Abstract

Inducing periodicity from chaotic oscillations of a compound-cavity laser diode is numerically demonstrated through the Van der Pol model. The deep modulation beyond the limit of the perturbation approximation leads to both turbulence and periodicity. Bifurcation against the modulation frequencies and modulation depth occurs under proper conditions. Therefore it is found that the different periodicity can be induced from chaos by a small change in the frequency and the amplitude of the periodic modulations.

© 1995 Optical Society of America

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References

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  1. E. Ott, C. Grebogi, J. York, Phys. Rev. Lett. 64, 1196 (1990).
    [CrossRef] [PubMed]
  2. E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1991).
    [CrossRef] [PubMed]
  3. R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
    [CrossRef]
  4. Y. Braiman, I. Goldhirsch, Phys. Rev. Lett. 66, 2545 (1991).
    [CrossRef] [PubMed]
  5. R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
    [CrossRef]
  6. J. Mørk, B. Tromborg, J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
    [CrossRef]

1992 (2)

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
[CrossRef]

J. Mørk, B. Tromborg, J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

1991 (2)

Y. Braiman, I. Goldhirsch, Phys. Rev. Lett. 66, 2545 (1991).
[CrossRef] [PubMed]

E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1991).
[CrossRef] [PubMed]

1990 (1)

E. Ott, C. Grebogi, J. York, Phys. Rev. Lett. 64, 1196 (1990).
[CrossRef] [PubMed]

1980 (1)

R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

Braiman, Y.

Y. Braiman, I. Goldhirsch, Phys. Rev. Lett. 66, 2545 (1991).
[CrossRef] [PubMed]

Gills, Z.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
[CrossRef]

Goldhirsch, I.

Y. Braiman, I. Goldhirsch, Phys. Rev. Lett. 66, 2545 (1991).
[CrossRef] [PubMed]

Grebogi, C.

E. Ott, C. Grebogi, J. York, Phys. Rev. Lett. 64, 1196 (1990).
[CrossRef] [PubMed]

Hunt, E. R.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
[CrossRef]

E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1991).
[CrossRef] [PubMed]

Kobayashi, K.

R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

Lang, R.

R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

Maier, T. D.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
[CrossRef]

Mark, J.

J. Mørk, B. Tromborg, J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Mørk, J.

J. Mørk, B. Tromborg, J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Murphy, T. W.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
[CrossRef]

Ott, E.

E. Ott, C. Grebogi, J. York, Phys. Rev. Lett. 64, 1196 (1990).
[CrossRef] [PubMed]

Roy, R.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
[CrossRef]

Tromborg, B.

J. Mørk, B. Tromborg, J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

York, J.

E. Ott, C. Grebogi, J. York, Phys. Rev. Lett. 64, 1196 (1990).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (2)

R. Lang, K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
[CrossRef]

J. Mørk, B. Tromborg, J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Phys. Rev. Lett. (4)

E. Ott, C. Grebogi, J. York, Phys. Rev. Lett. 64, 1196 (1990).
[CrossRef] [PubMed]

E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1991).
[CrossRef] [PubMed]

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, E. R. Hunt, Phys. Rev. Lett. 64, 1259 (1992).
[CrossRef]

Y. Braiman, I. Goldhirsch, Phys. Rev. Lett. 66, 2545 (1991).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Bifurcation diagram and leading Lyapunov exponent λ1 as a function of the modulation depth. The curve joining crosses at the top indicates the leading Lyapunov exponent λ1, and the dots below show distribution of the extrema of the normalized light output [I (t) = |E0(t)|2/|E0(sol)(t)| −1] of the compound-cavity LD. Calculation conditions are α = 2.0, τ = 4 ns, injection current I = 1.05Ith, modulation frequency 1.182 GHz, τp = 2 ps, τs = 0.5 ns, τin = 8 ps, Gn = 1.1 × 10−12 m3 s−1, and κ = 0.0112.

Fig. 2
Fig. 2

(a) Influence of the modulation depth on the power spectra of the light output from the sinusoidally modulated compound-cavity LD and (b) the return maps of the time series I (t) = |E0(t)|2/|E0(sol)(t)| −1. The numbers indicated in each graph in (b) are the modulation depth and λ1. Calculation conditions are the same as for Fig. 1, except for the modulation depth, which is indicated for each graph.

Fig. 3
Fig. 3

Bifurcation diagram and leading Lyapunov exponent λ1 as a function of modulation frequency. The modulation depth is 2.0%; other calculation conditions are the same as for Fig. 1.

Fig. 4
Fig. 4

(a) Influence of the modulation frequency on the power spectra of light output from the sinusoidally modulated compound-cavity LD and (b) the return maps of the time series. The numbers indicated in each graph in (b) are the modulation depth and λ1. Calculation conditions are the same as for Fig. 1, except for the modulation depth, which is indicated for each graph.

Equations (6)

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d E 0 ( t ) d t = 1 2 [ G ( N , E 0 ) - 1 τ p ] E 0 ( t ) + κ τ in E 0 ( t - τ ) × cos [ ω 0 τ + ϕ ( t ) - ϕ ( t - τ ) ] + R 2 V E 0 ( t ) ,
d ϕ ( t ) d t = 1 2 α [ G ( N , E 0 ) - 1 τ p ] - κ τ in E 0 ( t - τ ) E 0 ( t ) × sin [ ω 0 τ + ϕ ( t ) - ϕ ( t - τ ) ] ,
d N ( t ) d t = J 0 [ 1 + m sin ( ω i t ) ] - N ( t ) τ s - G ( N , E 0 ) E 0 ( t ) 2 .
G ( N , E 0 ) = G n ( N - N 0 ) [ 1 - E 0 ( t ) 2 ] .
λ 1 = Re ln ( η + f n ) ,
λ 1 ln η - 2 2 η + O ( 3 ) ,

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