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Soliton is a universal format of nonlinear wave propagation in nature. Soliton can maintain its shape during propagation. This unique property has been widely observed in plasma physics, high energy electromagnetics, hydrodynamics, and nonlinear optics. Soliton interactions can reflect collective dynamic behaviors in complex nonlinear systems, showing significant basic research value. Passive mode-locked laser is an ideal platform for studying soliton interaction. The attraction and repulsion between two optical solitons can form soliton molecules. Their properties have been intensively studied by optical spectral analysis. However, conventional optical spectrum analyzers show low resolution and long average time. Time-stretched dispersive Fourier transformation (TS-DFT) is an emerging-powerful measurement technology, which can map the spectrum of an optical pulse to a temporal waveform under sufficient dispersion. The TS-DFT makes it possible to detect the dynamics of the solitons in real time. Based on TS-DFT, the internal dynamics of the solitons in Ti:sapphire femtosecond laser is studied in experiment. By changing the pump power, the stable soliton molecules with a separation of 180 fs and the weak phase oscillatory soliton molecules with a separation of 105 fs are observed. The amplitude in the weak oscillation state is merely 0.05 rad. We also find that the soliton molecules in stable state can transform into phase sliding state under environmental perturbation. These optical soliton molecules with a binding separation of 100 fs are of great significance for studying the short-range nonlinear interactions of solitons.
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图 1 基于钛宝石激光器的TS-DFT实验装置图 (OC, 输出耦合镜; P, 棱镜对; CM, 啁啾镜对; L, 透镜; Ti:S, 钛宝石晶体; M, 平面镜; FC, 光纤耦合器; SMF, 单模光纤; PD, 光电探测器; OSC, 高速示波器)
Figure 1. TS-DFT experimental setup based on Ti: sapphire laser (OC, output coupler; P, prim; CM, chirped mirror; L, lens; Ti:S, Ti:sapphire; M, mirror; FC, fiber coupler; SMF, single-mode fiber; PD, photodetecter; OSC, high-speed oscilloscope).
图 2 实时观察105 fs时间间隔孤子分子参数图 (a) 孤子分子的光谱演化图样和对应的单帧光谱图; (b) 自相关的演化图和对应的单帧自相关曲线; (c) 相对相位演化图; (d) 光谱仪与DFT测到的光谱对比图
Figure 2. Experimental real-time observation soliton molecules with a separation of 105 fs: (a) Interferograms of a soliton bound state and its single-shot spectrum; (b) the field autocorrelations of the momentary bound state; (c) relative phase evolution diagram; (d) optical spectrum measured by OSA and DFT.
图 3 实时观察180 fs时间间隔孤子分子数据图 (a) 孤子分子的光谱演化图; (b)自相关演化图; (c)相对相位演化图; (d)光谱仪与DFT测到的光谱对比图
Figure 3. Experimental real-time observation stable soliton molecules with a separate of 180 fs: (a) Interferograms of a soliton bound state; (b) the field autocorrelations of the momentary bound state; (c) relative phase evolution diagram; (d) optical spectra measured by OSA and DFT.
图 4 滑动相位孤子分子数据图 (a) 光谱演化图; (b) 自相关演化图; (c) 相对相位演化图; (d)光谱仪与DFT测到的光谱对比图
Figure 4. Experimental real-time observation soliton molecules with a sliding phase: (a) Interferograms of a soliton bound state; (b) the field autocorrelations of the momentary bound state; (c) relative phase evolution diagram; (d) optical spectra measured by OSA and DFT.
图 5 孤子分子的动态互作用平面
Figure 5. Dynamics of the SM mapped into the interaction plane.
