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三阶非线性效应对边界限制的自聚焦振荡型响应函数系统中二次孤子的影响

陈娟 胡巍 陆大全

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三阶非线性效应对边界限制的自聚焦振荡型响应函数系统中二次孤子的影响

陈娟, 胡巍, 陆大全

Influence of cubic nonlinearity effect on quadratic solitons in boundary-constrained self-focusing oscillatory response function system

Chen Juan, Hu Wei, Lu Da-Quan
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  • 面向边界限制的自聚焦振荡型响应函数系统建立了二阶和三阶非线性共同作用时的二次孤子理论模型. 在此基础上, 通过数值模拟对有三阶非线性效应和无三阶非线性效应情况下的孤子解进行对比, 研究了三阶非线性效应对该系统中二次孤子的影响. 结果表明, 在孤子波形方面, 三阶非线性影响较小, 仅轻微改变其横向分布. 但是在孤子的存在区间方面, 三阶非线性导致孤子只存在于强非局域和一般非局域情况, 在弱非局域情况下找不到相应的孤子. 此外, 三阶非线性的存在还减少了孤子拓展半周期的个数. 而在孤子稳定性方面, 三阶非线性则缩减了孤子的稳定区间.
    In this paper, we theoretically study the influence of cubic nonlinearity effect on quadratic solitons in the boundary-constrained self-focusing oscillatory response function system. Based on the Newton iteration approach, we numerically solve the nonlinear coupled-wave equations with both quadratic and cubic nonlinearity. Moreover, a family of quadratic solitons is obtained. By comparing the quadratic solitons with both quadratic and cubic nonlinearity with those with only quadratic nonlinearity, we find that the cubic nonlinearity changes the transverse distribution of the soliton profiles only slightly. However, because of the existence of the cubic nonlinearity, quadratic solitons can be found only in the strongly nonlocal case and general nonlocal case, and they cannot be found in the weakly nonlocal case, in which the quadratic solitons with only quadratic nonlinearity can be found. In addition, the existence of cubic nonlinearity reduces the number of extended half-periods of the quadratic solitons. Through the linear stability analysis of the obtained soliton solutions, it is found that the stability intervals of solitons are also shrunk due to the existence of the cubic nonlinearity. The results of the linear stability analysis are verified by the numerical simulations of soliton propagations through using the split-step Fourier method.
      通信作者: 胡巍, huwei@scnu.edu.cn ; 陆大全, ludq@scnu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11174090, 11174091, 61575068)和广东省基础与应用基础研究基金(批准号: 2020A1515010632, 2020A1515010930)资助的课题.
      Corresponding author: Hu Wei, huwei@scnu.edu.cn ; Lu Da-Quan, ludq@scnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11174090, 11174091, 61575068) and the Guangdong Basic and Applied Basic Research Foundation, China (Grant Nos. 2020A1515010632, 2020A1515010930)
    [1]

    陈险峰 2014 非线性光学研究前沿 (上海: 上海交通大学出版社) 第257—333页

    Chen X F 2014 Advances in Nonlinear Optics (Shanghai: Shanghai Jiao Tong University Press) pp257–333 (in Chinese)

    [2]

    Kivshar Y S, Agrawal G P 2003 Optical Solitons: from Fibers to Photonic Crystals (New York: Elsevier) pp1–15

    [3]

    Mitchell M, Segev M, Christodoulides D N 1998 Phys. Rev. Lett. 80 4657Google Scholar

    [4]

    Hu W, Zhang T, Guo Q, Xuan L, Lan S 2006 Appl. Phys. Lett. 89 071111Google Scholar

    [5]

    Conti C, Fratalocchi A, Peccianti M, Ruocco G, Trillo S 2009 Phys. Rev. Lett. 102 083902Google Scholar

    [6]

    Skupin S, Saffman M, Królikowski W 2007 Phys. Rev. Lett. 98 263902Google Scholar

    [7]

