Abstract

We study the nonlinear dynamics of solitary and optically injected two-element laser arrays with a range of waveguide structures. The analysis is performed with a detailed direct numerical simulation, where high-resolution dynamic maps are generated to identify regions of dynamic instability in the parameter space of interest. Our combined one- and two-parameter bifurcation analysis uncovers globally diverse dynamical regimes (steady-state, oscillation, and chaos) in the solitary laser arrays, which are greatly influenced by static design waveguiding structures, the amplitude-phase coupling factor of the electric field, i.e. the linewidth-enhancement factor, as well as the control parameter, e.g. the pump rate. When external optical injection is introduced to one element of the arrays, we show that the whole system can be either injection-locked simultaneously or display rich, different dynamics outside the locking region. The effect of optical injection is to significantly modify the nature and the regions of nonlinear dynamics from those found in the solitary case. We also show similarities and differences (asymmetry) between the oscillation amplitude of the two elements of the array in specific well-defined regions, which hold for all the waveguiding structures considered. Our findings pave the way to a better understanding of dynamic instability in large arrays of lasers.

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References

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2018 (1)

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Locking bandwidth of two laterally-coupled lasers subjected to optical injection,” Sci. Rep. 8(109), 1–10 (2018).
[PubMed]

2017 (8)

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

Y. Kominis, V. Kovanis, and T. Bountis, “Controllable asymmetric phase-locked states of the fundamental active photonic dimer,” Phys. Rev. A 96(4), 043836 (2017).
[Crossref]

M. J. Adams, N. Q. Li, B. R. Cemlyn, H. Susanto, and I. D. Henning, “Effects of detuning, gain-guiding and index antiguiding on the dynamics of two laterally-coupled semiconductor lasers,” Phys. Rev. A 95(5), 053869 (2017).
[Crossref]

S. T. M. Fryslie, Z. Gao, H. Dave, B. J. Thompson, K. Lakomy, S. Lin, P. Decker, D. McElfresh, J. E. Schutt-Aine, and K. D. Choquette, “Modulation of coherently-coupled phased photonic crystal vertical cavity laser arrays,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1700409 (2017).
[Crossref]

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

J. P. Toomey, A. Argyris, C. McMahon, D. Syvridis, and D. M. Kane, “Time-scale independent permutation entropy of a photonic integrated Device,” J. Lightwave Technol. 35(1), 88–95 (2017).
[Crossref]

Z. Gao, S. T. M. Fryslie, B. J. Thompson, P. Scott Carney, and K. D. Choquette, “Parity-time symmetry in coherently coupled vertical cavity laser arrays,” Optica 4(3), 323–329 (2017).
[Crossref]

P. Li, J. Zhang, L. Sang, X. Liu, Y. Guo, X. Guo, A. Wang, K. Alan Shore, and Y. Wang, “Real-time online photonic random number generation,” Opt. Lett. 42(14), 2699–2702 (2017).
[Crossref] [PubMed]

2016 (1)

D. Rontani, D. Choi, C.-Y. Chang, A. Locquet, and D. S. Citrin, “Compressive sensing with optical chaos,” Sci. Rep. 6(1), 35206 (2016).
[Crossref] [PubMed]

2015 (2)

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23(17), 22619–22627 (2015).
[Crossref] [PubMed]

2014 (2)

C. M. Long, L. Mutter, B. Dwir, A. Mereuta, A. Caliman, A. Sirbu, V. Iakovlev, and E. Kapon, “Optical injection locking of transverse modes in 1.3-µm wavelength coupled-VCSEL arrays,” Opt. Express 22(18), 21137–21144 (2014).
[Crossref] [PubMed]

N. Blackbeard, S. Wieczorek, H. Erzgräber, and P. S. Dutta, “From synchronization to persistent optical turbulence in laser arrays,” Physica D 286–287, 43–58 (2014).
[Crossref]

2011 (1)

N. Blackbeard, H. Erzgräber, and S. Wieczorek, “Shear-induced bifurcations and chaos in models of three coupled lasers,” SIAM J. Appl. Dyn. Syst. 10(2), 469–509 (2011).
[Crossref]

2010 (1)

2009 (4)

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
[Crossref] [PubMed]

