Abstract

The effect of periodic phase modulation of light on a chaotic external-cavity semiconductor laser working in a regime of low-frequency fluctuations (LFFs) is studied numerically. It is observed that the phase modulation changes the time period between consecutive dropouts in the emitted laser intensity. A new variable ΦL is defined as the phase of the laser’s LFFs, which increases in time with 2π after each power dropout event. The phase ΦPM of the periodic phase modulator is unfolded on the real axis and increases linearly at a rate given by the modulating frequency. It is shown that the phase difference between the laser and the modulator ΔΦ(t)=mΦL(t)nΦPM(t), where m and n are integers, remains constant in time, leading to phase-synchronized states, for specific values of the modulating frequency and amplitude.

© 2006 Optical Society of America

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  1. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
    [CrossRef]
  2. C. Henry and R. F. Kazarinov, 'Instability of semiconductor lasers due to optical feedback from distant reflectors,' IEEE J. Quantum Electron. 22, 294-301 (1986).
    [CrossRef]
  3. G. H. M. Tartwijk, A. M. Levine, and D. Lenstra, 'Sisyphus effect in semiconductor lasers with optical feedback,' IEEE J. Sel. Top. Quantum Electron. 1, 175-183 (1995).
  4. A. Hohl and A. Gavrielidis, 'Bifurcation cascade in a semiconductor laser subject to optical feedback,' Phys. Rev. Lett. 82, 1148-1151 (1999).
    [CrossRef]
  5. V. Ahlers, U. Parlitz, and W. Lauterborn, 'Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,' Phys. Rev. E 58, 7208-7213 (1998).
    [CrossRef]
  6. D. W. Sukow and D. J. Gauthier, 'Entraining power-dropout events in an external cavity semiconductor laser using weak modulation of the injection current,' IEEE J. Quantum Electron. 36, 175-183 (2000).
    [CrossRef]
  7. J. S. Lawrence and D. B. Kane, 'Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,' IEEE J. Quantum Electron. 38, 185-192 (2002).
    [CrossRef]
  8. T. Heil, I. Fischer, W. Elsässer, and A. Gavrielides, 'Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,' Phys. Rev. Lett. 87, 243901 (2001).
    [CrossRef] [PubMed]
  9. E. F. Stone, 'Frequency entrainment of a phase coherent attractor,' Phys. Lett. A 163, 367-374 (1992).
    [CrossRef]
  10. E. Rosa Jr., E. Ott, and M. H. Hess, 'Transition to phase synchronization of chaos,' Phys. Rev. Lett. 80, 1642-1645 (1998).
    [CrossRef]
  11. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
    [CrossRef]
  12. C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
    [CrossRef] [PubMed]
  13. E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
    [CrossRef]
  14. W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
    [CrossRef]
  15. I. Z. Kiss and J. L. Hudson, 'Phase synchronization and suppresion of chaos through intermittency in forcing of an electrochemical oscillator,' Phys. Rev. E 64, 046215 (2001).
    [CrossRef]
  16. M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
    [CrossRef]
  17. W.-S. Lam, P. N. Guzdar, and R. Roy, 'Hilbert phase analysis of the dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 67, 025604(R) (2003).
    [CrossRef]
  18. T. Yalçinkaia and Y.-C. Lai, 'Phase characterization of chaos,' Phys. Rev. Lett. 79, 3885-3888 (1997).
    [CrossRef]
  19. K. V. Volodchenko, V. N. Ivanov, S.-H. Gong, M. Choi, Y.-J. Park, and C.-M. Kim, 'Phase synchronization in coupled Nd:YAG lasers,' Opt. Lett. 26, 1406-1408 (2001).
    [CrossRef]
  20. D. J. DeShazer, R. Breban, E. Ott, and R. Roy, 'Detecting phase synchronization in a chaotic laser array,' Phys. Rev. Lett. 87, 044101 (2001).
    [CrossRef] [PubMed]
  21. A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, 'Phase synchronization of chaotic oscillators by external driving,' Physica D 104, 219-238 (1997).
    [CrossRef]
  22. P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
    [CrossRef]
  23. R. Lang and K. Kobayashi, 'External optical feedback effects on semiconductor injection laser properties,' IEEE J. Quantum Electron. 16, 347-355 (1980).
    [CrossRef]
  24. A. Sánchez-Díaz, C. R. Mirasso, P. Colet, and P. García-Fernández, 'Encoded Gbits/s digital communications with synchronized chaotic semiconductor lasers,' IEEE J. Quantum Electron. 35, 292-297 (1999).
    [CrossRef]
  25. I. Wallace, D. Yu, W. Lu, and R. G. Harrison, 'Synchronization of power dropouts in coupled semiconductor lasers with external feedback,' Phys. Rev. A 63, 013809 (2001).
    [CrossRef]
  26. I. Z. Kiss, Q. Lv, and J. L. Hudson, 'Synchronization of non-phase-coherent chaotic electrochemical oscillations,' Phys. Rev. E 71, 035201 (2005).
    [CrossRef]
  27. T. Sano, 'Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback,' Phys. Rev. A 50, 2719-2726 (1994).
    [CrossRef] [PubMed]
  28. B. Tromborg, J. H. Osmundsen, and H. Olesen, 'Stability analysis for a semiconductor laser in an external cavity,' IEEE J. Quantum Electron. 20, 1023-1031 (1984).
    [CrossRef]
  29. J. Ohtsubo, Semiconductor Lasers, Stability, Instability and Chaos (Springer, 2006).
  30. W. Ray, W.-S. Lam, P. N. Guzdar, and R. Roy, 'Observation of chaotic itinerancy in the light and carrier dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 73, 026219 (2006).
    [CrossRef]

