Abstract

The nonlinear dynamics of two semiconductor laser systems: (i) with optical feedback, and (ii) with optical feedback and direct current modulation are evaluated from multi-GHz-bandwidth output power time-series. Animations of compilations of the RF spectrum (from the FFT of the time-series) as a function of optical feedback level, injection current and modulation signal strength is demonstrated as a new tool to give insight into the dynamics. The results are contrasted with prior art and new observations include fine structure in the RF spectrum at low levels of optical feedback and non-stationary switching between periodic and chaotic dynamics for some sets of laser system parameters. Correlation dimension analysis successfully identifies periodic dynamics but most of the dynamical states are too complex to be extracted using standard algorithms.

© 2010 OSA

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2010 (1)

D. M. Kane and C. J. McMahon, “Instantaneous frequency measurement applied to semiconductor laser relaxation oscillations,” Appl. Phys. B 98(4), 759–765 (2010).
[CrossRef]

2009 (1)

2006 (1)

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

2004 (2)

T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. 242(4-6), 613–622 (2004).
[CrossRef]

J. Paul, S. Sivaprakasam, and K. A. Shore, “Dual-channel chaotic optical communications using external-cavity semiconductor lasers,” J. Opt. Soc. Am. B 21(3), 514–521 (2004).
[CrossRef]

2003 (1)

K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
[CrossRef]

2002 (6)

S. Sivaprakasam, P. S. Spencer, P. Rees, and K. A. Shore, “Regimes of chaotic synchronization in external-cavity laser diodes,” IEEE J. Quantum Electron. 38(9), 1155–1161 (2002).
[CrossRef]

S. Donati and C. R. Mirasso, “Introduction to the feature section on Optical Chaos and Applications to Cryptography,” IEEE J. Quantum Electron. 38(9), 1138–1140 (2002).
[CrossRef]

J. S. Lawrence and D. M. Kane, “Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,” IEEE J. Quantum Electron. 38(2), 185–192 (2002).
[CrossRef]

S. Eriksson, “Dependence of the experimental stability diagram of an optically injected semiconductor laser on the laser current,” Opt. Commun. 210(3-6), 343–353 (2002).
[CrossRef]

S. Wieczorek, T. B. Simpson, B. Krauskopf, and D. Lenstra, “Global quantitative predictions of complex laser dynamics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2A), 045207 (2002).
[CrossRef] [PubMed]

S. Eriksson and A. M. Lindberg, “Observations on the dynamics of semiconductor lasers subjected to external optical injection,” J. Opt. B Quantum Semiclassical Opt. 4(2), 149–154 (2002).
[CrossRef]

2000 (1)

I. Pierce, P. Rees, and P. S. Spencer, “Multimode dynamics in laser diodes with optical feedback,” Phys. Rev. A 61(5), 053801 (2000).
[CrossRef]

1999 (3)

S. Sivaprakasam and K. A. Shore, “Demonstration of optical synchronization of chaotic external-cavity laser diodes,” Opt. Lett. 24(7), 466–468 (1999).
[CrossRef]

T. Heil, I. Fischer, and W. Elsasser, “Influence of amplitude-phase coupling on the dynamics of semiconductor lasers subject to optical feedback,” Phys. Rev. A 60(1), 634–641 (1999).
[CrossRef]

R. Hegger, H. Kantz, and T. Schreiber, “Practical implementation of nonlinear time series methods: The TISEAN package,” Chaos 9(2), 413–435 (1999).
[CrossRef]

1998 (1)

T. Heil, I. Fischer, and W. Elsasser, “Coexistence of low-frequency fluctuations and stable emission on a single high-gain mode in semiconductor lasers with external optical feedback,” Phys. Rev. A 58(4), R2672–R2675 (1998).
[CrossRef]

1997 (2)

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” J. Opt. B Quantum Semiclassical Opt. 9(5), 765–784 (1997).
[CrossRef]

S. Bennett, C. M. Snowden, and S. Iezekiel, “Nonlinear dynamics in directly modulated multiple-quantum-well laser diodes,” IEEE J. Quantum Electron. 33(11), 2076–2083 (1997).
[CrossRef]

