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As an important part of optical sources, nanolasers have a prominent influence in photonic circuit integration, and their nonlinear dynamics has become one of the research hotspots in recent years. In this work, we investigate the time-delay signature and bandwidth characteristics in a cascade-coupled nanolaser system, in which the master nanolaser is connected to an external feedback cavity and injected into the intermediate nanolaser and the slave nanolaser sequentially. The 0-1 chaos test is introduced to quantify the dynamics of the nanolaser, which can accurately distinguish whether the laser is in a chaotic state, and the autocorrelation function is used to analyze the time-delay characteristics in the laser output signal. This type of calculation has the advantages of fast operation speed, high accuracy and anti-noise robustness. The lower the autocorrelation value, the more difficult it is to extract useful information from the chaotic random sequence. The bandwidth is defined as a value where the range between DC and frequency contains 80% of the spectral power, a value that is only applicable to chaotic states. In the simulation, we compare and analyze the two cases of whether the intermediate nanolaser has a peak with obvious time-delay signature. The research results show that by selecting appropriate system parameters, the slave nanolaser can always output a broadband chaotic signal without obvious time-delay signature. Under the condition of a certain injection intensity, by changing the frequency detuning parameter, the intermediate nanolaser has an obvious time-delay signature, and then the slave nano-laser can output chaotic signals which can suppress time-delay signature and enhance bandwidth in a small parameter interval. When the time-delay signal of the intermediate nanolaser is completely hidden, the slave nanolaser can achieve the suppression of the time-delay signature in a larger parameter plane, meanwhile the bandwidth is significantly enhanced. In addition, by plotting the two-dimensional spatial distribution diagram and bandwidth line diagram of the output from the nanolaser under frequency detuning and injection intensity, it is determined that the nanolaser can simultaneously suppress the delay characteristics and enhance the bandwidth under chaotic signals. This provides an important theoretical basis for realizing the practical applications in secrecy-enhanced chaotic optical communication.
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Keywords:
- nanolaser /
- chaos /
- time delay signature /
- bandwidth
[1] Li N Q, Susanto H, Cemlyn B, Henning I D, Adams M J 2017 Opt. Lett. 42 3494
Google Scholar
[2] Li N Q, Pan W, Yan L S, Luo B, Xu M F, Tang Y L, Jiang N, Xiang S Y, Zhang Q 2012 J. Opt. Soc. Am. B 29 101
Google Scholar
[3] Lin F Y, Liu J M 2004 IEEE J. Select. Topics Quantum Electron. 10 991
Google Scholar
[4] Tsay H L, Wang C Y, Chen J D, Lin F Y 2020 Opt. Express 28 24037
Google Scholar
[5] Qin J, Zhao Q C, Xu D J, Yin H X, Chang Y, Huang D G 2016 Mod. Phys. Lett. B 30 1650199
Google Scholar
[6] Uchida A, Amano K, Inoue M, et al. 2008 Nat. Photon 2 728
Google Scholar
[7] Rasmussen T S, Mork J 2021 Opt. Express 29 14182
Google Scholar
[8] Simpson T B, Liu J M, Gavrielides A, Kovanis V, Alsing P M 1995 Phys. Rev. A 51 4181
Google Scholar
[9] Zhang Y, Chen Q, Li S, Yu J, Xu H, Yin F, Dai Y, Xu K 2023 2023 21st International Conference on Optical Communications and Networks (ICOCN) Qufu, China, 2023-07-31 pp1–3
[10] Wu J G, Xia G Q, Tang X, Lin X D, Deng T, Fan L, Wu Z M 2010 Opt. Express 18 6661
Google Scholar
[11] Jafari A, Sedghi H, Mabhouti K, Behnia S 2011 Opt. Commun. 284 3018
Google Scholar
[12] Udaltsov V S, Goedgebuer J P, Larger L, Cuenot J B, Levy P, Rhodes W T 2003 Phys. Lett. A 308 54
Google Scholar
[13] Tartwijk jk G H M V, Lenstra D 1995 Quantum Semiclass. Opt. 7 87
Google Scholar
