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本文基于非线性薛定谔方程的Peregrine怪波解, 讨论有理分式的脉冲动力学, 基于其特性并利用谱过滤方法, 提出一种光脉冲串的放大方法. 连续波泵浦与频谱过滤器相结合, 能够实现光放大器作用. 这一思路被应用到光脉冲串的长距离传输, 以4级放大为例, 实现了光脉冲串的级联放大, 并且通过矩形脉冲截断, 能够实现有限个数脉冲的放大. 其次, 以实验上可控的周期调制的平面波作为初始输入, 能够产生放大脉冲串, 且最大放大脉冲串产生的位置与调制强度有关. 改变调制强度的大小, 能够影响最大放大脉冲串所产生的位置. 研究结果表明, 对于不同频率的输入脉冲串, 利用此方法可以实现放大, 并且通过改变调制强度的大小, 能够实现两路不同频率信号的同时放大.In this paper, we discuss the pulse dynamics of rational fraction based on the Peregrine rogue wave solution of nonlinear Schrödinger equation. Based on its properties and using the spectral filtering, the amplification of optical pulse train is proposed. The results show that the combination of a continuous-wave pump and a spectral filter positioned in fiber can act as an amplifier. And the idea is applied to the long-haul transmission of optical pulse train and four amplification periods are demonstrated. Particularly, the amplification of limited number of pulses can be realized by rectangular pulse truncation and the number of pulses can be adjusted by changing the parameters. The periodically modulated plane wave that can be controlled experimentally is taken as an input which can produce the maximumly amplified zero background pulse train and the location of maximumly amplified pulse train relates to the modulation intensity. The location of the maximumly amplified zero background pulse train changes with the modulation intensity. The results show that for two input signals with different frequencies, they can realize the amplification with the above method. By changing the modulation intensity the simultaneous amplification for two signals with different frequencies can be realized.
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Keywords:
- Rogue wave /
- Optical amplification /
- Spectral filtering /
- Cascade amplification
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Yang G Y 2014 Ph. D. Dissertation (Taiyuan: Shanxi University) (in Chinese)
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图 1 光脉冲串稳定传输图 (a) 零背景脉冲串; (b) 零背景脉冲串稳定传输, 其中
$ a = 0.4, $ P0 = 0.7 WFig. 1. Stable transmission of the pulse trains: (a) Zero background pulse trains; (b) stable transmission of the pulse trains, where
$ a = 0.4, $ P0 = 0.7 W.
图 2 光脉冲串的放大 (a) 初始入射脉冲串; (b) 衰减脉冲串; (c) 放大脉冲串与初始脉冲串的比较, 其中
$ a = 0.45, $ P0 = 0.7 W.Fig. 2. Amplification of the pulse trains: (a) Initial input pulse trains; (b) attenuated pulse trains; (c) comparison of the amplified and the initial pulse trains, where
$ a = 0.4, $ P0 = 0.7 W.
图 3 光脉冲串4级放大, 其中
$ a = 0.45, $ ${P_0} = 0.7\;{\text{W}}$ Fig. 3. 4-cascade amplification of optical pulse trains, where
$ a = 0.4{\text{5}}, $ P0 = 0.7 W.
图 4 有限个数脉冲的放大, 其中
$k = 3$ , P0 = 0.7 W$a = 0.4$ Fig. 4. Amplification for limited number of pulses, where
$k = 3,$ P0 = 0.7 W$a = 0.4.$ 
图 5 调制强度A与最大放大脉冲串位置的关系
Fig. 5. Location relationship between modulation intensity A and the maximum amplified pulse trains.
图 6 不同频率脉冲串的放大 (a) 初始输入; (b)零背景的放大脉冲串, 其中
${\varOmega _1}=0.3,$ ${\varOmega _2}=0.6,$ ${A_1}=0.2,$ ${A_2}=0.07$ Fig. 6. Amplification for different frequencies of the pulse trains: (a) Initial input pulse trains; (b) amplified pulse trains of zero background, where
${\varOmega _1}=0.3, $ ${\varOmega _2}=0.6, $ ${A_1}=0.2, $ ${A_2}=0.07$ 
图 7 Lmax1与Lmax2之间放大脉冲的功率
Fig. 7. Power of the amplified pulse trains between Lmax1 and Lmax2.
