Abstract

The linewidth enhancement factor α of a semiconductor laser under the influences of optical feedback with different feedback strengths, external cavity lengths, and feedback phases are studied both experimentally and theoretically. The value of α is determined from the minimum of the Hopf bifurcation curve when the laser is subject to both optical feedback and optical injection. In the experiment, a pellicle beamsplitter mounted on a PZT stage placed on a linear translation stage is used as the reflector, where the external cavity length can be adjusted continuously from the long cavity regime to the short cavity regime with phase accuracy. With a moderate feedback strength, α is found to increase as the feedback strength increases. Moreover, while α is insensitive to the feedback phase in the long cavity regime, it can be tuned continuously in the short cavity regime when varying the phase. A normalized variation range of 21.59% is obtained experimentally at an external cavity length of 1.5 cm, which can be further enhanced by shortening the external cavity. To the best of our knowledge, this is the first detailed study of α from the long to the short cavity regime in a semiconductor laser subject to optical feedback. More particularly, the continuous tuning of α under phase variation is demonstrated the first time.

© 2014 Optical Society of America

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  1. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
    [CrossRef]
  2. H. Li, J. Ye, J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29, 2421–2432 (1993).
    [CrossRef]
  3. T. Zhang, N. H. Zhu, B. H. Zhang, X. Zhang, “Measurement of chirp parameter and modulation index of a semiconductor laser based on optical spectrum analysis,” IEEE Photon. Technol. Lett. 19, 227–229 (2007).
    [CrossRef]
  4. M. Osinski, J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
    [CrossRef]
  5. U. Kruger, K. Kruger, “Simultaneous measurement of the linewidth enhancement factor α, and FM and AM response of a semiconductor laser,” J. Lightwave Technol. 13, 592–597 (1995).
    [CrossRef]
  6. J. G. Provost, F. Grillot, “Measuring the Chirp and the Linewidth Enhancement Factor of Optoelectronic Devices with a Mach-Zehnder Interferometer,” IEEE Photon. J. 3, 476–488 (2011).
    [CrossRef]
  7. C. H. Lin, H. H. Lin, F. Y. Lin, “Four-wave mixing analysis of quantum dot semiconductor lasers for linewidth enhancement factor extraction,” Opt. Express 20, 101–110 (2012).
    [CrossRef] [PubMed]
  8. K. E. Chlouverakis, K. M. Al-Aswad, I. D. Henning, M. J. Adams, “Determining laser linewidth parameter from Hopf bifurcation minimum in lasers subject to optical injection,” Electron. Lett. 39, 1185–1187 (2003).
    [CrossRef]
  9. Y. Yu, G. Giuliani, S. Donati, “Measurement of the linewidth enhancement factor of a semiconductor laser based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16, 1185–1187 (2004).
  10. G. P. Agrawal, C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: Its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 227–229 (1993).
  11. R. J. Jones, P. S. Spencer, J. Lawrence, D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEE Proc. Optoelectron. 148, 7–12 (2001).
    [CrossRef]
  12. B. Lingnau, K. Lüdge, W. W. Chow, E. SchÖll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E 86, 065201 (2012).
    [CrossRef]
  13. C. H. Lin, F. Y. Lin, “Four-wave mixing analysis on injection-locked quantum dot semiconductor lasers,” Opt. Express 21, 21242–21253 (2013).
    [CrossRef] [PubMed]
  14. Y. Yu, J. Xi, “Influence of external optical feedback on the alpha factor of semiconductor lasers,” Opt. Lett. 38, 1781–1783 (2013).
    [CrossRef] [PubMed]
  15. K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express 20, 26062–26074 (2012).
    [CrossRef] [PubMed]
  16. A. Gavrielides, V. Kovanis, T. Erneux, “Analytical stability boundaries for a semiconductor laser subject to optical injection,” Opt. Commun. 136, 253–256, (1997).
    [CrossRef]
  17. F. Y. Lin, J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180, (2003).
    [CrossRef]
  18. D. Lenstra, B. H. Verbeek, A. J. D. Boef, “Coherence collapse in single-Mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21, 674–679 (1985).
    [CrossRef]
  19. F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
    [CrossRef]

2013

2012

2011

J. G. Provost, F. Grillot, “Measuring the Chirp and the Linewidth Enhancement Factor of Optoelectronic Devices with a Mach-Zehnder Interferometer,” IEEE Photon. J. 3, 476–488 (2011).
[CrossRef]

2008

F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
[CrossRef]

2007

T. Zhang, N. H. Zhu, B. H. Zhang, X. Zhang, “Measurement of chirp parameter and modulation index of a semiconductor laser based on optical spectrum analysis,” IEEE Photon. Technol. Lett. 19, 227–229 (2007).
[CrossRef]

2004

Y. Yu, G. Giuliani, S. Donati, “Measurement of the linewidth enhancement factor of a semiconductor laser based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16, 1185–1187 (2004).