1994 (1)

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[CrossRef]

1993 (4)

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 um InGaAsP distributed-feedback semiconductor-laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[CrossRef]

Y. H. Kao and H. T. Lin, “Virtual Hopf precursor of period-doubling route in directly modulated semiconductor-lasers,” IEEE J. Quantum Electron. 29(6), 1617–1623 (1993).
[CrossRef]

H. Li, J. Ye, and J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29(9), 2421–2432 (1993).
[CrossRef]

1992 (2)

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[CrossRef]

J. Sacher, D. Baums, P. Panknin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45(3), 1893–1905 (1992).
[CrossRef] [PubMed]

1990 (2)

E. Hemery, L. Chusseau, and J. M. Lourtioz, “Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 mm,” IEEE J. Quantum Electron. 26(4), 633–641 (1990).
[CrossRef]

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65(16), 1999–2002 (1990).
[CrossRef] [PubMed]

1989 (2)

T. H. Yoon, C. H. Lee, and S. Y. Shin, “Perturbation analysis of bistability and period doubling bifurcations in directly-modulated laser-diodes,” IEEE J. Quantum Electron. 25(9), 1993–2000 (1989).
[CrossRef]

L. Chusseau, E. Hemery, and J. M. Lourtioz, “Period doubling in directly modulated InGaAsP semiconductor-lasers,” Appl. Phys. Lett. 55(9), 822–824 (1989).
[CrossRef]

1988 (1)

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24(12), 2441–2447 (1988).
[CrossRef]

1986 (3)

Y. Cho and T. Umeda, “Observation of chaos in a semiconductor laser with delayed feedback,” Opt. Commun. 59(2), 131–136 (1986).
[CrossRef]

H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986).
[CrossRef]

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

1985 (2)

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[CrossRef]

C. H. Lee, T. H. Yoon, and S. Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46(1), 95–97 (1985).
[CrossRef]

1983 (1)

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[CrossRef]

1980 (1)

R. Lang and K. Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

Adams, M. J.

K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
[CrossRef]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Baums, D.

J. Sacher, D. Baums, P. Panknin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45(3), 1893–1905 (1992).
[CrossRef] [PubMed]

Bennett, S.

S. Bennett, C. M. Snowden, and S. Iezekiel, “Nonlinear dynamics in directly modulated multiple-quantum-well laser diodes,” IEEE J. Quantum Electron. 33(11), 2076–2083 (1997).
[CrossRef]

Chlouverakis, K. E.

K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
[CrossRef]

Cho, Y.

Y. Cho and T. Umeda, “Observation of chaos in a semiconductor laser with delayed feedback,” Opt. Commun. 59(2), 131–136 (1986).
[CrossRef]

Chraplyvy, A. R.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

Chusseau, L.

E. Hemery, L. Chusseau, and J. M. Lourtioz, “Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 mm,” IEEE J. Quantum Electron. 26(4), 633–641 (1990).
[CrossRef]

L. Chusseau, E. Hemery, and J. M. Lourtioz, “Period doubling in directly modulated InGaAsP semiconductor-lasers,” Appl. Phys. Lett. 55(9), 822–824 (1989).
[CrossRef]

Colet, P.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Collins, J. J.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

Deluca, C. J.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

Den Boef, A.

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[CrossRef]

Dente, G. C.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24(12), 2441–2447 (1988).
[CrossRef]

Donati, S.

S. Donati and C. R. Mirasso, “Introduction to the feature section on Optical Chaos and Applications to Cryptography,” IEEE J. Quantum Electron. 38(9), 1138–1140 (2002).
[CrossRef]

Durkin, P. S.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24(12), 2441–2447 (1988).
[CrossRef]

Elsasser, W.

T. Heil, I. Fischer, and W. Elsasser, “Influence of amplitude-phase coupling on the dynamics of semiconductor lasers subject to optical feedback,” Phys. Rev. A 60(1), 634–641 (1999).
[CrossRef]

T. Heil, I. Fischer, and W. Elsasser, “Coexistence of low-frequency fluctuations and stable emission on a single high-gain mode in semiconductor lasers with external optical feedback,” Phys. Rev. A 58(4), R2672–R2675 (1998).
[CrossRef]

Elsässer, W.