图 8 不同频率脉冲串的同时放大 (a)初始输入; (b)零背景的放大脉冲串, 其中
${\varOmega _1}{\text{ = }}0.3, $ ${\varOmega _2}=0.6, $ ${A_1}=0.2,$ ${A_2}=$ 0.07Fig. 8. Simultaneous amplification for different frequencies of the pulse trains: (a) Initial input pulse trains; (b) amplified pulse trains of zero background, where
${\varOmega _1}=0.3, $ ${\varOmega _2}=0.6, $ ${A_1}=0.2, $ ${A_2}=0.07$ -
[1] Kharif C, Pelinovsky E, Slunyaev A 2009 Rogue Waves in the Ocean (Springer)
[2] Akhmediev N, Soto-Crespo J M, Ankiewicz A 2009 Phys. Lett. A 373 2137
Google Scholar
[3] Peregrine D H 1983 J. Austral. Math. Soc. Ser. B 25 16
Google Scholar
[4] Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47
Google Scholar
[5] Dai C Q, Wang Y Y 2015 Nonlinear Dyn 80 715
Google Scholar
[6] Malomed B A, Mihalache D 2019 Rom. J. Phys. 64 106
[7] Baronio F 2017 Opt. Lett. 42 1756
Google Scholar
[8] 李淑青, 杨光晔, 李禄 2014 物理学报 63 104215
Google Scholar
[9] Dai C Q, Wang Y Y, Zhang J F 2020 Nonlinear Dyn. 102 379
Google Scholar
[10] Dai C Q, Liu J, Fan Y, Yu D G 2017 Nonlinear Dyn. 88 1373
Google Scholar
[11] AuDo F, Kibler B, Fatome J, Finot C 2018 Opt. Lett. 43 2864
Google Scholar
[12] Frostig H, Vidal I, Fischer R, Sheinfux H H, Silberberg Y 2020 Optica 7 864
Google Scholar
[13] Agafontsev D S, Randoux S, Suret P 2021 Phys. Rev. E 103 032209
Google Scholar
[14] Sun Z Y, Yu X 2021 Phys. Rev. E 103 062203
Google Scholar
[15] Bonatto C, Feyereisen M, Barland S, Giudici M, Masoller C, Leite J R R, Tredicce J R 2011 Phys. Rev. Lett. 107 053901
Google Scholar
[16] Dudley J M, Dias F, Erkintalo M, Genty G 2014 Nat. Photon. 8 755
Google Scholar
[17] Yan Z Y 2010 Commun. Theor. Phys. 54 947
Google Scholar
[18] Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503
Google Scholar
[19] Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502
Google Scholar
[20] Dematteis G, Grafke T, Onorato M, Eijnden E V 2019 Phys. Rev. X 9 041057
Google Scholar
[21] Bailung H S, Sharma K, Nakamura Y 2011 Phys. Rev. Lett. 107 255005
Google Scholar
[22] 陈智敏, 段文山 2020 物理学报 69 014701
Google Scholar
Chen Z M, Duan W S 2020 Acta Phys. Sin. 69 014701
Google Scholar
[23] 李再东, 郭奇奇 2020 物理学报 69 017501
Google Scholar
Li Z D, Guo Q Q 2020 Acta Phys. Sin. 69 017501
Google Scholar
[24] Kibler B, Fatome J, Finot C, Millot G, Dias F, Genty G, Akhmediev N, Dudley J M 2010 Nat. Phys. 6 790
Google Scholar
[25] Hammani K, Kibler B, Finot C, Morin P, Fatome J, Dudley J M, Millot G 2011 Opt. Lett. 36 112
Google Scholar
[26] 李淑青, 程永喜, 刘阳 2020 量子光学学报 26 180
Google Scholar
Li S Q, Chen Y X, Liu Y 2020 J. Quantum Opt. 26 180
Google Scholar
[27] 李敏, 王博婷, 许韬, 水涓涓 2020 物理学报 69 010502
Google Scholar
Li M, Wang B T, Xu T, Shui J J 2020 Acta Phys. Sin. 69 010502
Google Scholar
[28] Yang G Y, Li L, Jia S T 2012 Phys. Rev. E 85 046608
Google Scholar
[29] 张解放, 金美贞 2020 物理学报 69 214203
Google Scholar
Zhang J F, Jin M Z 2020 Acta Phys. Sin. 69 214203
Google Scholar
[30] Sgrignuoli F, Chen Y, Gorsky S, Britton W A, Negro L D 2021 Phys. Rev. B 103 195403
Google Scholar
[31] He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914
Google Scholar
[32] Wang Q, Liu D, Li X 2019 Commun. Nonlinear Sci. 75 302
Google Scholar
[33] Gao P, Zhao L C, Yang Z Y, Li X H, Yang W L 2020 Opt. Lett. 45 2399
Google Scholar
[34] Wang L H, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217
Google Scholar
[35] Yang G Y, Li L, Jia S T, Mihalache D 2013 Rom. Rep. Phys. 65 902
[36] Yang G Y, Wang Y, Qin Z, Malomed B A, Mihalache D, Li L 2014 Phys. Rev. E 90 062909
Google Scholar
[37] Yang G Y, Li L, Jia S T, Mihalache D 2013 Rom. Rep. Phys. 65 391
Google Scholar
[38] Fatome J, Kibler B, Finot C 2013 Opt. Lett. 38 1663
Google Scholar
[39] Wang Y, Song L J, Li L, Malomed B A 2015 J. Opt. Soc. Am. B 32 2257
Google Scholar
[40] Jia H P, Yang R C, Tian J P, Zhang W M 2019 Appl. Opt. 58 912
Google Scholar
[41] 杨光晔 2014 博士学位论文 (太原: 山西大学)
Yang G Y 2014 Ph. D. Dissertation (Taiyuan: Shanxi University) (in Chinese)
[42] Wang Y, Song L J, Li L 2016 Appl. Opt. 55 7241
Google Scholar
[43] Wang Y, Lu L 2017 Rom. Rep. Phys. 62 205
[44] Yang G Y, Li L, Tian J P 2016 Acta Opt. Sin. 36 0619002
Google Scholar
[45] Yang G Y, Wu F O, Helena E, Lopez A, Christodoulides D N 2020 Opt. Commun. 473 125899
Google Scholar
[46] Agrawal G P 著 (贾东方, 余震虹 译) 2010 (北京: 电子工业出版社) 第103−104页
Agrawal G P (translated by Jia D F, Yu Z H) 2010 Applications of Nonlinear Fiber Optics (Beijing: Publishing House of Electronics Industry) pp103−104 (in Chinese)
[47] Hammani K, Wetzel B, Kibler B, Fatome J, Finot C, Millot G, Akhmediev N, Dudley J M 2011 Opt. Lett. 36 2140
Google Scholar
[48] Dudley J M, Genty G, Dias F. Kibler B, Akhmediev N 2009 Opt. Express 17 21497
Google Scholar
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