耦合广义非线性薛定谔方程的相互作用表象龙格库塔算法及其误差分析

李磐; 时雷; 毛庆和

doi:10.7498/aps.62.154205

摘要

本文通过表象变换, 将耦合广义非线性薛定谔方程 (C-GNLSE) 变换成相互作用表象中的向量方程, 再利用向量形式的4阶龙格-库塔迭代格式, 建立了一种在频域内求解C-GNLSE的同步更新迭代算法. 通过将该向量形式的相互作用表象中的4阶龙格-库塔 (V-JH-RK4IP) 算法应用于高双折射光子晶体光纤中超连续谱产生的数值模拟, 验证了算法的有效性, 通过与现有其他典型算法的比较, 表明以V-JH-RK4IP算法求解C-GNLSE具有最高的计算精度和计算效率.

关键词:

Abstract

The numerical simulation method for accurately solving the coupled generalized nonlinear Schrödinger equations (C-GNLSE) is essential for describing the dynamic behavior of ultrashort pulse propagating in optical fiber and developing the corresponding nonlinear fiber-optic devices. C-GNLSE in the normal picture is first mapped into the interaction picture by the representation transformation, and then, the two coupled nonlinear partial differential equations in the interaction picture are solved in frequency domain, with synchronous data updating in each iteration step, by using the vector form of Hult’s fourth-order Runge-Kutta iterative scheme. The proposed vector form algorithm of fourth-order Runge-Kutta in interaction picture (V-JH-RK4IP) is verified by using it in simulating the supercontinuum generation in high birefringence photonic crystal fiber. Moreover, the V-JH-RK4IP algorithm also exhibits the highest accuracy and computational efficiency as compared to other classical algorithms.

Keywords:

作者及机构信息

1.
中国科学院安徽光学精密机械研究所, 安徽省光子器件与材料重点实验室, 合肥 230031

基金项目: 国家自然科学基金 (批准号: 61250017, 11104282)和中科院重要方向性项目(批准号: KJZD-EW-W02)资助的课题.

Authors and contacts

1.
Anhui Provincial Key Laboratory of Photonics Devices and Materials, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61250017, 11104282), and the Key Research Program of the Chinese Academy of Sciences (Grant No. KJZD-EW-W02).

参考文献

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[2]	Agrawal G P 2007 Nonlinear Fiber Optics (Academic Press)
[3]	Povazay B, Bizheva K, Unterhuber A, Hermann B, Sattmann H, Fercher A F, Drexler W, Apolonski A, Wadsworth W J, Knight J C, Russell P S,Vetterlein M, Scherzer E 2002 Opt. Lett. 27 1800
[4]	Paulsen H N, Hilligse K M, Thogersen J, Keiding S R, Larsen J J 2003 Opt. Lett. 28 1123
[5]	Dudley M, Coen S 2006 Rev. Mod. Phys. 78 1135
[6]	Zakharov V E, Shabat A B 1972 Sov. Phys. JETP 34 62
[7]	Fermann M E, Kruglov V I, Thomsen B C, Dudley J M, Harvey J D 2000 Phys. Rev. Lett. 84 6010
[8]	Hohage T, Schmidt F, Konrad-Zuse-Zentrum fr Informationstechnik, Berlin, Germany, 2002 Tech. Rep. ZIB-Report Jan 02-04, 2002
[9]	Zhao L, Sui Z, Zhu Q H, Zhang Y, Zuo Y L 2009 Acta Phys. Sin. 58 4731 (in Chinese) [赵磊, 隋展, 朱启华, 张颖, 左言磊 2009 58 4731]
[10]	Reeves W H, Skyabin D V, Biancalana F, Knight J C, Omenetto F G, Efimov A, Taylor A J 2003 Nature 424 511
[11]	Hillingsoe K M, Paulsen H Thogersen N J, Keiding S R, Larsen J J 2003 J. Opt. Soc. Am. B 20 1887
[12]	Siederdissen T H Z, Nielsen N C, Kuhl J, Giessen H 2006 J. Opt. Soc. Am. B 23 1360
[13]	Cristiani I, Tediosi R, Tartara L, Degiorgio V 2003 Opt. Express 12 124
[14]	Liu X M, Byoungho L 2003 IEEE Photon. Technol. Lett. 15 1549
[15]	Blow K J, D Wood 1989 IEEE J. Quantum Electron 25 2665
[16]	Hult J 2007 J. Lightw. Technol 25 3770
[17]	Sinkin O V, Holzlöhner R, Zweck J, Menyuk C R 2003 J. Lightw. Technol 21 61
[18]	Heidt A 2009 J. Lightwave Technol 27 3984
[19]	Rieznik A A, Heidt A M, Konig P G, Bettachini V A, Grosz D F 2012 IEEE Photonics Journal 4 552
[20]	Lu F, Lin Q, Knox W H, Agrawal G P 2004 Phys. Rev. Lett. 93 183901
[21]	TU H H, Liu Y, Liu X M, Turchinovich D, Laegsgaard J, Stephen A B 2012 Opt. Express 20 1113
[22]	Nishizawa N, Goto T 2003 Opt. Express 11 359
[23]	Voronin A A, Fedotov I V, Kobelke J, Jager M, Schuster K, Fedotov A B, Bartelt H, Zheltikov A M 2012 Opt. Lett. 37 5163
[24]	Liu X M 2011 Phys. Rev. A 84 053828
[25]	Liu X M 2011 Phys. Rev. A 84 023835
[26]	Trillo S, Wabnitz S 1992 J. Opt. Soc. Am. B 9 1061
[27]	Martins E R, Spadoti D H, Romero M A, B V Borges 2007 Opt. Express 15 14335
[28]	Lin Q, Agrawal G P 2006 Opt. Lett. 31 3086
[29]	Menyuk C R, Islam M N, Gordon J P 1991 Opt. Lett. 16 566
[30]	Chick B J, Chon J W M, Gu M 2007 Opt. Express 16 20099

