Abstract

We study numerically the response of a semiconductor laser to a small current modulation added to the dc-bias current. The laser is subjected to either optical feedback from an external reflector or optical injection from another laser. We characterize the nonlinear amplification near the Hopf bifurcation leading to the onset of undamped relaxation oscillations.

© 2004 Optical Society of America

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References

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  1. J. Ohtsubo, “Chaotic dynamics in semiconductor lasers with optical feedback,” in Progress in Optics, E. Wolf, ed., Vol. 44 (Elsevier, North Holland, Amsterdam, 2002), pp. 1–84.
  2. J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
    [CrossRef] [PubMed]
  3. Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995).
    [CrossRef]
  4. Y. Takiguchi, Y. Liu, and J. Ohtsubo, “Low-frequency fluctuation induced by injection-current modulation in semiconductor lasers with optical feedback,” Opt. Lett. 23, 1369–1371 (1998).
    [CrossRef]
  5. 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]
  6. J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002).
    [CrossRef]
  7. M. S. Torre, C. Masoller, P. Mandel, and K. A. Shore, “Transverse-mode dynamics in directly modulated vertical-cavity surface-emitting lasers with optical feedback,” submitted to IEEE J. Quantum Electron.
  8. T. B. Simpson, “Phase-locked microwave-frequency modulations in optically-injected laser diodes,” Opt. Commun. 170, 93–98 (1999).
    [CrossRef]
  9. T. B. Simpson and F. Doft, “Double-locked laser diode for miscrowave photonics applications,” IEEE Photon. Technol. Lett. 11, 1476–1478 (1999).
    [CrossRef]
  10. M. Nizette, T. Erneux, A. Gavrielides, and V. Kovanis, “Stability and bifurcations of periodically modulated, optically injected laser diodes,” Phys. Rev. E 63, 026212 (2001).
    [CrossRef]
  11. M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
    [CrossRef]
  12. M. Nizette, “Temporal dynamics of driven nonlinear optical systems,” Ph.D. dissertation (Universite Libre de Bruxelles, Bruxelles, Belgium, 2003).
  13. M. C. Eguia, G. B. Mindlin, and M. Giudici, “Low-frequency fluctuations in semiconductor lasers with optical feedback are induced with noise,” Phys. Rev. E 58, 2636–2639 (1998).
    [CrossRef]
  14. G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
    [CrossRef]
  15. T. Erneux, S. M. Baer, and P. Mandel, “Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser,” Phys. Rev. A 35, 1165–1171 (1987).
    [CrossRef] [PubMed]
  16. P. Mandel, P. Nardone, and T. Erneux, “Periodic loss modulation in a ring laser: influence of inhomogeneous broadening and detuning,” J. Opt. Soc. Am. B 5, 1113–1120 (1988).
    [CrossRef]
  17. C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
    [CrossRef]
  18. J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of tranverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. 9, 737–747 (1997).
    [CrossRef]

2002 (3)

J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002).
[CrossRef]

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
[CrossRef]

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

2001 (2)

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. Nizette, T. Erneux, A. Gavrielides, and V. Kovanis, “Stability and bifurcations of periodically modulated, optically injected laser diodes,” Phys. Rev. E 63, 026212 (2001).
[CrossRef]

1999 (2)

T. B. Simpson, “Phase-locked microwave-frequency modulations in optically-injected laser diodes,” Opt. Commun. 170, 93–98 (1999).
[CrossRef]

T. B. Simpson and F. Doft, “Double-locked laser diode for miscrowave photonics applications,” IEEE Photon. Technol. Lett. 11, 1476–1478 (1999).
[CrossRef]

1998 (2)

M. C. Eguia, G. B. Mindlin, and M. Giudici, “Low-frequency fluctuations in semiconductor lasers with optical feedback are induced with noise,” Phys. Rev. E 58, 2636–2639 (1998).
[CrossRef]

Y. Takiguchi, Y. Liu, and J. Ohtsubo, “Low-frequency fluctuation induced by injection-current modulation in semiconductor lasers with optical feedback,” Opt. Lett. 23, 1369–1371 (1998).
[CrossRef]

1997 (1)

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of tranverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. 9, 737–747 (1997).
[CrossRef]

1995 (2)

Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995).
[CrossRef]

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

1992 (1)

J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

T. Erneux, S. M. Baer, and P. Mandel, “Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser,” Phys. Rev. A 35, 1165–1171 (1987).
[CrossRef] [PubMed]

Agrawal, G. P.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of tranverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. 9, 737–747 (1997).
[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]

Baer, S. M.

T. Erneux, S. M. Baer, and P. Mandel, “Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser,” Phys. Rev. A 35, 1165–1171 (1987).
[CrossRef] [PubMed]

Baums, D.

J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
[CrossRef] [PubMed]

Buldú, J. M.