2006 (1)

W. Ray, W.-S. Lam, P. N. Guzdar, and R. Roy, 'Observation of chaotic itinerancy in the light and carrier dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 73, 026219 (2006).
[CrossRef]

2005 (1)

I. Z. Kiss, Q. Lv, and J. L. Hudson, 'Synchronization of non-phase-coherent chaotic electrochemical oscillations,' Phys. Rev. E 71, 035201 (2005).
[CrossRef]

2003 (2)

M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
[CrossRef]

W.-S. Lam, P. N. Guzdar, and R. Roy, 'Hilbert phase analysis of the dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 67, 025604(R) (2003).
[CrossRef]

2002 (1)

J. S. Lawrence and D. B. Kane, 'Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,' IEEE J. Quantum Electron. 38, 185-192 (2002).
[CrossRef]

2001 (6)

T. Heil, I. Fischer, W. Elsässer, and A. Gavrielides, 'Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,' Phys. Rev. Lett. 87, 243901 (2001).
[CrossRef] [PubMed]

K. V. Volodchenko, V. N. Ivanov, S.-H. Gong, M. Choi, Y.-J. Park, and C.-M. Kim, 'Phase synchronization in coupled Nd:YAG lasers,' Opt. Lett. 26, 1406-1408 (2001).
[CrossRef]

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, 'Detecting phase synchronization in a chaotic laser array,' Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
[CrossRef]

I. Z. Kiss and J. L. Hudson, 'Phase synchronization and suppresion of chaos through intermittency in forcing of an electrochemical oscillator,' Phys. Rev. E 64, 046215 (2001).
[CrossRef]

I. Wallace, D. Yu, W. Lu, and R. G. Harrison, 'Synchronization of power dropouts in coupled semiconductor lasers with external feedback,' Phys. Rev. A 63, 013809 (2001).
[CrossRef]

2000 (3)

C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
[CrossRef] [PubMed]

E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
[CrossRef]

D. W. Sukow and D. J. Gauthier, 'Entraining power-dropout events in an external cavity semiconductor laser using weak modulation of the injection current,' IEEE J. Quantum Electron. 36, 175-183 (2000).
[CrossRef]

1999 (2)

A. Hohl and A. Gavrielidis, 'Bifurcation cascade in a semiconductor laser subject to optical feedback,' Phys. Rev. Lett. 82, 1148-1151 (1999).
[CrossRef]

A. Sánchez-Díaz, C. R. Mirasso, P. Colet, and P. García-Fernández, 'Encoded Gbits/s digital communications with synchronized chaotic semiconductor lasers,' IEEE J. Quantum Electron. 35, 292-297 (1999).
[CrossRef]

1998 (3)

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

V. Ahlers, U. Parlitz, and W. Lauterborn, 'Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,' Phys. Rev. E 58, 7208-7213 (1998).
[CrossRef]