    Pedri P, Santos L 2005 Phys. Rev. Lett. 95 200404Google Scholar

    [8]

    Liang G, Liu J L, Hu W, Guo Q 2022 Appl. Sci. 12 2386Google Scholar

    [9]

    Królikowski W, Bang O, Rasmussen J J, Wyller J 2001 Phys. Rev. E 64 016612Google Scholar

    [10]

    Bang O, Królikowski W, Wyller J, Rasmussen J J 2002 Phys. Rev. E 66 046619Google Scholar

    [11]

    Conti C, Peccianti M, Assanto G 2004 Phys. Rev. Lett. 92 113902Google Scholar

    [12]

    Piccardi A, Alberucci A, Buchnev O, Kaczmarek M, Khoo I C, Assanto G 2012 Appl. Phys. Lett. 101 081112Google Scholar

    [13]

    Laudyn U A, Kwasny M, Karpierz M A, Assanto G 2019 Opt. Lett. 44 167Google Scholar

    [14]

    Rotschild C, Cohen O, Manela O, Segev M, Carmon T 2005 Phys. Rev. Lett. 95 213904Google Scholar

    [15]

    Shou Q, Zhang X, Hu W, Guo Q 2011 Opt. Lett. 36 4194Google Scholar

    [16]

    Nikolov N I, Neshev D, Bang O, Królikowski W Z 2003 Phys. Rev. E 68 036614Google Scholar

    [17]

    Torruellas W E, Wang Z, Hagan D J, VanStryland E W, Stegeman G I, Torner L, Menyuk C R 1995 Phys. Rev. Lett. 74 5036Google Scholar

    [18]

    Torner L, Menyuk C R, Torruellas W E, Stegeman G I 1995 Opt. Lett. 20 13Google Scholar

    [19]

    Chen J, Ge J W, Lu D Q, Hu W 2020 Appl. Math. Lett. 102 106108Google Scholar

    [20]

    Karamzin Y N, Sukhorukov A P 1974 JETP Lett. 20 339

    [21]

    Karamzin Y N, Sukhorukov A P 1975 Sov. Phys. JETP 41 414

    [22]

    Buryak A V, Kivshar Y S 1995 Phys. Lett. A 197 407Google Scholar

    [23]

    Buryak A V, Di Trapani P, Skryabin D V, Trillo S 2002 Phys. Rep. 370 63Google Scholar

    [24]

    Torner L, Mihalache D, Mazilu D, Akhmediev N N 1995 Opt. Lett. 20 2183Google Scholar

    [25]

    Schiek R, Baek Y, Stegeman G I 1996 Phys. Rev. E 53 1138Google Scholar

    [26]

    Esbensen B K, Bache M, Królikowski W, Bang O 2012 Phys. Rev. A 86 023849Google Scholar

    [27]

    Wang J, Li Y H, Guo Q, Hu W 2014 Opt. Lett. 39 405Google Scholar

    [28]

    Wang J, Ma Z L, Li Y H, Lu D Q, Guo Q, Hu W 2015 Phys. Rev. A 91 033801Google Scholar

    [29]

    Zheng Y Z, Gao Y, Wang J, Lv F, Lu D Q, Hu W 2017 Phys. Rev. A 95 013808Google Scholar

    [30]

    Chen M N, Ping X R, Liang G, Guo Q, Lu D Q, Hu W 2018 Phys. Rev. A 97 013829Google Scholar

    [31]

    Agrawal G P 2007 Nonlinear Fiber Optics (4th Ed.) (New York: Elsevier) pp25–50

  • 图 1  强非局域和一般非局域情况下二阶和三阶非线性共同作用时的孤子波形(红色: 基频波; 绿色: 二次谐波) (a), (b)强非局域情况$ \alpha=0.04 $, 其中图(b)的波形是在图(a)的基础上向两边各拓展了半个周期; (c), (d)一般非局域情况$ \alpha=10 $, 其中图(c)中$ l=18.0974 $, 图(d)中$ l=18.7214 $. 图中虚线为二阶和三阶非线性共同作用时的孤子波形; 实线为只有二阶非线性作用时的二次孤子波形; $ |d_2/d_1 |=1 $

