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通过数值方法研究了自延迟光、电反馈作用下分布式反馈半导体激光器(DFB-SL)的各种非线性动力学行为. 结果表明, 在不同光反馈强度下DFB-SL输出呈现出单周期、准周期、多周期等多种非线性动力学态. 当外部光反馈达到一定强度后, 激光器输出表现为混沌态; 当光反馈强度较小时, 在不同的电反馈强度下DFB-SL输出也会出现多种非线性动力学态; 当光反馈强度较大时, 改变电反馈强度无法得到单周期的动力学态. 光反馈与电反馈延迟时间也对DFB-SL非线性动力学态有重要影响. 当二者的延迟时间相匹配时激光器的弛豫振荡被增强, 表现为单周期状态, 而在延迟时间不匹配的情况下, 可能引发混沌或不稳定状态. 偏置电流也会对动力学态产生影响, 但随着电流大小单向变化, 动力学态的演化方向不是单一的; 当DFB-SL处于单周期态时, 改变偏置电流会改变单周期振荡频率. 这些发现为自延迟反馈DFB-SL在微波光子信号处理和保密光通信等应用方面提供了重要理论基础, 也为各种非线性科学研究提供了实验手段.
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关键词:
- 分布式反馈半导体激光器 /
- 自延迟反馈 /
- 非线性动力学态
In this paper, various nonlinear dynamic behaviors of distributed feedback semiconductor laser (DFB-SL) subjected to self-delayed optical and electrical feedback are studied numerically. The results show that the DFB-SL output presents a variety of nonlinear dynamic states such as single-period, quasi-period, and multi-period under different optical feedback intensities. When the external light feedback reaches a certain intensity, the laser output enters a chaotic regime. When the optical feedback intensity is small, a variety of nonlinear dynamic states will appear in the DFB-SL output under different electrical feedback intensities. When the optical feedback intensity is large, the single-period dynamic state cannot be obtained by changing the electrical feedback intensity. The optical feedback and electrical feedback delay time also have a significant influence on the nonlinearity of DFB-SL. When their time delays match, the relaxation oscillation of the laser is enhanced and exhibits a single-period state. And time mismatch may lead to chaos or instability. The bias current also affects the dynamic state, however, the direction of evolution of the dynamic states is not unidirectional as the current changes unidirectionally. When the DFB-SL is in a single-period state, changing the bias current will result in the change of the single-cycle oscillation frequency. These findings provide an important theoretical basis for applying the self-delayed feedback DFB-SL to microwave photonic signal processing and secure optical communication, as well as experimental means for conducting various nonlinear scientific researches.[1] Olesen H, Osmundsen J, Tromborg B 1986 IEEE J. Quantum Electron. 22 762
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Cao L P, Deng T, Lin X D, Wu J G, Xia G Q, Wu Z M 2010 Chin. Lasers 37 939
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[3] 刘凌锋, 妙锁霞 2015 光电子·激光 26 86
Google Scholar
Liu L F, Miao S X 2015 J. Optoelectron. Laser 26 86
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[4] Lang R, Kobayashi K 1980 IEEE J. Quantum Electron. 16 347
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[5] Soriano M C, Garcíaojalvo J, Mirasso C R, Fischer I, Soriano M C 2013 Rev. Mod. Phys. 85 421
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[6] Saboureau P, Foing J P, Schanne P 1997 IEEE J. Quantum Electron. 33 1582
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[7] Tang S, Liu J M 2001 IEEE J. Quantum Electron. 37 329
Google Scholar
[8] Hizanidis J, Deligiannidis S, Bogris A, Syvridis D 2010 IEEE J. Quantum Electron. 46 1642
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[9] Chen G C, Lu D, Guo L, Zhao W, Huang Y G, Zhao L J 2019 Optik 180 313
Google Scholar
[10] 包琪 2020 硕士学位论文 (杭州: 杭州电子科技大学)
Bao Q 2020 M. S. Thesis (Hangzhou: Hangzhou Dianzi University
[11] 张梦雨 2022 硕士学位论文 (郑州: 郑州大学)
Zhang M Y 2022 M.S. Thesis (Zhengzhou: Zhengzhou University
[12] Argyris A, Syvridis D, Larger L 2005 Nature 438 343
Google Scholar
[13] 孔慧君, 吴正茂, 吴加贵, 林晓东, 谢瑛珂, 夏光琼 2006 中国激光 17 1490
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Kong H J, Wu Z M, Wu J G, Lin X D, Xie Y K, Xia G Q 2006 Chin. Lasers 17 1490
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[14] Argyris A, Hamacher M, Chlouverakis K E 2008 Phys. Rev. Lett. 100 194101
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[15] Vicente R, Dauden J, Colet P, Toral R 2005 IEEE J. Quantum Electron. 41 541
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[16] 蔺玉雪, 高慧, 王龙生, 李腾龙, 赵彤, 常朋发, 王安帮, 王云才 2024 中国激光 51 0606002
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[17] Heil T, Fischer I, Elsasser W, Gavrielides A 2001 Phys. Rev. Lett. 87 243901
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[18] Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815
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[19] Lin F Y, Liu J M 2003 IEEE J. Quantum Electron. 39 562
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[21] Mork J, Tromborg B, Mark J 1992 IEEE J. Quantum Electron. 28 93
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[22] 李增, 冯玉玲, 王晓茜, 姚治海 2018 67 140501
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Li Z, Feng Y L, Wang X Q, Yao Z H 2018 Acta Phys. Sin. 67 140501
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[23] 戈杉杉, 王腾午, 戈静怡, 周沛, 李念强 2023 72 164201
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Ge S S, Wang T W, Ge J Y, Zhou P, Li N Q 2023 Acta Phys. Sin. 72 164201
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[24] 张依宁, 徐艾诗, 冯玉玲, 赵振明, 姚治海 2020 光学学报 40 125
Zhang Y N, Xu A S, Feng Y L, Zhao Z M, Yao Z H 2020 Acta Opt. Sin. 40 125
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图 1 DFB-SL自延迟光电反馈结构图
Fig. 1. Diagram of the DFB-SL self-delay photoelectric feedback structure.