J. Sacher, D. Baums, P. Panknin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45(3), 1893–1905 (1992).
[CrossRef] [PubMed]

Eriksson, S.

S. Eriksson, “Dependence of the experimental stability diagram of an optically injected semiconductor laser on the laser current,” Opt. Commun. 210(3-6), 343–353 (2002).
[CrossRef]

S. Eriksson and A. M. Lindberg, “Observations on the dynamics of semiconductor lasers subjected to external optical injection,” J. Opt. B Quantum Semiclassical Opt. 4(2), 149–154 (2002).
[CrossRef]

Fischer, I.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

T. Heil, I. Fischer, and W. Elsasser, “Influence of amplitude-phase coupling on the dynamics of semiconductor lasers subject to optical feedback,” Phys. Rev. A 60(1), 634–641 (1999).
[CrossRef]

T. Heil, I. Fischer, and W. Elsasser, “Coexistence of low-frequency fluctuations and stable emission on a single high-gain mode in semiconductor lasers with external optical feedback,” Phys. Rev. A 58(4), R2672–R2675 (1998).
[CrossRef]

Fordell, T.

T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. 242(4-6), 613–622 (2004).
[CrossRef]

García-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Göbel, E. O.

J. Sacher, D. Baums, P. Panknin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45(3), 1893–1905 (1992).
[CrossRef] [PubMed]

Grassberger, P.

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[CrossRef]

Hegger, R.

R. Hegger, H. Kantz, and T. Schreiber, “Practical implementation of nonlinear time series methods: The TISEAN package,” Chaos 9(2), 413–435 (1999).
[CrossRef]

Heil, T.

T. Heil, I. Fischer, and W. Elsasser, “Influence of amplitude-phase coupling on the dynamics of semiconductor lasers subject to optical feedback,” Phys. Rev. A 60(1), 634–641 (1999).
[CrossRef]

T. Heil, I. Fischer, and W. Elsasser, “Coexistence of low-frequency fluctuations and stable emission on a single high-gain mode in semiconductor lasers with external optical feedback,” Phys. Rev. A 58(4), R2672–R2675 (1998).
[CrossRef]

Hemery, E.

E. Hemery, L. Chusseau, and J. M. Lourtioz, “Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 mm,” IEEE J. Quantum Electron. 26(4), 633–641 (1990).
[CrossRef]

L. Chusseau, E. Hemery, and J. M. Lourtioz, “Period doubling in directly modulated InGaAsP semiconductor-lasers,” Appl. Phys. Lett. 55(9), 822–824 (1989).
[CrossRef]

Huang, K. F.

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” J. Opt. B Quantum Semiclassical Opt. 9(5), 765–784 (1997).
[CrossRef]

Iezekiel, S.

S. Bennett, C. M. Snowden, and S. Iezekiel, “Nonlinear dynamics in directly modulated multiple-quantum-well laser diodes,” IEEE J. Quantum Electron. 33(11), 2076–2083 (1997).
[CrossRef]

Kane, D. M.

D. M. Kane and C. J. McMahon, “Instantaneous frequency measurement applied to semiconductor laser relaxation oscillations,” Appl. Phys. B 98(4), 759–765 (2010).
[CrossRef]

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
[CrossRef] [PubMed]

D. M. Kane, J. P. Toomey, M. W. Lee, and K. A. Shore, “Correlation dimension signature of wideband chaos synchronization of semiconductor lasers,” Opt. Lett. 31(1), 20–22 (2006).
[CrossRef] [PubMed]

J. S. Lawrence and D. M. Kane, “Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,” IEEE J. Quantum Electron. 38(2), 185–192 (2002).
[CrossRef]

Kantz, H.

R. Hegger, H. Kantz, and T. Schreiber, “Practical implementation of nonlinear time series methods: The TISEAN package,” Chaos 9(2), 413–435 (1999).
[CrossRef]

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[CrossRef]

Kao, Y. H.