E. Rosa Jr., E. Ott, and M. H. Hess, 'Transition to phase synchronization of chaos,' Phys. Rev. Lett. 80, 1642-1645 (1998).
[CrossRef]

1997 (2)

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, 'Phase synchronization of chaotic oscillators by external driving,' Physica D 104, 219-238 (1997).
[CrossRef]

T. Yalçinkaia and Y.-C. Lai, 'Phase characterization of chaos,' Phys. Rev. Lett. 79, 3885-3888 (1997).
[CrossRef]

1995 (1)

G. H. M. Tartwijk, A. M. Levine, and D. Lenstra, 'Sisyphus effect in semiconductor lasers with optical feedback,' IEEE J. Sel. Top. Quantum Electron. 1, 175-183 (1995).

1994 (1)

T. Sano, 'Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback,' Phys. Rev. A 50, 2719-2726 (1994).
[CrossRef] [PubMed]

1992 (1)

E. F. Stone, 'Frequency entrainment of a phase coherent attractor,' Phys. Lett. A 163, 367-374 (1992).
[CrossRef]

1986 (1)

C. Henry and R. F. Kazarinov, 'Instability of semiconductor lasers due to optical feedback from distant reflectors,' IEEE J. Quantum Electron. 22, 294-301 (1986).
[CrossRef]

1984 (1)

B. Tromborg, J. H. Osmundsen, and H. Olesen, 'Stability analysis for a semiconductor laser in an external cavity,' IEEE J. Quantum Electron. 20, 1023-1031 (1984).
[CrossRef]

1980 (1)

R. Lang and K. Kobayashi, 'External optical feedback effects on semiconductor injection laser properties,' IEEE J. Quantum Electron. 16, 347-355 (1980).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
[CrossRef]

Ahlers, V.

V. Ahlers, U. Parlitz, and W. Lauterborn, 'Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,' Phys. Rev. E 58, 7208-7213 (1998).
[CrossRef]

Baptista, M. S.

M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
[CrossRef]

Breban, R.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, 'Detecting phase synchronization in a chaotic laser array,' Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

Caldas, I. L.

M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
[CrossRef]

Choi, M.

Colet, P.

A. Sánchez-Díaz, C. R. Mirasso, P. Colet, and P. García-Fernández, 'Encoded Gbits/s digital communications with synchronized chaotic semiconductor lasers,' IEEE J. Quantum Electron. 35, 292-297 (1999).
[CrossRef]

DeShazer, D. J.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, 'Detecting phase synchronization in a chaotic laser array,' Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

Elsässer, W.

T. Heil, I. Fischer, W. Elsässer, and A. Gavrielides, 'Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,' Phys. Rev. Lett. 87, 243901 (2001).
[CrossRef] [PubMed]

Fischer, I.

T. Heil, I. Fischer, W. Elsässer, and A. Gavrielides, 'Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,' Phys. Rev. Lett. 87, 243901 (2001).
[CrossRef] [PubMed]

Freund, H. J.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

García-Fernández, P.

A. Sánchez-Díaz, C. R. Mirasso, P. Colet, and P. García-Fernández, 'Encoded Gbits/s digital communications with synchronized chaotic semiconductor lasers,' IEEE J. Quantum Electron. 35, 292-297 (1999).
[CrossRef]

Gauthier, D. J.

D. W. Sukow and D. J. Gauthier, 'Entraining power-dropout events in an external cavity semiconductor laser using weak modulation of the injection current,' IEEE J. Quantum Electron. 36, 175-183 (2000).
[CrossRef]

Gavrielides, A.

T. Heil, I. Fischer, W. Elsässer, and A. Gavrielides, 'Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,' Phys. Rev. Lett. 87, 243901 (2001).
[CrossRef] [PubMed]

Gavrielidis, A.

A. Hohl and A. Gavrielidis, 'Bifurcation cascade in a semiconductor laser subject to optical feedback,' Phys. Rev. Lett. 82, 1148-1151 (1999).
[CrossRef]

Gong, S.-H.

Guzdar, P. N.

W. Ray, W.-S. Lam, P. N. Guzdar, and R. Roy, 'Observation of chaotic itinerancy in the light and carrier dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 73, 026219 (2006).
[CrossRef]

W.-S. Lam, P. N. Guzdar, and R. Roy, 'Hilbert phase analysis of the dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 67, 025604(R) (2003).
[CrossRef]

Harrison, R. G.

