288)和浙江省教育厅科研项目(批准号:Y201534211)资助的课题.
      Corresponding author: Wu Qiong, wuqiong@zjnu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11674288) and the Education Department Program of Zhejiang Province, China (Grant No. Y201534211).
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  • [1]

    Durnin J, Miceli J J Jr, Eberly J H 1987 Phys. Rev. Lett. 58 1499

    [2]

    Durnin J 1987 J. Opt. Soc. Am. A 4 651

    [3]

    Rajesh K B, Anbarasan P M 2008 Chin. Opt. Lett. 6 785

    [4]

    Dudley A, Lavery M, Padgett M, Forbes A 2013 Opt. Photonics News 24 22

    [5]

    Lorenser D, Singe C C, Curatolo A, Sampson D D 2014 Opt. Lett. 39 548

    [6]

    Planchon T A, Gao L, Milkie D E, Davidson M W, Galbraith J A, Galbraith C G, Betzig E 2011 Nat. Methods 8 417

    [7]

    Yan Z, Jureller J E, Sweet J, Guffey M J, Pelton M, Schere N F 2012 Nano Lett. 12 5155

    [8]

    Gutirrez-Vega J C, Iturbe-Castillo M D, Chvez-Cerda S 2000 Opt. Lett. 25 1493

    [9]

    Gutirrez-Vega J C, Iturbe-Castillo M D, Ramreza G A, Tepichna E, Rodrguez-Dagninob R M, Chvez-Cerdac S, Newc G H C 2001 Opt. Commun. 195 35

    [10]

    Chvez-Cerda S, Padgett M J, Allison I, New G H C 2002 J. Opt. B 4 S52

    [11]

    Bandres M A, Gutirrez-Vega J C, Chvez-Cerda S 2004 Opt. Lett. 29 44

    [12]

    Lpez-Mariscal C, Bandres M, Gutirrez-Vega J, Chvez-Cerda S 2005 Opt. Express 13 2364

    [13]

    Chafiq A, Hricha Z, Belafhal A 2006 Opt. Commun. 265 594

    [14]

    Gutirrez-Vega J C, Bandres M A 2007 J. Opt. Soc. Am. A 24 215

    [15]

    Alvarez-Elizondo M B, Rodrguez-Masegosa R,Gutirrez-Vega J C 2008 Opt. Express 16 18770

    [16]

    Deschamps G A 1971 Electron. Lett. 7 684

    [17]

    Felsen L B 1976 J. Opt. Soc. Am. A 66 751

    [18]

    Shin S Y, Felsen L B 1977 J. Opt. Soc. Am. 67 699

    [19]

    Seshadri S R 2002 Opt. Lett. 27 1872

    [20]

    Seshadri S R 2002 Opt. Lett. 27 998

    [21]

    Deng D, Guo Q 2008 Opt. Lett. 33 1225

    [22]

    Deng D, Chen C, Zhao X, Chen B, Peng X, Zheng Y 2014 Opt. Lett. 39 2703

    [23]

    Gutirrez-Vega J C, Rodrguez-Dagnino R M 2003 Am. J. Phys. 71 233

    [24]

    Li D, Wu F T, Xie X X, Sun C 2015 Acta Phys. Sin. 64 014201 (in Chinese)[李冬, 吴逢铁, 谢晓霞, 孙川2015物理学报64 014201]

    [25]

    Li D, Wu F T, Xie X X, Wu M 2014 Acta Phys. Sin. 63 152401 (in Chinese)[李冬, 吴逢铁, 谢晓霞, 吴敏2014物理学报63 152401]

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出版历程
  • 收稿日期:  2017-04-11
  • 修回日期:  2017-05-19
  • 刊出日期:  2017-10-05

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