H. Erzgräber, S. Wieczorek, and B. Krauskopf, “Locking behavior of three coupled laser oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(2), 026212 (2009).
[Crossref] [PubMed]

R. Santos and H. Lamela, “Experimental observation of chaotic dynamics in two coupled diode lasers through lateral model locking,” IEEE J. Quantum Electron. 45(11), 1490–1494 (2009).
[Crossref]

G. A. Gottwald and I. Melbourne, “On the implementation of the 0–1 test for chaos,” SIAM J. Appl. Dyn. Syst. 8(1), 129–145 (2009).
[Crossref]

2008 (2)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

H. Erzgräber, S. Wieczorek, and B. Krauskopf, “Dynamics of two laterally coupled semiconductor lasers: strong- and weak-coupling theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066201 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (3)

S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A 72(3), 033810 (2005).
[Crossref]

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

2004 (2)

F. Rogister and M. Blondel, “Dynamics of two mutually delay-coupled semiconductor lasers,” Opt. Commun. 239(1), 173–180 (2004).
[Crossref]

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: instantaneous coupling limit,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(5), 056221 (2004).
[Crossref] [PubMed]

2003 (2)

F. Rogister and J. García-Ojalvo, “Symmetry breaking and high-frequency periodic oscillations in mutually coupled laser diodes,” Opt. Lett. 28(14), 1176–1178 (2003).
[Crossref] [PubMed]

K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
[Crossref]

2001 (1)

H. Lamela, M. Leones, G. Carpintero, C. Simmendinger, and O. Hess, “Analysis of the dynamics behavior and short-pulse modulation scheme for laterally coupled diode lasers,” IEEE J. Sel. Top. Quantum Electron. 7(2), 192–200 (2001).
[Crossref]

1999 (1)

A. Hohl and A. Gavrielides, “Bifurcation cascade in a semiconductor laser subject to optical feedback,” Phys. Rev. Lett. 82(6), 1148–1151 (1999).
[Crossref]

1997 (1)

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclassic. Opt. 9(5), 765–784 (1997).
[Crossref]

1994 (3)

O. Hess and E. Scholl, “Spatio-temporal dynamics in twin-stripe semiconductor lasers,” Physica D 70(1–2), 165–177 (1994).
[Crossref]

L. Rahman and H. G. Winful, “Nonlinear dynamics of semiconductor laser arrays: a mean field model,” IEEE J. Quantum Electron. 30(6), 1405–1416 (1994).
[Crossref]

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994).
[Crossref]

1993 (1)

1991 (1)

G. A. Wilson, R. K. DeFreez, and H. G. Winful, “Modulation of phased-array semiconductor lasers at K-band frequencies,” IEEE J. Quantum Electron. 27(6), 1696–1704 (1991).
[Crossref]

1990 (1)

H. G. Winful and L. Rahman, “Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers,” Phys. Rev. Lett. 65(13), 1575–1578 (1990).
[Crossref] [PubMed]

1988 (2)

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52(21), 1774–1776 (1988).
[Crossref]

H. G. Winful and S. S. Wang, “Stability of phase-locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
[Crossref]

1987 (1)

1984 (2)

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9(4), 125–127 (1984).
[Crossref] [PubMed]

E. Marom, O. G. Ramer, and S. Ruschin, “Relation between normal-mode and coupled-mode analyses of parallel waveguides,” IEEE J. Quantum Electron. 20(12), 1311–1319 (1984).
[Crossref]

Adams, M. J.

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Locking bandwidth of two laterally-coupled lasers subjected to optical injection,” Sci. Rep. 8(109), 1–10 (2018).
[PubMed]

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

M. J. Adams, N. Q. Li, B. R. Cemlyn, H. Susanto, and I. D. Henning, “Effects of detuning, gain-guiding and index antiguiding on the dynamics of two laterally-coupled semiconductor lasers,” Phys. Rev. A 95(5), 053869 (2017).
[Crossref]

K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
[Crossref]

Alan Shore, K.

Amano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Argyris, A.

J. P. Toomey, A. Argyris, C. McMahon, D. Syvridis, and D. M. Kane, “Time-scale independent permutation entropy of a photonic integrated Device,” J. Lightwave Technol. 35(1), 88–95 (2017).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Blackbeard, N.

