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Analysis of photonic crystal transmission properties by the precise integration time domain

Yang Hong-Wei Meng Shan-Shan Gao Ran-Ran Peng Shuo

Analysis of photonic crystal transmission properties by the precise integration time domain

Yang Hong-Wei, Meng Shan-Shan, Gao Ran-Ran, Peng Shuo
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  • Photonic crystals are materials patterned with a periodicity in the dielectric constant, which can create a range of forbidden frequencies called as a photonic band gap. The photonic band gap of the photonic crystal indicates its primary property, which is the basis of its application. In recent years, photonic crystals have been widely used to design optical waveguides, filters, microwave circuits and other functional devices. Therefore, the study on the transmission properties in photonic crystal is significantly important for constructing the optical devices. The finite difference time domain (FDTD) is a very useful numerical simulation technique for solving the transmission properties of the photonic crystals. However, as the FDTD method is based on the second order central difference algorithm, its accuracy is relatively low and the Courant stability condition must be satisfied when this method is used, which may restrict its application. To increase the accuracy and the stability, considerable scientific interest has been attracted to explore the schemes to improve the performance of the FDTD. The fourth order Ronge-Kutta (RK4) method has been applied to the FDTD method, which improves the accuracy and eliminates the influence of accumulation errors of the results, but the stability remains very poor if the time step is large. An effective time domain algorithm based on the high precision integration is proposed to solve the transmission properties of photonic crystals. The Yee cell differential technique is used to discretize the first order Maxwell equations in the spatial domain. Then the discretized Maxwell equations with the absorption boundary conditions and the expression of excitation source are rewritten in the standard form of the first order ordinary differential equation. According to the precise division of the time step and the additional theorem of exponential matrix, the high precision integration is used to obtain the homogeneous solution. To obtain the discretized electric and magnetic fields, the particular solution must be solved based on the excitation and then be added to the homogeneous solution. The transmission properties of photonic crystals are obtained by the Fourier transform. Practical calculation of photonic crystals is carried out by the precise integration time domain, and the accuracy and the stability are compared with those from the FDTD and the RK4 methods. The numerical results show that the precise integration time domain has a higher calculation precision and overcomes the restriction of stability conditions on the time step, which provides an effective analytical method of studying the transmission properties of photonic crystals.
      Corresponding author: Yang Hong-Wei, yanghongwei@bjut.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11172008, 11272020).
    [1]

    Sun H T, Song Z X, Weng Z K, Wang D P, Jiang Y S, Yu Y 2011 Acta Photon. Sin. 40 1

    [2]

    Su A 2009 M. S. Dissertation (Nanning: Guangxi University) (in Chinese) [苏安 2009 硕士学位论文 (南宁: 广西大学)]

    [3]

    Lin M 2010 Ph. D. Dissertation (Chengdu: University of Electronic Science and Technology of China) (in Chinese) [林密 2010 博士学位论文 (成都: 电子科技大学)]

    [4]

    Wang Q 2009 Ph. D. Dissertation (Nanjing: Southeast University) (in Chinese) [王琼 2009 博士学位论文 (南京: 东南大学)]

    [5]

    Liu P 2012 M. S. Dissertation (Tianjin: Tianjin University of Technology) (in Chinese) [刘佩 2012 硕士学位论文 (天津: 天津理工大学)]

    [6]

    Ding T 2011 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese) [丁涛 2011 博士学位论文 (北京: 中国科学院大学)]

    [7]

    Zhao N S, Guan J M 2014 Laser Optoelectron. Prog. 51 042302 (in Chinese) [赵年顺, 官骏鸣 2014 激光与光电子学进展 51 042302]

    [8]

    Chen H M, Meng Q 2011 Acta Phys. Sin. 60 014202 (in Chinese) [陈鹤鸣, 孟晴 2011 物理学报 60 014202]

    [9]

    Pan W, Yu H J, Zhang X G, Xi L X 2012 Acta Phys. Sin. 61 034209 (in Chinese) [潘伟, 余和军, 张晓光, 席丽霞 2012 物理学报 61 034209]

    [10]

    Cheng J, Wang W Y, Xiong Y B, Tan W J 2012 Electro-Opt. Technol. Appl. 27 34 (in Chinese) [承军, 王玮钰, 熊耀兵, 谭文疆 2012 光电技术应用 27 34]