Y. H. Kao and H. T. Lin, “Virtual Hopf precursor of period-doubling route in directly modulated semiconductor-lasers,” IEEE J. Quantum Electron. 29(6), 1617–1623 (1993).
[CrossRef]

Kobayashi, K.

R. Lang and K. Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

Krauskopf, B.

S. Wieczorek, T. B. Simpson, B. Krauskopf, and D. Lenstra, “Global quantitative predictions of complex laser dynamics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2A), 045207 (2002).
[CrossRef] [PubMed]

Lang, R.

R. Lang and K. Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Lawrence, J. S.

J. S. Lawrence and D. M. Kane, “Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,” IEEE J. Quantum Electron. 38(2), 185–192 (2002).
[CrossRef]

Lee, C. H.

T. H. Yoon, C. H. Lee, and S. Y. Shin, “Perturbation analysis of bistability and period doubling bifurcations in directly-modulated laser-diodes,” IEEE J. Quantum Electron. 25(9), 1993–2000 (1989).
[CrossRef]

C. H. Lee, T. H. Yoon, and S. Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46(1), 95–97 (1985).
[CrossRef]

Lee, M. W.

Lenstra, D.

S. Wieczorek, T. B. Simpson, B. Krauskopf, and D. Lenstra, “Global quantitative predictions of complex laser dynamics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2A), 045207 (2002).
[CrossRef] [PubMed]

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[CrossRef]

Li, H.

H. Li, J. Ye, and J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29(9), 2421–2432 (1993).
[CrossRef]

Lin, H. T.

Y. H. Kao and H. T. Lin, “Virtual Hopf precursor of period-doubling route in directly modulated semiconductor-lasers,” IEEE J. Quantum Electron. 29(6), 1617–1623 (1993).
[CrossRef]

Lindberg, A. M.

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009).
[CrossRef] [PubMed]

T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. 242(4-6), 613–622 (2004).
[CrossRef]

S. Eriksson and A. M. Lindberg, “Observations on the dynamics of semiconductor lasers subjected to external optical injection,” J. Opt. B Quantum Semiclassical Opt. 4(2), 149–154 (2002).
[CrossRef]

Liu, H. F.

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 um InGaAsP distributed-feedback semiconductor-laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[CrossRef]

Liu, J. M.

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” J. Opt. B Quantum Semiclassical Opt. 9(5), 765–784 (1997).
[CrossRef]

Lourtioz, J. M.

E. Hemery, L. Chusseau, and J. M. Lourtioz, “Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 mm,” IEEE J. Quantum Electron. 26(4), 633–641 (1990).
[CrossRef]

L. Chusseau, E. Hemery, and J. M. Lourtioz, “Period doubling in directly modulated InGaAsP semiconductor-lasers,” Appl. Phys. Lett. 55(9), 822–824 (1989).
[CrossRef]

Mark, J.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[CrossRef]

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65(16), 1999–2002 (1990).
[CrossRef] [PubMed]

McInerney, J. G.

H. Li, J. Ye, and J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29(9), 2421–2432 (1993).
[CrossRef]

McMahon, C. J.

D. M. Kane and C. J. McMahon, “Instantaneous frequency measurement applied to semiconductor laser relaxation oscillations,” Appl. Phys. B 98(4), 759–765 (2010).
[CrossRef]

Mirasso, C. R.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

S. Donati and C. R. Mirasso, “Introduction to the feature section on Optical Chaos and Applications to Cryptography,” IEEE J. Quantum Electron. 38(9), 1138–1140 (2002).
[CrossRef]

Moeller, C. E.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24(12), 2441–2447 (1988).
[CrossRef]

Mork, J.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[CrossRef]

Mørk, J.

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65(16), 1999–2002 (1990).
[CrossRef] [PubMed]

Ngai, W. F.

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 um InGaAsP distributed-feedback semiconductor-laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[CrossRef]

Olesen, H.

H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986).
[CrossRef]

Osmundsen, J.

H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986).
[CrossRef]

Panknin, P.

J. Sacher, D. Baums, P. Panknin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45(3), 1893–1905 (1992).
[CrossRef] [PubMed]

Paul, J.

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Pierce, I.

