Time delay concealment and unpredictability enhancement of nanolasers under external cavity regulation

Jiang Pei; Zhou Pei; Li Nian-Qiang; Mu Peng-Hua; Li Xiao-Feng

doi:10.7498/aps.70.20210049

Abstract

As an important optical element of the optical integration in the future, nanolasers has been a research hotspot in recent years, and the corresponding structural engineering and output characteristics have been widely investigated. However, the nonlinear dynamical performances of nanolasers are rarely reported. Only some preliminary analyses of the dynamic behavior under the optical feedback, optical injection and mutual injection can be found. Some researches pointed out the future prospect of nanolasers, however, some chaos-based applications have not been explored. Therefore, we numerically investigate chaos dynamics in a nanolaser subjected to optical feedback and in another nanolaser subjected to chaotic injection from the former structure by using single mode rate equation, which includes the Purcell cavity-enhanced spontaneous emission factor F and spontaneous emission coupling factor β. The F denotes the ratio of the spontaneous emission rate into the cavity mode to the total spontaneous emission rate in the bulk medium in the absence of a cavity and β represents the fraction of spontaneous emitted photons which are coupled into cavity mode. Specifically, chaos time delay signature (TDS) and unpredictability are evaluated by the peak size of autocorrelation function (ACF) and permutation entropy (PE) respectively. Such kinds of calculations have the advantage of fast operation speed and anti-noise robustness. The results show that the increasing of bias current and the decreasing of gain saturation factor ε, F and β are beneficial to improving the unpredictability and suppressing TDS because the weak damping of the relaxation oscillation leads to strong oscillation. Large linewidth enhancement factor α will increase the number of laser oscillating modes, sideband modes, the spectral components, and enhance the dispersion effect, which will also weaken the information about outer cavity and improve the complexity of chaos. In addition, the above-mentioned chaos properties can be enhanced by injecting the chaos output from a nanolaser subjected to optical feedback into another (slave) nanolaser, which is due to the nonlinear interaction between the driving chaotic signal and the internal electric field of the slave nanolaser. Finally, two-dimensional maps depicting high unpredictability and TDS concealment in the parameter space of the frequency detuning and the injection strength are obtained. It can be found that unpredictability degree can be enhanced by choosing high detuning frequency and intermediate injection strength in the non-injection locking area. The numerical results pave the way for generating the high-quality chaotic sources on a chip or the photonic integrated circuits based on novel semiconductor nanolaser and its related applications.

Keywords:

Authors and contacts

1.
School of Optoelectronic Science and Engineering, Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
2.
Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province, Key Lab of Modern Optical Technologies of Education Ministry of China, Suzhou 215006, China
3.
Institute of Science and Technology for Opto-Electronic Information, Yantai University, Yantai 264005, China

Corresponding author: Li Nian-Qiang, wan_103301@163.com ; Li Xiao-Feng, xfli@suda.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 62004135, 62001317, 61875143), the Natural Science Research Project of Jiangsu Higher Education Institutions, China (Grant No. 20KJA416001), the Startup Funding of Soochow University, China (Grant No. Q415900119), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20180042), and the Superior Discipline Construction Project of Jiangsu Higher Education Institutions, China