I. Wallace, D. Yu, W. Lu, and R. G. Harrison, 'Synchronization of power dropouts in coupled semiconductor lasers with external feedback,' Phys. Rev. A 63, 013809 (2001).
[CrossRef]

Heil, T.

T. Heil, I. Fischer, W. Elsässer, and A. Gavrielides, 'Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,' Phys. Rev. Lett. 87, 243901 (2001).
[CrossRef] [PubMed]

Henry, C.

C. Henry and R. F. Kazarinov, 'Instability of semiconductor lasers due to optical feedback from distant reflectors,' IEEE J. Quantum Electron. 22, 294-301 (1986).
[CrossRef]

Hess, M. H.

E. Rosa Jr., E. Ott, and M. H. Hess, 'Transition to phase synchronization of chaos,' Phys. Rev. Lett. 80, 1642-1645 (1998).
[CrossRef]

Hohl, A.

A. Hohl and A. Gavrielidis, 'Bifurcation cascade in a semiconductor laser subject to optical feedback,' Phys. Rev. Lett. 82, 1148-1151 (1999).
[CrossRef]

Hudson, J. L.

I. Z. Kiss, Q. Lv, and J. L. Hudson, 'Synchronization of non-phase-coherent chaotic electrochemical oscillations,' Phys. Rev. E 71, 035201 (2005).
[CrossRef]

I. Z. Kiss and J. L. Hudson, 'Phase synchronization and suppresion of chaos through intermittency in forcing of an electrochemical oscillator,' Phys. Rev. E 64, 046215 (2001).
[CrossRef]

Ivanov, V. N.

Kane, D. B.

J. S. Lawrence and D. B. Kane, 'Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,' IEEE J. Quantum Electron. 38, 185-192 (2002).
[CrossRef]

Kazarinov, R. F.

C. Henry and R. F. Kazarinov, 'Instability of semiconductor lasers due to optical feedback from distant reflectors,' IEEE J. Quantum Electron. 22, 294-301 (1986).
[CrossRef]

Kim, C.-M.

Kiss, I. Z.

I. Z. Kiss, Q. Lv, and J. L. Hudson, 'Synchronization of non-phase-coherent chaotic electrochemical oscillations,' Phys. Rev. E 71, 035201 (2005).
[CrossRef]

I. Z. Kiss and J. L. Hudson, 'Phase synchronization and suppresion of chaos through intermittency in forcing of an electrochemical oscillator,' Phys. Rev. E 64, 046215 (2001).
[CrossRef]

Kobayashi, K.

R. Lang and K. Kobayashi, 'External optical feedback effects on semiconductor injection laser properties,' IEEE J. Quantum Electron. 16, 347-355 (1980).
[CrossRef]

Kurths, J.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, 'Phase synchronization of chaotic oscillators by external driving,' Physica D 104, 219-238 (1997).
[CrossRef]

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

Lai, Y.-C.

T. Yalçinkaia and Y.-C. Lai, 'Phase characterization of chaos,' Phys. Rev. Lett. 79, 3885-3888 (1997).
[CrossRef]

Lam, W.-S.

W. Ray, W.-S. Lam, P. N. Guzdar, and R. Roy, 'Observation of chaotic itinerancy in the light and carrier dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 73, 026219 (2006).
[CrossRef]

W.-S. Lam, P. N. Guzdar, and R. Roy, 'Hilbert phase analysis of the dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 67, 025604(R) (2003).
[CrossRef]

Lang, R.

R. Lang and K. Kobayashi, 'External optical feedback effects on semiconductor injection laser properties,' IEEE J. Quantum Electron. 16, 347-355 (1980).
[CrossRef]

Lauterborn, W.

V. Ahlers, U. Parlitz, and W. Lauterborn, 'Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,' Phys. Rev. E 58, 7208-7213 (1998).
[CrossRef]

Lawrence, J. S.

J. S. Lawrence and D. B. Kane, 'Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,' IEEE J. Quantum Electron. 38, 185-192 (2002).
[CrossRef]

Lenstra, D.

