Abstract

The cluster synchronization of mutually coupled vertical-cavity surface-emitting lasers (VCSELs) networks subject to variable-polarization optical feedback (VPOF) with symmetric structure is theoretically investigated. Zero-lag synchronization could be achieved between different VCSELs within same cluster in such networks, which is solely derived from the intrinsic symmetry of network topology. The influences of significant parameters of VCSELs networks on the stability of cluster synchronization are further discussed. Moreover, it is shown that the polarizer angle of optical feedback in VCSELs plays a particularly important role on the formation of cluster.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2017 (7)

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95, 052212 (2017).
[Crossref] [PubMed]

L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118, 174102 (2017).
[Crossref] [PubMed]

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7, 42116 (2017).
[Crossref] [PubMed]

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96, 032215 (2017).
[Crossref]

N. Jiang, C. Xue, D. Liu, Y. Lv, and K. Qiu, “Secure key distribution based on chaos synchronization of vcsels subject to symmetric random-polarization optical injection,” Opt. Lett. 42, 1055–1058 (2017).
[Crossref] [PubMed]

D. Zhong, G. Xu, W. Luo, and Z. Xiao, “Real-time multi-target ranging based on chaotic polarization laser radars in the drive-response vcsels,” Opt. Express 25, 21684–21704 (2017).
[Crossref] [PubMed]

N. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-vcsels,” Opt. Lett. 42, 3494–3497 (2017).
[Crossref] [PubMed]

2016 (6)

A. Argyris, M. Bourmpos, and D. Syvridis, “Experimental synchrony of semiconductor lasers in coupled networks,” Opt. Express 24, 5600–5614 (2016).
[Crossref] [PubMed]

M. Virte, M. Sciamanna, and K. Panajotov, “Synchronization of polarization chaos from a free-running vcsel,” Opt. Lett. 41, 4492–4495 (2016).
[Crossref] [PubMed]

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2, e1501737 (2016).
[Crossref] [PubMed]

A. Argyris, E. Pikasis, and D. Syvridis, “Highly correlated chaotic emission from bidirectionally coupled semiconductor lasers,” IEEE Photon. Tech. Lett. 28, 1819–1822 (2016).
[Crossref]

S. Xiang, A. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photon. Tech. Lett. 28, 1988–1991 (2016).
[Crossref]

A. Röhm, F. Böhm, and K. Lüdge, “Small chimera states without multistability in a globally delay-coupled network of four lasers,” Phys. Rev. E 94, 042204 (2016).
[Crossref] [PubMed]

2014 (1)

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

2013 (3)

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photon. Tech. Lett. 25, 817–820 (2013).
[Crossref]

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy, “Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays,” Chaos. 23, 043117 (2013).
[Crossref]

2012 (5)

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108, 214101 (2012).
[Crossref] [PubMed]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E 86, 016202 (2012).
[Crossref]

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20, 4352–4359 (2012).
[Crossref] [PubMed]

2011 (3)

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223901 (2011).
[Crossref] [PubMed]

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref] [PubMed]

2009 (2)

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled vcsels with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17, 12619–12626 (2009).
[Crossref] [PubMed]

2007 (2)

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32, 1629–1631 (2007).
[Crossref] [PubMed]

M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213 (2007).
[Crossref]

2004 (1)

2003 (2)

Y. C. Kouomou and P. Woafo, “Cluster synchronization in coupled chaotic semiconductor lasers and application to switching in chaos-secured communication networks,” Opt. Commun. 223, 283–293 (2003).
[Crossref]

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B 6, 97–105 (2003).
[Crossref]

2002 (1)

J. White, M. Matus, and J. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E 65, 036229 (2002).
[Crossref]

1994 (1)

Adams, M. J.

Argyris, A.