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References

[1]	Argyris A, Syvridis D, Larger L, Annovazzi-Lodi V, Colet P, Fischer I, García-Ojalvo J, Mirasso C R, Pesquera L, Shore K A 2005 Nature 438 343 Google Scholar
[2]	Sciamanna M, Shore K A 2015 Nat. Photonics 9 151 Google Scholar
[3]	Uchida A, Amano K, Inoue M, Hirano K, Naito S, Someya H, Oowada I, Kurashige T, Shiki M, Yoshimori S, Yoshimura K, Davis P 2008 Nat. Photonics 2 728 Google Scholar
[4]	Li N Q, Kim B, Chizhevsky V N, Locquet A, Bloch M, Citrin D S, Pan W 2014 Opt. Express 22 6634 Google Scholar
[5]	Lin F Y, Liu J M 2004 IEEE J. Sel. Top. Quantum Electron. 10 991 Google Scholar
[6]	Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815 Google Scholar
[7]	Locquet A, Kim B, Choi D, Li N Q, Citrin D S 2017 Phy. Rev. A 95 023801 Google Scholar
[8]	Rontani D, Locquet A, Sciamanna M, Citrin D S 2007 Opt. Lett. 32 2960 Google Scholar
[9]	Zhao Q C, Wang Y C, Wang A B 2009 Appl. Opt. 48 3515 Google Scholar
[10]	Kanno K, Uchida A, Bunsen M 2016 Phy. Rev. E 93 032206 Google Scholar
[11]	Li N Q, Pan W, Xiang S Y, Zhao Q C, Zhang L Y, Mu P H 2014 IEEE J. Quantum Electron. 50 766 Google Scholar
[12]	Keller K, Sinn M 2010 Phy. D 239 997 Google Scholar
[13]	Bandt C, Pompe B 2002 Phy. Rev. Lett. 88 174102 Google Scholar
[14]	Xiang S Y, Pan W, Yan L S, Luo B, Zou X H, Jiang N, Wen K H 2011 Opt. Lett. 36 310 Google Scholar
[15]	Xiang S Y, Pan W, Luo B, Yan L S, Zou X H, Jiang N, Yang L, Zhu H N 2011 J. Lightwave Technol. 29 2173 Google Scholar
[16]	Xiang S Y, Pan W, Luo B, Yan L S, Zou X H, Jiang N, Yang L, Li N Q 2011 IEEE J. Quantum Electron. 47 1354 Google Scholar
[17]	Xiang S Y, Pan W, Luo B, Yan L S, Zou X H, Jiang N, Zhu H N 2012 IEEE J. Quantum Electron. 48 1069 Google Scholar
[18]	Xiang S Y, Pan W, Li N Q, Zhang L Y, Zhu H N 2013 Opt. Commun. 311 294 Google Scholar
[19]	Guo X X, Xiang S Y, Zhang Y H, Wen A J, Hao Y 2018 IEEE J. Quantum Electron. 54 2000308 Google Scholar
[20]	Zhang H, Xiang S Y, Zhang Y H, Guo X X 2017 Appl. Opt. 56 6728 Google Scholar
[21]	Li N Q, Pan W, Locquet A, Citrin D S 2015 Opt. Lett. 40 4416 Google Scholar
[22]	Li N Q, Pan W, Xiang S Y, Yan L S, Luo B, Zou X H, Zhang L Y 2013 Opt. Laser Technol. 53 45 Google Scholar
[23]	Hill M T, Oei Y S, Smalbrugge B, Zhu Y C, Vries T D, Veldhoven P J V, Otten F W M V, Eijkemans T J, Turkiewicz J P, Waardt H D, Geluk E J, Kwon S H, Lee Y H, Nötzel R, Smit M K 2007 Nat. Photonics 1 589 Google Scholar
[24]	Lau E K, Lakhani A, Tucker R S, Wu M C 2009 Opt. Express 17 7790 Google Scholar
[25]	Sattar Z A, Shore K A 2015 J. Lightwave Technol. 33 3028 Google Scholar
[26]	Sattar Z A, Shore K A 2015 IEEE J. Sel. Top. Quantum Electron. 21 1800106 Google Scholar
[27]	Sattar Z A, Shore K A 2016 IEEE J. Quantum Electron. 52 1100108 Google Scholar
[28]	Sattar Z A, Kamel N A, Shore K A 2016 IEEE J. Quantum Electron. 52 1200108 Google Scholar
[29]	Han H, Shore K A 2018 IET Optoelectron. 12 25 Google Scholar
[30]	Han H, Shore K A 2016 IEEE J. Quantum Electron. 52 2000306 Google Scholar
[31]	Elsonbaty A, Hegazy S F, Obayya S S A 2018 Opt. Laser. Eng. 107 342 Google Scholar
[32]	Qu Y, Xiang S Y, Wang Y, Lin L, Wen A J, Hao Y 2019 IEEE J. Quantum Electron. 55 2000407 Google Scholar
[33]	Jiang P, Zhou P, Li N Q, Mu P H, Li X F 2020 Opt. Express 28 26421 Google Scholar
[34]	Chao M, Wang D M, Wang L S, Sun Y C, Han H, Guo Y Y, Jia Z W, Wang Y C, Wang A B 2020 Opt. Commun. 456 124702 Google Scholar
[35]	阎娟, 潘炜, 李念强, 张力月, 刘庆喜 2016 65 204203 Google Scholar Yan J, Pan W, Li N Q, Zhang L Y, Liu Q X 2016 Acta Phys. Sin. 65 204203 Google Scholar
[36]	Zhou P, Fang Q, Li N Q 2020 Opt. Lett. 45 399 Google Scholar
[37]	苏斌斌, 陈建军, 吴正茂, 夏光琼 2017 66 244206 Google Scholar Su B B, Chen J J, Wu Z M, Xia G Q 2017 Acta. Phys. Sin. 66 244206 Google Scholar
[38]	Uchida A 2012 Optical Communication with Chaotic Lasers: Applications of Nonlinear Dynamics and Synchronization (Berlin: Wiley) p195