I. Pierce, P. Rees, and P. S. Spencer, “Multimode dynamics in laser diodes with optical feedback,” Phys. Rev. A 61(5), 053801 (2000).
[CrossRef]

Procaccia, I.

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[CrossRef]

Rees, P.

S. Sivaprakasam, P. S. Spencer, P. Rees, and K. A. Shore, “Regimes of chaotic synchronization in external-cavity laser diodes,” IEEE J. Quantum Electron. 38(9), 1155–1161 (2002).
[CrossRef]

I. Pierce, P. Rees, and P. S. Spencer, “Multimode dynamics in laser diodes with optical feedback,” Phys. Rev. A 61(5), 053801 (2000).
[CrossRef]

Rosenstein, M. T.

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

Sacher, J.

J. Sacher, D. Baums, P. Panknin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45(3), 1893–1905 (1992).
[CrossRef] [PubMed]

Schreiber, T.

R. Hegger, H. Kantz, and T. Schreiber, “Practical implementation of nonlinear time series methods: The TISEAN package,” Chaos 9(2), 413–435 (1999).
[CrossRef]

Shin, S. Y.

T. H. Yoon, C. H. Lee, and S. Y. Shin, “Perturbation analysis of bistability and period doubling bifurcations in directly-modulated laser-diodes,” IEEE J. Quantum Electron. 25(9), 1993–2000 (1989).
[CrossRef]

C. H. Lee, T. H. Yoon, and S. Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46(1), 95–97 (1985).
[CrossRef]

Shore, K. A.

D. M. Kane, J. P. Toomey, M. W. Lee, and K. A. Shore, “Correlation dimension signature of wideband chaos synchronization of semiconductor lasers,” Opt. Lett. 31(1), 20–22 (2006).
[CrossRef] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

J. Paul, S. Sivaprakasam, and K. A. Shore, “Dual-channel chaotic optical communications using external-cavity semiconductor lasers,” J. Opt. Soc. Am. B 21(3), 514–521 (2004).
[CrossRef]

S. Sivaprakasam, P. S. Spencer, P. Rees, and K. A. Shore, “Regimes of chaotic synchronization in external-cavity laser diodes,” IEEE J. Quantum Electron. 38(9), 1155–1161 (2002).
[CrossRef]

S. Sivaprakasam and K. A. Shore, “Demonstration of optical synchronization of chaotic external-cavity laser diodes,” Opt. Lett. 24(7), 466–468 (1999).
[CrossRef]

Simpson, T. B.

S. Wieczorek, T. B. Simpson, B. Krauskopf, and D. Lenstra, “Global quantitative predictions of complex laser dynamics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2A), 045207 (2002).
[CrossRef] [PubMed]

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” J. Opt. B Quantum Semiclassical Opt. 9(5), 765–784 (1997).
[CrossRef]

Sivaprakasam, S.

Snowden, C. M.

S. Bennett, C. M. Snowden, and S. Iezekiel, “Nonlinear dynamics in directly modulated multiple-quantum-well laser diodes,” IEEE J. Quantum Electron. 33(11), 2076–2083 (1997).
[CrossRef]

Spencer, P. S.

S. Sivaprakasam, P. S. Spencer, P. Rees, and K. A. Shore, “Regimes of chaotic synchronization in external-cavity laser diodes,” IEEE J. Quantum Electron. 38(9), 1155–1161 (2002).
[CrossRef]

I. Pierce, P. Rees, and P. S. Spencer, “Multimode dynamics in laser diodes with optical feedback,” Phys. Rev. A 61(5), 053801 (2000).
[CrossRef]

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Tai, K.

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” J. Opt. B Quantum Semiclassical Opt. 9(5), 765–784 (1997).
[CrossRef]

Tkach, R. W.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

Toomey, J. P.

Tromborg, B.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[CrossRef]

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65(16), 1999–2002 (1990).
[CrossRef] [PubMed]

H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986).
[CrossRef]

Umeda, T.

Y. Cho and T. Umeda, “Observation of chaos in a semiconductor laser with delayed feedback,” Opt. Commun. 59(2), 131–136 (1986).
[CrossRef]

Valling, S.

Verbeek, B.

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[CrossRef]

Wieczorek, S.

