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非局域非线性耦合器中暗孤子的传输

李森清 张肖 林机

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非局域非线性耦合器中暗孤子的传输

李森清, 张肖, 林机

Propagation of dark soliton in nonlocal nonlinear coupler

Li Sen-Qing, Zhang Xiao, Lin Ji
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  • 研究了空间暗孤子在非局域非线性耦合器中的新解和传输稳定性. 发现非局域非线性耦合器中存在稳定的基态暗孤子和多极暗孤子的束缚态. 分析了非局域程度、非线性参数、传播常数和耦合系数变化对基态暗孤子的峰值、束宽和功率的影响. 讨论了基态暗孤子和多极暗孤子的传输稳定性, 发现基态暗孤子在其存在的区域总是稳定的, 偶极以及多极暗孤子存在不稳定区间, 稳定区间取决于传播常数和介质的非局域程度, 并且多极暗孤子的稳定传输还受孤子间距的影响.
    The characteristics of fundamental and mutipole dark solitons in the nonlocal nonlinear couplers are studied through numerical simulation in this work. Firstly, the fundamental dark solitons with different parameters are obtained by the Newton iteration. It is found that the amplitude and beam width of the ground state dark soliton increase with the enhancement of the nonlocality degree. As the nonlinear parameters increase or the propagation constant decreases, the amplitude of the fundamental dark soliton increases and the beam width decreases. The power of the fundamental dark soliton increases with the nonlocality degree and nonlinear parameters increasing, and decreases with the propagation constant increasing. The refractive index induced by the light field decreases with the nonlocality degree increasing and the propagation constant decreasing. The amplitudes of the two components of the fundamental dark soliton can be identical by adjusting the coupling coefficient. These numerical results are also verified in the case of multipole dark solitons. Secondly, the transmission stability of fundamental and mutipole dark solitons are studied. The stability of dark soliton is verified by the linear stability analysis and fractional Fourier evolution. It is found that the fundamental dark solitons are stable in their existing regions, while the stable region of the multipolar dark solitons depends on the nonlocality degree and the propagation constant. Finally, these different types of dark dipole solitons and dark tripole solitons are obtained by changing different parameters, and their structures affect the stability of dark soliton. It is found that the multipole dark soliton with potential well is more stable than that with potential barrier. The refractive-index distribution dependent spacing between the adjacent multipole dark solitons favors their stability.
      通信作者: 林机, linji@zjnu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11835011, 12004338)资助的课题
      Corresponding author: Lin Ji, linji@zjnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11835011, 12004338)
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    Wu Y D 2004 Fiber Integr. Opt. 23 405Google Scholar

    [3]

    Nistazakis H E, Frantzeskakis D J, Atai J, Malomed A B, Efremidis N, Hizanidis K 2002 Phys. Rev. E 65 036605Google Scholar

    [4]

    Malomed B A, Peng G D, Chu P L 1996 Opt. Lett. 21 330Google Scholar

    [5]

    Harel A, Malomed B A 2014 Phys. Rev. A 89 043809Google Scholar

    [6]

    Mak W C, Malomed B A, Chu P L 2004 Phys. Rev. E 69 279

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    Trillo S, Wabnitz S 1988 J. Opt. Soc. Am. B 5 483Google Scholar

    [8]

    Smirnova D A, Gorbach A V, Iorsh I V, Shadrivov I V, Kivshar Y S 2013 Phys. Rev. B 88 045443Google Scholar

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    Albuch L, Malomed B A 2007 Math. Comput. Simul. 74 312Google Scholar

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    黄光桥, 林机 2017 物理学报 66 054208Google Scholar

    Huang G Q, Lin J 2017 Acta Phys. Sin. 66 054208Google Scholar

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    Gubeskys A, Malomed B A 2007 Phys. Rev. A 75 063602Google Scholar

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    Zhao X, Tian B, Qu Q X, Yuan Y Q, Du X X, Chu M X 2020 Mod. Phys. Lett. B 34 2050282

