Abstract

Phase-locked solutions are found numerically in a semiconductor laser with one and two external cavities. Different periodic, quasiperiodic, chaotic, and steady-state regimes form Arnold’s tongues in bi-dimensional parameter spaces of the length and feedback strengths of the external cavities and the pump parameter. This rich structure gives additional possibility for controlling complex dynamics and chaos in a semiconductor laser with external cavities by properly adjusting their lengths and feedback strengths.

© 2006 Optical Society of America

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  1. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, Springer Series in Optical Sciences, vol. 111 (Springer-Verlag, Berlin, 2005).
  2. P. Besnard, B. Meziane, and G. N. Stephan, "Feedback phenomena in a semiconductor laser induced by distant reflectors," IEEE J. Quantum Electron. 29, 1271 (1993).
    [CrossRef]
  3. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
    [CrossRef]
  4. R. P. Salathe, "Diode lasers coupled to external resonators," Appl. Phys. 20, 1 (1979).
    [CrossRef]
  5. D. Lenstra, B. H. Verbeek, and A. J. den Boef, "Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback," IEEE J. Quantum Electron. 21, 674 (1985).
    [CrossRef]
  6. M. Jesper, T. Bjarne, and M. Jannik, "Chaos in semiconductor laser with optical feedback: theory and experiment," IEEE J. Quantum Electron. 28, 93 (1992).
    [CrossRef]
  7. T. K. Sano, "Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback," Phys. Rev. A 50, 2719 (1994).
    [CrossRef] [PubMed]
  8. Y. Liu and J. Ohtsubo, "Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer," IEEE J. Quantum Electron. 33, 1163 (1997).
    [CrossRef]
  9. F. Rogister, P. Mégret, O. Deparis, M. Blondel, and T. Erneux, "Suppression of low frequency fluctuations and stabilization of a semiconductor laser subject to optical feedback from a double cavity: theoretical results," Opt. Lett. 24, 1218 (1999).
    [CrossRef]
  10. F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, "Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external cavity laser diode," Opt. Lett. 25, 808 (2000).
    [CrossRef]
  11. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1989).
  12. J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
    [CrossRef]
  13. R. Lang and K Kobayashi, "External Optical Feedback Effects on Semiconductor Injection Laser Properties," IEEE J. Quantum Electron. 16, 347-355 (1980).
    [CrossRef]
  14. S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, "Experimental verification of the synchronization condition for chaotic external cavity diode lasers," Phys. Rev. E 62, 7505-7507 (2000).
    [CrossRef]
  15. T.W. Carr, "Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback," Eur. Phys. J. D 19, 245-255 (2002).
    [CrossRef]
  16. T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
    [CrossRef]
  17. H. G. Shuster, Deterministic Chaos (VCH Verlag, Weinheim, 1988).

2003 (1)

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

2002 (1)

T.W. Carr, "Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback," Eur. Phys. J. D 19, 245-255 (2002).
[CrossRef]

2001 (1)

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
[CrossRef]

2000 (2)

F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, "Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external cavity laser diode," Opt. Lett. 25, 808 (2000).
[CrossRef]

S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, "Experimental verification of the synchronization condition for chaotic external cavity diode lasers," Phys. Rev. E 62, 7505-7507 (2000).
[CrossRef]

1999 (1)

1997 (2)

Y. Liu and J. Ohtsubo, "Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer," IEEE J. Quantum Electron. 33, 1163 (1997).
[CrossRef]

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
[CrossRef]

1994 (1)

T. K. Sano, "Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback," Phys. Rev. A 50, 2719 (1994).
[CrossRef] [PubMed]

1993 (1)

P. Besnard, B. Meziane, and G. N. Stephan, "Feedback phenomena in a semiconductor laser induced by distant reflectors," IEEE J. Quantum Electron. 29, 1271 (1993).
[CrossRef]

1992 (1)

M. Jesper, T. Bjarne, and M. Jannik, "Chaos in semiconductor laser with optical feedback: theory and experiment," IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

1985 (1)

D. Lenstra, B. H. Verbeek, and A. J. den Boef, "Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback," IEEE J. Quantum Electron. 21, 674 (1985).
[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]

1979 (1)

R. P. Salathe, "Diode lasers coupled to external resonators," Appl. Phys. 20, 1 (1979).
[CrossRef]

Aliaga, J.