    [11]

    Zhang H J 2007 J. Ankang Univ. 19 74 (in Chinese) [张洪江 2007 安康学院学报 19 74]

    [12]

    Song Q, Gao J S, Wang X Y, Wang T T, Chen H, Zheng X M, Shen Z F, Ling W 2006 Opt. Instrum. 28 37 (in Chinese) [宋琦, 高劲松, 王笑夷, 王彤彤, 陈红, 郑宣鸣, 申振峰, 凌伟 2006 光学仪器 28 37]

    [13]

    Namiki T 1999 IEEE Trans. Microw. Theory Tech. 47 2003

    [14]

    Young J L, Gaitonde D, Shang J S 1997 IEEE Trans. Antennas Propag. 45 1573

    [15]

    Zhong W X 2002 Dual System in Applied Mechanics (1st Edition) (Beijing: Science Press) pp4-10 (in Chinese) [钟万勰 2002 应用力学对偶体系 (1 版) (北京: 科学出版社) 第4-10页]

    [16]

    Liu P J, Gu L C, Ren J H 2008 J. Anhui Univ. Nat. Sci. Ed. 32 61 (in Chinese) [刘沛津, 谷立臣, 任继红 2008 安徽大学学报(自然科学版) 32 61]

    [17]

    Ge D B, Yan Y B 2011 Finite Difference Time Domain Method for Electromagnetic Waves (3rd Edition) (Xi'an: Xidian University Press) pp9-21 (in Chinese) [葛德彪, 闫玉波 2011 电磁波时域有限差分方法 (3版) (西安: 西安电子科技大学出版社) 第9-21页]

    [18]

    Chen J, Lee J H, Liu Q H 2009 IEEE Trans. Antennas Propag. 57 3223

    [19]

    Liu Q H 1997 Microw. Opt. Technol. Lett. 14 134

    [20]

    Lin J, Shen W, Williams F W 1995 Comput. Struct. 56 113

    [21]

    Liu W S, Li J L 1981 Sampling Technology Principle and Application (1st Ed.) (Beijing: Science Press) pp10-360 (in Chinese) [刘文生, 李锦林 1981 取样技术原理与应用 (第1版) (北京: 科学出版社) 第10-360页]

  • [1]

    Sun H T, Song Z X, Weng Z K, Wang D P, Jiang Y S, Yu Y 2011 Acta Photon. Sin. 40 1

    [2]

    Su A 2009 M. S. Dissertation (Nanning: Guangxi University) (in Chinese) [苏安 2009 硕士学位论文 (南宁: 广西大学)]

    [3]

    Lin M 2010 Ph. D. Dissertation (Chengdu: University of Electronic Science and Technology of China) (in Chinese) [林密 2010 博士学位论文 (成都: 电子科技大学)]

    [4]

    Wang Q 2009 Ph. D. Dissertation (Nanjing: Southeast University) (in Chinese) [王琼 2009 博士学位论文 (南京: 东南大学)]

    [5]

    Liu P 2012 M. S. Dissertation (Tianjin: Tianjin University of Technology) (in Chinese) [刘佩 2012 硕士学位论文 (天津: 天津理工大学)]

    [6]

    Ding T 2011 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese) [丁涛 2011 博士学位论文 (北京: 中国科学院大学)]

    [7]

    Zhao N S, Guan J M 2014 Laser Optoelectron. Prog. 51 042302 (in Chinese) [赵年顺, 官骏鸣 2014 激光与光电子学进展 51 042302]

    [8]

    Chen H M, Meng Q 2011 Acta Phys. Sin. 60 014202 (in Chinese) [陈鹤鸣, 孟晴 2011 物理学报 60 014202]

    [9]

    Pan W, Yu H J, Zhang X G, Xi L X 2012 Acta Phys. Sin. 61 034209 (in Chinese) [潘伟, 余和军, 张晓光, 席丽霞 2012 物理学报 61 034209]

    [10]

    Cheng J, Wang W Y, Xiong Y B, Tan W J 2012 Electro-Opt. Technol. Appl. 27 34 (in Chinese) [承军, 王玮钰, 熊耀兵, 谭文疆 2012 光电技术应用 27 34]

    [11]

    Zhang H J 2007 J. Ankang Univ. 19 74 (in Chinese) [张洪江 2007 安康学院学报 19 74]