Cited By

图 1 (a1), (a2)光反馈下的纳米激光器时序; (b1), (b2) ACF曲线; (c1), (c2) PE曲线. 反馈耦合因子: (a1)—(c1)$f = 0.02$; (a2)—(c2)$ f = 0.06$

Figure 1. (a1), (a2) Time series; (b1), (b2) ACF curve; (c1), (c2) PE curve of a nanolaser under optical feedback. The feedback coupling fraction: (a1)–(c1) $ f = 0.02 $; (a2)–(c2) $ f = 0.06 $.

DownLoad: Full-Size Img PowerPoint

图 2 光反馈纳米激光器　(a) TDS与 (b) H[P ]随着$ \alpha $, $ \varepsilon $的变化

Figure 2. (a) TDS and (b) H[P ] of a nanolaser subjected to optical feedback as functions of $ \alpha $, $ \varepsilon $.

DownLoad: Full-Size Img PowerPoint

图 3 光反馈纳米激光器　(a) TDS与 (b) H[P] 随着$ F $的变化

Figure 3. (a) TDS and (b) H[P] of a nanolaser subjected to optical feedback as functions of $ F$.

DownLoad: Full-Size Img PowerPoint

图 4 光反馈纳米激光器　(a) TDS与 (b) H[P] 随着偏置电流的变化

Figure 4. (a) TDS and (b) H[P] of a nanolaser subjected to optical feedback as functions of the bias current.

DownLoad: Full-Size Img PowerPoint

图 5 混沌光注入下从纳米激光器的　(a1), (a2)时序; (b1), (b2)ACF曲线; (c1), (c2) PE曲线. 注入强度(a1)—(c1)$ {k_{\rm{r}}} = 100\;{\rm{n}}{{\rm{s}}^{ - 1}} $; (a2)—(c2)$ {k_{\rm{r}}} = 200\;{\rm{n}}{{\rm{s}}^{ - 1}} $; $ \Delta f = 25\;{\rm{GHz}} $

Figure 5. (a1), (a2) Time series; (b1), (b2) ACF curve; (c1), (c2) PE curve of the slave nanolaser under chaotic optical injection. The injection strength (a1)–(c1) $ {k_{\rm{r}}} = 100\;{\rm{n}}{{\rm{s}}^{ - 1}} $; (a2)–(c2) $ {k_{\rm{r}}} = 200\;{\rm{n}}{{\rm{s}}^{ - 1}} $;$ \Delta f = 25\;{\rm{GHz}} $.

