Abstract

Sustained periodic relaxation oscillations (soft mode) and periodic spiking oscillations (hard mode) leading to chaotic oscillations with different bifurcation situations are found to coexist in lasers with incoherent delayed feedback that can be modeled by particle motion in a Toda-like potential with retarded interaction. Bistability of the soft-mode attractor (quasi periodicity) and the hard-mode attractor (period doubling) and their combined attraction are predicted theoretically.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Mork, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65, 1999–2002 (1990).
    [CrossRef] [PubMed]
  2. J. Sacher, W. Elsasser, and E. O. Gobel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
    [CrossRef] [PubMed]
  3. F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Dynamics of a CO2 laser with delayed feedback: the short-delay regime,” Phys. Rev. A 43, 4997–5004 (1991).
    [CrossRef] [PubMed]
  4. K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. 16, 1759–1761 (1991).
    [CrossRef] [PubMed]
  5. J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Opt. Commun. 96, 259–266 (1993).
    [CrossRef]
  6. F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
    [CrossRef]
  7. D. Dangoisse, P. Glorieux, and D. Hennequin, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1986).
    [CrossRef] [PubMed]
  8. D. K. Bandy, L. M. Narducci, and L. A. Lugiato, “Coexisting attractors in a laser with an injected signal,” J. Opt. Soc. Am. B 2, 148–155 (1985).
    [CrossRef]
  9. J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
    [CrossRef]
  10. F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
    [CrossRef]
  11. G. L. Oppo and A. Politi, “Toda potential in laser equations,” Z. Phys. B 59, 111–115 (1985).
    [CrossRef]
  12. T. Ogawa, “Quasiperiodic instability and chaos in the bad-cavity laser with modulated inversion: numerical analysis of a Toda oscillator system,” Phys. Rev. A 37, 4286–4302 (1988).
    [CrossRef] [PubMed]
  13. T. Ogawa, “Modulation properties of the single-mode laser with a low cavity,” Jpn. J. Appl. Phys. 27, 2992–2309 (1988).
    [CrossRef]
  14. M. Toda, “Vibration of a chain with nonlinear interaction,” J. Phys. Soc. Jpn. 22, 431–436 (1967).
    [CrossRef]
  15. K. Kubodera and K. Otsuka, “Spike-mode oscillation in laser-diode pumped LiNdP4O12 lasers,” IEEE J. Quantum Electron. QE-17, 1139–1144 (1981).
    [CrossRef]

1993 (1)

J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Opt. Commun. 96, 259–266 (1993).
[CrossRef]

1991 (2)

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Dynamics of a CO2 laser with delayed feedback: the short-delay regime,” Phys. Rev. A 43, 4997–5004 (1991).
[CrossRef] [PubMed]

K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. 16, 1759–1761 (1991).
[CrossRef] [PubMed]

1990 (1)

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

1989 (1)

J. Sacher, W. Elsasser, and E. O. Gobel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

1988 (2)

T. Ogawa, “Quasiperiodic instability and chaos in the bad-cavity laser with modulated inversion: numerical analysis of a Toda oscillator system,” Phys. Rev. A 37, 4286–4302 (1988).
[CrossRef] [PubMed]

T. Ogawa, “Modulation properties of the single-mode laser with a low cavity,” Jpn. J. Appl. Phys. 27, 2992–2309 (1988).
[CrossRef]

1986 (1)

D. Dangoisse, P. Glorieux, and D. Hennequin, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1986).
[CrossRef] [PubMed]

1985 (3)

1984 (1)

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[CrossRef]

1982 (1)

F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

1981 (1)

K. Kubodera and K. Otsuka, “Spike-mode oscillation in laser-diode pumped LiNdP4O12 lasers,” IEEE J. Quantum Electron. QE-17, 1139–1144 (1981).
[CrossRef]

1967 (1)

M. Toda, “Vibration of a chain with nonlinear interaction,” J. Phys. Soc. Jpn. 22, 431–436 (1967).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Dynamics of a CO2 laser with delayed feedback: the short-delay regime,” Phys. Rev. A 43, 4997–5004 (1991).
[CrossRef] [PubMed]

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[CrossRef]

F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Bandy, D. K.

