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Rare ultrahigh pulses, classified as rogue waves (RWs), are inevitable and catastrophic in many different systems. Considering the damage they may produce, it is meaningful to understand the formation mechanism of these pulses and, if possible, control them. However, the rarity of RW and the difficulty in implementing the experiment are major limitations to understanding their formation. In 2007, Solli et al. (Solli D R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054 ) introduced the concept of optical RW, i.e. extreme event (EE) by comparing the appearance of oceanic RWs with the propagation of light fields in optical fibers. After that, the research of EEs entered into a flourishing period and different optical systems were proposed to analyze the generation and origin of EEs. Linear system is one of the most widely studied EE systems, such as linear light propagation in glass fibers, random media, and linear interference models. In addition to the linear systems mentioned above, efforts have also been made to produce nonlinear systems of EEs, such as microstructure fibers and tapered gradient exponential nonlinear fibers. In these nonlinear systems, the formation mechanism of EE is studied by using the nonlinear Schrödinger equation. Recently, the EEs in semiconductor laser systems have received a great deal of attention. On the one hand, semiconductor lasers with rich dynamic properties provide a cheap and controllable platform for understanding and predicting EE. The behavior of EE, on the other hand, is a powerful tool for understanding the fundamental mechanism of different laser systems.In this work, based on the EEs generated in a semiconductor laser with phase-conjugate optical feedback (the master laser, ML), we inject its output into another free-running semiconductor laser (the slave laser, SL) and discuss the evolution of EEs in the system by numerical simulation. Herein, we analyze the influence of injection parameters on EEs through the two-dimensional maps of the relative number of EEs in the injection-parameter space. It can be concluded that in an area of high correlation, the relative number of EEs in SL tends to be a stationary value close to that in ML, while it may be enhanced in some weakly correlated regions. The results demonstrate the possibility of controlling EEs by optical injection, which is beneficial to optimizing the performance of chaotic laser systems or expanding their application scope. -
Keywords:
- semiconductor laser /
- extreme event /
- chaotic optical injection /
- phase-conjugate optical feedback
[1] Talouneh K, Rimoldi C, Kheradmand R, Tissoni G, Eslami M 2020 Phys. Rev. A 102 033508
Google Scholar
[2] Solli D R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054
Google Scholar
[3] Arecchi F T, Bortolozzo U, Montina A, Residori S 2011 Phys. Rev. Lett. 106 153901
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[4] Metzger J J, Fleischmann R, Geisel T 2014 Phys. Rev. Lett. 112 203903
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[5] Birkholz S, Bree C, Veselic I, Demircan A, Steinmeyer G 2016 Sci. Rep. 6 35207
Google Scholar
[6] Kumar C N, Gupta R, Goyal A, Loomba S, Raju T S, Panigrahi P K 2012 Phys. Rev. A 86 025802
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[7] Dal Bosco A K, Wolfersberger D, Sciamanna M 2013 Opt. Lett. 38 703
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[9] Zamora-Munt J, Garbin B, Barland S, Giudici M, Leite J R R, Masoller C, Tredicce J R 2013 Phys. Rev. A 87 035802
Google Scholar
[10] Ahuja J, Nalawade D B, Zamora-Munt J, Vilaseca R, Masoller C 2014 Opt. Express 22 28377
Google Scholar
[11] Perrone S, Vilaseca R, Zamora-Munt J, Masoller C 2014 Phys. Rev. A 89 033804
Google Scholar
[12] Mercier E, Even A, Mirisola E, Wolfersberger D, Sciamanna M 2015 Phys. Rev. E 91 042914
Google Scholar
[13] Choi D, Wishon M J, Barnoud J, Chang C Y, Bouazizi Y, Locquet A, Citrin D S 2016 Phys. Rev. E 93 042216
Google Scholar
[14] Lee M W, Baladi F, Burie J R, Bettiati M A, Boudrioua A, Fischer A P 2016 Opt. Lett. 41 4476
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[15] Jin T, Siyu C, Masoller C 2017 Opt. Express 25 31326
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[16] Uy C H, Rontani D, Sciamanna M 2017 Opt. Lett. 42 2177
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Google Scholar
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Google Scholar
[20] Jiang N, Zhao A, Xue C, Tang J, Qiu K 2019 Opt. Lett. 44 1536
Google Scholar
[21] Lawrence J S, Kane D M 2001 Phys. Rev. A 63 033805
Google Scholar
[22] Rontani D, Mercier E, Wolfersberger D, Sciamanna M 2016 Opt. Lett. 41 4637
Google Scholar
[23] Feinberg J 1982 Opt. Lett. 7 486
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 ML的时序图和峰值强度的统计分布图(红色虚线和竖线代表极端事件的阈值AI = 2) (a) kf = 9 ns–1; (b) kf = 15 ns–1; (c) kf = 30 ns–1
Figure 1. Time series and statistical distributions of peak intensity in ML (Red dashed lines and vertical lines represent the threshold of EEs, AI = 2): (a) kf = 9 ns–1; (b) kf = 15 ns–1; (c) kf = 30 ns–1.