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    Stegeman G I, Assanto G, Zanoni R, Seaton C T, Garmire E, Maradudin A A, Reinisch R, Vitrant G 1988 Appl. Phys. Lett. 52 869Google Scholar

    [14]

    Krölikowski W, Bang O, Nikolov N I, Neshev D, Wyller J, Rasmussen J J, Edmundson D 2004 J. Opt. B: Quantum Semiclassical Opt. 6 S288Google Scholar

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    Conti C, Peccianti M, Assanto G 2003 Phys. Rev. Lett. 91 073902Google Scholar

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    Jia J, Lin J 2012 Opt. Express 20 7469Google Scholar

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    Liang G, Dang D L, Li W, Li H G, Guo Q 2020 New J. Phys. 22 073024Google Scholar

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

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

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    Dreischuh A, Neshev D N, Petersen D E, Bang O, Królikowski W 2006 Phys. Rev. Lett. 96 043901Google Scholar

    [21]

    Lin Y Y, Lee R K, Malomed B A 2009 Phys. Rev. A 80 013838Google Scholar

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

    [23]

    高星辉, 杨振军, 周罗红, 郑一周, 陆大全, 胡巍 2011 物理学报 60 084213Google Scholar

    Gao X H, Yang Z J, Zhou L H, Zheng Y Z, Lu D Q, Hu W 2011 Acta Phys. Sin. 60 084213Google Scholar

    [24]

    Kong Q, Wei N, Fan C Z, Shi J L, Shen M 2017 Phys. Rep. 7 4198Google Scholar

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    Briedis D, Petersen D E, Edmundson D, Krolikowski W, Bang O 2005 Opt. Express 13 435Google Scholar

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    Rotschild C, Cohen O, Manela O, Segev M, Carmon T 2005 Phys. Rev. Lett. 95 2365

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    曹觉能, 郭旗 2005 物理学报 54 3688Google Scholar

    Cao J N, Guo Q 2005 Acta Phys. Sin. 54 3688Google Scholar

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    Ye F, Kartashov Y, Hu B, Torner L 2010 Opt. Lett. 35 628Google Scholar

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    Sun C Z, Liang G 2019 Chin. Phys. B 28 074206Google Scholar

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    Ye F, Dong L, Hu B 2009 Opt. Lett. 34 584Google Scholar

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    Chen W, Wang Q, Shi J L, Shen M 2017 Opt. Commun. 403 22Google Scholar

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    郑一凡, 黄光侨, 林机 2018 物理学报 67 214207Google Scholar

    Zheng Y F, Huang G Q, Lin J 2018 Acta Phys Sin. 67 214207Google Scholar

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    Wang Q, Deng Z Z 2020 Results Phys. 17 103056Google Scholar

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    Akhmediev N, Ankiewicz A 1993 Phys. Rev. Lett. 70 2395Google Scholar

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    Shi X L, Malomed B A, Ye F W, Chen X F 2012 Phys. Rev. A 85 053839Google Scholar

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    Dang Y L, Li H J, Lin J 2017 Nonlinear Dynam. 88 489Google Scholar

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    Islam M J, Atai J 2018 J. Mod. Opt. 65 1499980

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    Gao Z J, Dang Y L, Lin J 2018 Opt. Commun. 44 302

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    方乒乒, 金新伟, 林机 2019 光子学报 48 51Google Scholar

    Fang P P, Jin X W, Lin J 2019 Acta Photon. Sin. 48 51Google Scholar

    [40]

    Mahato D K, Govindarajan A, Sarma A K 2020 J. Opt. Soc. Am. B 37 3443Google Scholar

    [41]

    Kartashov Y V, Konotop V V, Malomed B A 2015 Opt. Lett. 40 004126Google Scholar

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    Safaei L, Zarandi M B, Hatami M 2018 Opt. Quantum Electron. 50 382Google Scholar

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    Zakery A, Hatami M 2007 J. Phys. D: Appl. Phys. 40 1010Google Scholar

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    Govindaraji A, Mahalingam A, Uthayakumar A 2015 Appl. Phys. B 120 341