A. Argyris, E. Pikasis, and D. Syvridis, “Highly correlated chaotic emission from bidirectionally coupled semiconductor lasers,” IEEE Photon. Tech. Lett. 28, 1819–1822 (2016).
[Crossref]

A. Argyris, M. Bourmpos, and D. Syvridis, “Experimental synchrony of semiconductor lasers in coupled networks,” Opt. Express 24, 5600–5614 (2016).
[Crossref] [PubMed]

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photon. Tech. Lett. 25, 817–820 (2013).
[Crossref]

Aviad, Y.

Böhm, F.

A. Röhm, F. Böhm, and K. Lüdge, “Small chimera states without multistability in a globally delay-coupled network of four lasers,” Phys. Rev. E 94, 042204 (2016).
[Crossref] [PubMed]

Bourmpos, M.

A. Argyris, M. Bourmpos, and D. Syvridis, “Experimental synchrony of semiconductor lasers in coupled networks,” Opt. Express 24, 5600–5614 (2016).
[Crossref] [PubMed]

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photon. Tech. Lett. 25, 817–820 (2013).
[Crossref]

Bunsen, M.

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95, 052212 (2017).
[Crossref] [PubMed]

Cemlyn, B.

Cohen, A. B.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref] [PubMed]

Colet, P.

Dahms, T.

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E 86, 016202 (2012).
[Crossref]

Davidson, N.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108, 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223901 (2011).
[Crossref] [PubMed]

Deng, T.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

Fischer, I.

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B 6, 97–105 (2003).
[Crossref]

Fridman, M.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108, 214101 (2012).
[Crossref] [PubMed]

Friedman, M.

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223901 (2011).
[Crossref] [PubMed]

Friesem, A. A.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108, 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223901 (2011).
[Crossref] [PubMed]

Gatare, I.

M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213 (2007).
[Crossref]

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32, 1629–1631 (2007).
[Crossref] [PubMed]

Hagerstrom, A. M.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2, e1501737 (2016).
[Crossref] [PubMed]

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

Heil, T.

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B 6, 97–105 (2003).
[Crossref]

Henning, I. D.

Hida, T.

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95, 052212 (2017).
[Crossref] [PubMed]

Hizanidis, J.

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7, 42116 (2017).
[Crossref] [PubMed]

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96, 032215 (2017).
[Crossref]

Hong, Y.

Hövel, P.

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96, 032215 (2017).
[Crossref]

Jiang, N.

N. Jiang, C. Xue, D. Liu, Y. Lv, and K. Qiu, “Secure key distribution based on chaos synchronization of vcsels subject to symmetric random-polarization optical injection,” Opt. Lett. 42, 1055–1058 (2017).
[Crossref] [PubMed]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Kanno, K.

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95, 052212 (2017).
[Crossref] [PubMed]

Kanter, I.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108, 214101 (2012).
[Crossref] [PubMed]

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20, 4352–4359 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223901 (2011).
[Crossref] [PubMed]

Kouomou, Y. C.

Y. C. Kouomou and P. Woafo, “Cluster synchronization in coupled chaotic semiconductor lasers and application to switching in chaos-secured communication networks,” Opt. Commun. 223, 283–293 (2003).
[Crossref]

Kovanis, V.

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7, 42116 (2017).
[Crossref] [PubMed]

Lee, M. W.

Lehnert, J.

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E 86, 016202 (2012).
[Crossref]

Li, N.

N. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-vcsels,” Opt. Lett. 42, 3494–3497 (2017).
[Crossref] [PubMed]

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

Li, N. Q.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Lin, X. D.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

Liu, D.

Liu, J.

Locquet, A.

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32, 1629–1631 (2007).
[Crossref] [PubMed]

M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213 (2007).
[Crossref]

Lüdge, K.

A. Röhm, F. Böhm, and K. Lüdge, “Small chimera states without multistability in a globally delay-coupled network of four lasers,” Phys. Rev. E 94, 042204 (2016).
[Crossref] [PubMed]

Luo, B.

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Luo, W.

