Abstract

We theoretically examine the consequences of modulating an external-cavity semiconductor laser around its mode-locking resonant frequency. When the modulation frequency is below resonance, the laser exhibits a three-frequency route to chaos. When the modulation frequency is above resonance, the laser oscillates in two- and three-frequency states. The chaotic instability is a result of the nonlinear interaction of three periodic modes of the laser system. These modes are dynamical manifestations of the composite cavity mode-locking resonance, the applied field that is due to the modulation, and the laser relaxation oscillation.

© 1995 Optical Society of America

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  1. B. C. Lam, A. L. Kellner, M. M. Sushchik, H. D. I. Abarbanel, and P. K. L. Yu, "Observation of chaotic instability in the active mode locking of a semiconductor laser," J. Opt. Soc. Am. B 10, 2065–2070 (1993).
    [CrossRef]
  2. H. Olesen, J. H. Osmundsen, and B. Tromborg, "Nonlinear dynamics and spectral behavior for an external cavity laser," IEEE J. Quantum Electron. QE-22, 762–773 (1986).
    [CrossRef]
  3. A. Ritter and H. Haug, "Theory of laser diodes with weak optical feedback, Parts I and II," J. Opt. Soc. Am. B 10, 130–153 (1993).
    [CrossRef]
  4. J. A. Glazier and A. Libchaber, "Quasi-periodicity and dynamical systems: an experimentalist's view," IEEE Trans. Circuits Syst. 35, 790–809 (1988).
    [CrossRef]
  5. P. Bryant and C. Jeffries, "The dynamics of phase locking and points of resonance in a forced magnetic oscillator," Physica D 25D, 196–232 (1987).
    [CrossRef]
  6. J. Sacher, D. Baums, P. Panknin, W. Elsasser, and E. Gobel, "Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback," Phys. Rev. A 45, 1893–1905 (1992).
    [CrossRef] [PubMed]
  7. H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, "The analysis of observed chaotic data in physical systems," Rev. Mod. Phys. 65, 1331–1392 (1993).
    [CrossRef]
  8. R. Lang and K. Kobayashi, "External optical feedback effects on semiconductor injection laser properties," IEEE J. Quantum Electron. QE-16, 347–355 (1980).
    [CrossRef]
  9. B. Tromborg, J. H. Osmundsen, and H. Olesen, "Stability analysis for a semiconductor laser in an external cavity," IEEE J. Quantum Electron. QE-20, 1023–1032 (1984).
    [CrossRef]
  10. K. Y. Lau and A. Yariv, "High-frequency current modulation of semiconductor lasers," in Semiconductor and Semimetals, W.T. Tsang, ed. (Academic, Orlando, Fla., 1985), Vol. 22, Part B.
    [CrossRef]
  11. G. P. Agrawal and N. K. Dutta, Long Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), p. 220.
    [CrossRef]
  12. T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
    [CrossRef]
  13. F. Takens, Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics No. 898 (Springer-Verlag, New York, 1980).
  14. A. M. Fraser and H. L. Swinney, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A 33, 1134–1140 (1986).
    [CrossRef] [PubMed]
  15. P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D 9D, 189–208, 1983.
    [CrossRef]
  16. J. Theiler, "Estimating fractal dimension," J. Opt. Soc. Am. A 7, 1055–1073 (1990).
    [CrossRef]
  17. J. P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617–656 (1985).
    [CrossRef]
  18. R. Brown, P. B. Bryant, and H. D. I. Abarbanel, "Computing the Lyapunov spectrum of a dynamical system from observed time series," Phys. Rev. A 43, 2787–2806 (1991).
    [CrossRef] [PubMed]

1993 (3)

1992 (1)

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

1991 (1)

R. Brown, P. B. Bryant, and H. D. I. Abarbanel, "Computing the Lyapunov spectrum of a dynamical system from observed time series," Phys. Rev. A 43, 2787–2806 (1991).
[CrossRef] [PubMed]

1990 (1)

1988 (1)