    Fig. 1.  Soliton profiles (red: FW; green: SH) with quadratic and cubic nonlinearities for strongly nonlocal case and general nonlocal case: (a), (b) Strongly nonlocal case $ \alpha=0.04 $; the soliton in panel (b) is an extension of that in panel (a) toward the right and left sides; (c), (d) general nonlocal case $ \alpha=10 $, where (c) $ l=18.0974 $, (d) $ l=18.7214 $. Dotted line: soliton profiles with quadratic and cubic nonlinearities; Solid line: the profiles of quadratic solitons with only quadratic nonlinearity. $ |d_2/d_1 |=1 $

    图 2  基频波的束宽(蓝色)和功率(红色)随样品宽度l 的变化规律 (a)强非局域情况$ \alpha=0.04 $; (b)一般非局域情况$ \alpha=10 $. 虚线为二阶和三阶非线性共同作用时的结果; 实线为只有二阶非线性作用时的结果. $ |d_2/d_1 |=0.8 $

    Fig. 2.  Soliton width (blue line, right ordinate) and power (red line, left ordinate) of the FW versus the sample size l: (a) Strongly nonlocal case $ \alpha=0.04 $; (b) general nonlocal case $ \alpha=10 $. Dotted line: results with quadratic and cubic nonlinearities; Solid line: results with only quadratic nonlinearity. $ |d_2/d_1 |=0.8 $

    图 3  (a), (b)对应图2的孤子不稳定增长率曲线; (c)孤子的不稳定增长率随非局域程度α的变化规律, $ l=18 $, $ |d_2/d_1 |=2 $; (d)孤子的不稳定增长率随$ d_2/d_1 $的变化规律, $ \alpha=0.11 $, $ l=18 $. 虚线为二阶和三阶非线性共同作用时的结果; 实线为只有二阶非线性作用时的结果

    Fig. 3.  (a), (b) Curves of unstable growth rate corresponding to Fig. 2; (c) unstable growth rate versus the nonlocal degree α. $ l=18 $, $ |d_2/d_1 |=2 $; (d) unstable growth rate versus $ d_2/d_1 $. $ \alpha=0.11 $, $ l=18 $. Dotted line: results with quadratic and cubic nonlinearities; Solid line: results with only quadratic nonlinearity

    图 4  图3(d)中两个圆点所对应孤子解的数值模拟传输结果 (a), (b)基频波; (c), (d)二次谐波. 第一行中$ d_2/d_1=-0.8 $; 第二行中$ d_2/d_1=-0.4 $

    Fig. 4.  Propagation of the two solitons corresponding to the two dots in Fig. 3(d): (a), (b) The FW; (c), (d) the SH. Upper row: $ d_2/d_1=-0.8 $; Bottom row: $ d_2/d_1=-0.4 $

  • [1]

    陈险峰 2014 非线性光学研究前沿 (上海: 上海交通大学出版社) 第257—333页

    Chen X F 2014 Advances in Nonlinear Optics (Shanghai: Shanghai Jiao Tong University Press) pp257–333 (in Chinese)

    [2]

    Kivshar Y S, Agrawal G P 2003 Optical Solitons: from Fibers to Photonic Crystals (New York: Elsevier) pp1–15

    [3]

    Mitchell M, Segev M, Christodoulides D N 1998 Phys. Rev. Lett. 80 4657Google Scholar

    [4]

    Hu W, Zhang T, Guo Q, Xuan L, Lan S 2006 Appl. Phys. Lett. 89 071111Google Scholar

    [5]

    Conti C, Fratalocchi A, Peccianti M, Ruocco G, Trillo S 2009 Phys. Rev. Lett. 102 083902Google Scholar