N. Blackbeard, S. Wieczorek, H. Erzgräber, and P. S. Dutta, “From synchronization to persistent optical turbulence in laser arrays,” Physica D 286–287, 43–58 (2014).
[Crossref]

N. Blackbeard, H. Erzgräber, and S. Wieczorek, “Shear-induced bifurcations and chaos in models of three coupled lasers,” SIAM J. Appl. Dyn. Syst. 10(2), 469–509 (2011).
[Crossref]

Blondel, M.

F. Rogister and M. Blondel, “Dynamics of two mutually delay-coupled semiconductor lasers,” Opt. Commun. 239(1), 173–180 (2004).
[Crossref]

Bountis, T.

Y. Kominis, V. Kovanis, and T. Bountis, “Controllable asymmetric phase-locked states of the fundamental active photonic dimer,” Phys. Rev. A 96(4), 043836 (2017).
[Crossref]

Caliman, A.

Carney, P. S.

Carpintero, G.

H. Lamela, M. Leones, G. Carpintero, C. Simmendinger, and O. Hess, “Analysis of the dynamics behavior and short-pulse modulation scheme for laterally coupled diode lasers,” IEEE J. Sel. Top. Quantum Electron. 7(2), 192–200 (2001).
[Crossref]

Cemlyn, B. R.

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Locking bandwidth of two laterally-coupled lasers subjected to optical injection,” Sci. Rep. 8(109), 1–10 (2018).
[PubMed]

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

M. J. Adams, N. Q. Li, B. R. Cemlyn, H. Susanto, and I. D. Henning, “Effects of detuning, gain-guiding and index antiguiding on the dynamics of two laterally-coupled semiconductor lasers,” Phys. Rev. A 95(5), 053869 (2017).
[Crossref]

Chan, S. C.

Chang, C.-Y.

D. Rontani, D. Choi, C.-Y. Chang, A. Locquet, and D. S. Citrin, “Compressive sensing with optical chaos,” Sci. Rep. 6(1), 35206 (2016).
[Crossref] [PubMed]

Chen, Y.

Chlouverakis, K. E.

K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
[Crossref]

Choi, D.

D. Rontani, D. Choi, C.-Y. Chang, A. Locquet, and D. S. Citrin, “Compressive sensing with optical chaos,” Sci. Rep. 6(1), 35206 (2016).
[Crossref] [PubMed]

Choquette, K. D.

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H. Erzgräber, S. Wieczorek, and B. Krauskopf, “Locking behavior of three coupled laser oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(2), 026212 (2009).
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A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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R. Santos and H. Lamela, “Experimental observation of chaotic dynamics in two coupled diode lasers through lateral model locking,” IEEE J. Quantum Electron. 45(11), 1490–1494 (2009).
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N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Locking bandwidth of two laterally-coupled lasers subjected to optical injection,” Sci. Rep. 8(109), 1–10 (2018).
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J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
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D. Rontani, D. Choi, C.-Y. Chang, A. Locquet, and D. S. Citrin, “Compressive sensing with optical chaos,” Sci. Rep. 6(1), 35206 (2016).
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A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
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F. Rogister and J. García-Ojalvo, “Symmetry breaking and high-frequency periodic oscillations in mutually coupled laser diodes,” Opt. Lett. 28(14), 1176–1178 (2003).
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D. Rontani, D. Choi, C.-Y. Chang, A. Locquet, and D. S. Citrin, “Compressive sensing with optical chaos,” Sci. Rep. 6(1), 35206 (2016).
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S. T. M. Fryslie, Z. Gao, H. Dave, B. J. Thompson, K. Lakomy, S. Lin, P. Decker, D. McElfresh, J. E. Schutt-Aine, and K. D. Choquette, “Modulation of coherently-coupled phased photonic crystal vertical cavity laser arrays,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1700409 (2017).
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J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7, 42116 (2017).
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A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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H. Lamela, M. Leones, G. Carpintero, C. Simmendinger, and O. Hess, “Analysis of the dynamics behavior and short-pulse modulation scheme for laterally coupled diode lasers,” IEEE J. Sel. Top. Quantum Electron. 7(2), 192–200 (2001).
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S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
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N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Locking bandwidth of two laterally-coupled lasers subjected to optical injection,” Sci. Rep. 8(109), 1–10 (2018).
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Z. Gao, S. T. M. Fryslie, B. J. Thompson, P. Scott Carney, and K. D. Choquette, “Parity-time symmetry in coherently coupled vertical cavity laser arrays,” Optica 4(3), 323–329 (2017).
[Crossref]

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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).
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J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
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Wang, S. S.