J. A. Glazier and A. Libchaber, "Quasi-periodicity and dynamical systems: an experimentalist's view," IEEE Trans. Circuits Syst. 35, 790–809 (1988).
[CrossRef]

1987 (1)

P. Bryant and C. Jeffries, "The dynamics of phase locking and points of resonance in a forced magnetic oscillator," Physica D 25D, 196–232 (1987).
[CrossRef]

1986 (2)

A. M. Fraser and H. L. Swinney, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A 33, 1134–1140 (1986).
[CrossRef] [PubMed]

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

1985 (1)

J. P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617–656 (1985).
[CrossRef]

1984 (1)

B. Tromborg, J. H. Osmundsen, and H. Olesen, "Stability analysis for a semiconductor laser in an external cavity," IEEE J. Quantum Electron. QE-20, 1023–1032 (1984).
[CrossRef]

1983 (1)

P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D 9D, 189–208, 1983.
[CrossRef]

1980 (1)

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

Abarbanel, H. D. I.

B. C. Lam, A. L. Kellner, M. M. Sushchik, H. D. I. Abarbanel, and P. K. L. Yu, "Observation of chaotic instability in the active mode locking of a semiconductor laser," J. Opt. Soc. Am. B 10, 2065–2070 (1993).
[CrossRef]

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, "The analysis of observed chaotic data in physical systems," Rev. Mod. Phys. 65, 1331–1392 (1993).
[CrossRef]

R. Brown, P. B. Bryant, and H. D. I. Abarbanel, "Computing the Lyapunov spectrum of a dynamical system from observed time series," Phys. Rev. A 43, 2787–2806 (1991).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Long Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), p. 220.
[CrossRef]

Baums, D.

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

Brown, R.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, "The analysis of observed chaotic data in physical systems," Rev. Mod. Phys. 65, 1331–1392 (1993).
[CrossRef]

R. Brown, P. B. Bryant, and H. D. I. Abarbanel, "Computing the Lyapunov spectrum of a dynamical system from observed time series," Phys. Rev. A 43, 2787–2806 (1991).
[CrossRef] [PubMed]

Bryant, P.

P. Bryant and C. Jeffries, "The dynamics of phase locking and points of resonance in a forced magnetic oscillator," Physica D 25D, 196–232 (1987).
[CrossRef]

Bryant, P. B.

R. Brown, P. B. Bryant, and H. D. I. Abarbanel, "Computing the Lyapunov spectrum of a dynamical system from observed time series," Phys. Rev. A 43, 2787–2806 (1991).
[CrossRef] [PubMed]

Chua, L. O.

T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
[CrossRef]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Long Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), p. 220.
[CrossRef]

Eckmann, J. P.

J. P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617–656 (1985).
[CrossRef]

Elsasser, W.

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

Fraser, A. M.

A. M. Fraser and H. L. Swinney, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A 33, 1134–1140 (1986).
[CrossRef] [PubMed]

Glazier, J. A.

J. A. Glazier and A. Libchaber, "Quasi-periodicity and dynamical systems: an experimentalist's view," IEEE Trans. Circuits Syst. 35, 790–809 (1988).
[CrossRef]

Gobel, E.

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

Grassberger, P.

P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D 9D, 189–208, 1983.
[CrossRef]

Haug, H.

Jeffries, C.

P. Bryant and C. Jeffries, "The dynamics of phase locking and points of resonance in a forced magnetic oscillator," Physica D 25D, 196–232 (1987).
[CrossRef]

Kellner, A. L.

Kobayashi, K.

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

Lam, B. C.

Lang, R.

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

Lau, K. Y.

K. Y. Lau and A. Yariv, "High-frequency current modulation of semiconductor lasers," in Semiconductor and Semimetals, W.T. Tsang, ed. (Academic, Orlando, Fla., 1985), Vol. 22, Part B.
[CrossRef]

Libchaber, A.

J. A. Glazier and A. Libchaber, "Quasi-periodicity and dynamical systems: an experimentalist's view," IEEE Trans. Circuits Syst. 35, 790–809 (1988).
[CrossRef]

Olesen, H.