图 2 kf = 15 ns–1, kinj = 100 ns–1时, SL的时序图和峰值强度的统计分布图 (红色虚线和竖线代表极端事件的阈值AI = 2) (a) Δf = 15 GHz; (b) Δf = 0 GHz; (c) Δf = –20 GHz
Figure 2. Time series and statistical distributions of peak intensity in SL at kf = 15 ns–1 and kinj = 100 ns–1 (Red dashed lines and vertical lines represent the threshold of EEs, AI = 2): (a) Δf = 15 GHz; (b) Δf = 0 GHz; (c) Δf = –20 GHz.
图 3 SL中EE的相对数量随着注入强度变化情况(灰色虚线代表ML中EE的相对数量) (a) kf = 9 ns–1; (b) kf = 15 ns–1; (c) kf = 30 ns–1
Figure 3. Relative number of EEs versus injection strength (Gray dashed lines represent the relative number of EEs in ML): (a) kf = 9 ns–1; (b) kf = 15 ns–1; (c) kf = 30 ns–1.
图 4 COIS-PCF中EE相对数量在注入参数空间内的二维图(a1)—(c1)以及对应的主从激光器互相关系数的二维映射图(a2)—(c2) (a) kf = 9 ns–1; (b) kf = 15 ns–1; (c) kf = 30 ns–1
Figure 4. Maps of relative number of EEs generated from the COIS-PCF in the injection-parameter space (a1)–(c1) and corresponding CC between the ML and SL (a2)–(c2): (a) kf = 9 ns–1; (b) kf = 15 ns–1; (c) kf = 30 ns–1.
图 5 kf = 30 ns–1时, ML的时序图和峰值强度的统计分布图 (红色虚线代表极端事件的阈值AI = 2) (a) JM = 1.05Jth; (b) JM = 1.30Jth; (c) JM = 1.50Jth; (d) JM = 2.00Jth
Figure 5. Time series and statistical distributions of peak intensity in ML at kf = 30 ns–1 (Red dashed lines and vertical lines represent the threshold of EEs, AI = 2): (a) JM = 1.05Jth; (b) JM = 1.30Jth; (c) JM = 1.50Jth; (d) JM = 2.00Jth.
图 6 COIS-PCF中EE相对数量在注入参数空间内的二维图(a1)—(d1)以及对应的主从激光器互相关系数(a2)—(d2) (a) JM = 1.05Jth; (b) JM = 1.30Jth; (c) JM = 1.50Jth; (d) JM = 2.00Jth
Figure 6. Maps of relative number of EEs generated from the COIS-PCF in the injection-parameter space (a1)–(d1) and corresponding CC between the ML and SL (a2)–(d2): (a) JM = 1.05Jth; (b) JM = 1.30Jth; (c) JM = 1.50Jth; (d) JM = 2.00Jth.