图 2 DFB-SL自延迟光反馈结构图
Fig. 2. Diagram of the DFB-SL self-delay optical feedback structure.
图 3 不同光反馈强度下DFB激光器的S-N图及其输出的时序图和频谱图 (a1)—(a3) $\kappa = 0.018$; (b1)—(b3) $\kappa = 0.044$; (c1)—(c3) $\kappa = 0.059$
Fig. 3. Under different optical feedback intensities, S-N plots of DFB lasers and their output timing plots and spectrograms: (a1)–(a3) $\kappa = 0.018$; (b1)–(b3) $\kappa = 0.044$; (c1)–(c3) $\kappa = 0.059$.
图 4 不同光反馈时间下DFB激光器的S-N图及其输出的时序图和频谱图 (a1)—(a3) $\tau = 2.18 {\text{ ns}}$; (b1)—(b3) $\tau = 2.21 {\text{ ns}}$
Fig. 4. Under different optical feedback time, S-N plots of DFB lasers and their output timing plots and spectrograms: (a1)–(a3) $\tau = 2.18 {\text{ ns}}$; (b1)–(b3) $\tau = 2.21 {\text{ ns}}$.
图 5 DFB-SL自延迟电反馈结构图
Fig. 5. Diagram of the DFB-SL self-delay electricity feedback structure.
图 6 不同光电反馈强度下DFB激光器的S-N图及其输出的时序图和频谱图 (a1)—(a3) $\zeta = 0.045$; (b1)—(b3) $\zeta = 0.0555$; (c1)—(c3) $\zeta = 0.0973$; (d1)—(d3) $\zeta = 0.19$
Fig. 6. Under different feedback intensity, S-N plots of DFB lasers and their output timing plots and spectrograms: (a1)–(a3) $\zeta = 0.045$; (b1)–(b3) $\zeta = 0.0555$; (c1)–(c3) $\zeta = 0.0973$; (d1)–(d3) $\zeta = 0.19$.
图 7 不同光-电反馈强度下的动力学态相图
Fig. 7. Effect of photo-electrical feedback intensity on laser dynamics.
图 8 不同光-电反馈延迟时间条件下的动力学态相图
Fig. 8. Effect of optical-electrical feedback delay time on laser dynamics.
图 9 不同偏置电流下DFB激光器的S-N图及激光器输出的时序图和频谱图 (a1)—(a3) $I = 40 {\text{ mA}}$; (b1)—(b3) $I = 36 {\text{ mA}}$; (c1)—(c3) $I = 20 {\text{ mA}}$
Fig. 9. Under different bias current, S-N plots of DFB lasers and their output timing plots and spectrograms: (a1)–(a3) $I = 40 {\text{ mA}}$; (b1)–(b3) $I = 36 {\text{ mA}}$; (c1)–(c3) $I = 20 {\text{ mA}}$.
表 1 参数符号及取值
Table 1. Parameter symbols and their values.