[14] Vicente R, Daudén J, Colet P, Toral R 2005 IEEE J. Quantum Electron. 41 541
Google Scholar
[15] Jiang N, Wang Y J, Zhao A K, Liu S Q, Zhang Y Q, Chen L, Li B C, Qiu K 2020 Opt. Express 28 1999
Google Scholar
[16] Zhao Q C, Wang Y C, Wang A B 2009 Appl. Opt. 48 3515
Google Scholar
[17] Rontani D, Locquet A, Sciamanna M, Citrin D S 2007 Opt. Lett. 32 2960
Google Scholar
[18] Wu J G, Xia G Q, Wu Z M 2009 Opt. Express 17 20124
Google Scholar
[19] Sakuraba R, Kanno K, Iwakawa K, Uchida A 2013 Frontiers in Optics Orlando, Florida, 2013 pFM3C.4
[20] Hong Y, Quirce A, Wang B, Ji S, Panajotov K, Spencer P S 2016 IEEE J. Quantum Electron. 52 2400508
Google Scholar
[21] Mu P H, He P F, Li N Q 2019 IEEE Access 7 11041
Google Scholar
[22] Sattar Z A, Shore K A 2015 J. Lightwave Technol. 33 3028
Google Scholar
[23] Sattar Z A, Shore K A 2015 IEEE J. Sel. Top. Quantum Electron. 21 1800106
Google Scholar
[24] Sattar Z A, Kamel N A, Shore K A 2016 IEEE J. Quantum Electron. 52 1200108
Google Scholar
[25] Han H, Shore K A 2016 IEEE J. Quantum Electron. 52 2000306
Google Scholar
[26] Elsonbaty A, Hegazy S F, Obayya S S A 2018 Opt. Laser Eng. 107 342
Google Scholar
[27] Qu Y, Xiang S Y, Wang Y, Lin L, Wen A J, Hao Y 2019 IEEE J. Quantum Electron. 55 2000407
Google Scholar
[28] Sattar Z A, Shore K A 2016 IEEE J. Quantum Electron. 52 1100108
Google Scholar
[29] Nguimdo R M, Soriano M C, Colet P 2011 Opt. Lett. 36 4332
Google Scholar
[30] Nguimdo R M, Colet P, Larger L, Pesquera L 2011 Phys. Rev. Lett. 107 034103
Google Scholar
[31] Rontani D, Locquet A, Sciamanna M, Citrin D, Ortin S 2009 IEEE J. Quantum Electron. 45 879
Google Scholar
[32] Priyadarshi S, Hong Y, Pierce I, Shore K A 2013 IEEE J. Sel. Top. Quantum Electron. 19 1700707
Google Scholar
[33] Li X F, Pan W, Luo B, Ma D 2006 IEEE J. Quantum Electron. 42 953
Google Scholar
[34] Jiang P, Zhou P, Li N Q, Mu P H, Li X F 2020 Opt. Express 28 26421
Google Scholar
[35] Gottwald G A, Melbourne I 2009 SIAM J. Appl. Dyn. Syst. 8 129
Google Scholar
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图 1 三级联耦合NL的示意图. MNL, 主纳米激光器; INL, 中间纳米激光器; SNL, 从纳米激光器; FM, 反射镜; VA, 可变衰减器; OI, 光隔离器
Figure 1. Schematic diagram of three cascade-coupled NLs. MNL, master NL; INL, intermediate NL; SNL, slave NL; FM, reflector; VA, variable attenuator; OI, optical isolator.
图 2 $ {\tau }_{{\mathrm{d}}}=0.2\;{\mathrm{n}}{\mathrm{s}} $时, 不同反馈参数下MNL的时序图 (a) 0.001; (b) 0.003; (c) 0.005; (d) 0.014
Figure 2. Timeseries of the MNL with different feedback parameters at $ {\tau }_{{\mathrm{d}}}=0.2\;{\mathrm{n}}{\mathrm{s}} $: (a) 0.001; (b) 0.003; (c) 0.005; (d) 0.014.
图 3 反馈时延和反馈耦合因子下MNL混沌0-1测试的二维彩图
Figure 3. Two-dimensional color map of the 0-1 test for chaos of MNL with feedback delay-time and feedback coupling factor.
图 4 MNL (a1)—(a3), INL (b1)—(b3)和SNL (c1)—(c3)的时间序列、功率谱和ACF, 参数设置为$ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1, $ {{{\Delta }}f}_{2} $= 15 GHz和$ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $= 85 ns–1
Figure 4. Timeseries, power spectrum and ACF of the MNL (a1)–(a3), INL (b1)–(b3) and SNL (c1)–(c3), parameter settings are $ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1, $ {{{\Delta }}f}_{2} $= 15 GHz and $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $= 85 ns–1.
图 5 情况1条件下SNL的0-1混沌二维彩图, 参数设置为$ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1
Figure 5. Two-dimensional color map of the 0-1 test for chaos of MNL under the condition of case1, parameter settings are $ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1.
图 6 在情况1的条件下, 参数平面($ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $, $ {{\Delta }}{f}_{2} $)中SNL的TDS值二维图, 参数设置为$ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1
Figure 6. Two-dimensional map of TDS value of SNL in the parameter plane ($ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $, $ {{\Delta }}{f}_{2} $) under the condition of case1, parameter settings are $ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1.