G. H. M. Tartwijk, A. M. Levine, and D. Lenstra, 'Sisyphus effect in semiconductor lasers with optical feedback,' IEEE J. Sel. Top. Quantum Electron. 1, 175-183 (1995).

Levine, A. M.

G. H. M. Tartwijk, A. M. Levine, and D. Lenstra, 'Sisyphus effect in semiconductor lasers with optical feedback,' IEEE J. Sel. Top. Quantum Electron. 1, 175-183 (1995).

Lu, W.

I. Wallace, D. Yu, W. Lu, and R. G. Harrison, 'Synchronization of power dropouts in coupled semiconductor lasers with external feedback,' Phys. Rev. A 63, 013809 (2001).
[CrossRef]

Lv, Q.

I. Z. Kiss, Q. Lv, and J. L. Hudson, 'Synchronization of non-phase-coherent chaotic electrochemical oscillations,' Phys. Rev. E 71, 035201 (2005).
[CrossRef]

Mirasso, C. R.

A. Sánchez-Díaz, C. R. Mirasso, P. Colet, and P. García-Fernández, 'Encoded Gbits/s digital communications with synchronized chaotic semiconductor lasers,' IEEE J. Quantum Electron. 35, 292-297 (1999).
[CrossRef]

Monti, M.

W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
[CrossRef]

E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
[CrossRef]

C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
[CrossRef] [PubMed]

Ohtsubo, J.

J. Ohtsubo, Semiconductor Lasers, Stability, Instability and Chaos (Springer, 2006).

Olesen, H.

B. Tromborg, J. H. Osmundsen, and H. Olesen, 'Stability analysis for a semiconductor laser in an external cavity,' IEEE J. Quantum Electron. 20, 1023-1031 (1984).
[CrossRef]

Osipov, G. V.

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, 'Phase synchronization of chaotic oscillators by external driving,' Physica D 104, 219-238 (1997).
[CrossRef]

Osmundsen, J. H.

B. Tromborg, J. H. Osmundsen, and H. Olesen, 'Stability analysis for a semiconductor laser in an external cavity,' IEEE J. Quantum Electron. 20, 1023-1031 (1984).
[CrossRef]

Ott, E.

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, 'Detecting phase synchronization in a chaotic laser array,' Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

E. Rosa Jr., E. Ott, and M. H. Hess, 'Transition to phase synchronization of chaos,' Phys. Rev. Lett. 80, 1642-1645 (1998).
[CrossRef]

Pardo, W. B.

W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
[CrossRef]

C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
[CrossRef] [PubMed]

E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
[CrossRef]

Park, Y.-J.

Parlitz, U.

V. Ahlers, U. Parlitz, and W. Lauterborn, 'Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,' Phys. Rev. E 58, 7208-7213 (1998).
[CrossRef]

Pikovsky, A.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

Pikovsky, A. S.

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, 'Phase synchronization of chaotic oscillators by external driving,' Physica D 104, 219-238 (1997).
[CrossRef]

Ray, W.

W. Ray, W.-S. Lam, P. N. Guzdar, and R. Roy, 'Observation of chaotic itinerancy in the light and carrier dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 73, 026219 (2006).
[CrossRef]

Rosa, E.

M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
[CrossRef]

W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
[CrossRef]

E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
[CrossRef]

C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
[CrossRef] [PubMed]

E. Rosa Jr., E. Ott, and M. H. Hess, 'Transition to phase synchronization of chaos,' Phys. Rev. Lett. 80, 1642-1645 (1998).
[CrossRef]

Rosenblum, M.

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

Rosenblum, M. G.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, 'Phase synchronization of chaotic oscillators by external driving,' Physica D 104, 219-238 (1997).
[CrossRef]

Roy, R.

W. Ray, W.-S. Lam, P. N. Guzdar, and R. Roy, 'Observation of chaotic itinerancy in the light and carrier dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 73, 026219 (2006).
[CrossRef]

W.-S. Lam, P. N. Guzdar, and R. Roy, 'Hilbert phase analysis of the dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 67, 025604(R) (2003).
[CrossRef]

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, 'Detecting phase synchronization in a chaotic laser array,' Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

Sánchez-Díaz, A.