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[1] Kivshar Y S, Agrawal G P 2003 Optical Solitons: from Fibers to Photonic Crystals (SanDiego: Academic Press) p5
[2] Wu Y, Deng L 2004 Phys. Rev. Lett. 93 143904
Google Scholar
[3] Melo F, Douady S 1993 Phys. Rev. Lett. 71 3283
Google Scholar
[4] Denschlag J 2000 Science 287 97
Google Scholar
[5] Kibler B, Fatome J, Finot C, Millot G, Dias F, Genty G 2010 Nat. Phys. 6 790
Google Scholar
[6] 黄诗盛, 王勇刚, 李会权, 林荣勇, 闫培光 2014 63 084202
Google Scholar
Huang S S, Wang Y G, Li H Q, Lin R Y, Yan P G 2014 Acta Phys. Sin. 63 084202
Google Scholar
[7] Shi H S, Song Y J, Wang Q Y, Zhao L M, Hu M L 2018 Opt. Lett. 43 1623
Google Scholar
[8] 徐佳, 吴思达, 刘江, 孙若愚, 王璞 2013 中国激光 40 0702003
Google Scholar
Xu J, Wu S D, Liu J, Sun R Y, Wang P 2013 Chin. J. Lasers 40 0702003
Google Scholar
[9] Pang M, He W, Jiang X, Russell P S J 2016 Nat. Photonics 10 454
Google Scholar
[10] 王光斗, 杨光, 刘艳格, 王志 2017 中国激光 44 83
Wang G D, Yang G, Liu Y G, Wang Z 2017 Chin. J. Lasers 44 83
[11] Huang Q, Wang T, Zou C, AlAraimi M, Rozhin A, Mou C 2018 Chin. Opt. Lett. 16 020019
Google Scholar
[12] Zhao C, Huang Q, Al Araimi M, Rozhin A, Sergeyev S, Mou C 2019 Chin. Opt. Lett. 17 020012
Google Scholar
[13] Hause A, Hartwig H, Böhm M, Mitschke F 2008 Phys. Rev. A 78 63817
Google Scholar
[14] Wang Z Q, Nithyanandan K, Coillet A, Tchofo-Dinda P, Grelu P 2019 Nat. Commun. 10 830
Google Scholar
[15] 王志, 贺瑞敬, 刘艳格 2019 中国激光 46 13
Wang Z, He R J, Liu Y G 2019 Chin. J. Lasers 46 13
[16] 魏志伟, 刘萌, 崔虎, 罗爱平, 徐文成, 罗智超 2019 激光与光电子学进展 56 79
Wei Z W, Liu M, Cui H, Luo A P, Xu W C, Luo Z C 2019 Laser & Optoelectronics Progress 56 79
[17] He W, Pang M, Yeh D H, Huang J, Menyuk C R, Russell P S J 2019 Nat. Commun. 10 1
Google Scholar
[18] Liu X M, Popa D, Akhmediev N 2019 Phys. Rev. Lett. 123 093901
Google Scholar
[19] Liu X M, Pang M 2019 Laser Photonics Rev. 13 1800333
Google Scholar
[20] Peng J, Boscolo S, Zhao Z, Zeng H 2019 Sci. Adv. 5 eaax1110
Google Scholar
[21] Goda K, Jalali B 2013 Nat. Photonics 7 102
Google Scholar
[22] Goda K, Solli D R, Tsia K K, Jalali B 2009 Phys. Rev. A 80 043821
Google Scholar
[23] Malomed BA 1991 Phys. Rev. A 44 6954
Google Scholar
[24] Krupa K, Nithyanandan K, Andral U, Tchofo-Dinda P, Grelu P 2017 Phys. Rev. Lett. 118 243901
Google Scholar
[25] Liu X M, Yao X K, Cui Y D 2018 Phys. Rev. Lett. 121 023905
Google Scholar
[26] 廖睿, 文锦辉, 刘智刚, 邓莉, 赖天树, 林位株 2002 中国激光 29 53
Liao R, Wen J H, Liu Z G, Deng L, Lai T S, Lin W Z 2002 Chin. J. Lasers 29 53
[27] 王胭脂, 邵建达, 董洪成, 晋云霞, 王清月 2011 60 018101
Google Scholar
Wang Y Z, Shao J D, Dong H C, Jin Y X, Wang Q Y 2011 Acta Phys. Sin. 60 018101
Google Scholar
[28] 范海涛, 王胭脂, 王兆华, 叶蓬, 胡国行, 秦爽, 何会军, 易葵, 邵建达, 魏志义 2015 64 144204
Google Scholar
Fan H T, Wang Y Z, Wang Z H, Ye P, Hu G H, Qin S, He H J, Yi K, Shao J D, Wei Z Y 2015 Acta Phys. Sin. 64 144204
Google Scholar
[29] Herink G, Kurtz F, Jalali B, Solli D R, Ropers C 2017 Science 356 50
Google Scholar
[30] Lederer M J, Luther-Davies B, Tan H H, Jagadish C, Soto-Crespo J M 1999 J. Opt. Soc. Am. B 16 895
Google Scholar
[31] Han X, Wu J, Zeng H 2008 Opt. Express 16 3686
Google Scholar
[32] Kurtz F, Ropers C, Herink G 2020 Nat. Photonics 14 9
Google Scholar
[33] Zavyalov A, Iliew R, Egorov O, Lederer F 2009 Phys. Rev. A 79 053841
Google Scholar
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