DownLoad: Full-Size Img PowerPoint

图 6 在不同F, β及偏置电流下, 从纳米激光器混沌输出的H[P ]随失谐频率及注入强度变化的二维映射　(a1)—(a3)${I_{{\rm{dc}}}} = $$ 2{I_{{\rm{th}}}}$; (b1)—(b3)$ {I_{{\rm{dc}}}} = 4{I_{{\rm{th}}}} $. (a1), (b1)F = 14, β = 0.05; (a2), (b2)F = 14, β = 0.1; (a3), (b3)F = 30, β = 0.1

Figure 6. Two-dimensional maps of H[P ] in the parameter space of the frequency detuning and injection strength under different values of F, β and injection current for the slave nanolaser: (a1)–(a3) $ {I_{{\rm{dc}}}} = 2{I_{{\rm{th}}}} $; (b1)–(b3) $ {I_{{\rm{dc}}}} = 4{I_{{\rm{th}}}} $. (a1), (b1) F = 14, β = 0.05; (a2), (b2) F = 14, β = 0.1; (a3), (b3) F = 30, β = 0.1.

DownLoad: Full-Size Img PowerPoint

图 7 在不同线宽增强因子下, 从纳米激光器混沌输出的　(a1)—(a3)H[P ]及(b1)—(b3)时延特征峰值随失谐频率及注入强度变化的二维映射. (a1), (b1)$ \alpha=4 $; (a2), (b2)$ \alpha=5 $; (a3), (b3)$ \alpha=6 $

Figure 7. Two-dimensional maps of (a1)–(a3) H[P ] and (b1)–(b3) TDS in the parameter space of the frequency detuning and injection strength under different values of the linewidth enhancement factor for the slave nanolaser. (a1), (b1) $ \alpha=4 $; (a2), (b2) $ \alpha=5 $; (a3), (b3) $ \alpha=6 $.

