Abstract

In this paper we study the conditions for achieving almost perfect phase locking in large arrays of semiconductor diodes. We show that decayed non-local coupling of diode lasers can provide the necessary conditions for robust phase synchronization of an entire diode laser array. Perfect global coupling is known to allow for robust synchronization, however it is often physically impossible or impractical to achieve. We show that when diodes are coupled via the decayed non-local coupling layout, the dominant transverse mode of the laser array has a uniform phase across the lasers and can be stable. This state is robust to noise and frequency disorder and can be realized under periodic (fixed-intensity limit cycle) continuous-wave and chaotic behavior of lasers.

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

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  38. T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  44. J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
    [Crossref]
  45. D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
    [Crossref] [PubMed]

2016 (2)

A. Argyris, M. Bourmpos, and D. Syvridis, “Experimental synchrony of semiconductor lasers in coupled networks,” Opt. Express 24(5), 5600–5614 (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(4), e1501737 (2016).
[Crossref] [PubMed]

2015 (2)

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
[Crossref] [PubMed]

2014 (4)

C. J. Corcoran and F. Durville, “Passive coherent combination of a diode laser array with 35 elements,” Opt. Express 22(7), 8420–8425 (2014).
[Crossref] [PubMed]

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

B. Kim, N. Li, A. Locquet, and D. S. Citrin, “Experimental bifurcation-cascade diagram of an external-cavity semiconductor laser,” Opt. Express 22(3), 2348–2357 (2014).
[Crossref] [PubMed]

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

2013 (3)

2012 (2)

2011 (2)

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

D. Pabœuf, D. Vijayakumar, O. B. Jensen, B. Thestrup, J. Lim, S. Sujecki, E. Larkins, G. Lucas-Leclin, and P. Georges, “Volume Bragg grating external cavities for the passive phase locking of high-brightness diode laser arrays: theoretical and experimental study,” J. Opt. Soc. Am. B 28(5), 1289 (2011).
[Crossref]

2010 (1)

2009 (1)

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

2007 (1)

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

2005 (3)

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

2004 (1)

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

2003 (1)

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

2001 (1)

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

2000 (2)

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

1999 (1)

J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
[Crossref]

1998 (2)

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

1997 (2)

C. Masoller, “Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model,” IEEE J. Quantum Electron. 33(5), 795–803 (1997).
[Crossref]

C. Masoller, “Comparison of the effects of nonlinear gain and weak optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 33(5), 804–814 (1997).
[Crossref]

1993 (1)

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

1992 (1)

H. G. Winful, “Instability threshold for an array of coupled semiconductor lasers,” Phys. Rev. A 46(9), 6093–6094 (1992).
[Crossref] [PubMed]

1991 (2)

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

P. K. Jakobsen, H. G. Winful, L. Raman, R. A. Indik, J. V. Moloney, and A. C. Newell, “Diode-laser array modes: discrete and continuous models and their stability,” J. Opt. Soc. Am. B 8(8), 1674 (1991).
[Crossref]

1988 (2)

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

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

1984 (1)

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Abraham, N. B.

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

Acebrón, J. A.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Aceves, A. B.

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Argyris, A.

Atsuki, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Berkvens, T.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Bochove, E.

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

Bochove, E. J.

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Bonilla, L. L.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Bourmpos, M.

Braiman, Y.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

B. Liu and Y. Braiman, “Coherent beam combining of high power broad-area laser diode array with near diffraction limited beam quality and high power conversion efficiency,” Opt. Express 21(25), 31218–31228 (2013).
[Crossref] [PubMed]

B. Liu, Y. Liu, and Y. Braiman, “Coherent beam combining of high power broad-area laser diode array with a closed-V-shape external Talbot cavity,” Opt. Express 18(7), 7361–7368 (2010).
[Crossref] [PubMed]

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

Brunner, D.

Butkovski, M.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Citrin, D. S.

Colet, P.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Connors, M. K.

Corcoran, C. J.

Creedon, K. J.

Dahms, T.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Danckaert, J.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Donnelly, J. P.

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Durville, F.

Elsässer, W.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

Eng, L.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Englert, A.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Fan, T. Y.

Fischer, I.

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
[Crossref] [PubMed]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

Flunkert, V.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Garcia-Ojalvo, J.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

García-Ojalvo, J.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

Georges, P.

Gopinath, J. T.