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Wang, Y.

Wieczorek, S.

N. Blackbeard, S. Wieczorek, H. Erzgräber, and P. S. Dutta, “From synchronization to persistent optical turbulence in laser arrays,” Physica D 286–287, 43–58 (2014).
[Crossref]

N. Blackbeard, H. Erzgräber, and S. Wieczorek, “Shear-induced bifurcations and chaos in models of three coupled lasers,” SIAM J. Appl. Dyn. Syst. 10(2), 469–509 (2011).
[Crossref]

H. Erzgräber, S. Wieczorek, and B. Krauskopf, “Locking behavior of three coupled laser oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(2), 026212 (2009).
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H. Erzgräber, S. Wieczorek, and B. Krauskopf, “Dynamics of two laterally coupled semiconductor lasers: strong- and weak-coupling theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066201 (2008).
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S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
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Wilson, G. A.

G. A. Wilson, R. K. DeFreez, and H. G. Winful, “Modulation of phased-array semiconductor lasers at K-band frequencies,” IEEE J. Quantum Electron. 27(6), 1696–1704 (1991).
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Winful, H. G.

L. Rahman and H. G. Winful, “Nonlinear dynamics of semiconductor laser arrays: a mean field model,” IEEE J. Quantum Electron. 30(6), 1405–1416 (1994).
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G. A. Wilson, R. K. DeFreez, and H. G. Winful, “Modulation of phased-array semiconductor lasers at K-band frequencies,” IEEE J. Quantum Electron. 27(6), 1696–1704 (1991).
[Crossref]

H. G. Winful and L. Rahman, “Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers,” Phys. Rev. Lett. 65(13), 1575–1578 (1990).
[Crossref] [PubMed]

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52(21), 1774–1776 (1988).
[Crossref]

H. G. Winful and S. S. Wang, “Stability of phase-locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
[Crossref]

Xu, J.

Yanchuk, S.

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: instantaneous coupling limit,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(5), 056221 (2004).
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Yariv, A.

Yoshimori, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Yoshimura, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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Zhang, J.

Appl. Phys. Lett. (2)

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52(21), 1774–1776 (1988).
[Crossref]

H. G. Winful and S. S. Wang, “Stability of phase-locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
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G. A. Wilson, R. K. DeFreez, and H. G. Winful, “Modulation of phased-array semiconductor lasers at K-band frequencies,” IEEE J. Quantum Electron. 27(6), 1696–1704 (1991).
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L. Rahman and H. G. Winful, “Nonlinear dynamics of semiconductor laser arrays: a mean field model,” IEEE J. Quantum Electron. 30(6), 1405–1416 (1994).
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IEEE J. Sel. Top. Quantum Electron. (2)

S. T. M. Fryslie, Z. Gao, H. Dave, B. J. Thompson, K. Lakomy, S. Lin, P. Decker, D. McElfresh, J. E. Schutt-Aine, and K. D. Choquette, “Modulation of coherently-coupled phased photonic crystal vertical cavity laser arrays,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1700409 (2017).
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H. Lamela, M. Leones, G. Carpintero, C. Simmendinger, and O. Hess, “Analysis of the dynamics behavior and short-pulse modulation scheme for laterally coupled diode lasers,” IEEE J. Sel. Top. Quantum Electron. 7(2), 192–200 (2001).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Nat. Photonics (2)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
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Opt. Commun. (2)

F. Rogister and M. Blondel, “Dynamics of two mutually delay-coupled semiconductor lasers,” Opt. Commun. 239(1), 173–180 (2004).
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K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
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Opt. Express (4)

Opt. Lett. (3)

Optica (1)

Phys. Rep. (1)

S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected lasers,” Phys. Rep. 416(1-2), 1–128 (2005).
[Crossref]

Phys. Rev. A (4)