参数 符号 取值 单位 光子数 $ S\left(t\right) $ — — 载流子数 $ N\left(t\right) $ — — 光功率 $ P\left(t\right) $ — — 微分增益 $ {G}_{{\mathrm{N}}} $ $ (3—4) \times {10}^{4}$ s–1 透明载流子数 $ {N}_{0} $ $ 1.36\times{10}^{8} $ — 阈值光子数 $ {S}_{0} $ $ 4.04\times{10}^{4} $ — 激光腔内反馈时间 $ {\tau }_{{\mathrm{i}}{\mathrm{n}}} $ 9 $ {\mathrm{p}}{\mathrm{s}} $ 光子寿命 $ {\tau }_{{\mathrm{p}}} $ 2 $ {\mathrm{p}}{\mathrm{s}} $ 载流子寿命 $ {\tau }_{{\mathrm{e}}} $ 2 $ {\mathrm{n}}{\mathrm{s}} $ 电子电荷 $ e $ $ 1.6\times{10}^{-19}$ C 限制因子 $ \varGamma $ 0.5 — 自发辐射因子 $ \beta $ $1\times {10}^{-5} $ — 饱和增益因子 $ \varepsilon $ $ (7—8)\times{10}^{-8} $ — 激光器的中心频率 $ {\omega }_{0} $ $ 1.938 \times {10}^{14}~{\mathrm{Hz}} $ — 线宽增强因子 $ a $ 4.5 — 
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[1] Olesen H, Osmundsen J, Tromborg B 1986 IEEE J. Quantum Electron. 22 762
Google Scholar
[2] 操良平, 邓涛, 林晓东, 吴加贵, 夏光琼, 吴正茂 2010 中国激光 37 939
Google Scholar
Cao L P, Deng T, Lin X D, Wu J G, Xia G Q, Wu Z M 2010 Chin. Lasers 37 939
Google Scholar
[3] 刘凌锋, 妙锁霞 2015 光电子·激光 26 86
Google Scholar
Liu L F, Miao S X 2015 J. Optoelectron. Laser 26 86
Google Scholar
[4] Lang R, Kobayashi K 1980 IEEE J. Quantum Electron. 16 347
Google Scholar
[5] Soriano M C, Garcíaojalvo J, Mirasso C R, Fischer I, Soriano M C 2013 Rev. Mod. Phys. 85 421
Google Scholar
[6] Saboureau P, Foing J P, Schanne P 1997 IEEE J. Quantum Electron. 33 1582
Google Scholar
[7] Tang S, Liu J M 2001 IEEE J. Quantum Electron. 37 329
Google Scholar
[8] Hizanidis J, Deligiannidis S, Bogris A, Syvridis D 2010 IEEE J. Quantum Electron. 46 1642
Google Scholar
[9] Chen G C, Lu D, Guo L, Zhao W, Huang Y G, Zhao L J 2019 Optik 180 313
Google Scholar
[10] 包琪 2020 硕士学位论文 (杭州: 杭州电子科技大学)
Bao Q 2020 M. S. Thesis (Hangzhou: Hangzhou Dianzi University
[11] 张梦雨 2022 硕士学位论文 (郑州: 郑州大学)
Zhang M Y 2022 M.S. Thesis (Zhengzhou: Zhengzhou University
[12] Argyris A, Syvridis D, Larger L 2005 Nature 438 343
Google Scholar
[13] 孔慧君, 吴正茂, 吴加贵, 林晓东, 谢瑛珂, 夏光琼 2006 中国激光 17 1490
Google Scholar
Kong H J, Wu Z M, Wu J G, Lin X D, Xie Y K, Xia G Q 2006 Chin. Lasers 17 1490
Google Scholar
[14] Argyris A, Hamacher M, Chlouverakis K E 2008 Phys. Rev. Lett. 100 194101
Google Scholar
[15] Vicente R, Dauden J, Colet P, Toral R 2005 IEEE J. Quantum Electron. 41 541
Google Scholar
[16] 蔺玉雪, 高慧, 王龙生, 李腾龙, 赵彤, 常朋发, 王安帮, 王云才 2024 中国激光 51 0606002
Google Scholar
Lin Y X, Gao H, Wang L S, Li T L, Zhao T, Chang P F, Wang A B, Wang Y C 2024 Chin. Lasers 51 0606002
Google Scholar
[17] Heil T, Fischer I, Elsasser W, Gavrielides A 2001 Phys. Rev. Lett. 87 243901
Google Scholar
[18] Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815
Google Scholar
[19] Lin F Y, Liu J M 2003 IEEE J. Quantum Electron. 39 562
Google Scholar
[20] 韩韬, 刘香莲, 李璞, 郭晓敏, 郭龑强, 王云才 2017 66 124203
Google Scholar
Han T, Liu X L, Li P, Guo X M, Guo Y Q, Wang Y C 2017 Acta Phys. Sin. 66 124203
Google Scholar
[21] Mork J, Tromborg B, Mark J 1992 IEEE J. Quantum Electron. 28 93
Google Scholar
[22] 李增, 冯玉玲, 王晓茜, 姚治海 2018 67 140501
Google Scholar
Li Z, Feng Y L, Wang X Q, Yao Z H 2018 Acta Phys. Sin. 67 140501
Google Scholar
[23] 戈杉杉, 王腾午, 戈静怡, 周沛, 李念强 2023 72 164201
Google Scholar
Ge S S, Wang T W, Ge J Y, Zhou P, Li N Q 2023 Acta Phys. Sin. 72 164201
Google Scholar
[24] 张依宁, 徐艾诗, 冯玉玲, 赵振明, 姚治海 2020 光学学报 40 125
Zhang Y N, Xu A S, Feng Y L, Zhao Z M, Yao Z H 2020 Acta Opt. Sin. 40 125
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