Guo, Y.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

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(4), e1501737 (2016).
[Crossref] [PubMed]

Harris, C. T.

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Heil, T.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

Hirosawa, K.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Huang, R. K.

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Indik, R. A.

Jakobsen, P. K.

Jensen, O. B.

Jones, A. M.

Kannari, F.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Kansky, J. E.

Kanter, I.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Kapon, E.

Katz, J.

Kawashima, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Kim, B.

Kinzel, W.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Kozyreff, G.

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

Krauskopf, B.

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Larkins, E.

Leger, J. R.

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

Lehnert, J.

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Lenstra, D.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Li, N.

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Lim, J.

Liu, B.

Liu, Y.

Locquet, A.

Lu, Y.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Lucas-Leclin, G.

Mandel, P.

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

Marshall, W. K.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Masoller, C.

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

C. Masoller, “Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model,” IEEE J. Quantum Electron. 33(5), 795–803 (1997).
[Crossref]

C. Masoller, “Comparison of the effects of nonlinear gain and weak optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 33(5), 804–814 (1997).
[Crossref]

Mehuys, D.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Mirasso, C. R.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Missaggia, L. J.

K. J. Creedon, S. M. Redmond, G. M. Smith, L. J. Missaggia, M. K. Connors, J. E. Kansky, T. Y. Fan, G. W. Turner, and A. Sanchez-Rubio, “High efficiency coherent beam combining of semiconductor optical amplifiers,” Opt. Lett. 37(23), 5006–5008 (2012).
[Crossref] [PubMed]

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Mitsunaga, K.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Moloney, J. V.

Mowry, G.

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

Mulet, J.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Murakami, A.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (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(4), e1501737 (2016).
[Crossref] [PubMed]

Nair, N.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Newell, A. C.

Ohtsubo, J.

J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
[Crossref]

Pabœuf, D.

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(4), e1501737 (2016).
[Crossref] [PubMed]

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Petermann, K.

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

Raman, L.

Redmond, S. M.

Ritort, F.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Rottschafer, V.

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

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(4), e1501737 (2016).
[Crossref] [PubMed]

Sanchez-Rubio, A.

Schöll, E.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Shohda, F.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Smith, G. M.

Soriano, M.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Soriano, M. C.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

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(4), e1501737 (2016).
[Crossref] [PubMed]

Spigler, R.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Sujecki, S.

Syvridis, D.

Thestrup, B.

Turner, G. W.

K. J. Creedon, S. M. Redmond, G. M. Smith, L. J. Missaggia, M. K. Connors, J. E. Kansky, T. Y. Fan, G. W. Turner, and A. Sanchez-Rubio, “High efficiency coherent beam combining of semiconductor optical amplifiers,” Opt. Lett. 37(23), 5006–5008 (2012).
[Crossref] [PubMed]

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Van der Sande, G.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Vemuri, G.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Verschaffelt, G.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Vicente, C. J. P.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Vijayakumar, D.

Vladimirov, A. G.

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

Wang, J.

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

Wang, S.

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

Wardlaw, M.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Winful, H.

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

Winful, H. G.

Yanagisawa, T.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Yanchuk, S.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

Yariv, A.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

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]

Yousefi, G. V. M.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Yousefi, M.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Zigzag, M.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Zunoubi, M. R.

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Appl. Phys. Lett. (3)

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

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

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

EPL (1)

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

IEEE J. Quantum Electron. (6)

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

C. Masoller, “Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model,” IEEE J. Quantum Electron. 33(5), 795–803 (1997).
[Crossref]

C. Masoller, “Comparison of the effects of nonlinear gain and weak optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 33(5), 804–814 (1997).
[Crossref]

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

IEEE Photonics Technol. Lett. (1)

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Int. J. Bifurcat. Chaos (1)

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

J. Math. Sci. (1)

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

J. Opt. B Quantum Semiclass. Opt. (1)

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

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

Opt. Commun. (1)

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Opt. Rev. (1)

J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
[Crossref]

Phys. Rev. A (2)

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

H. G. Winful, “Instability threshold for an array of coupled semiconductor lasers,” Phys. Rev. A 46(9), 6093–6094 (1992).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

Proc. SPIE (1)

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Rev. Mod. Phys. (2)

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

Sci. Adv. (1)

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(4), e1501737 (2016).
[Crossref] [PubMed]

Other (5)