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
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S. Valling, T. Fordell, and A. M. Lindberg, “Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A 72(3), 033810 (2005).
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Y. Kominis, V. Kovanis, and T. Bountis, “Controllable asymmetric phase-locked states of the fundamental active photonic dimer,” Phys. Rev. A 96(4), 043836 (2017).
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M. J. Adams, N. Q. Li, B. R. Cemlyn, H. Susanto, and I. D. Henning, “Effects of detuning, gain-guiding and index antiguiding on the dynamics of two laterally-coupled semiconductor lasers,” Phys. Rev. A 95(5), 053869 (2017).
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Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: instantaneous coupling limit,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(5), 056221 (2004).
[Crossref] [PubMed]

H. Erzgräber, S. Wieczorek, and B. Krauskopf, “Locking behavior of three coupled laser oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(2), 026212 (2009).
[Crossref] [PubMed]

H. Erzgräber, S. Wieczorek, and B. Krauskopf, “Dynamics of two laterally coupled semiconductor lasers: strong- and weak-coupling theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066201 (2008).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

H. G. Winful and L. Rahman, “Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers,” Phys. Rev. Lett. 65(13), 1575–1578 (1990).
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A. Hohl and A. Gavrielides, “Bifurcation cascade in a semiconductor laser subject to optical feedback,” Phys. Rev. Lett. 82(6), 1148–1151 (1999).
[Crossref]

Physica D (2)

N. Blackbeard, S. Wieczorek, H. Erzgräber, and P. S. Dutta, “From synchronization to persistent optical turbulence in laser arrays,” Physica D 286–287, 43–58 (2014).
[Crossref]

O. Hess and E. Scholl, “Spatio-temporal dynamics in twin-stripe semiconductor lasers,” Physica D 70(1–2), 165–177 (1994).
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Quantum Semiclassic. Opt. (1)

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclassic. Opt. 9(5), 765–784 (1997).
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Sci. Rep. (3)

D. Rontani, D. Choi, C.-Y. Chang, A. Locquet, and D. S. Citrin, “Compressive sensing with optical chaos,” Sci. Rep. 6(1), 35206 (2016).
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J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7, 42116 (2017).
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N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Locking bandwidth of two laterally-coupled lasers subjected to optical injection,” Sci. Rep. 8(109), 1–10 (2018).
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SIAM J. Appl. Dyn. Syst. (2)

G. A. Gottwald and I. Melbourne, “On the implementation of the 0–1 test for chaos,” SIAM J. Appl. Dyn. Syst. 8(1), 129–145 (2009).
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N. Blackbeard, H. Erzgräber, and S. Wieczorek, “Shear-induced bifurcations and chaos in models of three coupled lasers,” SIAM J. Appl. Dyn. Syst. 10(2), 469–509 (2011).
[Crossref]

Other (2)

E. J. Doedel, A. R. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Pfaffenroth, B. Sandastede, X. Wang, and C. Zhang, AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations (Concordia University, Montreal, 2008).

J. Ohtsubo, Semiconductor Lasers: Stability, Instability, and Chaos (Springer, 2012).

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Figures (10)