    [12]

    Song Q, Gao J S, Wang X Y, Wang T T, Chen H, Zheng X M, Shen Z F, Ling W 2006 Opt. Instrum. 28 37 (in Chinese) [宋琦, 高劲松, 王笑夷, 王彤彤, 陈红, 郑宣鸣, 申振峰, 凌伟 2006 光学仪器 28 37]

    [13]

    Namiki T 1999 IEEE Trans. Microw. Theory Tech. 47 2003

    [14]

    Young J L, Gaitonde D, Shang J S 1997 IEEE Trans. Antennas Propag. 45 1573

    [15]

    Zhong W X 2002 Dual System in Applied Mechanics (1st Edition) (Beijing: Science Press) pp4-10 (in Chinese) [钟万勰 2002 应用力学对偶体系 (1 版) (北京: 科学出版社) 第4-10页]

    [16]

    Liu P J, Gu L C, Ren J H 2008 J. Anhui Univ. Nat. Sci. Ed. 32 61 (in Chinese) [刘沛津, 谷立臣, 任继红 2008 安徽大学学报(自然科学版) 32 61]

    [17]

    Ge D B, Yan Y B 2011 Finite Difference Time Domain Method for Electromagnetic Waves (3rd Edition) (Xi'an: Xidian University Press) pp9-21 (in Chinese) [葛德彪, 闫玉波 2011 电磁波时域有限差分方法 (3版) (西安: 西安电子科技大学出版社) 第9-21页]

    [18]

    Chen J, Lee J H, Liu Q H 2009 IEEE Trans. Antennas Propag. 57 3223

    [19]

    Liu Q H 1997 Microw. Opt. Technol. Lett. 14 134

    [20]

    Lin J, Shen W, Williams F W 1995 Comput. Struct. 56 113

    [21]

    Liu W S, Li J L 1981 Sampling Technology Principle and Application (1st Ed.) (Beijing: Science Press) pp10-360 (in Chinese) [刘文生, 李锦林 1981 取样技术原理与应用 (第1版) (北京: 科学出版社) 第10-360页]

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  • Received Date:  10 November 2016
  • Accepted Date:  16 January 2017
  • Published Online:  20 April 2017

Analysis of photonic crystal transmission properties by the precise integration time domain

    Corresponding author: Yang Hong-Wei, yanghongwei@bjut.edu.cn
  • 1. College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 11172008, 11272020).

Abstract: Photonic crystals are materials patterned with a periodicity in the dielectric constant, which can create a range of forbidden frequencies called as a photonic band gap. The photonic band gap of the photonic crystal indicates its primary property, which is the basis of its application. In recent years, photonic crystals have been widely used to design optical waveguides, filters, microwave circuits and other functional devices. Therefore, the study on the transmission properties in photonic crystal is significantly important for constructing the optical devices. The finite difference time domain (FDTD) is a very useful numerical simulation technique for solving the transmission properties of the photonic crystals. However, as the FDTD method is based on the second order central difference algorithm, its accuracy is relatively low and the Courant stability condition must be satisfied when this method is used, which may restrict its application. To increase the accuracy and the stability, considerable scientific interest has been attracted to explore the schemes to improve the performance of the FDTD. The fourth order Ronge-Kutta (RK4) method has been applied to the FDTD method, which improves the accuracy and eliminates the influence of accumulation errors of the results, but the stability remains very poor if the time step is large. An effective time domain algorithm based on the high precision integration is proposed to solve the transmission properties of photonic crystals. The Yee cell differential technique is used to discretize the first order Maxwell equations in the spatial domain. Then the discretized Maxwell equations with the absorption boundary conditions and the expression of excitation source are rewritten in the standard form of the first order ordinary differential equation. According to the precise division of the time step and the additional theorem of exponential matrix, the high precision integration is used to obtain the homogeneous solution. To obtain the discretized electric and magnetic fields, the particular solution must be solved based on the excitation and then be added to the homogeneous solution. The transmission properties of photonic crystals are obtained by the Fourier transform. Practical calculation of photonic crystals is carried out by the precise integration time domain, and the accuracy and the stability are compared with those from the FDTD and the RK4 methods. The numerical results show that the precise integration time domain has a higher calculation precision and overcomes the restriction of stability conditions on the time step, which provides an effective analytical method of studying the transmission properties of photonic crystals.

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