Lv, Y.

Matus, M.

J. White, M. Matus, and J. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E 65, 036229 (2002).
[Crossref]

Mihara, T.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

Mirasso, C.

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B 6, 97–105 (2003).
[Crossref]

Moloney, J.

J. White, M. Matus, and J. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E 65, 036229 (2002).
[Crossref]

Motter, A. E.

L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118, 174102 (2017).
[Crossref] [PubMed]

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref] [PubMed]

Mulet, J.

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B 6, 97–105 (2003).
[Crossref]

Murphy, T. E.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2, e1501737 (2016).
[Crossref] [PubMed]

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy, “Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays,” Chaos. 23, 043117 (2013).
[Crossref]

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref] [PubMed]

Nishikawa, T.

L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118, 174102 (2017).
[Crossref] [PubMed]

Nixon, M.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108, 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223901 (2011).
[Crossref] [PubMed]

Ozaki, M.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

Pan, W.

S. Xiang, A. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photon. Tech. Lett. 28, 1988–1991 (2016).
[Crossref]

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Panajotov, K.

M. Virte, M. Sciamanna, and K. Panajotov, “Synchronization of polarization chaos from a free-running vcsel,” Opt. Lett. 41, 4492–4495 (2016).
[Crossref] [PubMed]

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213 (2007).
[Crossref]

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32, 1629–1631 (2007).
[Crossref] [PubMed]

A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

Pecora, L. M.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2, e1501737 (2016).
[Crossref] [PubMed]

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

Pikasis, E.

A. Argyris, E. Pikasis, and D. Syvridis, “Highly correlated chaotic emission from bidirectionally coupled semiconductor lasers,” IEEE Photon. Tech. Lett. 28, 1819–1822 (2016).
[Crossref]

Qiu, K.

Ravoori, B.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref] [PubMed]

Reidler, I.

Röhm, A.

A. Röhm, F. Böhm, and K. Lüdge, “Small chimera states without multistability in a globally delay-coupled network of four lasers,” Phys. Rev. E 94, 042204 (2016).
[Crossref] [PubMed]

Ronen, E.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108, 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106, 223901 (2011).
[Crossref] [PubMed]

Rosenbluh, M.

Roy, R.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2, e1501737 (2016).
[Crossref] [PubMed]

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy, “Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays,” Chaos. 23, 043117 (2013).
[Crossref]

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref] [PubMed]

P. Colet and R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19, 2056–2058 (1994).
[Crossref] [PubMed]

Schöll, E.

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E 86, 016202 (2012).
[Crossref]

Sciamanna, M.

M. Virte, M. Sciamanna, and K. Panajotov, “Synchronization of polarization chaos from a free-running vcsel,” Opt. Lett. 41, 4492–4495 (2016).
[Crossref] [PubMed]

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213 (2007).
[Crossref]

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32, 1629–1631 (2007).
[Crossref] [PubMed]

A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

Shena, J.

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96, 032215 (2017).
[Crossref]

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7, 42116 (2017).
[Crossref] [PubMed]

Shore, K. A.

Someya, H.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

Sorrentino, F.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2, e1501737 (2016).
[Crossref] [PubMed]

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy, “Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays,” Chaos. 23, 043117 (2013).
[Crossref]

Spencer, P. S.

Sun, J.

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy, and R. Roy, “Robustness of optimal synchronization in real networks,” Phys. Rev. Lett. 107, 034102 (2011).
[Crossref] [PubMed]

Susanto, H.

Syvridis, D.

A. Argyris, E. Pikasis, and D. Syvridis, “Highly correlated chaotic emission from bidirectionally coupled semiconductor lasers,” IEEE Photon. Tech. Lett. 28, 1819–1822 (2016).
[Crossref]

A. Argyris, M. Bourmpos, and D. Syvridis, “Experimental synchrony of semiconductor lasers in coupled networks,” Opt. Express 24, 5600–5614 (2016).
[Crossref] [PubMed]

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photon. Tech. Lett. 25, 817–820 (2013).
[Crossref]

Tanaka, K.