图 7 INL和SNL在不同注入强度和频率失谐下的带宽, 参数设置为$ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1
Figure 7. Bandwidth of INL and SNL with different injection strength and frequency detuning, parameter settings are $ {{\Delta }}{f}_{1} $= 0 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1.
图 8 MNL (a1)—(a3), INL (b1)—(b3)和SNL (c1)—(c3)的时间序列、功率谱和ACF, 参数设置为$ {{\Delta }}{f}_{1} $= 15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1, $ {{{\Delta }}f}_{2} $= 15 GHz和$ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $= 85 ns–1
Figure 8. Timeseries, power spectrum and ACF of the MNL (a1)–(a3), INL (b1)–(b3) and SNL (c1)–(c3), parameter settings are $ {{\Delta }}{f}_{1} $= 15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1, $ {{{\Delta }}f}_{2} $= 15 GHz and $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $= 85 ns–1.
图 9 情况2条件下SNL的0-1混沌二维彩图, 参数设置为$ {{\Delta }}{f}_{1} $= 15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1}$= 120 ns–1
Figure 9. Two-dimensional color map of the 0-1 test for chaos of MNL under the condition of case2, parameter settings are $ {{\Delta }}{f}_{1} $= 15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $= 120 ns–1.
图 10 在情况二条件下, 参数平面($ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $, $ {{{\Delta }}f}_{2} $)中SNL的TDS值二维图, 参数设置为: $ {{\Delta }}{f}_{1} $=15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $=120 ns–1
Figure 10. Two-dimensional map of TDS value of SNL in the parameter plane ($ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}2} $, $ {{\Delta }}{f}_{2} $) under the condition of case 2, parameter settings are $ {{\Delta }}{f}_{1} $=15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $=120 ns–1.
图 11 INL和SNL在不同注入强度和频率失谐下的带宽, 参数设置为$ {{\Delta }}{f}_{1} $=15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $=120 ns–1
Figure 11. Bandwidth of INL and SNL with different injection strength and frequency detuning, parameter settings are $ {{\Delta }}{f}_{1} $=15 GHz, $ {k}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}1} $=120 ns–1.
参数 符号 参考值 工作波长/nm $ \lambda $ $ 1591 $ 阈值电流/mA $ {I}_{{\mathrm{t}}{\mathrm{h}}} $ $ 1.1 $ 反馈腔长/μm $ L $ $ 1.39 $ 活性层体积/cm3 $ {V}_{{\mathrm{a}}} $ $ 3.96\times {10}^{-13} $ 局限因子 $ \varGamma $ $ 0.645 $ 品质因子 $ Q $ $ 428 $ Purcell因子 $ F $ $ 14 $ 自发辐射耦合因子/($ {{\mathrm{c}}{\mathrm{m}}}^{3}\cdot {{\mathrm{s}}}^{-1} $) $ \beta $ $ 0.05 $ 微分增益/($ {{\mathrm{c}}{\mathrm{m}}}^{3}\cdot {{\mathrm{s}}}^{-1} $) $ {g}_{{\mathrm{n}}} $ $ 1.65\times {10}^{-6} $ 光子寿命/ps $ {\tau }_{{\mathrm{p}}} $ $ 0.36 $ 反馈时延/ns $ {\tau }_{{\mathrm{d}}} $ $ 0.2 $ 载流子寿命/ns $ {\tau }_{{\mathrm{n}}} $ $ 1 $ 透明载流子数/$ {{\mathrm{c}}{\mathrm{m}}}^{-3} $ $ {N}_{0} $ $ 1.1\times {10}^{18} $ 增益饱和因子/$ {{\mathrm{c}}{\mathrm{m}}}^{3} $ $ \varepsilon $ $ 2.3\times {10}^{-17} $ 折射率 $ n $ $ 3.4 $ 线宽增强因子 $ \alpha $ $ 5 $ 镜面反射率 $ {R}_{{\mathrm{e}}{\mathrm{x}}{\mathrm{t}}} $ $ 0.95 $ 激光器腔面反射率 $ R $ $ 0.85 $ 自由空间中的光速/(m·s–1) $ c $ $ 3\times {10}^{8} $ 注入系数 $ {R}_{{\mathrm{i}}{\mathrm{n}}{\mathrm{j}}} $ $ 0-0.1 $ 反馈耦合因子 $ f $ $ 0-0.9 $ 
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[1] Li N Q, Susanto H, Cemlyn B, Henning I D, Adams M J 2017 Opt. Lett. 42 3494
Google Scholar