J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002).
[CrossRef]

Doft, F.

T. B. Simpson and F. Doft, “Double-locked laser diode for miscrowave photonics applications,” IEEE Photon. Technol. Lett. 11, 1476–1478 (1999).
[CrossRef]

Eguia, M. C.

M. C. Eguia, G. B. Mindlin, and M. Giudici, “Low-frequency fluctuations in semiconductor lasers with optical feedback are induced with noise,” Phys. Rev. E 58, 2636–2639 (1998).
[CrossRef]

Elsässer, W.

J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
[CrossRef] [PubMed]

Erneux, T.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
[CrossRef]

M. Nizette, T. Erneux, A. Gavrielides, and V. Kovanis, “Stability and bifurcations of periodically modulated, optically injected laser diodes,” Phys. Rev. E 63, 026212 (2001).
[CrossRef]

P. Mandel, P. Nardone, and T. Erneux, “Periodic loss modulation in a ring laser: influence of inhomogeneous broadening and detuning,” J. Opt. Soc. Am. B 5, 1113–1120 (1988).
[CrossRef]

T. Erneux, S. M. Baer, and P. Mandel, “Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser,” Phys. Rev. A 35, 1165–1171 (1987).
[CrossRef] [PubMed]

Garcia-Ojalvo, J.

J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002).
[CrossRef]

Gavrielides, A.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
[CrossRef]

M. Nizette, T. Erneux, A. Gavrielides, and V. Kovanis, “Stability and bifurcations of periodically modulated, optically injected laser diodes,” Phys. Rev. E 63, 026212 (2001).
[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. C. Eguia, G. B. Mindlin, and M. Giudici, “Low-frequency fluctuations in semiconductor lasers with optical feedback are induced with noise,” Phys. Rev. E 58, 2636–2639 (1998).
[CrossRef]

Göbel, E. O.

J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
[CrossRef] [PubMed]

Kikuchi, N.

Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995).
[CrossRef]

Kovanis, V.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
[CrossRef]

M. Nizette, T. Erneux, A. Gavrielides, and V. Kovanis, “Stability and bifurcations of periodically modulated, optically injected laser diodes,” Phys. Rev. E 63, 026212 (2001).
[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]

Law, J. Y.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of tranverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. 9, 737–747 (1997).
[CrossRef]

Lenstra, D.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

Liu, Y.

Y. Takiguchi, Y. Liu, and J. Ohtsubo, “Low-frequency fluctuation induced by injection-current modulation in semiconductor lasers with optical feedback,” Opt. Lett. 23, 1369–1371 (1998).
[CrossRef]

Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995).
[CrossRef]

Mandel, P.

P. Mandel, P. Nardone, and T. Erneux, “Periodic loss modulation in a ring laser: influence of inhomogeneous broadening and detuning,” J. Opt. Soc. Am. B 5, 1113–1120 (1988).
[CrossRef]

T. Erneux, S. M. Baer, and P. Mandel, “Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser,” Phys. Rev. A 35, 1165–1171 (1987).
[CrossRef] [PubMed]

Mayol, C.

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

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]

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]

M. C. Eguia, G. B. Mindlin, and M. Giudici, “Low-frequency fluctuations in semiconductor lasers with optical feedback are induced with noise,” Phys. Rev. E 58, 2636–2639 (1998).
[CrossRef]

Mirasso, C. R.

J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002).
[CrossRef]

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

Nardone, P.

Nizette, M.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
[CrossRef]

M. Nizette, T. Erneux, A. Gavrielides, and V. Kovanis, “Stability and bifurcations of periodically modulated, optically injected laser diodes,” Phys. Rev. E 63, 026212 (2001).
[CrossRef]

Ohtsubo, J.

Y. Takiguchi, Y. Liu, and J. Ohtsubo, “Low-frequency fluctuation induced by injection-current modulation in semiconductor lasers with optical feedback,” Opt. Lett. 23, 1369–1371 (1998).
[CrossRef]

Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995).
[CrossRef]

Pankin, P.

J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
[CrossRef] [PubMed]

Pesquera, L.

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

Sacher, J.

J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
[CrossRef] [PubMed]

Simpson, T. B.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
[CrossRef]

T. B. Simpson, “Phase-locked microwave-frequency modulations in optically-injected laser diodes,” Opt. Commun. 170, 93–98 (1999).
[CrossRef]

T. B. Simpson and F. Doft, “Double-locked laser diode for miscrowave photonics applications,” IEEE Photon. Technol. Lett. 11, 1476–1478 (1999).
[CrossRef]

Takiguchi, Y.

Toral, R.

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

Torrent, M. C.

J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002).
[CrossRef]

Turovets, S. I.