2003

K. E. Chlouverakis, K. M. Al-Aswad, I. D. Henning, M. J. Adams, “Determining laser linewidth parameter from Hopf bifurcation minimum in lasers subject to optical injection,” Electron. Lett. 39, 1185–1187 (2003).
[CrossRef]

F. Y. Lin, J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180, (2003).
[CrossRef]

2001

R. J. Jones, P. S. Spencer, J. Lawrence, D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEE Proc. Optoelectron. 148, 7–12 (2001).
[CrossRef]

1997

A. Gavrielides, V. Kovanis, T. Erneux, “Analytical stability boundaries for a semiconductor laser subject to optical injection,” Opt. Commun. 136, 253–256, (1997).
[CrossRef]

1995

U. Kruger, K. Kruger, “Simultaneous measurement of the linewidth enhancement factor α, and FM and AM response of a semiconductor laser,” J. Lightwave Technol. 13, 592–597 (1995).
[CrossRef]

1993

G. P. Agrawal, C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: Its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 227–229 (1993).

H. Li, J. Ye, J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29, 2421–2432 (1993).
[CrossRef]

1987

M. Osinski, J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

1985

D. Lenstra, B. H. Verbeek, A. J. D. Boef, “Coherence collapse in single-Mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21, 674–679 (1985).
[CrossRef]

1982

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Adams, M. J.

K. E. Chlouverakis, K. M. Al-Aswad, I. D. Henning, M. J. Adams, “Determining laser linewidth parameter from Hopf bifurcation minimum in lasers subject to optical injection,” Electron. Lett. 39, 1185–1187 (2003).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: Its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 227–229 (1993).

Al-Aswad, K. M.

K. E. Chlouverakis, K. M. Al-Aswad, I. D. Henning, M. J. Adams, “Determining laser linewidth parameter from Hopf bifurcation minimum in lasers subject to optical injection,” Electron. Lett. 39, 1185–1187 (2003).
[CrossRef]

Boef, A. J. D.

D. Lenstra, B. H. Verbeek, A. J. D. Boef, “Coherence collapse in single-Mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21, 674–679 (1985).
[CrossRef]

Bowden, C. M.

G. P. Agrawal, C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: Its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 227–229 (1993).

Buus, J.

M. Osinski, J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

Chlouverakis, K. E.

K. E. Chlouverakis, K. M. Al-Aswad, I. D. Henning, M. J. Adams, “Determining laser linewidth parameter from Hopf bifurcation minimum in lasers subject to optical injection,” Electron. Lett. 39, 1185–1187 (2003).
[CrossRef]

Chow, W. W.

B. Lingnau, K. Lüdge, W. W. Chow, E. SchÖll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E 86, 065201 (2012).
[CrossRef]

Dagens, B.

F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
[CrossRef]

Donati, S.

Y. Yu, G. Giuliani, S. Donati, “Measurement of the linewidth enhancement factor of a semiconductor laser based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16, 1185–1187 (2004).

Erasme, D.

Erneux, T.

A. Gavrielides, V. Kovanis, T. Erneux, “Analytical stability boundaries for a semiconductor laser subject to optical injection,” Opt. Commun. 136, 253–256, (1997).
[CrossRef]

Gavrielides, A.

A. Gavrielides, V. Kovanis, T. Erneux, “Analytical stability boundaries for a semiconductor laser subject to optical injection,” Opt. Commun. 136, 253–256, (1997).
[CrossRef]

Giuliani, G.

Y. Yu, G. Giuliani, S. Donati, “Measurement of the linewidth enhancement factor of a semiconductor laser based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16, 1185–1187 (2004).

Grillot, F.

K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express 20, 26062–26074 (2012).
[CrossRef] [PubMed]

J. G. Provost, F. Grillot, “Measuring the Chirp and the Linewidth Enhancement Factor of Optoelectronic Devices with a Mach-Zehnder Interferometer,” IEEE Photon. J. 3, 476–488 (2011).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
[CrossRef]

Henning, I. D.

K. E. Chlouverakis, K. M. Al-Aswad, I. D. Henning, M. J. Adams, “Determining laser linewidth parameter from Hopf bifurcation minimum in lasers subject to optical injection,” Electron. Lett. 39, 1185–1187 (2003).
[CrossRef]

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Jones, R. J.

R. J. Jones, P. S. Spencer, J. Lawrence, D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEE Proc. Optoelectron. 148, 7–12 (2001).
[CrossRef]

Kane, D. M.