[2] Li N Q, Pan W, Yan L S, Luo B, Xu M F, Tang Y L, Jiang N, Xiang S Y, Zhang Q 2012 J. Opt. Soc. Am. B 29 101
Google Scholar
[3] Lin F Y, Liu J M 2004 IEEE J. Select. Topics Quantum Electron. 10 991
Google Scholar
[4] Tsay H L, Wang C Y, Chen J D, Lin F Y 2020 Opt. Express 28 24037
Google Scholar
[5] Qin J, Zhao Q C, Xu D J, Yin H X, Chang Y, Huang D G 2016 Mod. Phys. Lett. B 30 1650199
Google Scholar
[6] Uchida A, Amano K, Inoue M, et al. 2008 Nat. Photon 2 728
Google Scholar
[7] Rasmussen T S, Mork J 2021 Opt. Express 29 14182
Google Scholar
[8] Simpson T B, Liu J M, Gavrielides A, Kovanis V, Alsing P M 1995 Phys. Rev. A 51 4181
Google Scholar
[9] Zhang Y, Chen Q, Li S, Yu J, Xu H, Yin F, Dai Y, Xu K 2023 2023 21st International Conference on Optical Communications and Networks (ICOCN) Qufu, China, 2023-07-31 pp1–3
[10] Wu J G, Xia G Q, Tang X, Lin X D, Deng T, Fan L, Wu Z M 2010 Opt. Express 18 6661
Google Scholar
[11] Jafari A, Sedghi H, Mabhouti K, Behnia S 2011 Opt. Commun. 284 3018
Google Scholar
[12] Udaltsov V S, Goedgebuer J P, Larger L, Cuenot J B, Levy P, Rhodes W T 2003 Phys. Lett. A 308 54
Google Scholar
[13] Tartwijk jk G H M V, Lenstra D 1995 Quantum Semiclass. Opt. 7 87
Google Scholar
[14] Vicente R, Daudén J, Colet P, Toral R 2005 IEEE J. Quantum Electron. 41 541
Google Scholar
[15] Jiang N, Wang Y J, Zhao A K, Liu S Q, Zhang Y Q, Chen L, Li B C, Qiu K 2020 Opt. Express 28 1999
Google Scholar
[16] Zhao Q C, Wang Y C, Wang A B 2009 Appl. Opt. 48 3515
Google Scholar
[17] Rontani D, Locquet A, Sciamanna M, Citrin D S 2007 Opt. Lett. 32 2960
Google Scholar
[18] Wu J G, Xia G Q, Wu Z M 2009 Opt. Express 17 20124
Google Scholar
[19] Sakuraba R, Kanno K, Iwakawa K, Uchida A 2013 Frontiers in Optics Orlando, Florida, 2013 pFM3C.4
[20] Hong Y, Quirce A, Wang B, Ji S, Panajotov K, Spencer P S 2016 IEEE J. Quantum Electron. 52 2400508
Google Scholar
[21] Mu P H, He P F, Li N Q 2019 IEEE Access 7 11041
Google Scholar
[22] Sattar Z A, Shore K A 2015 J. Lightwave Technol. 33 3028
Google Scholar
[23] Sattar Z A, Shore K A 2015 IEEE J. Sel. Top. Quantum Electron. 21 1800106
Google Scholar
[24] Sattar Z A, Kamel N A, Shore K A 2016 IEEE J. Quantum Electron. 52 1200108
Google Scholar
[25] Han H, Shore K A 2016 IEEE J. Quantum Electron. 52 2000306
Google Scholar
[26] Elsonbaty A, Hegazy S F, Obayya S S A 2018 Opt. Laser Eng. 107 342
Google Scholar
[27] Qu Y, Xiang S Y, Wang Y, Lin L, Wen A J, Hao Y 2019 IEEE J. Quantum Electron. 55 2000407
Google Scholar
[28] Sattar Z A, Shore K A 2016 IEEE J. Quantum Electron. 52 1100108
Google Scholar
[29] Nguimdo R M, Soriano M C, Colet P 2011 Opt. Lett. 36 4332
Google Scholar
[30] Nguimdo R M, Colet P, Larger L, Pesquera L 2011 Phys. Rev. Lett. 107 034103
Google Scholar
[31] Rontani D, Locquet A, Sciamanna M, Citrin D, Ortin S 2009 IEEE J. Quantum Electron. 45 879
Google Scholar
[32] Priyadarshi S, Hong Y, Pierce I, Shore K A 2013 IEEE J. Sel. Top. Quantum Electron. 19 1700707
Google Scholar
[33] Li X F, Pan W, Luo B, Ma D 2006 IEEE J. Quantum Electron. 42 953
Google Scholar
[34] Jiang P, Zhou P, Li N Q, Mu P H, Li X F 2020 Opt. Express 28 26421
Google Scholar
[35] Gottwald G A, Melbourne I 2009 SIAM J. Appl. Dyn. Syst. 8 129
Google Scholar
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