中国物理B ›› 2009, Vol. 18 ›› Issue (9): 3893-3899.doi: 10.1088/1674-1056/18/9/046

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A quasi-discrete Hankel transform for nonlinear beam propagation

胡勇华1, 陈列尊2, 王友文2, 游开明3, 文双春4   

  1. (1)Key Laboratory of Micro/Nano Optoelectronic Devices of Ministry of Education, School of Computer and Communication, Hunan University, Changsha 410082, China; (2)Key Laboratory of Micro/Nano Optoelectronic Devices of Ministry of Education, School of Computer and Communication, Hunan University, Changsha 410082, China;Department of Physics and Electronic Information Science, Hengyang Normal University, Hengyang 4; (3)School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China;Department of Physics and Electronic Information Science, Hengyang Normal University, Hengyang 421008, China; (4)School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China;Key Laboratory of Micro/Nano Optoelectronic Devices of Ministry of Education, School of Computer and Communication, Hunan University, Changsha 410082, China
  • 收稿日期:2008-12-16 修回日期:2009-01-08 出版日期:2009-09-20 发布日期:2009-09-20
  • 基金资助:
    Project partially supported by the National Natural Science Foundation of China (Grant Nos 10674045 and 60538010), and the National Natural Science Foundation of Hunan Province, China (Grant No 08jj3001).

A quasi-discrete Hankel transform for nonlinear beam propagation

You Kai-Ming(游开明)a)c), Wen Shuang-Chun(文双春)a)b)†, Chen Lie-Zun(陈列尊)b)c), Wang You-Wen(王友文)b)c), and Hu Yong-Hua(胡勇华)b)   

  1. a School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China; b Key Laboratory of Micro/Nano Optoelectronic Devices of Ministry of Education, School of Computer and Communication, Hunan University, Changsha 410082, China; c Department of Physics and Electronic Information Science, Hengyang Normal University, Hengyang 421008, China
  • Received:2008-12-16 Revised:2009-01-08 Online:2009-09-20 Published:2009-09-20
  • Supported by:
    Project partially supported by the National Natural Science Foundation of China (Grant Nos 10674045 and 60538010), and the National Natural Science Foundation of Hunan Province, China (Grant No 08jj3001).

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摘要： We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion (DQDHT) in this paper. By making use of the property that the zero-order Bessel function derivative J' 0(0)=0, the DQDHT can be used to calculate the values on the symmetry axis directly. In addition, except for the truncated treatment of the input function, no other approximation is made, thus the DQDHT satisfies the discrete Parseval theorem for energy conservation, implying that it has a high numerical accuracy. Further, we have performed several numerical tests. The test results show that the DQDHT has a very high numerical accuracy and keeps energy conservation even after thousands of times of repeating the transform either in a spatial domain or in a frequency domain. Finally, as an example, we have applied the DQDHT to the nonlinear propagation of a Gaussian beam through a Kerr medium system with cylindrical symmetry. The calculated results are found to be in excellent agreement with those based on the conventional 2D-FFT algorithm, while the simulation based on the proposed DQDHT takes much less computing time.

Abstract: We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion (DQDHT) in this paper. By making use of the property that the zero-order Bessel function derivative $J'_0(0)=0$, the DQDHT can be used to calculate the values on the symmetry axis directly. In addition, except for the truncated treatment of the input function, no other approximation is made, thus the DQDHT satisfies the discrete Parseval theorem for energy conservation, implying that it has a high numerical accuracy. Further, we have performed several numerical tests. The test results show that the DQDHT has a very high numerical accuracy and keeps energy conservation even after thousands of times of repeating the transform either in a spatial domain or in a frequency domain. Finally, as an example, we have applied the DQDHT to the nonlinear propagation of a Gaussian beam through a Kerr medium system with cylindrical symmetry. The calculated results are found to be in excellent agreement with those based on the conventional 2D-FFT algorithm, while the simulation based on the proposed DQDHT takes much less computing time.

Key words: Hankel transform, Kerr medium, nonlinear propagation

中图分类号:  (Phase conjugation; photorefractive and Kerr effects)

  • 42.65.Hw
02.30.Gp (Special functions) 02.30.Uu (Integral transforms) 42.25.Bs (Wave propagation, transmission and absorption)