    [6]

    Skupin S, Saffman M, Królikowski W 2007 Phys. Rev. Lett. 98 263902Google Scholar

    [7]

    Pedri P, Santos L 2005 Phys. Rev. Lett. 95 200404Google Scholar

    [8]

    Liang G, Liu J L, Hu W, Guo Q 2022 Appl. Sci. 12 2386Google Scholar

    [9]

    Królikowski W, Bang O, Rasmussen J J, Wyller J 2001 Phys. Rev. E 64 016612Google Scholar

    [10]

    Bang O, Królikowski W, Wyller J, Rasmussen J J 2002 Phys. Rev. E 66 046619Google Scholar

    [11]

    Conti C, Peccianti M, Assanto G 2004 Phys. Rev. Lett. 92 113902Google Scholar

    [12]

    Piccardi A, Alberucci A, Buchnev O, Kaczmarek M, Khoo I C, Assanto G 2012 Appl. Phys. Lett. 101 081112Google Scholar

    [13]

    Laudyn U A, Kwasny M, Karpierz M A, Assanto G 2019 Opt. Lett. 44 167Google Scholar

    [14]

    Rotschild C, Cohen O, Manela O, Segev M, Carmon T 2005 Phys. Rev. Lett. 95 213904Google Scholar

    [15]

    Shou Q, Zhang X, Hu W, Guo Q 2011 Opt. Lett. 36 4194Google Scholar

    [16]

    Nikolov N I, Neshev D, Bang O, Królikowski W Z 2003 Phys. Rev. E 68 036614Google Scholar

    [17]

    Torruellas W E, Wang Z, Hagan D J, VanStryland E W, Stegeman G I, Torner L, Menyuk C R 1995 Phys. Rev. Lett. 74 5036Google Scholar

    [18]

    Torner L, Menyuk C R, Torruellas W E, Stegeman G I 1995 Opt. Lett. 20 13Google Scholar

    [19]

    Chen J, Ge J W, Lu D Q, Hu W 2020 Appl. Math. Lett. 102 106108Google Scholar

    [20]

    Karamzin Y N, Sukhorukov A P 1974 JETP Lett. 20 339

    [21]

    Karamzin Y N, Sukhorukov A P 1975 Sov. Phys. JETP 41 414

    [22]

    Buryak A V, Kivshar Y S 1995 Phys. Lett. A 197 407Google Scholar

    [23]

    Buryak A V, Di Trapani P, Skryabin D V, Trillo S 2002 Phys. Rep. 370 63Google Scholar

    [24]

    Torner L, Mihalache D, Mazilu D, Akhmediev N N 1995 Opt. Lett. 20 2183Google Scholar

    [25]

    Schiek R, Baek Y, Stegeman G I 1996 Phys. Rev. E 53 1138Google Scholar

    [26]

    Esbensen B K, Bache M, Królikowski W, Bang O 2012 Phys. Rev. A 86 023849Google Scholar

    [27]

    Wang J, Li Y H, Guo Q, Hu W 2014 Opt. Lett. 39 405Google Scholar

    [28]

    Wang J, Ma Z L, Li Y H, Lu D Q, Guo Q, Hu W 2015 Phys. Rev. A 91 033801Google Scholar

    [29]

    Zheng Y Z, Gao Y, Wang J, Lv F, Lu D Q, Hu W 2017 Phys. Rev. A 95 013808Google Scholar

    [30]

    Chen M N, Ping X R, Liang G, Guo Q, Lu D Q, Hu W 2018 Phys. Rev. A 97 013829Google Scholar

    [31]

    Agrawal G P 2007 Nonlinear Fiber Optics (4th Ed.) (New York: Elsevier) pp25–50

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出版历程
  • 收稿日期:  2022-05-02
  • 修回日期:  2022-07-26
  • 上网日期:  2022-10-18
  • 刊出日期:  2022-11-05

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