A. Sánchez-Díaz, C. R. Mirasso, P. Colet, and P. García-Fernández, 'Encoded Gbits/s digital communications with synchronized chaotic semiconductor lasers,' IEEE J. Quantum Electron. 35, 292-297 (1999).
[CrossRef]

Sano, T.

T. Sano, 'Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback,' Phys. Rev. A 50, 2719-2726 (1994).
[CrossRef] [PubMed]

Sartorelli, J. C.

M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
[CrossRef]

Schnitzler, A.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

Silva, T. P.

M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
[CrossRef]

Stone, E. F.

E. F. Stone, 'Frequency entrainment of a phase coherent attractor,' Phys. Lett. A 163, 367-374 (1992).
[CrossRef]

Sukow, D. W.

D. W. Sukow and D. J. Gauthier, 'Entraining power-dropout events in an external cavity semiconductor laser using weak modulation of the injection current,' IEEE J. Quantum Electron. 36, 175-183 (2000).
[CrossRef]

Tartwijk, G. H. M.

G. H. M. Tartwijk, A. M. Levine, and D. Lenstra, 'Sisyphus effect in semiconductor lasers with optical feedback,' IEEE J. Sel. Top. Quantum Electron. 1, 175-183 (1995).

Tass, P.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

Ticos, C. M.

W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
[CrossRef]

C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
[CrossRef] [PubMed]

E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
[CrossRef]

Tromborg, B.

B. Tromborg, J. H. Osmundsen, and H. Olesen, 'Stability analysis for a semiconductor laser in an external cavity,' IEEE J. Quantum Electron. 20, 1023-1031 (1984).
[CrossRef]

Volkmann, J.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

Volodchenko, K. V.

Walkenstein, J. A.

W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
[CrossRef]

E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
[CrossRef]

C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
[CrossRef] [PubMed]

Wallace, I.

I. Wallace, D. Yu, W. Lu, and R. G. Harrison, 'Synchronization of power dropouts in coupled semiconductor lasers with external feedback,' Phys. Rev. A 63, 013809 (2001).
[CrossRef]

Weule, J.

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

Yalçinkaia, T.

T. Yalçinkaia and Y.-C. Lai, 'Phase characterization of chaos,' Phys. Rev. Lett. 79, 3885-3888 (1997).
[CrossRef]

Yu, D.

I. Wallace, D. Yu, W. Lu, and R. G. Harrison, 'Synchronization of power dropouts in coupled semiconductor lasers with external feedback,' Phys. Rev. A 63, 013809 (2001).
[CrossRef]

IEEE J. Quantum Electron. (6)

C. Henry and R. F. Kazarinov, 'Instability of semiconductor lasers due to optical feedback from distant reflectors,' IEEE J. Quantum Electron. 22, 294-301 (1986).
[CrossRef]

D. W. Sukow and D. J. Gauthier, 'Entraining power-dropout events in an external cavity semiconductor laser using weak modulation of the injection current,' IEEE J. Quantum Electron. 36, 175-183 (2000).
[CrossRef]

J. S. Lawrence and D. B. Kane, 'Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,' IEEE J. Quantum Electron. 38, 185-192 (2002).
[CrossRef]

R. Lang and K. Kobayashi, 'External optical feedback effects on semiconductor injection laser properties,' IEEE J. Quantum Electron. 16, 347-355 (1980).
[CrossRef]

A. Sánchez-Díaz, C. R. Mirasso, P. Colet, and P. García-Fernández, 'Encoded Gbits/s digital communications with synchronized chaotic semiconductor lasers,' IEEE J. Quantum Electron. 35, 292-297 (1999).
[CrossRef]

B. Tromborg, J. H. Osmundsen, and H. Olesen, 'Stability analysis for a semiconductor laser in an external cavity,' IEEE J. Quantum Electron. 20, 1023-1031 (1984).
[CrossRef]

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

G. H. M. Tartwijk, A. M. Levine, and D. Lenstra, 'Sisyphus effect in semiconductor lasers with optical feedback,' IEEE J. Sel. Top. Quantum Electron. 1, 175-183 (1995).