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    Armaroli A, Trillo S, Fratalocchi A 2009 Phys. Rev. A 80 053803Google Scholar

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    高星辉, 张承云, 唐冬, 郑晖, 陆大全, 胡巍 2013 物理学报 62 044214Google Scholar

    Gao X H, Zhang C Y, Tang D, Zheng H, Lu D Q, Hu W 2013 Acta Phys. Sin. 62 044214Google Scholar

  • 图 1  基态暗孤子数值解 (a)不同d对应的波形($ d $分别取2.0 (蓝线), 4.5 (红线)和7.5 (绿线)); (b)功率与传播常数的关系图; (c) $ b = -1 $, $ d = 2 $, $ \beta_{2} = -1 $时, 不同$ \beta_{1} $对应的波形($ \beta_{1} $分别取–2 (蓝线), –5 (红线)和–7 (绿线)); (d) $ d $ = 2时, 不同$ b $对应的波形($ b $分别取–0.2 (蓝线), –0.5 (红线)和–1.0 (绿线)); (e)不同耦合系数比值下, $ U_{1} $$ U_{2} $的幅值图

    Fig. 1.  Numerical solution of the ground state dark soliton: (a) Waveform corresponds to different $ d $ ($ d $ selected as 2.0 (blue lines), 4.5 (red lines) and 7.5 (green lines), respectively); (b) the power versus the propagation constant; (c) waveform corresponds to different $ \beta_{1} $ when $ b $ = –1, $ d $ = 2, $\beta_{2} = -1$ ($ \beta_{1} $ selected as –2 (blue lines), –5 (red lines) and –7 (green lines), respectively); (d) waveform corresponds to different $ b $ ($ b $ selected as –0.2 (blue lines), –0.5 (red lines) and –1.0 (green lines), respectively); (e) amplitude diagram of $ U_{1} $ and $ U_{2} $ for different coupling coefficient ratios.

    图 2  基态暗孤子的传输特征及稳定性 (a)$ d $ = 2, $ b $ = –1时, $ U_{1} $的加噪传输图; (b)稳定性图谱

    Fig. 2.  Propagation characteristics and stability of the ground state dark soliton: (a) Propagation of $ U_{1} $ with the white noise when $ d $ = 2 and $ b $ = –1; (b) stability profile

    图 3  偶极暗孤子的特性  (a) $ d $ = 1, $ b $ = –1时偶极暗孤子的强度分布图; (b) 线性稳定性区域图; (c) $ d $ = 4.2, $ b $ = –1.5时, $ U_{1} $传输图; (d) $ d $ = 1.5, $ b $ = –2时, $ U_{1} $传输图

    Fig. 3.  Characteristics of the dipole dark soliton: (a) Intensity profiles of dipole dark soliton when $ d $ = 1 and $ b $ = –1; (b) linear stability region diagram; (c) transmission diagram of $ U_{1} $ when $ d $ = 4.2 and $ b $ = –1.5; (d) transmission diagram of $ U_{1} $ when $ d $ = 1.5 and $ b $ = –2

    图 4  不同类型的偶极暗孤子的强度分布和传播特性 (a)−(c)分别为$ d $ = 4, $ b $ = –1.5时, 偶极暗孤子的强度分布图、传输图和稳定性图谱; (d)−(f) 分别为$ b $ = –1.3, $ d $ = 1时, 偶极暗孤子的强度分布图、传输图和稳定性图谱; (g)和(h)分别为$ b $ = –1.7, $ d $ = 2时, 两个偶极暗孤子的强度分布和$ U_{1} $的传输图; (i) 和(j)分别为$ b $ = –1.4, $ d $ = 5时, 两个偶极暗孤子的强度分布和$ U_{1} $的传输图

    Fig. 4.  Intensity profiles and propagating characteristics of different types of dipole dark soliton: (a)– (c) The intensity distribution diagram, the transmission diagram and the stability diagram of the dipole dark soliton when $ d $ = 4 and $ b $ = –1.5 respectively; (d)−(f) the intensity distribution diagram, the transmission diagram and the stability diagram of the dipole dark soliton when $ b $ = –1.3 and $ d $ = 1 respectively; (g) and (h) the intensity distribution of dipole-dipole dark solitons and the transmission diagram of $ U_{1} $ when $ b $ = –1.7 and $ d $ = 2, respectively; (i) and (j) the intensity distribution of dipole-dipole dark solitons and the transmission diagram of $ U_{1} $ when $ b $ = –1.4 and $ d $ = 5, respectively.