T. Erneux and P. Glorieux, Laser Dynamics (Cambridge University, 2010).

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

J. L. Levy and K. Roh, “Coherent array of 900 semiconductor laser amplifiers,” in Photonics West ’95, 2382, pp. 58–69. (1995)

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

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Figures (4)

Fig. 1
Fig. 1 Poincare sections of (ϕ(t)ϕ(tτ))/τ are plotted for a single laser and a 10-laser array as a function of feedback strength. Each trajectory was generated from a continuation simulation started with random initial conditions and κ f =0.01 ns 1 . Every 1000ns, κ f was increased by 0.01 ns 1 . The yellow line in each plot is the central solution frequency Ω from Eq. (11) for a single laser with κ f = κ .It is clear from these diagrams that the dynamics are almost identical except for the scaling.
Fig. 2
Fig. 2 Average phase synchrony of large arrays of lasers. Each point corresponds to a simulation of an array with randomized initial conditions (we started averaging process after 800ns simulation for convergence to occur). Lines denoting regions of effective coupling are related to the degree of synchronization in the system. For very large arrays achieving phase synchronization requires either weak or very strong (though possibly unrealistic) coupling κ f between the diodes in the array.
Fig. 3
Fig. 3 cos ϕ i are plotted for 10-laser arrays of identical lasers with varying feedback strength in the presence of carrier and phase noise. When κ f =5 ns 1 , the effective coupling κ =3.65 ns 1 and the behavior is CW as predicted in Fig. 1. When κ f =20 ns 1 , the effective coupling κ =14.6 ns 1 , and the behavior is chaotic with a high degree of synchrony. When κ f =30 ns 1 , the effective coupling κ =21.9 ns 1 , the behavior is chaotic, and the lasers begin to de-synchronize.
Fig. 4
Fig. 4 (a-c) The inner products of the first four eigenvectors V i T E (t) ( i=1,2,3,4) are plotted for an array of 30 lasers with d x =.8 at various values of coupling strength. The simulations have realistic amounts of noise and a frequency disorder of στ=0.3. The synchronization level S is also given, showing that there is indeed synchronization in the quasiperiodic (b) and chaotic (c) regimes. (d) The central far-field lobes of an array of 100 lasers subject to noise and disorder are plotted. For σ=0, the synchrony level is S=1.0. For σ=0.1/τ, S=.97. For σ=0.5/τ, S=.63. For σ=1.0/τ, S=.39.

Equations (10)

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X . i (t)=F( X i (t))+η( X i (t),t)+ κ f M j=1 M K ij C( X j (tτ), X i (t))
r ˙ i (t)= 1 2 (g N i (t) N 0 1+s r i 2 (t) γ) r i (t)+ R sp ( η E e i ϕ i (t) ) + κ f M j=1 M K ij r j (tτ)cos( ϕ j (tτ) ϕ i (t))
ϕ ˙ i (t)= α 2 (g N i (t) N 0 1+s r i 2 (t) γ)+ ω i R sp r i (t) ( η E e i ϕ i (t) ) + κ f M j=1 M K ij r j (tτ) r i (t) sin( ϕ j (tτ) ϕ i (t))
N ˙ i (t)= J 0 γ n N i (t)g N i (t) N 0 1+s r i 2 (t) r i 2 (t)+ γ n N i (t) η N (t)
K ij = d x |ij|
ξ ˙ (t)=[ I M F( X i (t)) X i (t) | X i (t)= X * (t) + κ f M Γ C( X i (t), X j (tτ)) X i (t) | X i (t)= X * (t) ] ξ (t) +[ κ f M K C( X i (t), X j (tτ)) X j (tτ) | X i (t)= X * (t) ] ξ (tτ)
δ ˙ i (t)=[ X i (t) F+ κ f M λ 1 X i (t) C] δ i (t) +[ κ f M λ i X j (tτ) C] δ i (tτ)
δ ˙ 1 (t)=[ X i (t) F+ κ X i (t) C] δ i (t) +[ κ X j (tτ) C] δ i (tτ)
ξ ˙ (t)=[ X i (t) F+ κ f X i (t) C]ξ(t) +[ κ f X j (tτ) C]ξ(tτ)
Ω= κ f αcos(Ωτ) κ f sin(Ωτ) r= g J 0 γ n N 0 (s γ n +g)(γ2 κ f cos(Ωτ)) γ n s γ n +g N= J 0 + g N 0 r 2 1+s r 2 γ n + g r 2 1+s r 2

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