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[1] Talouneh K, Rimoldi C, Kheradmand R, Tissoni G, Eslami M 2020 Phys. Rev. A 102 033508
Google Scholar
[2] Solli D R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054
Google Scholar
[3] Arecchi F T, Bortolozzo U, Montina A, Residori S 2011 Phys. Rev. Lett. 106 153901
Google Scholar
[4] Metzger J J, Fleischmann R, Geisel T 2014 Phys. Rev. Lett. 112 203903
Google Scholar
[5] Birkholz S, Bree C, Veselic I, Demircan A, Steinmeyer G 2016 Sci. Rep. 6 35207
Google Scholar
[6] Kumar C N, Gupta R, Goyal A, Loomba S, Raju T S, Panigrahi P K 2012 Phys. Rev. A 86 025802
Google Scholar
[7] Dal Bosco A K, Wolfersberger D, Sciamanna M 2013 Opt. Lett. 38 703
Google Scholar
[8] Reinoso J A, Zamora-Munt J, Masoller C 2013 Phys. Rev. E 87 062913
Google Scholar
[9] Zamora-Munt J, Garbin B, Barland S, Giudici M, Leite J R R, Masoller C, Tredicce J R 2013 Phys. Rev. A 87 035802
Google Scholar
[10] Ahuja J, Nalawade D B, Zamora-Munt J, Vilaseca R, Masoller C 2014 Opt. Express 22 28377
Google Scholar
[11] Perrone S, Vilaseca R, Zamora-Munt J, Masoller C 2014 Phys. Rev. A 89 033804
Google Scholar
[12] Mercier E, Even A, Mirisola E, Wolfersberger D, Sciamanna M 2015 Phys. Rev. E 91 042914
Google Scholar
[13] Choi D, Wishon M J, Barnoud J, Chang C Y, Bouazizi Y, Locquet A, Citrin D S 2016 Phys. Rev. E 93 042216
Google Scholar
[14] Lee M W, Baladi F, Burie J R, Bettiati M A, Boudrioua A, Fischer A P 2016 Opt. Lett. 41 4476
Google Scholar
[15] Jin T, Siyu C, Masoller C 2017 Opt. Express 25 31326
Google Scholar
[16] Uy C H, Rontani D, Sciamanna M 2017 Opt. Lett. 42 2177
Google Scholar
[17] Huang Y, Zhou P, Zeng Y, Zhang R H, Li N Q 2022 Phys. Rev. A 105 043521
Google Scholar
[18] Li X Z, Zhou X Q, Gu Y Y, Zhao M S 2022 IEEE J. Sel. Top. Quantum Electron. 28 0600108
Google Scholar
[19] Bonatto C, Feyereisen M, Barland S, Giudici M, Masoller C, Leite J R, Tredicce J R 2011 Phys. Rev. Lett. 107 053901
Google Scholar
[20] Jiang N, Zhao A, Xue C, Tang J, Qiu K 2019 Opt. Lett. 44 1536
Google Scholar
[21] Lawrence J S, Kane D M 2001 Phys. Rev. A 63 033805
Google Scholar
[22] Rontani D, Mercier E, Wolfersberger D, Sciamanna M 2016 Opt. Lett. 41 4637
Google Scholar
[23] Feinberg J 1982 Opt. Lett. 7 486
Google Scholar
[24] Jiang N, Zhao A, Liu S, Zhang Y, Peng J, Qiu K 2020 Opt. Express 28 9477
Google Scholar
[25] Mu P, Pan W, Yan L, Luo B, Zou X 2017 IEEE Photonics J. 9 1
Google Scholar
[26] Zhang R, Zhou P, Yang Y, Fang Q, Mu P, Li N 2020 Opt. Express 28 7197
Google Scholar
[27] Mu P H, Pan W, Li N Q 2018 Opt. Express 26 15642
Google Scholar
[28] Huang Y, Zhou P, Li N Q 2021 Opt. Express 29 19675
Google Scholar
[29] Zhao A, Jiang N, Peng J, Liu S, Zhang Y, Qiu K 2022 Opto-Electron. Adv. 5 200026
Google Scholar
[30] Zeng Y, Zhou P, Huang Y, Li N 2022 Opt. Lett. 47 142
Google Scholar
[31] Rimoldi C, Barland S, Prati F, Tissoni G 2017 Phys. Rev. A 95 023841
Google Scholar
[32] Tlidi M, Panajotov K 2017 Chaos 27 013119
Google Scholar
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