S. Wieczorek, T. B. Simpson, B. Krauskopf, and D. Lenstra, “Global quantitative predictions of complex laser dynamics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2A), 045207 (2002).
[CrossRef] [PubMed]

Wilson, K. A.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24(12), 2441–2447 (1988).
[CrossRef]

Ye, J.

H. Li, J. Ye, and J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29(9), 2421–2432 (1993).
[CrossRef]

Yoon, T. H.

T. H. Yoon, C. H. Lee, and S. Y. Shin, “Perturbation analysis of bistability and period doubling bifurcations in directly-modulated laser-diodes,” IEEE J. Quantum Electron. 25(9), 1993–2000 (1989).
[CrossRef]

C. H. Lee, T. H. Yoon, and S. Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46(1), 95–97 (1985).
[CrossRef]

Appl. Phys. B (1)

D. M. Kane and C. J. McMahon, “Instantaneous frequency measurement applied to semiconductor laser relaxation oscillations,” Appl. Phys. B 98(4), 759–765 (2010).
[CrossRef]

Appl. Phys. Lett. (2)

C. H. Lee, T. H. Yoon, and S. Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46(1), 95–97 (1985).
[CrossRef]

L. Chusseau, E. Hemery, and J. M. Lourtioz, “Period doubling in directly modulated InGaAsP semiconductor-lasers,” Appl. Phys. Lett. 55(9), 822–824 (1989).
[CrossRef]

Chaos (1)

R. Hegger, H. Kantz, and T. Schreiber, “Practical implementation of nonlinear time series methods: The TISEAN package,” Chaos 9(2), 413–435 (1999).
[CrossRef]

IEEE J. Quantum Electron. (14)

S. Sivaprakasam, P. S. Spencer, P. Rees, and K. A. Shore, “Regimes of chaotic synchronization in external-cavity laser diodes,” IEEE J. Quantum Electron. 38(9), 1155–1161 (2002).
[CrossRef]

E. Hemery, L. Chusseau, and J. M. Lourtioz, “Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 mm,” IEEE J. Quantum Electron. 26(4), 633–641 (1990).
[CrossRef]

S. Bennett, C. M. Snowden, and S. Iezekiel, “Nonlinear dynamics in directly modulated multiple-quantum-well laser diodes,” IEEE J. Quantum Electron. 33(11), 2076–2083 (1997).
[CrossRef]

H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 um InGaAsP distributed-feedback semiconductor-laser,” IEEE J. Quantum Electron. 29(6), 1668–1675 (1993).
[CrossRef]

D. Lenstra, B. Verbeek, and A. Den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. 21(6), 674–679 (1985).
[CrossRef]

H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986).
[CrossRef]

R. Lang and K. Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[CrossRef]

T. H. Yoon, C. H. Lee, and S. Y. Shin, “Perturbation analysis of bistability and period doubling bifurcations in directly-modulated laser-diodes,” IEEE J. Quantum Electron. 25(9), 1993–2000 (1989).
[CrossRef]

Y. H. Kao and H. T. Lin, “Virtual Hopf precursor of period-doubling route in directly modulated semiconductor-lasers,” IEEE J. Quantum Electron. 29(6), 1617–1623 (1993).
[CrossRef]

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24(12), 2441–2447 (1988).
[CrossRef]

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor-lasers with optical feedback - Theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[CrossRef]

H. Li, J. Ye, and J. G. McInerney, “Detailed analysis of coherence collapse in semiconductor lasers,” IEEE J. Quantum Electron. 29(9), 2421–2432 (1993).
[CrossRef]

J. S. Lawrence and D. M. Kane, “Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,” IEEE J. Quantum Electron. 38(2), 185–192 (2002).
[CrossRef]

S. Donati and C. R. Mirasso, “Introduction to the feature section on Optical Chaos and Applications to Cryptography,” IEEE J. Quantum Electron. 38(9), 1138–1140 (2002).
[CrossRef]

J. Lightwave Technol. (1)

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5μm distributed feedback lasers,” J. Lightwave Technol. 4(11), 1655–1661 (1986).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt. (2)