Fig. 1
Fig. 1 Intensity time series of a solitary two-element laser array where the laser separation ratio varies from (a) d / a = 1.0 , (b) 1.3, (c) 1.6 to (d) 2.5. Other parameters are P = 1.1 P t h , offset Δ Ω / 2 π = 0 GHz , and α H = 2 . Solid red lines represent laser A, and broken blue lines stand for laser B.
Fig. 2
Fig. 2 Examples of antiphase and in-phase solutions in the solitary two-element laser array. (a) structure of first line in Table 1 with offset Δ Ω / 2 π = 0 GHz and d / a = 1.85; (b) structure of second line in Table 1 with Δ Ω / 2 π = 5 GHz and d / a = 1.85. Other parameters are P = 1.1 P t h and α H = 2 .
Fig. 3
Fig. 3 One-parameter BDs as a function of laser separation d / a , calculated from laser A for offset Δ Ω / 2 π = 0 GHz and α H = 2 in the solitary two-element laser array. Those for laser B are similar. (a1-d1) P = 1.1 P t h ; (a2-d2) P = 2 P t h .
Fig. 4
Fig. 4 One-parameter BDs as a function of laser separation d / a , calculated from laser A for different frequency offsets in the structure in the last line of Table 1. (a) Δ Ω / 2 π = 5 GHz , (b) Δ Ω / 2 π = 3 GHz , (c) Δ Ω / 2 π = 1 GHz , and (d) Δ Ω / 2 π = 0 GHz . Those for laser B are similar. Other parameters are P = 1.1 P t h and α H = 2 .
Fig. 5
Fig. 5 Two-parameter BDs in the ( d / a , Δ Ω / 2 π ) plane in the solitary two-element laser array, where α H = 2. From top to bottom: cases described in sequence of lines from top to bottom of Table 1. From left to right: P = 1.1 P t h , 2 P t h and 5 P t h . The color code marks the number of extremal values (maxima and minima) of the intensity time series in laser A. The white indicates phase-locked solutions, dark blue shows period-one oscillation, light blue marks period-two oscillation, and other colors represent complex dynamics.
Fig. 6
Fig. 6 Comparison between the (a, b) bifurcation diagrams and (c, d) maps obtained by 0-1 test for chaos in the ( d / a , Δ Ω / 2 π ) plane. (a,c) case of second line in Table 1; (b,d) case of last line in Table 1. Other parameters are α H = 6, P = 1.1 P t h . On the left panel: the white indicates phase-locked solutions, dark blue shows period-one oscillation, light blue marks period-two oscillation, and other colors represent complex dynamics; on the right panel: red stands for chaos and others nonchaotic.
Fig. 7
Fig. 7 Oscillation frequency as a function of d/a; (a) case described in the second line of Table 1, Δ Ω / 2 π = 6 GHz ; (b) case described in the last line of Table 1: Δ Ω / 2 π = 20 GHz . Other parameters are P = 1.1 P t h α H = 2.
Fig. 8
Fig. 8 (a) Oscillation frequency (GHz) and (b) normalized amplitude in the ( d / a , Δ Ω / 2 π ) plane for the case described in the last line of Table 1. Parameters are P = 1.1 P t h , α H = 2. On the left panel: white color means the non-period one region.
Fig. 9
Fig. 9 Two-parameter BDs in the ( K , Δ ω i n j / 2 π ) plane, with case described in the second row of Table 2. (a) α H = 2; (b) α H = 3. Other parameters are P = 1.1 P t h , Δ Ω / 2 π = 6 GHz , d / a = 1.2 . (a,c) results for laser A and (b,d) laser B.
Fig. 10
Fig. 10 Amplitude comparison in terms of the contrast ratio C between the two lasers in the ( K , Δ ω i n j / 2 π ) plane; (a) Case of line 1 of Table 1, d / a = 1.2; (b) Case of line 2 of Table 1, d / a = 1.5; (c) Case of line 3 of Table 1, d / a = 2.2; (d) Case of line 4 of Table 1, d / a = 3.2. Other parameters are P = 1.1 P t h and α H = 2. Hopf and saddle-node bifurcation curves for a single laser subjected to injection are shown.

Tables (1)

Tables Icon

Table 1 Values of key parameters for modelling, using material parameter values given in [38].

Equations (8)

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d Y A d t = 1 2 τ p ( M A 1 ) Y A + Y B ( η r sin ϕ η i cos ϕ ) + K i n j τ N cos ϕ A ,
d Y B d t = 1 2 τ p ( M B 1 ) Y B Y A ( η r sin ϕ + η i cos ϕ ) ,
d ϕ A d t = α H 2 τ p ( M A 1 ) ( ω Ω A ) Y B Y A ( η r cos ϕ + η i sin ϕ ) K i n j τ N Y A sin ϕ A Δ ω ,
d ϕ d t = α H 2 τ p ( M B M A ) + Δ Ω η r cos ϕ ( Y A Y B Y B Y A ) + η i sin ϕ ( Y A Y B + Y B Y A ) + K i n j τ N Y A sin ϕ A ,
d M A , B d t = 1 τ N [ Q A , B M A , B ( 1 + Y A , B 2 ) ] ,
M A , B = 1 + c n g Γ a d i f f τ p ( N A , B N A , B t h ) ,
Q A , B = 1 + c n g Γ a d i f f τ p ( P A , B τ N N A , B t h ) .
K i n j = c a d i f f τ N n k i n j E i n j τ N ,

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