DownLoad: Full-Size Img PowerPoint

map

[1]	Argyris A, Syvridis D, Larger L, Annovazzi-Lodi V, Colet P, Fischer I, García-Ojalvo J, Mirasso C R, Pesquera L, Shore K A 2005 Nature 438 343 Google Scholar
[2]	Sciamanna M, Shore K A 2015 Nat. Photonics 9 151 Google Scholar
[3]	Uchida A, Amano K, Inoue M, Hirano K, Naito S, Someya H, Oowada I, Kurashige T, Shiki M, Yoshimori S, Yoshimura K, Davis P 2008 Nat. Photonics 2 728 Google Scholar
[4]	Li N Q, Kim B, Chizhevsky V N, Locquet A, Bloch M, Citrin D S, Pan W 2014 Opt. Express 22 6634 Google Scholar
[5]	Lin F Y, Liu J M 2004 IEEE J. Sel. Top. Quantum Electron. 10 991 Google Scholar
[6]	Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815 Google Scholar
[7]	Locquet A, Kim B, Choi D, Li N Q, Citrin D S 2017 Phy. Rev. A 95 023801 Google Scholar
[8]	Rontani D, Locquet A, Sciamanna M, Citrin D S 2007 Opt. Lett. 32 2960 Google Scholar
[9]	Zhao Q C, Wang Y C, Wang A B 2009 Appl. Opt. 48 3515 Google Scholar
[10]	Kanno K, Uchida A, Bunsen M 2016 Phy. Rev. E 93 032206 Google Scholar
[11]	Li N Q, Pan W, Xiang S Y, Zhao Q C, Zhang L Y, Mu P H 2014 IEEE J. Quantum Electron. 50 766 Google Scholar
[12]	Keller K, Sinn M 2010 Phy. D 239 997 Google Scholar
[13]	Bandt C, Pompe B 2002 Phy. Rev. Lett. 88 174102 Google Scholar
[14]	Xiang S Y, Pan W, Yan L S, Luo B, Zou X H, Jiang N, Wen K H 2011 Opt. Lett. 36 310 Google Scholar
[15]	Xiang S Y, Pan W, Luo B, Yan L S, Zou X H, Jiang N, Yang L, Zhu H N 2011 J. Lightwave Technol. 29 2173 Google Scholar
[16]	Xiang S Y, Pan W, Luo B, Yan L S, Zou X H, Jiang N, Yang L, Li N Q 2011 IEEE J. Quantum Electron. 47 1354 Google Scholar
[17]	Xiang S Y, Pan W, Luo B, Yan L S, Zou X H, Jiang N, Zhu H N 2012 IEEE J. Quantum Electron. 48 1069 Google Scholar
[18]	Xiang S Y, Pan W, Li N Q, Zhang L Y, Zhu H N 2013 Opt. Commun. 311 294 Google Scholar
[19]	Guo X X, Xiang S Y, Zhang Y H, Wen A J, Hao Y 2018 IEEE J. Quantum Electron. 54 2000308 Google Scholar
[20]	Zhang H, Xiang S Y, Zhang Y H, Guo X X 2017 Appl. Opt. 56 6728 Google Scholar
[21]	Li N Q, Pan W, Locquet A, Citrin D S 2015 Opt. Lett. 40 4416 Google Scholar
[22]	Li N Q, Pan W, Xiang S Y, Yan L S, Luo B, Zou X H, Zhang L Y 2013 Opt. Laser Technol. 53 45 Google Scholar
[23]	Hill M T, Oei Y S, Smalbrugge B, Zhu Y C, Vries T D, Veldhoven P J V, Otten F W M V, Eijkemans T J, Turkiewicz J P, Waardt H D, Geluk E J, Kwon S H, Lee Y H, Nötzel R, Smit M K 2007 Nat. Photonics 1 589 Google Scholar
[24]	Lau E K, Lakhani A, Tucker R S, Wu M C 2009 Opt. Express 17 7790 Google Scholar
[25]	Sattar Z A, Shore K A 2015 J. Lightwave Technol. 33 3028 Google Scholar
[26]	Sattar Z A, Shore K A 2015 IEEE J. Sel. Top. Quantum Electron. 21 1800106 Google Scholar
[27]	Sattar Z A, Shore K A 2016 IEEE J. Quantum Electron. 52 1100108 Google Scholar
[28]	Sattar Z A, Kamel N A, Shore K A 2016 IEEE J. Quantum Electron. 52 1200108 Google Scholar
[29]	Han H, Shore K A 2018 IET Optoelectron. 12 25 Google Scholar
[30]	Han H, Shore K A 2016 IEEE J. Quantum Electron. 52 2000306 Google Scholar
[31]	Elsonbaty A, Hegazy S F, Obayya S S A 2018 Opt. Laser. Eng. 107 342 Google Scholar
[32]	Qu Y, Xiang S Y, Wang Y, Lin L, Wen A J, Hao Y 2019 IEEE J. Quantum Electron. 55 2000407 Google Scholar
[33]	Jiang P, Zhou P, Li N Q, Mu P H, Li X F 2020 Opt. Express 28 26421 Google Scholar
[34]	Chao M, Wang D M, Wang L S, Sun Y C, Han H, Guo Y Y, Jia Z W, Wang Y C, Wang A B 2020 Opt. Commun. 456 124702 Google Scholar
[35]	阎娟, 潘炜, 李念强, 张力月, 刘庆喜 2016 65 204203 Google Scholar Yan J, Pan W, Li N Q, Zhang L Y, Liu Q X 2016 Acta Phys. Sin. 65 204203 Google Scholar
[36]	Zhou P, Fang Q, Li N Q 2020 Opt. Lett. 45 399 Google Scholar
[37]	苏斌斌, 陈建军, 吴正茂, 夏光琼 2017 66 244206 Google Scholar Su B B, Chen J J, Wu Z M, Xia G Q 2017 Acta. Phys. Sin. 66 244206 Google Scholar
[38]	Uchida A 2012 Optical Communication with Chaotic Lasers: Applications of Nonlinear Dynamics and Synchronization (Berlin: Wiley) p195

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