Chern, J.-L.

J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Opt. Commun. 96, 259–266 (1993).
[CrossRef]

K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. 16, 1759–1761 (1991).
[CrossRef] [PubMed]

Dangoisse, D.

D. Dangoisse, P. Glorieux, and D. Hennequin, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1986).
[CrossRef] [PubMed]

Elsasser, W.

J. Sacher, W. Elsasser, and E. O. Gobel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Dynamics of a CO2 laser with delayed feedback: the short-delay regime,” Phys. Rev. A 43, 4997–5004 (1991).
[CrossRef] [PubMed]

Glorieux, P.

D. Dangoisse, P. Glorieux, and D. Hennequin, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1986).
[CrossRef] [PubMed]

Gobel, E. O.

J. Sacher, W. Elsasser, and E. O. Gobel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Hennequin, D.

D. Dangoisse, P. Glorieux, and D. Hennequin, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1986).
[CrossRef] [PubMed]

Ishiyama, F.

J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Opt. Commun. 96, 259–266 (1993).
[CrossRef]

Kubodera, K.

K. Kubodera and K. Otsuka, “Spike-mode oscillation in laser-diode pumped LiNdP4O12 lasers,” IEEE J. Quantum Electron. QE-17, 1139–1144 (1981).
[CrossRef]

Lapucci, A.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Dynamics of a CO2 laser with delayed feedback: the short-delay regime,” Phys. Rev. A 43, 4997–5004 (1991).
[CrossRef] [PubMed]

Lippi, G. L.

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[CrossRef]

Lugiato, L. A.

Mark, J.

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

Meucci, R.

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Dynamics of a CO2 laser with delayed feedback: the short-delay regime,” Phys. Rev. A 43, 4997–5004 (1991).
[CrossRef] [PubMed]

F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Mork, J.

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

Narducci, L. M.

Ogawa, T.

T. Ogawa, “Quasiperiodic instability and chaos in the bad-cavity laser with modulated inversion: numerical analysis of a Toda oscillator system,” Phys. Rev. A 37, 4286–4302 (1988).
[CrossRef] [PubMed]

T. Ogawa, “Modulation properties of the single-mode laser with a low cavity,” Jpn. J. Appl. Phys. 27, 2992–2309 (1988).
[CrossRef]

Oppo, G. L.

G. L. Oppo and A. Politi, “Toda potential in laser equations,” Z. Phys. B 59, 111–115 (1985).
[CrossRef]

Otsuka, K.

J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Opt. Commun. 96, 259–266 (1993).
[CrossRef]

K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. 16, 1759–1761 (1991).
[CrossRef] [PubMed]

K. Kubodera and K. Otsuka, “Spike-mode oscillation in laser-diode pumped LiNdP4O12 lasers,” IEEE J. Quantum Electron. QE-17, 1139–1144 (1981).
[CrossRef]

Politi, A.

G. L. Oppo and A. Politi, “Toda potential in laser equations,” Z. Phys. B 59, 111–115 (1985).
[CrossRef]

Puccioni, G.

F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Puccioni, G. P.

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[CrossRef]

Sacher, J.

J. Sacher, W. Elsasser, and E. O. Gobel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Toda, M.

M. Toda, “Vibration of a chain with nonlinear interaction,” J. Phys. Soc. Jpn. 22, 431–436 (1967).
[CrossRef]

Tredicce, J.

F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

Tredicce, J. R.

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[CrossRef]

Tromborg, B.