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

B. Tromborg, J. H. Osmundsen, and H. Olesen, "Stability analysis for a semiconductor laser in an external cavity," IEEE J. Quantum Electron. QE-20, 1023–1032 (1984).
[CrossRef]

Osmundsen, J. H.

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

B. Tromborg, J. H. Osmundsen, and H. Olesen, "Stability analysis for a semiconductor laser in an external cavity," IEEE J. Quantum Electron. QE-20, 1023–1032 (1984).
[CrossRef]

Panknin, P.

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

Parker, T. S.

T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
[CrossRef]

Procaccia, I.

P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D 9D, 189–208, 1983.
[CrossRef]

Ritter, A.

Ruelle, D.

J. P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617–656 (1985).
[CrossRef]

Sacher, J.

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

Sidorowich, J. J.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, "The analysis of observed chaotic data in physical systems," Rev. Mod. Phys. 65, 1331–1392 (1993).
[CrossRef]

Sushchik, M. M.

Swinney, H. L.

A. M. Fraser and H. L. Swinney, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A 33, 1134–1140 (1986).
[CrossRef] [PubMed]

Takens, F.

F. Takens, Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics No. 898 (Springer-Verlag, New York, 1980).

Theiler, J.

Tromborg, B.

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

B. Tromborg, J. H. Osmundsen, and H. Olesen, "Stability analysis for a semiconductor laser in an external cavity," IEEE J. Quantum Electron. QE-20, 1023–1032 (1984).
[CrossRef]

Tsimring, L. S.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, "The analysis of observed chaotic data in physical systems," Rev. Mod. Phys. 65, 1331–1392 (1993).
[CrossRef]

Yariv, A.

K. Y. Lau and A. Yariv, "High-frequency current modulation of semiconductor lasers," in Semiconductor and Semimetals, W.T. Tsang, ed. (Academic, Orlando, Fla., 1985), Vol. 22, Part B.
[CrossRef]

Yu, P. K. L.

IEEE J. Quantum Electron. (3)

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

B. Tromborg, J. H. Osmundsen, and H. Olesen, "Stability analysis for a semiconductor laser in an external cavity," IEEE J. Quantum Electron. QE-20, 1023–1032 (1984).
[CrossRef]

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

IEEE Trans. Circuits Syst. (1)

J. A. Glazier and A. Libchaber, "Quasi-periodicity and dynamical systems: an experimentalist's view," IEEE Trans. Circuits Syst. 35, 790–809 (1988).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Phys. Rev. A (3)

R. Brown, P. B. Bryant, and H. D. I. Abarbanel, "Computing the Lyapunov spectrum of a dynamical system from observed time series," Phys. Rev. A 43, 2787–2806 (1991).
[CrossRef] [PubMed]

A. M. Fraser and H. L. Swinney, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A 33, 1134–1140 (1986).
[CrossRef] [PubMed]

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

Physica D (2)

P. Bryant and C. Jeffries, "The dynamics of phase locking and points of resonance in a forced magnetic oscillator," Physica D 25D, 196–232 (1987).
[CrossRef]

P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D 9D, 189–208, 1983.
[CrossRef]

Rev. Mod. Phys. (2)

J. P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617–656 (1985).
[CrossRef]

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, "The analysis of observed chaotic data in physical systems," Rev. Mod. Phys. 65, 1331–1392 (1993).
[CrossRef]

Other (4)

K. Y. Lau and A. Yariv, "High-frequency current modulation of semiconductor lasers," in Semiconductor and Semimetals, W.T. Tsang, ed. (Academic, Orlando, Fla., 1985), Vol. 22, Part B.
[CrossRef]

G. P. Agrawal and N. K. Dutta, Long Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), p. 220.
[CrossRef]

T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, New York, 1989).
[CrossRef]

F. Takens, Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics No. 898 (Springer-Verlag, New York, 1980).

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Figures (6)

Fig. 1
Fig. 1

External-cavity setup used in the theoretical modeling.