S. Eriksson and A. M. Lindberg, “Observations on the dynamics of semiconductor lasers subjected to external optical injection,” J. Opt. B Quantum Semiclassical Opt. 4(2), 149–154 (2002).
[CrossRef]

T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” J. Opt. B Quantum Semiclassical Opt. 9(5), 765–784 (1997).
[CrossRef]

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

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
[CrossRef]

Opt. Commun. (4)

K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. 216(4-6), 405–412 (2003).
[CrossRef]

T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. 242(4-6), 613–622 (2004).
[CrossRef]

S. Eriksson, “Dependence of the experimental stability diagram of an optically injected semiconductor laser on the laser current,” Opt. Commun. 210(3-6), 343–353 (2002).
[CrossRef]

Y. Cho and T. Umeda, “Observation of chaos in a semiconductor laser with delayed feedback,” Opt. Commun. 59(2), 131–136 (1986).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Lett. A (1)

H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A 185(1), 77–87 (1994).
[CrossRef]

Phys. Rev. A (4)

J. Sacher, D. Baums, P. Panknin, W. Elsässer, and E. O. Göbel, “Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback,” Phys. Rev. A 45(3), 1893–1905 (1992).
[CrossRef] [PubMed]

T. Heil, I. Fischer, and W. Elsasser, “Coexistence of low-frequency fluctuations and stable emission on a single high-gain mode in semiconductor lasers with external optical feedback,” Phys. Rev. A 58(4), R2672–R2675 (1998).
[CrossRef]

T. Heil, I. Fischer, and W. Elsasser, “Influence of amplitude-phase coupling on the dynamics of semiconductor lasers subject to optical feedback,” Phys. Rev. A 60(1), 634–641 (1999).
[CrossRef]

I. Pierce, P. Rees, and P. S. Spencer, “Multimode dynamics in laser diodes with optical feedback,” Phys. Rev. A 61(5), 053801 (2000).
[CrossRef]

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

S. Wieczorek, T. B. Simpson, B. Krauskopf, and D. Lenstra, “Global quantitative predictions of complex laser dynamics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2A), 045207 (2002).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65(16), 1999–2002 (1990).
[CrossRef] [PubMed]

Physica D (2)

M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65(1-2), 117–134 (1993).
[CrossRef]

P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D 9(1-2), 189–208 (1983).
[CrossRef]

Other (8)

J. C. Sprott, “Chaos Data Analyzer The Professional Version,” (Physics Academic Software, 2003).

F. Sporleder, “Proceedings of the URSI International Symposium on Electromagnetic Theory,” in Proceedings of the URSI International Symposium on Electromagnetic Theory(Brussels, Belgium, 1983), p. 585.

P. Spencer, P. Rees, and I. Pierce, “Theoretical Analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, D. M. Kane, and K. A. Shore, eds. (John Wiley & Sons, West Sussex, 2005), pp. 23–54.

D. Lenstra, G. Vemuri, and M. Yousefi, “Generalized Optical Feedback,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, D. M. Kane, and K. A. Shore, eds. (John Wiley & Sons, West Sussex, 2005), pp. 55–80.

J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos (Springer-Verlag, Berlin, 2006).

A. T. Gavrielides, and D. W. Sukow, “Experimental Observations,” in Unlocking Dynamical Diverstiy: Optical Feedback Effects on Semiconductor Lasers, D. M. Kane, and K. A. Shore, eds. (John Wiley & Sons, West Sussex, 2005), pp. 81–145.

C. McMahon, D. M. Kane, J. P. Toomey, and J. S. Lawrence, “High Accuracy Measurement of Relaxation Oscillation Frequency in Heavily Damped Quantum Well Lasers,” in Proceedings of the International Conference on Nanoscience and Nanotechnology, C. Jagadish, and G. Q. M. Lu, eds. (IEEE, Brisbane, 2006), pp. 497–500.

D. M. Kane, and J. P. Toomey, “Precision threshold current measurement for semiconductor lasers based on relaxation oscillation frequency,” J. Lightwave Tech. (2009).