R. J. Jones, P. S. Spencer, J. Lawrence, D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEE Proc. Optoelectron. 148, 7–12 (2001).
[CrossRef]

Kechaou, K.

Kovanis, V.

A. Gavrielides, V. Kovanis, T. Erneux, “Analytical stability boundaries for a semiconductor laser subject to optical injection,” Opt. Commun. 136, 253–256, (1997).
[CrossRef]

Kruger, K.

U. Kruger, K. Kruger, “Simultaneous measurement of the linewidth enhancement factor α, and FM and AM response of a semiconductor laser,” J. Lightwave Technol. 13, 592–597 (1995).
[CrossRef]

Kruger, U.

U. Kruger, K. Kruger, “Simultaneous measurement of the linewidth enhancement factor α, and FM and AM response of a semiconductor laser,” J. Lightwave Technol. 13, 592–597 (1995).
[CrossRef]

Lawrence, J.

R. J. Jones, P. S. Spencer, J. Lawrence, D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEE Proc. Optoelectron. 148, 7–12 (2001).
[CrossRef]

Lenstra, D.

D. Lenstra, B. H. Verbeek, A. J. D. Boef, “Coherence collapse in single-Mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21, 674–679 (1985).
[CrossRef]

Lester, L. F.

F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
[CrossRef]

Li, H.

H. Li, J. Ye, J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29, 2421–2432 (1993).
[CrossRef]

Lin, C. H.

Lin, F. Y.

Lin, H. H.

Lingnau, B.

B. Lingnau, K. Lüdge, W. W. Chow, E. SchÖll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E 86, 065201 (2012).
[CrossRef]

Liu, J. M.

F. Y. Lin, J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180, (2003).
[CrossRef]

Lüdge, K.

B. Lingnau, K. Lüdge, W. W. Chow, E. SchÖll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E 86, 065201 (2012).
[CrossRef]

McInerney, J. G.

H. Li, J. Ye, J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29, 2421–2432 (1993).
[CrossRef]

Osinski, M.

M. Osinski, J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

Provost, J. G.

K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express 20, 26062–26074 (2012).
[CrossRef] [PubMed]

J. G. Provost, F. Grillot, “Measuring the Chirp and the Linewidth Enhancement Factor of Optoelectronic Devices with a Mach-Zehnder Interferometer,” IEEE Photon. J. 3, 476–488 (2011).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
[CrossRef]

SchÖll, E.

B. Lingnau, K. Lüdge, W. W. Chow, E. SchÖll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E 86, 065201 (2012).
[CrossRef]

Spencer, P. S.

R. J. Jones, P. S. Spencer, J. Lawrence, D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEE Proc. Optoelectron. 148, 7–12 (2001).
[CrossRef]

Su, H.

F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
[CrossRef]

Thedrez, B.

Verbeek, B. H.

D. Lenstra, B. H. Verbeek, A. J. D. Boef, “Coherence collapse in single-Mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21, 674–679 (1985).
[CrossRef]

Xi, J.

Ye, J.

H. Li, J. Ye, J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29, 2421–2432 (1993).
[CrossRef]

Yu, Y.

Y. Yu, J. Xi, “Influence of external optical feedback on the alpha factor of semiconductor lasers,” Opt. Lett. 38, 1781–1783 (2013).
[CrossRef] [PubMed]

Y. Yu, G. Giuliani, S. Donati, “Measurement of the linewidth enhancement factor of a semiconductor laser based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16, 1185–1187 (2004).

Zhang, B. H.

T. Zhang, N. H. Zhu, B. H. Zhang, X. Zhang, “Measurement of chirp parameter and modulation index of a semiconductor laser based on optical spectrum analysis,” IEEE Photon. Technol. Lett. 19, 227–229 (2007).
[CrossRef]

Zhang, T.

T. Zhang, N. H. Zhu, B. H. Zhang, X. Zhang, “Measurement of chirp parameter and modulation index of a semiconductor laser based on optical spectrum analysis,” IEEE Photon. Technol. Lett. 19, 227–229 (2007).
[CrossRef]

Zhang, X.

T. Zhang, N. H. Zhu, B. H. Zhang, X. Zhang, “Measurement of chirp parameter and modulation index of a semiconductor laser based on optical spectrum analysis,” IEEE Photon. Technol. Lett. 19, 227–229 (2007).
[CrossRef]

Zhu, N. H.

T. Zhang, N. H. Zhu, B. H. Zhang, X. Zhang, “Measurement of chirp parameter and modulation index of a semiconductor laser based on optical spectrum analysis,” IEEE Photon. Technol. Lett. 19, 227–229 (2007).
[CrossRef]

Electron. Lett.