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
[CrossRef]

Besnard, P.

P. Besnard, B. Meziane, and G. N. Stephan, "Feedback phenomena in a semiconductor laser induced by distant reflectors," IEEE J. Quantum Electron. 29, 1271 (1993).
[CrossRef]

Bjarne, T.

M. Jesper, T. Bjarne, and M. Jannik, "Chaos in semiconductor laser with optical feedback: theory and experiment," IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Blondel, M.

Carr, T.W.

T.W. Carr, "Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback," Eur. Phys. J. D 19, 245-255 (2002).
[CrossRef]

den Boef, A. J.

D. Lenstra, B. H. Verbeek, and A. J. den Boef, "Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback," IEEE J. Quantum Electron. 21, 674 (1985).
[CrossRef]

Deparis, O.

ElsäBer, W

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

Erneux, T.

Fischer, I.

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

Gavrielides, A.

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, "Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external cavity laser diode," Opt. Lett. 25, 808 (2000).
[CrossRef]

Giacomelli, G.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
[CrossRef]

Giudici, M.

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
[CrossRef]

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
[CrossRef]

Green, C.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
[CrossRef]

Green, K

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

Heil, T.

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

Jannik, M.

M. Jesper, T. Bjarne, and M. Jannik, "Chaos in semiconductor laser with optical feedback: theory and experiment," IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Jesper, M.

M. Jesper, T. Bjarne, and M. Jannik, "Chaos in semiconductor laser with optical feedback: theory and experiment," IEEE J. Quantum Electron. 28, 93 (1992).
[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]

Krauskopf, B

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

Laje, R.

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
[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]

Lenstra, D.

D. Lenstra, B. H. Verbeek, and A. J. den Boef, "Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback," IEEE J. Quantum Electron. 21, 674 (1985).
[CrossRef]

Liu, Y.

Y. Liu and J. Ohtsubo, "Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer," IEEE J. Quantum Electron. 33, 1163 (1997).
[CrossRef]

Mégret, P.

Mendez, J. M.

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
[CrossRef]

Meziane, B.

P. Besnard, B. Meziane, and G. N. Stephan, "Feedback phenomena in a semiconductor laser induced by distant reflectors," IEEE J. Quantum Electron. 29, 1271 (1993).
[CrossRef]

Mindlin, G. B.

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
[CrossRef]

Nespolo, U.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
[CrossRef]

Ohtsubo, J.

Y. Liu and J. Ohtsubo, "Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer," IEEE J. Quantum Electron. 33, 1163 (1997).
[CrossRef]

Rogister, F.

Salathe, R. P.

R. P. Salathe, "Diode lasers coupled to external resonators," Appl. Phys. 20, 1 (1979).
[CrossRef]

Sano, T. K.

T. K. Sano, "Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback," Phys. Rev. A 50, 2719 (1994).
[CrossRef] [PubMed]

Shahverdi, E. M.

S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, "Experimental verification of the synchronization condition for chaotic external cavity diode lasers," Phys. Rev. E 62, 7505-7507 (2000).
[CrossRef]

Shore, K. A.

S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, "Experimental verification of the synchronization condition for chaotic external cavity diode lasers," Phys. Rev. E 62, 7505-7507 (2000).
[CrossRef]

Sivapakrasam, S.

S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, "Experimental verification of the synchronization condition for chaotic external cavity diode lasers," Phys. Rev. E 62, 7505-7507 (2000).
[CrossRef]

Stephan, G. N.

P. Besnard, B. Meziane, and G. N. Stephan, "Feedback phenomena in a semiconductor laser induced by distant reflectors," IEEE J. Quantum Electron. 29, 1271 (1993).
[CrossRef]

Sukow, D. W.

Tredicce, J. R.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
[CrossRef]

Verbeek, B. H.