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

IEEE J. Quantum Electron. (1)

K. Kubodera and K. Otsuka, “Spike-mode oscillation in laser-diode pumped LiNdP4O12 lasers,” IEEE J. Quantum Electron. QE-17, 1139–1144 (1981).
[CrossRef]

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

J. Phys. Soc. Jpn. (1)

M. Toda, “Vibration of a chain with nonlinear interaction,” J. Phys. Soc. Jpn. 22, 431–436 (1967).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Ogawa, “Modulation properties of the single-mode laser with a low cavity,” Jpn. J. Appl. Phys. 27, 2992–2309 (1988).
[CrossRef]

Opt. Commun. (2)

J.-L. Chern, K. Otsuka, and F. Ishiyama, “Coexistence of two attractors in lasers with delayed incoherent optical feedback,” Opt. Commun. 96, 259–266 (1993).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Dynamics of a CO2 laser with delayed feedback: the short-delay regime,” Phys. Rev. A 43, 4997–5004 (1991).
[CrossRef] [PubMed]

T. Ogawa, “Quasiperiodic instability and chaos in the bad-cavity laser with modulated inversion: numerical analysis of a Toda oscillator system,” Phys. Rev. A 37, 4286–4302 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett. 49, 1217–1220 (1982).
[CrossRef]

D. Dangoisse, P. Glorieux, and D. Hennequin, “Laser chaotic attractors in crisis,” Phys. Rev. Lett. 57, 2657–2660 (1986).
[CrossRef] [PubMed]

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

J. Sacher, W. Elsasser, and E. O. Gobel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Z. Phys. B (1)

G. L. Oppo and A. Politi, “Toda potential in laser equations,” Z. Phys. B 59, 111–115 (1985).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Asymmetric Toda potential at w=1.5, K=1000, and =1.2×10-7. (See text for explanation of abbreviations.)

Fig. 2
Fig. 2

Bifurcation diagrams at γ=0.21, K=1000, =1.2×10-7, and T=0.6 for the hard mode and the soft mode. Successive maxima are plotted in these figures. There are some monostable regions of the hard mode and the soft mode. In these regions the system goes into this attractor independent of the initial conditions. Other regions on the diagram are bistable. We can select which mode to go into by selecting the initial condition or by adding photons. The hard-mode window begins with period-3, doubles into a chaotic region, and then undergoes reverse period doubling of 3×2n(n=2, 1, 0). This window goes back into period-1, becomes unstable at a certain w, and the branch ends there.

Fig. 3
Fig. 3

Phase-space trajectories for coexisting hard mode and soft mode at w=1.5 (point a in Fig. 2). Other parameters are the same as in Fig. 2. The projected boundary between the two states on the plane is approximately on the hard-mode orbit. This figure shows that the force FD prevents the hard-mode oscillation from relaxing into the soft-mode oscillation.

Fig. 4
Fig. 4

Hard-mode period-doubling window corresponding to b in Fig. 2.

Fig. 5
Fig. 5

Time series (of intermittency at w=1.91; other parameters are as in Fig. 2) showing period-3 hard-mode spiking. Arrows show intermittent structures.

Fig. 6
Fig. 6

Quasi-periodic window in the soft-mode diagram shown in Fig. 2. This window begins with period-9, doubles into period-18, shows circle-like structures, and then ends with period-9. There are many narrow chaotic regions in this window.

Fig. 7
Fig. 7

Lorenz plots for various pump power w. (a) w=1.60, (b) w=1.75, (c) w=1.80, (d) w=1.86, (e) w=1.90), (f) w=1.91. Other parameters are as in Fig. 2. Quasi periodicity (a) gradually disappears and goes into the chaotic state [(b)–(d)]. Then three dark spots gradually appear [(e) and (f)]. This corresponds to the intermittent region. After that, only three dots are seen on the map. This is the period-3 region. Then the system goes into the period-doubling region.

Fig. 8
Fig. 8

Combined motion of the hard-mode and the soft-mode data at w=1.7 in Fig. 2.

Equations (12)

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

dn(t)dt=w-n(t)-n(t)[s(t)+γs(t-T)],
ds(t)dt=K{[n(t)-1]s(t)+n(t)},
n¯s¯s¯+1,
s¯w-11+γ+ww-1w-11+γ.
n¯w,
s¯w1-w0.
u(t)ln s(t)
d2udt2+κ dudt+Vu=FD,
V=K[w exp(-u)+exp(u)-(w-1)u-(w+1)],
κ=exp(u)+K+dudt+1+exp(u),
FD=-γK+dudtexp[u(t-T)].
K+dudtKn>0,

Metrics