Int. J. Bifurcation Chaos Appl. Sci. Eng. (1)

E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Phase synchronization of chaos in a plasma discharge tube,' Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2551-2563 (2000).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

W. B. Pardo, E. Rosa Jr., C. M. Ticos, J. A. Walkenstein, and M. Monti, 'Pacing a chaotic plasma with a music signal,' Phys. Lett. A 284, 259-265 (2001).
[CrossRef]

E. F. Stone, 'Frequency entrainment of a phase coherent attractor,' Phys. Lett. A 163, 367-374 (1992).
[CrossRef]

Phys. Rev. A (2)

T. Sano, 'Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback,' Phys. Rev. A 50, 2719-2726 (1994).
[CrossRef] [PubMed]

I. Wallace, D. Yu, W. Lu, and R. G. Harrison, 'Synchronization of power dropouts in coupled semiconductor lasers with external feedback,' Phys. Rev. A 63, 013809 (2001).
[CrossRef]

Phys. Rev. E (6)

I. Z. Kiss, Q. Lv, and J. L. Hudson, 'Synchronization of non-phase-coherent chaotic electrochemical oscillations,' Phys. Rev. E 71, 035201 (2005).
[CrossRef]

W. Ray, W.-S. Lam, P. N. Guzdar, and R. Roy, 'Observation of chaotic itinerancy in the light and carrier dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 73, 026219 (2006).
[CrossRef]

V. Ahlers, U. Parlitz, and W. Lauterborn, 'Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,' Phys. Rev. E 58, 7208-7213 (1998).
[CrossRef]

I. Z. Kiss and J. L. Hudson, 'Phase synchronization and suppresion of chaos through intermittency in forcing of an electrochemical oscillator,' Phys. Rev. E 64, 046215 (2001).
[CrossRef]

M. S. Baptista, T. P. Silva, J. C. Sartorelli, I. L. Caldas, and E. Rosa Jr., 'Phase synchronization in the perturbed Chua circuit,' Phys. Rev. E 67, 056212 (2003).
[CrossRef]

W.-S. Lam, P. N. Guzdar, and R. Roy, 'Hilbert phase analysis of the dynamics of a semiconductor laser with optical feedback,' Phys. Rev. E 67, 025604(R) (2003).
[CrossRef]

Phys. Rev. Lett. (7)

T. Yalçinkaia and Y.-C. Lai, 'Phase characterization of chaos,' Phys. Rev. Lett. 79, 3885-3888 (1997).
[CrossRef]

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, 'Detecting phase synchronization in a chaotic laser array,' Phys. Rev. Lett. 87, 044101 (2001).
[CrossRef] [PubMed]

C. M. Ticos, E. Rosa, Jr., W. B. Pardo, J. A. Walkenstein, and M. Monti, 'Experimental real-time phase synchronization of a paced chaotic plasma discharge,' Phys. Rev. Lett. 85, 2929-2932 (2000).
[CrossRef] [PubMed]

P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund, 'Detection of n:m phase locking from noisy data: application to magnetoencephalography,' Phys. Rev. Lett. 81, 3291-3294 (1998).
[CrossRef]

A. Hohl and A. Gavrielidis, 'Bifurcation cascade in a semiconductor laser subject to optical feedback,' Phys. Rev. Lett. 82, 1148-1151 (1999).
[CrossRef]

E. Rosa Jr., E. Ott, and M. H. Hess, 'Transition to phase synchronization of chaos,' Phys. Rev. Lett. 80, 1642-1645 (1998).
[CrossRef]

T. Heil, I. Fischer, W. Elsässer, and A. Gavrielides, 'Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,' Phys. Rev. Lett. 87, 243901 (2001).
[CrossRef] [PubMed]

Physica D (1)

A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, 'Phase synchronization of chaotic oscillators by external driving,' Physica D 104, 219-238 (1997).
[CrossRef]

Other (3)

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge U. Press, 2001).
[CrossRef]

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002).
[CrossRef]

J. Ohtsubo, Semiconductor Lasers, Stability, Instability and Chaos (Springer, 2006).

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Figures (10)

Fig. 1
Fig. 1

Schematics of the external-cavity semiconductor laser configuration with a phase modulator (PM) and an unidirectional optical filter (OI).

Fig. 2
Fig. 2

Dropouts in the chaotic output intensity of the laser: (a) without modulation and (b) with modulation at Ω PM = 1.97 G rad s and A = 1 .

Fig. 3
Fig. 3

Histogram of the power dropout events: (a) laser without modulation and (b) laser modulated at Ω PM = 1.97 G rad s and A = 1 .