Supplementary Material (3)

» Media 1: MOV (1402 KB)     
» Media 2: MOV (7850 KB)     
» Media 3: MOV (2613 KB)     

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Figures (14)

Fig. 1
Fig. 1

Experimental setup for measuring output power time-series from a semiconductor laser with varying injection current and feedback levels. LD = laser diode, BS = beam splitter, VA = variable attenuator, M = mirror, PM = power meter, OI = optical isolator, Det = fibre-coupled optical detector.

Fig. 2
Fig. 2

Single frame excerpts from animation (Media 1) showing compilations of the fast Fourier transforms (FFT) of experimental time-series for varying injection current. The subsequent frames show the evolution of the time-series dynamics with increasing optical feedback. Feedback level controlled by adjusting the optical density (O.D.) of a variable attenuator (a) O.D. = 2.2, (b) O.D. = 1.9, (c) O.D. = 1.2, (d) O.D. = 0.7.

Fig. 3
Fig. 3

Close up of the relaxation oscillation region of Fig. 2(a) (Media 1). The power scale has been modified to highlight the structure within the RO peak. A curve RO frequency has been overlaid to show the trend.

Fig. 4
Fig. 4

Experimental output power time-series showing switching between chaotic and periodic dynamics for low feedback levels (O.D. = 1.9, Iinj = 38.7 mA).

Fig. 5
Fig. 5

Switching observed between chaotic and periodic dynamics in a noise-free simulation of a semiconductor laser with optical feedback. Based on the Lang-Kobayashi equations [19]. Parameters used in the simulation were diode facet reflectivity (front and back) = 0.309, refractive index = 3.5, laser cavity roundtrip time = 9 ps, external cavity roundtrip time = 1 ns, linewidth enhancement factor = 4.5 and external mirror reflectivity = 4.8x10−4. The simulation sampling rate was 7.85 ps.

Fig. 6
Fig. 6

Semiconductor laser with weak optical feedback (attenuation O.D. = 1.2) and injection current 35.3 mA. (a) Experimental output power time-series, (b) phase diagram and (c) RF spectrum (FFT).

Fig. 7
Fig. 7

(a) Average output power and (b) RMS amplitude of the AC-coupled output power time-series as a function of injection current and feedback level.

Fig. 8
Fig. 8

Semiconductor laser with moderate optical feedback (attenuation O.D. = 0.7) and injection current = 39.2 mA. (a) Experimental output power time-series, (b) phase diagram and (c) radio frequency spectrum (FFT).

Fig. 9
Fig. 9

Single frame excerpts from direct modulation experiment animation (Media 2) with (a) O.D. = 0.7 and modulation level = −4 dBm, (b) O.D. = 0.7 and modulation level = + 4 dBm, (c) O.D. = 0.7 and modulation level = + 6 dBm and (d) O.D. = 0.08 and modulation level = + 5 dBm,

Fig. 10
Fig. 10

Single frame excerpts from direct modulation experiment animation (Media 2) with minimum optical feedback (O.D. = 1.2) and modulation level (a) −4 dBm, and (b) + 6 dBm.

Fig. 11
Fig. 11

With weak optical feedback (O.D. = 1.2) and f mod = 500 MHz. (a) Experimental output power time-series and (b) frequency spectrum for modulation level −4 dBm. (c) Output power time-series and (b) frequency spectrum for modulation level + 6 dBm.

Fig. 12
Fig. 12

With strong optical feedback (O.D. = 0.08) and modulation level = + 5 dBm. (a) Experimental output power time-series, (b) phase diagram and (c) frequency spectrum for modulation frequency 497 MHz. (d) Output power time-series, (e) phase diagram and (f) frequency spectrum for modulation frequency 530 MHz.

Fig. 13
Fig. 13

Single frame excerpts from direct modulation RMS amplitude animation (Media 3) with (a) O.D. = 1.1, (b) O.D. = 0.7, (c) O.D. = 0.34 and (d) O.D. = 0.08.

Fig. 14
Fig. 14

Single frame from the periodic map animation (Media 3) with for an optical feedback attenuation of O.D. = 0.34. Periodic regions (blue) are identified by correlation dimension analysis giving an estimate of ~1, indicating limit cycle dynamics. Black regions represent unstable outputs.

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