    图 5  不同类型的三极暗孤子的强度分布和传播特性 (a)和(b)分别为$ b $ = –4.45, $ d $ = 1时, 三极暗孤子的强度分布和$ U_{1} $传输图; (c)和(d)分别为$ b $ = –1.2, $ d $ = 1时, 三极暗孤子的强度分布和$ U_{1} $传输图

    Fig. 5.  Intensity profiles and propagating characteristics of tripole dark soliton: (a) and (b) The intensity distribution of the tripole dark soliton and the transmission diagram of $ U_{1} $ when $ b $ = –4.45 and $ d $ = 1, respectively; (c) and (d) the intensity distribution of the tripole dark soliton and the transmission diagram of $ U_{1} $ when $ b $ = –1.2 and $ d $ = 1, respectively.

  • [1]

    Lederer F, Stegeman G I, Christodoulides D N, Assanto G, Segev M, Silberg Y 2008 Phys. Rep. 463 1Google Scholar

    [2]

    Wu Y D 2004 Fiber Integr. Opt. 23 405Google Scholar

    [3]

    Nistazakis H E, Frantzeskakis D J, Atai J, Malomed A B, Efremidis N, Hizanidis K 2002 Phys. Rev. E 65 036605Google Scholar

    [4]

    Malomed B A, Peng G D, Chu P L 1996 Opt. Lett. 21 330Google Scholar

    [5]

    Harel A, Malomed B A 2014 Phys. Rev. A 89 043809Google Scholar

    [6]

    Mak W C, Malomed B A, Chu P L 2004 Phys. Rev. E 69 279

    [7]

    Trillo S, Wabnitz S 1988 J. Opt. Soc. Am. B 5 483Google Scholar

    [8]

    Smirnova D A, Gorbach A V, Iorsh I V, Shadrivov I V, Kivshar Y S 2013 Phys. Rev. B 88 045443Google Scholar

    [9]

    Albuch L, Malomed B A 2007 Math. Comput. Simul. 74 312Google Scholar

    [10]

    黄光桥, 林机 2017 物理学报 66 054208Google Scholar

    Huang G Q, Lin J 2017 Acta Phys. Sin. 66 054208Google Scholar

    [11]

    Gubeskys A, Malomed B A 2007 Phys. Rev. A 75 063602Google Scholar

    [12]

    Zhao X, Tian B, Qu Q X, Yuan Y Q, Du X X, Chu M X 2020 Mod. Phys. Lett. B 34 2050282

    [13]

    Stegeman G I, Assanto G, Zanoni R, Seaton C T, Garmire E, Maradudin A A, Reinisch R, Vitrant G 1988 Appl. Phys. Lett. 52 869Google Scholar

    [14]

    Krölikowski W, Bang O, Nikolov N I, Neshev D, Wyller J, Rasmussen J J, Edmundson D 2004 J. Opt. B: Quantum Semiclassical Opt. 6 S288Google Scholar

    [15]

    Conti C, Peccianti M, Assanto G 2003 Phys. Rev. Lett. 91 073902Google Scholar

    [16]

    Jia J, Lin J 2012 Opt. Express 20 7469Google Scholar

    [17]

    Liang G, Dang D L, Li W, Li H G, Guo Q 2020 New J. Phys. 22 073024Google Scholar

    [18]

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

    [19]

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

    [20]

    Dreischuh A, Neshev D N, Petersen D E, Bang O, Królikowski W 2006 Phys. Rev. Lett. 96 043901Google Scholar