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

van Tartwijk, G. H. M.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of tranverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. 9, 737–747 (1997).
[CrossRef]

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. B. Simpson and F. Doft, “Double-locked laser diode for miscrowave photonics applications,” IEEE Photon. Technol. Lett. 11, 1476–1478 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

T. B. Simpson, “Phase-locked microwave-frequency modulations in optically-injected laser diodes,” Opt. Commun. 170, 93–98 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

T. Erneux, S. M. Baer, and P. Mandel, “Subharmonic bifurcation and bistability of periodic solutions in a periodically modulated laser,” Phys. Rev. A 35, 1165–1171 (1987).
[CrossRef] [PubMed]

J. Sacher, D. Baums, P. Pankin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45, 1893–1905 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (6)

Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995).
[CrossRef]

M. Nizette, T. Erneux, A. Gavrielides, and V. Kovanis, “Stability and bifurcations of periodically modulated, optically injected laser diodes,” Phys. Rev. E 63, 026212 (2001).
[CrossRef]

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and T. B. Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65, 056610 (2002).
[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]

J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002).
[CrossRef]

M. C. Eguia, G. B. Mindlin, and M. Giudici, “Low-frequency fluctuations in semiconductor lasers with optical feedback are induced with noise,” Phys. Rev. E 58, 2636–2639 (1998).
[CrossRef]

Quantum Semiclass. Opt. (2)

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of tranverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. 9, 737–747 (1997).
[CrossRef]

Other (3)

J. Ohtsubo, “Chaotic dynamics in semiconductor lasers with optical feedback,” in Progress in Optics, E. Wolf, ed., Vol. 44 (Elsevier, North Holland, Amsterdam, 2002), pp. 1–84.

M. S. Torre, C. Masoller, P. Mandel, and K. A. Shore, “Transverse-mode dynamics in directly modulated vertical-cavity surface-emitting lasers with optical feedback,” submitted to IEEE J. Quantum Electron.

M. Nizette, “Temporal dynamics of driven nonlinear optical systems,” Ph.D. dissertation (Universite Libre de Bruxelles, Bruxelles, Belgium, 2003).

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Figures (6)

Fig. 1
Fig. 1

External-cavity laser under current modulation. Effect of the optical feedback strength. We plot the maximum, minimum, and average values of the intensity oscillations versus the modulation period for weak modulation (δ=0.01): (a) without feedback and (b)–(d) with feedback. The feedback strength is (b) 0.8 ns-1, (c) 0.88 ns-1, (d) 0.96 ns-1.

Fig. 2
Fig. 2

External-cavity laser under current modulation. Effect of the modulation period. We plot the local maxima and minima, and the average value of the intensity oscillations versus the optical feedback strength: (a) without injection current modulation and (b)–(f) with weak injection current modulation. The modulation amplitude is δ=0.01 and the period of Tmod is (b) 0.25 ns, (c) 0.26 ns, (d) 0.265 ns, (e) 0.27 ns, (f) 0.28 ns.

Fig. 3
Fig. 3

External-cavity laser under current modulation as a function of the modulation period and the optical feedback strength: (a) Imax-Imean and (b) Imean-Imin. The values are represented by gray tones: a dark tone represents a small value, and a light tone represents a large value.

Fig. 4
Fig. 4

Effect of the modulation amplitude. We plot the maximum, minimum, and average values of the intensity oscillations versus the modulation period. The modulation amplitude is δ equal to (a) 0.01, (b) 0.05, (c) 0.1, (d) 0.2.

Fig. 5
Fig. 5

Optically injected laser under weak current modulation. Effect of the frequency detuning. We plot the maximum, minimum, and average values of the intensity oscillations versus the modulation period. The modulation amplitude is δ=0.01: (a) without optical injection and (b)–(d) with weak optical injection. The optical injection strength is Einj=5 ns-1 and the detuning between the free-running oscillation frequency and the frequency of the injected field is (b) -10.8 GHz, (c) -10.6 GHz, (d) -10.2 GHz. The Hopf bifurcation occurs for Δω-10.9 GHz.

Fig. 6
Fig. 6

Optically injected laser under weak current modulation. Effect of the modulation period. We plot the maximum, minimum, and average values of the intensity oscillations versus detuning between the free-running oscillation frequency and the frequency of the injected signal. The optical injection strength Einj=5 ns-1: (a) without current modulation (δ=0) and (b)–(f) with weak injection current modulation of amplitude δ=0.01 and the period of Imod is (b) 0.24 ns, (c) 0.25 ns, (d) 0.255 ns, (e) 0.26 ns, (f) 0.27 ns.

Equations (2)

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

E˙=k(1+iα)(N-1)E(t)+γE(t-τ)exp(-iωτ)+iΔwE(t)+Einj+Dξ(t),
N˙={j[1+δ sin(2πt/Tmod)]-N-N|E|2}/τn.

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