K. E. Chlouverakis, K. M. Al-Aswad, I. D. Henning, M. J. Adams, “Determining laser linewidth parameter from Hopf bifurcation minimum in lasers subject to optical injection,” Electron. Lett. 39, 1185–1187 (2003).
[CrossRef]

IEE Proc. Optoelectron.

R. J. Jones, P. S. Spencer, J. Lawrence, D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEE Proc. Optoelectron. 148, 7–12 (2001).
[CrossRef]

IEEE J. Quantum Electron.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

H. Li, J. Ye, J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29, 2421–2432 (1993).
[CrossRef]

M. Osinski, J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

D. Lenstra, B. H. Verbeek, A. J. D. Boef, “Coherence collapse in single-Mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21, 674–679 (1985).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-um InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008).
[CrossRef]

IEEE Photon. J.

J. G. Provost, F. Grillot, “Measuring the Chirp and the Linewidth Enhancement Factor of Optoelectronic Devices with a Mach-Zehnder Interferometer,” IEEE Photon. J. 3, 476–488 (2011).
[CrossRef]

IEEE Photon. Technol. Lett.

T. Zhang, N. H. Zhu, B. H. Zhang, X. Zhang, “Measurement of chirp parameter and modulation index of a semiconductor laser based on optical spectrum analysis,” IEEE Photon. Technol. Lett. 19, 227–229 (2007).
[CrossRef]

Y. Yu, G. Giuliani, S. Donati, “Measurement of the linewidth enhancement factor of a semiconductor laser based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16, 1185–1187 (2004).

G. P. Agrawal, C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: Its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 227–229 (1993).

J. Lightwave Technol.

U. Kruger, K. Kruger, “Simultaneous measurement of the linewidth enhancement factor α, and FM and AM response of a semiconductor laser,” J. Lightwave Technol. 13, 592–597 (1995).
[CrossRef]

Opt. Commun.

A. Gavrielides, V. Kovanis, T. Erneux, “Analytical stability boundaries for a semiconductor laser subject to optical injection,” Opt. Commun. 136, 253–256, (1997).
[CrossRef]

F. Y. Lin, J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180, (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

B. Lingnau, K. Lüdge, W. W. Chow, E. SchÖll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E 86, 065201 (2012).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for measuring the α of a semiconductor laser subject to various feedback conditions.

Fig. 2
Fig. 2

(a) Stability map obtained numerically with Eqs. (2)(4) using the laser parameters extracted. HB: Hopf bifurcation; SNB: saddle-node bifurcation. (b) α calculated from the Hopf bifurcation minimum by using Eq. (1). (c) Hopf bifurcation curve obtained experimentally with the optically-injected SL.

Fig. 3
Fig. 3

Phase variations of α obtained (a) experimentally with ηfb = 0.28 and (b) numerically with ξfb = 0.27 for Lext = 2 cm, 5 cm, and 9 cm, respectively.

Fig. 4
Fig. 4

Phase variations of α obtained (a) experimentally and (c) numerically for different ηfb and ξfb, respectively. To show the variation ranges, α covering a range of 4π (two cycles) are plotted for different (b) ηfb and (d) ξfb. Here Lext is 2 cm in the short cavity regime. The dashed lines indicate the α at free-running.

Fig. 5
Fig. 5

(a) and (b) show α covering a range of 4π (two cycles) for different external cavity length obtained experimentally and numerically with ηfb = 0.28 and ξfb = 0.27, respectively. (c) and (d) show the respective normalized tuning ranges of α shown in (a) and (b). The red dashed lines indicate the boundary of the long and short cavity regime.

Fig. 6
Fig. 6

α obtained experimentally with ηfb = 0.17 and Lext = 1.0 cm, where α suppressed below its free-running value is demonstrated.

Equations (5)

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

ω min ( α 2 1 ) 3 32 α 2 .
d a d t = 1 2 [ γ c γ n γ s J n γ p ( 2 a + a 2 ) ] ( 1 + a ) + ξ i γ c cos ϕ + ξ fb γ c [ 1 + a ( t τ ) ] cos [ ϕ ( t τ ) ϕ ( t ) + ϕ fb ] ,
d ϕ d t = α set 2 [ γ c γ n γ s J n γ p ( 2 a + a 2 ) ] ξ i γ c 1 + a sin ϕ + 2 π f + ξ fb γ c [ 1 + a ( t τ ) ] 1 + a sin [ ϕ ( t τ ) ϕ ( t ) + ϕ fb ] ,
d n d t = γ s n γ n ( 1 + a 2 ) n γ s J ( 2 a + a 2 ) + γ s γ p γ c J ( 2 a + a 2 ) ( 1 + a ) 2 ,
α ( adjust ) = 0.039 α 2 + 0.655 α + 1.02

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