Fig. 2
Fig. 2

(a) Power spectrum of the mode-locked laser while the laser is in the periodic state. The modulation frequency is 1.97 GHz. (b) Power spectrum of the mode-locked laser in the two-frequency state. The modulation frequency is 1.90 GHz. (c) Power spectrum of the laser in the three-frequency state. The modulation frequency is 1.78 GHz. (d) Power spectrum of the laser in the nonperiodic state.The modulation frequency is 1.71 GHz.

Fig. 3
Fig. 3

Power spectrum of the laser with no modulation. The spectrum is that of the laser in an external cavity.

Fig. 4
Fig. 4

(a) Time history of the laser when the laser is modulated at 1.97 GHz. (b) Time history of the laser modulated at 1.78 GHz.

Fig. 5
Fig. 5

(a) Attractor of the mode-locked laser while the laser is in the periodic state. This attractor is associated with the spectrum shown in Fig. 2(a). (b) Attractor of the mode-locked laser in the two-frequency state. This attractor is associated with the spectrum shown in Fig. 2(b). (c) Attractor of the laser in the three-frequency state. This attractor is associated with the spectrum shown in Fig. 2(c). (d) Attractor of the laser in the nonperiodic state. This attractor is associated with the spectrum in Fig. 2(d).

Fig. 6
Fig. 6

Power spectrum of the laser when the modulation frequency is tuned to 2.29 GHz, which is above the mode-locking resonance.

Tables (1)

Tables Icon

Table 1 Laser Parameters Used in the Numerical Solution of Eqs. (14)(16)a

Equations (18)

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d E ( t ) d t ( i ω [ N ( t ) ] + 1 2 { [ N ( t ) N 0 ] G N 1 τ p } ) E ( t ) = η E ( t τ ) ,
ω [ N ( t ) ] ω o + α 2 [ N ( t ) N th ] G N ,
E r ( t ) = R 2 { E i ( t ) + R 3 / R 2 k = 0 ( R 2 R 3 ) k × ( 1 R 2 ) E i [ t ( k + 1 ) τ ] } ,
E r ( t ) = R 2 [ E i ( t ) + R 3 / R 2 ( 1 R 2 ) E i ( t τ ) ] .
η = c κ 2 n L int ,
κ = R 3 / R 2 ( 1 R 2 ) ,
d E 0 ( t ) d t = ½ { [ N ( t ) N 0 ] G N Γ 0 } E 0 ( t ) + η E 0 ( t τ ) × cos [ ω 0 τ + ϕ ( t ) ϕ ( t τ ) ] ,
d ϕ ( t ) d t = ω [ N ( t ) ] ω 0 η E 0 ( t τ ) E 0 ( t ) × sin [ ω 0 τ + ϕ ( t ) ϕ ( t τ ) ] ,
d N ( t ) d t = J N ( t ) τ s [ N ( t ) N 0 ] G N E 0 ( t ) 2 ,
t = t / τ ,
0 ( t ) = E 0 ( t ) G N τ s ,
Ñ ( t ) = N ( t ) G N τ p ,
J ( t ) = J ( t ) G N τ s τ p ,
d 0 ( t ) d t = 1 2 τ τ p { [ Ñ ( t ) Ñ 0 1 ] 0 ( t ) 2 + β Ñ ( t ) } + F 1 ,
d ϕ ( t ) d t = τ τ p α 2 [ Ñ ( t ) Ñ th ] F 2 ,
d Ñ ( t ) d t = τ τ s { J ( t ) Ñ ( t ) 2 ( Ñ ( t ) Ñ 0 ) 0 ( t ) 2 } ,
F 1 = 2 τ τ in R 3 / R 2 ( 1 R 2 ) k = 1 ( R 2 R 3 ) 0 ( t k ) × cos [ k ω 0 τ + ϕ ( t ) ϕ ( t k ) ] ,
F 2 = 2 τ τ in R 3 / R 2 ( 1 R 2 ) k = 1 ( R 2 R 3 ) k 1 0 ( t k ) 0 ( t ) × sin [ k ω 0 τ + ϕ ( t ) ϕ ( t k ) ] .

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