D. Lenstra, B. H. Verbeek, and A. J. den Boef, "Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback," IEEE J. Quantum Electron. 21, 674 (1985).
[CrossRef]

Appl. Phys. (1)

R. P. Salathe, "Diode lasers coupled to external resonators," Appl. Phys. 20, 1 (1979).
[CrossRef]

Eur. Phys. J. D (1)

T.W. Carr, "Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback," Eur. Phys. J. D 19, 245-255 (2002).
[CrossRef]

IEEE J. Quantum Electron. (5)

R. Lang and K Kobayashi, "External Optical Feedback Effects on Semiconductor Injection Laser Properties," IEEE J. Quantum Electron. 16, 347-355 (1980).
[CrossRef]

D. Lenstra, B. H. Verbeek, and A. J. den Boef, "Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback," IEEE J. Quantum Electron. 21, 674 (1985).
[CrossRef]

M. Jesper, T. Bjarne, and M. Jannik, "Chaos in semiconductor laser with optical feedback: theory and experiment," IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

P. Besnard, B. Meziane, and G. N. Stephan, "Feedback phenomena in a semiconductor laser induced by distant reflectors," IEEE J. Quantum Electron. 29, 1271 (1993).
[CrossRef]

Y. Liu and J. Ohtsubo, "Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer," IEEE J. Quantum Electron. 33, 1163 (1997).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

T. K. Sano, "Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback," Phys. Rev. A 50, 2719 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (4)

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997).
[CrossRef]

T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003).
[CrossRef]

S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, "Experimental verification of the synchronization condition for chaotic external cavity diode lasers," Phys. Rev. E 62, 7505-7507 (2000).
[CrossRef]

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001).
[CrossRef]

Other (3)

J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, Springer Series in Optical Sciences, vol. 111 (Springer-Verlag, Berlin, 2005).

H. G. Shuster, Deterministic Chaos (VCH Verlag, Weinheim, 1988).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1989).

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Figures (8)

Fig. 1.
Fig. 1.

Optical scheme of a semiconductor laser with two external cavities. Ld, Lext1, and Lext2 are the lengths of the internal and two external cavities, r1, r2 and r3, r4 are the reflectivities of the laser facets and the external mirrors, respectively.

Fig. 2.
Fig. 2.

Chaotic time series in semiconductor laser with one external cavity represented LFF.

Fig. 3.
Fig. 3.

State diagram of semiconductor laser with one external cavity in parameter space of its length and feedback strength.

Fig. 4.
Fig. 4.

State diagram of semiconductor laser with one external cavity in space of cavity length and pump parameter. κ1=25 ns-1.

Fig. 5.
Fig. 5.

State diagram of semiconductor laser with one external cavity in space of feedback strength and pump parameter. l 1=3.3 cm.

Fig. 6.
Fig. 6.

Bifurcation diagram of laser peak intensity with τ 2/τ 1 as a control parameter. P=8+1024 m-3ns-1 and κ 2/κ 1=1. The arrows indicate different dynamical regimes: quasiperiodicity (qp), limit cycle (lc), fixed point (fp), small-amplitude chaos (sac), and large-amplitude chaos (chaos).

Fig. 7.
Fig. 7.

State diagram of semiconductor laser with two external cavities in space of ratios of their round trip times and feedback strengths. P=8×1024 m-3ns-1.

Fig. 8.
Fig. 8.

State diagram of semiconductor laser with two external cavities in space of the ratios of the round trip times of the two external cavities and pump parameter. κ 2/κ 1=1.

Equations (5)

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

d E . 0 ( t ) = ( 1 2 ) G [ ( N ( t ) N th ) ] E 0 ( t ) + κ 1 E 0 ( t τ 1 ) cos [ ψ 1 + φ ( t ) φ ( t τ 1 ) ]
+ κ 2 E 0 ( t τ 2 ) cos [ ψ 2 + φ ( t ) φ ( t τ 2 ) ] ,
d φ . ( t ) = ( α 2 ) G [ N ( t ) N th ] E 0 ( t ) κ 1 [ E 0 ( t τ 1 ) E 0 ( t ) ] sin [ ψ 1 + φ ( t ) φ ( t τ 1 ) ] ,
κ 2 [ E 0 ( t τ 1 ) E 0 ( t ) ] sin [ ψ 2 + φ ( t ) φ ( t τ 2 ) ] ,
d N . ( t ) = P N ( t ) τ s G [ N ( t ) N 0 ] E 0 2 ( t ) .

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