Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

Citation:

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
PDF
Get Citation

(PLEASE TRANSLATE TO ENGLISH

BY GOOGLE TRANSLATE IF NEEDED.)

  • 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.
      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).
    [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]

  • [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]

  • [1] Zhao Yun-Jin, Tian Meng, Huang Yong-Gang, Wang Xiao-Yun, Yang Hong, Mi Xian-Wu. Renormalization of photon dyadic Green function by finite element method and its applications in the study of spontaneous emission rate and energy level shift. Acta Physica Sinica, 2018, 67(19): 193102. doi: 10.7498/aps.67.20180898
    [2] Deng Xuan-Bing, Deng Dong-Mei, Chen Chi-Dao, Liu Cheng-Yi. Analytical vectorial structure of Airy-Gaussian beam. Acta Physica Sinica, 2013, 62(17): 174201. doi: 10.7498/aps.62.174201
    [3] Zhang Hui-Yun, Liu Meng, Yin Yi-Heng, Wu Zhi-Xin, Shen Duan-Long, Zhang Yu-Ping. Study on scattering properties of the metal wire gating in a THz band based on Green function method. Acta Physica Sinica, 2013, 62(19): 194207. doi: 10.7498/aps.62.194207
    [4] Huo Bing-Zhong, Yuan Jing-He, Shang Ying, Meng Chun-Ning. Spherical superlens analysed by dyadic Green’s function. Acta Physica Sinica, 2010, 59(11): 8178-8183. doi: 10.7498/aps.59.8178
    [5] Yao Wen-Jie, Yu Zhong-Yuan, Liu Yu-Min, Lu Peng-Fei. Effect of wire width on strain distribution and bandgap in quantum-wire nanostructures based on continuum elasticity theory. Acta Physica Sinica, 2009, 58(2): 1185-1189. doi: 10.7498/aps.58.1185
    [6] Chen Xiong-Wen, Shi Zhen-Gang, Chen Bao-Ju, Song Ke-Hui. Asymmetric Kondo resonance splitting transition in T-shaped coupled-quantum-dot devices. Acta Physica Sinica, 2008, 57(4): 2421-2426. doi: 10.7498/aps.57.2421
    [7] Liu Chun-Xiang, Cheng Chuan-Fu, Ren Xiao-Rong, Liu Man, Teng Shu-Yun, Xu Zhi-Zhan. Green's function method of light scattering from random surfaces compares with Kirchhoff's approximation. Acta Physica Sinica, 2004, 53(2): 427-435. doi: 10.7498/aps.53.427
    [8] Guo Ru-Hai, Shi Hong-Yan, Sun Xiu-Dong. The calculation of strain distribution in quantum dots with Green method. Acta Physica Sinica, 2004, 53(10): 3487-3492. doi: 10.7498/aps.53.3487
    [9] WANG CHUN-LEI, QIN ZI-KAI, D. L. LIN. GREEN'S FUNCTION THEORY OF PHASE TRANSITION IN HYDROGEN-BOND FERROELECTRICS (Ⅱ). Acta Physica Sinica, 1990, 39(4): 547-554. doi: 10.7498/aps.39.547
    [10] WANG CHUN-LEI, ZHANG JIN-BO, QIN ZI-KAI, D. L. LIN. GREEN'S FUNCTION THEORY OF PHASE TRANSITIONS IN HYDROGEN-BOND FERROELECTRICS (Ⅰ). Acta Physica Sinica, 1989, 38(11): 1740-1747. doi: 10.7498/aps.38.1740
    [11] XU HONG-HUA. THE INTERACTION PICTURE IN THE CLOSED TIME PATH GREEN'S FUNCTIONS. Acta Physica Sinica, 1985, 34(10): 1359-1362. doi: 10.7498/aps.34.1359
    [12] SU ZHAO-BIN, YU LU, ZHOU GUANG-ZHAO. ON A SET OF COUPLED EQUATIONS FOR THE ORDER PARAMETER -STATISTICAL GREEN'S FUNCTIONS. Acta Physica Sinica, 1984, 33(6): 805-813. doi: 10.7498/aps.33.805
    [13] ZHOU MIN-YAO, CHEN LIAN-PAN, GUO HAN-YING. HORIZONS AND GENERATING FUNCTIONALS FOR THE TEMPERATURE GREEN'S FUNCTIONS. Acta Physica Sinica, 1983, 32(9): 1127-1138. doi: 10.7498/aps.32.1127
    [14] WANG WAI-TONG, LIN ZHONG-HENG, SU ZHAO-BIN, HAO BAI-LIN. THE CLOSED TIME PATH GREEN'S FUNCTIONS AND THE THEORY OF NONLINEAR RESPONSE (Ⅱ). Acta Physica Sinica, 1982, 31(11): 1493-1500. doi: 10.7498/aps.31.1493
    [15] WANG WAI-TONG, LIN ZHONG-HENG, SU ZHAO-BIN, HAO BAI-LIN. THE CLOSED TIME PATH GREEN'S FUNCTIONS AND THE THEORY OF NONLINEAR RESPONSE (Ⅰ). Acta Physica Sinica, 1982, 31(11): 1483-1492. doi: 10.7498/aps.31.1483
    [16] ZHOU GUANG-ZHAO, YU LU, HAO BAI-LIN. TRANSFORMATION PROPERTIES OF THREE SETS OF CLOSED TIME PATH GREEN'S FUNCTIONS. Acta Physica Sinica, 1980, 29(7): 878-888. doi: 10.7498/aps.29.878
    [17] . Acta Physica Sinica, 1966, 22(3): 377-380. doi: 10.7498/aps.22.377
    [18] KU FU-NIEN. ABOUT TENSOR GREEN FUNCTIONS IN WAVEGUIDES. Acta Physica Sinica, 1963, 19(10): 617-626. doi: 10.7498/aps.19.617
    [19] HU NING. THE DERIVATION OF ONE-PARTICAL GREEN'S FUNCTION BY THE METHOD OF DISPERSION RELATION. Acta Physica Sinica, 1962, 18(10): 509-513. doi: 10.7498/aps.18.509
    [20] LIN WEI-GANG. APPLICATIONS OF GREEN'S FUNCTIONS IN CALCULATING THE MUTUAL CAPACITANCE BETWEEN SMALL BODIES. Acta Physica Sinica, 1959, 15(1): 13-24. doi: 10.7498/aps.15.13
Metrics
  • Abstract views:  4393
  • PDF Downloads:  214
  • Cited By: 0
Publishing process
  • Received Date:  11 April 2017
  • Accepted Date:  19 May 2017
  • Published Online:  05 October 2017

/

返回文章
返回