Fig. 4
Fig. 4

Phases of the modulator Φ PM and laser Φ L and their instantaneous ratio r = Φ PM Φ L , for a modulation frequency Ω PM = 1.97 G rad s and A = 1 .

Fig. 5
Fig. 5

(a) σ and (b) r versus the modulator frequency, respectively, for A = 1 . Region (1) is characterized by chaos suppression and stabilization, region (2) by period-2 oscillations, and region (3) by chaotic intermittency between LFFs and low-amplitude chaotic oscillations. For σ 0.6 the LFFs are steady and well synchronized with the modulating signal.

Fig. 6
Fig. 6

(a) Chaos suppression for Ω PM = 1.91 G rad s , (b) output stabilization for Ω PM = 1.93 G rad s , (c) period-2 oscillations for Ω PM = 1.94 G rad s , and (d) chaos intermittency between LFFs and low-amplitude oscillations. The amplitude of the modulating signal is set to A = 1 .

Fig. 7
Fig. 7

Map of σ as a function of the modulator parameters Ω PM and A. The color bar indicates the local values of σ. The regions of synchronization between the laser’s LFFs and the modulator are marked with white and light gray. In black ( σ = 0 ) is the region of non-LFFs states, i.e., states in which the emission is stabilized, periodic, or chaotic but with a much lower amplitude.

Fig. 8
Fig. 8

Map of r as a function of the modulator parameters Ω PM and A. The color bar indicates the local values of r. Notice the region colored uniformly in gray with triangular shape between Ω PM = 1.95 and 2.02 G rad s and A 0.7 , which corresponds to values of r in the range 20 to 30. Within this region the LFFs are synchronized with the PM. In black ( r = 0 ) is the region of non-LFFs states.

Fig. 9
Fig. 9

Region of non-LFFs, which corresponds to the black zone of Figs. 7, 8. Here, the stable states are marked with black, the periodic states are marked with dark gray, and the low-amplitude chaos states, including the intermittency between low-amplitude chaos and LFFs, are marked with light gray.

Fig. 10
Fig. 10

(a). Trajectory in phase space of a solitary ECSL. The external-cavity modes and antimodes are situated on the ellipse marked with a dashed curve. (b) Modes and antimodes of solitary ECSL marked with (1) and (2), respectively. Modes and antimodes marked with (3) and (4) for A = 1 and with (5) and (6) for A = 1 , respectively. (c) Trajectory in phase space of a modulated laser at Φ PM = 1.97 G rad s . (d) Trajectory in phase space of a modulated laser at Φ PM = 1.94 G rad s .

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

d E ( t ) d t = ( 1 i α ) [ G ( t ) 1 τ p ] E ( t ) 2 + γ E ( t τ r ) × exp { i [ ω 0 τ r + A sin ( Ω PM t ) ] } ,
d N ( t ) d t = I e N ( t ) τ n G ( t ) E ( t ) 2 ,
with G ( t ) = g [ N ( t ) N 0 ] 1 + s E ( s ) 2 .
Δ ϕ PM = A sin Ω PM t ,
exp [ i A sin ( Ω PM t ) ] = J 0 ( A ) + i k = 0 2 J 2 k + 1 ( A ) sin [ ( 2 k + 1 ) Ω PM t ] + k = 1 2 J 2 k ( A ) cos ( 2 k Ω PM t ) .
Φ L ( t ) = 2 π t t j t j + 1 t j + 2 π j , with j = 1 , 2 ,
Ω L lim t 2 π N t = 1 N j Ω L ( j ) .
lim t Φ L ( t ) = Ω L t .
Φ PM ( t ) = Ω PM ( t t k ) + 2 π k = Ω PM t ,
Δ Φ ( t ) = m Φ L ( t ) n Φ PM ( t ) ,
lim t Δ Φ ( t ) = m Ω L t n Ω PM t .
Ω PM Ω L = m n .
r ( t ) = Φ PM ( t ) Φ L ( t ) ,
S = i = 1 M p i log p i .
η s = C sin ( arctan α + η s + ω 0 τ r + Φ PMs ) ,
Δ N s = 1 g [ 1 τ p 2 γ cos ( η s + ω 0 τ r + Φ PMs ) ] ,

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