    [21]

    Lin Y Y, Lee R K, Malomed B A 2009 Phys. Rev. A 80 013838Google Scholar

    [22]

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

    [23]

    高星辉, 杨振军, 周罗红, 郑一周, 陆大全, 胡巍 2011 物理学报 60 084213Google Scholar

    Gao X H, Yang Z J, Zhou L H, Zheng Y Z, Lu D Q, Hu W 2011 Acta Phys. Sin. 60 084213Google Scholar

    [24]

    Kong Q, Wei N, Fan C Z, Shi J L, Shen M 2017 Phys. Rep. 7 4198Google Scholar

    [25]

    Briedis D, Petersen D E, Edmundson D, Krolikowski W, Bang O 2005 Opt. Express 13 435Google Scholar

    [26]

    Rotschild C, Cohen O, Manela O, Segev M, Carmon T 2005 Phys. Rev. Lett. 95 2365

    [27]

    曹觉能, 郭旗 2005 物理学报 54 3688Google Scholar

    Cao J N, Guo Q 2005 Acta Phys. Sin. 54 3688Google Scholar

    [28]

    Ye F, Kartashov Y, Hu B, Torner L 2010 Opt. Lett. 35 628Google Scholar

    [29]

    Sun C Z, Liang G 2019 Chin. Phys. B 28 074206Google Scholar

    [30]

    Ye F, Dong L, Hu B 2009 Opt. Lett. 34 584Google Scholar

    [31]

    Chen W, Wang Q, Shi J L, Shen M 2017 Opt. Commun. 403 22Google Scholar

    [32]

    郑一凡, 黄光侨, 林机 2018 物理学报 67 214207Google Scholar

    Zheng Y F, Huang G Q, Lin J 2018 Acta Phys Sin. 67 214207Google Scholar

    [33]

    Wang Q, Deng Z Z 2020 Results Phys. 17 103056Google Scholar

    [34]

    Akhmediev N, Ankiewicz A 1993 Phys. Rev. Lett. 70 2395Google Scholar

    [35]

    Shi X L, Malomed B A, Ye F W, Chen X F 2012 Phys. Rev. A 85 053839Google Scholar

    [36]

    Dang Y L, Li H J, Lin J 2017 Nonlinear Dynam. 88 489Google Scholar

    [37]

    Islam M J, Atai J 2018 J. Mod. Opt. 65 1499980

    [38]

    Gao Z J, Dang Y L, Lin J 2018 Opt. Commun. 44 302

    [39]

    方乒乒, 金新伟, 林机 2019 光子学报 48 51Google Scholar

    Fang P P, Jin X W, Lin J 2019 Acta Photon. Sin. 48 51Google Scholar

    [40]

    Mahato D K, Govindarajan A, Sarma A K 2020 J. Opt. Soc. Am. B 37 3443Google Scholar

    [41]

    Kartashov Y V, Konotop V V, Malomed B A 2015 Opt. Lett. 40 004126Google Scholar

    [42]

    Safaei L, Zarandi M B, Hatami M 2018 Opt. Quantum Electron. 50 382Google Scholar

    [43]

    Zakery A, Hatami M 2007 J. Phys. D: Appl. Phys. 40 1010Google Scholar

    [44]

    Govindaraji A, Mahalingam A, Uthayakumar A 2015 Appl. Phys. B 120 341

    [45]

    Armaroli A, Trillo S, Fratalocchi A 2009 Phys. Rev. A 80 053803Google Scholar

    [46]

    高星辉, 张承云, 唐冬, 郑晖, 陆大全, 胡巍 2013 物理学报 62 044214Google Scholar

    Gao X H, Zhang C Y, Tang D, Zheng H, Lu D Q, Hu W 2013 Acta Phys. Sin. 62 044214Google Scholar

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出版历程
  • 收稿日期:  2021-02-05
  • 修回日期:  2021-04-15
  • 上网日期:  2021-06-07
  • 刊出日期:  2021-09-20

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