施引文献

[1]	Dudley J M, Taylor J R 2010 Supercontinuum Generation in Optical Fibers (Cambridge University Press)
[2]	Agrawal G P 2007 Nonlinear Fiber Optics (Academic Press)
[3]	Povazay B, Bizheva K, Unterhuber A, Hermann B, Sattmann H, Fercher A F, Drexler W, Apolonski A, Wadsworth W J, Knight J C, Russell P S,Vetterlein M, Scherzer E 2002 Opt. Lett. 27 1800
[4]	Paulsen H N, Hilligse K M, Thogersen J, Keiding S R, Larsen J J 2003 Opt. Lett. 28 1123
[5]	Dudley M, Coen S 2006 Rev. Mod. Phys. 78 1135
[6]	Zakharov V E, Shabat A B 1972 Sov. Phys. JETP 34 62
[7]	Fermann M E, Kruglov V I, Thomsen B C, Dudley J M, Harvey J D 2000 Phys. Rev. Lett. 84 6010
[8]	Hohage T, Schmidt F, Konrad-Zuse-Zentrum fr Informationstechnik, Berlin, Germany, 2002 Tech. Rep. ZIB-Report Jan 02-04, 2002
[9]	Zhao L, Sui Z, Zhu Q H, Zhang Y, Zuo Y L 2009 Acta Phys. Sin. 58 4731 (in Chinese) [赵磊, 隋展, 朱启华, 张颖, 左言磊 2009 58 4731]
[10]	Reeves W H, Skyabin D V, Biancalana F, Knight J C, Omenetto F G, Efimov A, Taylor A J 2003 Nature 424 511
[11]	Hillingsoe K M, Paulsen H Thogersen N J, Keiding S R, Larsen J J 2003 J. Opt. Soc. Am. B 20 1887
[12]	Siederdissen T H Z, Nielsen N C, Kuhl J, Giessen H 2006 J. Opt. Soc. Am. B 23 1360
[13]	Cristiani I, Tediosi R, Tartara L, Degiorgio V 2003 Opt. Express 12 124
[14]	Liu X M, Byoungho L 2003 IEEE Photon. Technol. Lett. 15 1549
[15]	Blow K J, D Wood 1989 IEEE J. Quantum Electron 25 2665
[16]	Hult J 2007 J. Lightw. Technol 25 3770
[17]	Sinkin O V, Holzlöhner R, Zweck J, Menyuk C R 2003 J. Lightw. Technol 21 61
[18]	Heidt A 2009 J. Lightwave Technol 27 3984
[19]	Rieznik A A, Heidt A M, Konig P G, Bettachini V A, Grosz D F 2012 IEEE Photonics Journal 4 552
[20]	Lu F, Lin Q, Knox W H, Agrawal G P 2004 Phys. Rev. Lett. 93 183901
[21]	TU H H, Liu Y, Liu X M, Turchinovich D, Laegsgaard J, Stephen A B 2012 Opt. Express 20 1113
[22]	Nishizawa N, Goto T 2003 Opt. Express 11 359
[23]	Voronin A A, Fedotov I V, Kobelke J, Jager M, Schuster K, Fedotov A B, Bartelt H, Zheltikov A M 2012 Opt. Lett. 37 5163
[24]	Liu X M 2011 Phys. Rev. A 84 053828
[25]	Liu X M 2011 Phys. Rev. A 84 023835
[26]	Trillo S, Wabnitz S 1992 J. Opt. Soc. Am. B 9 1061
[27]	Martins E R, Spadoti D H, Romero M A, B V Borges 2007 Opt. Express 15 14335
[28]	Lin Q, Agrawal G P 2006 Opt. Lett. 31 3086
[29]	Menyuk C R, Islam M N, Gordon J P 1991 Opt. Lett. 16 566
[30]	Chick B J, Chon J W M, Gu M 2007 Opt. Express 16 20099

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