A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

Tang, X.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

Tsironis, G.

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96, 032215 (2017).
[Crossref]

Tsironis, G. P.

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7, 42116 (2017).
[Crossref] [PubMed]

Uchida, A.

K. Kanno, T. Hida, A. Uchida, and M. Bunsen, “Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers,” Phys. Rev. E 95, 052212 (2017).
[Crossref] [PubMed]

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

Virte, M.

Wen, A.

S. Xiang, A. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photon. Tech. Lett. 28, 1988–1991 (2016).
[Crossref]

White, J.

J. White, M. Matus, and J. Moloney, “Achronal generalized synchronization in mutually coupled semiconductor lasers,” Phys. Rev. E 65, 036229 (2002).
[Crossref]

Williams, C. R. S.

C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy, “Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays,” Chaos. 23, 043117 (2013).
[Crossref]

Woafo, P.

Y. C. Kouomou and P. Woafo, “Cluster synchronization in coupled chaotic semiconductor lasers and application to switching in chaos-secured communication networks,” Opt. Commun. 223, 283–293 (2003).
[Crossref]

Wu, J. G.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

Wu, Z. M.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled vcsels with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17, 12619–12626 (2009).
[Crossref] [PubMed]

Xia, G. Q.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled vcsels with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17, 12619–12626 (2009).
[Crossref] [PubMed]

Xiang, S.

S. Xiang, A. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photon. Tech. Lett. 28, 1988–1991 (2016).
[Crossref]

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

Xiang, S. Y.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Xiao, Y.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

Xiao, Z.

Xu, G.

Xue, C.

Yan, L.

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

Yan, L. S.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Yang, L.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Yoshimori, S.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79, 026210 (2009).
[Crossref]

A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

Zeng, L. B.

Y. Xiao, T. Deng, Z. M. Wu, J. G. Wu, X. D. Lin, X. Tang, L. B. Zeng, and G. Q. Xia, “Chaos synchronization between arbitrary two response vcsels in a broadband chaos network driven by a bandwidth-enhanced chaotic signal,” Opt. Commun. 285, 1442–1448 (2012).
[Crossref]

Zhang, L. Y.

L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118, 174102 (2017).
[Crossref] [PubMed]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

Zhong, D.

Zigzag, M.

Zou, X.

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

Zou, X. H.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

Chaos. (1)

C. R. S. Williams, F. Sorrentino, T. E. Murphy, and R. Roy, “Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays,” Chaos. 23, 043117 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and N. Q. Li, “Synchronization of unpredictability-enhanced chaos in vcsels with variable-polarization optical feedback,” IEEE J. Quantum Electron. 47, 1354–1361 (2011).
[Crossref]

IEEE J. Sel. Top. Quan. Electron. (1)

N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quan. Electron. 19, 0600109 (2013).
[Crossref]

IEEE Photon. Tech. Lett. (4)

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photon. Tech. Lett. 25, 817–820 (2013).
[Crossref]

S. Xiang, A. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photon. Tech. Lett. 28, 1988–1991 (2016).
[Crossref]

L. Y. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, S. Xiang, and N. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photon. Tech. Lett. 24, 1693–1695 (2012).
[Crossref]

A. Argyris, E. Pikasis, and D. Syvridis, “Highly correlated chaotic emission from bidirectionally coupled semiconductor lasers,” IEEE Photon. Tech. Lett. 28, 1819–1822 (2016).
[Crossref]

J. Opt. B (1)

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B 6, 97–105 (2003).
[Crossref]

Nat. Commun. (1)

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

Opt. Commun. (2)

Y. C. Kouomou and P. Woafo, “Cluster synchronization in coupled chaotic semiconductor lasers and application to switching in chaos-secured communication networks,” Opt. Commun. 223, 283–293 (2003).
[Crossref]

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[Crossref]

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A. Uchida, H. Someya, M. Ozaki, K. Tanaka, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Synchronization of chaos in mutually coupled vertical-cavity surface-emitting lasers with time delay,” in “2009 IEEE/LEOS Winter Topicals Meeting Series,” (2009), pp. 126–127.

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Figures (6)

Fig. 1
Fig. 1 Schematic diagrams of VCSELs networks with different symmetric topologies. The nodes with the same color are defined in the same synchronized cluster.
Fig. 2
Fig. 2 The dynamics of VCSELs with different state, the panels correspond to different value of polarizer angle. (a) θp = 54°, (b) θp = 72° with σ = 120ns−1, μ = 2, γ = 6ns−1.
Fig. 3
Fig. 3 RMS and CCF as the function of polarizer angle and current factor for different clusters. (a), (d) cluster (2, 3), (b), (e) cluster (4, 5), (c), (f) cluster (6, 7) with coupling strength σ = 120ns−1 and self-feedback strength γ = 6ns−1.
Fig. 4
Fig. 4 The time series of cluster (2, 3) with different state and the power spectra of VCSEL 2, the panels correspond to different value of current factor. (a), (d) μ = 2, (b), (e) μ = 2.5 and (c), (f) μ = 3 with σ = 120ns−1, θp = 63°, γ = 6ns−1.
Fig. 5
Fig. 5 Two dimensional maps of RMS in the parameter space of coupling strength σ and self-feedback strength γ with θp = 63°, μ = 2.5. (a) cluster (2, 3), (b) cluster (4, 5), (c) cluster (6, 7); The RMS as function of polarizer angle θp and current factor μ with σ = 120ns−1 and γ = 6ns−1. (d) cluster (2, 3), (e) cluster (4, 5), (f) cluster (6, 7).
Fig. 6
Fig. 6 The dynamics of VCSELs with different values of polarizer angle. (a) θp1 = θp2 = θp3 = θp4 = 72°, (b) θp1 = θp2 = 63° and θp3 = θp4 = 72°, with σ = 120ns−1, μ = 2, γ = 12.5ns−1.

Equations (8)

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E m x . = ( 1 + i α ) [ ( N m 1 ) E m x + i n m E m y ] ( γ a + i γ p ) E m x + γ E m x ( t τ f ) cos 2 ( θ p m ) exp ( i w m τ f ) + γ E m y ( t τ f ) cos ( θ p m ) sin ( θ p m ) exp ( i w m τ f ) + σ n = 1 N s A m l E n x ( t τ r ) exp ( i w n τ r )
E m y . = ( 1 + i α ) [ ( N m 1 ) E m y + i n m E m x ] + ( γ a + i γ p ) E m y + γ E m x ( t τ f ) cos ( θ p m ) sin ( θ p m ) exp ( i w m τ f ) + γ E m y ( t τ f ) sin 2 exp ( i w m τ f ) + σ n = 1 N s A m l E n y ( t τ r ) exp ( i w n τ r )
N m . = γ N [ μ N m ( 1 + | E m x | 2 ) + | E m y | 2 ) + i n m ( E m x E m y * E m y E m x * ) ]
n m . = γ s n m γ N [ n m ( 1 + | E m x | 2 ) + | E m y | 2 ) + i N m ( E m y E m x * E m x E m y * ) ]
A = ( 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 )
RMS = n = 1 N c ( I Tn ( t ) I ^ T ( t ) ) / N e N c I ^ T ( t )
CCF = [ I Ti ( t ) I Ti ( t ) ] [ I Ti ( t + Δ ( t ) ) I Ti ( t + Δ ( t ) ) ] [ I Ti ( t ) I Ti ( t ) ] 2 [ I Tj ( t + Δ t ) ) I Tj ( t + Δ t ) ) ] 2
A = ( 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 )

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