Abstract

In this paper we report studies on the dynamical behaviors of erbium-doped fiber lasers with one polarized mode. The chaotic behaviors, as well as the coexistent scenario of the period doubling and the intermittently chaotic routes to chaos, are observed experimentally in this fiber laser with pump modulation. Optical bistability and a simple bifurcation are also observed in different modulation frequencies. A simple theoretical model has been developed to obtain almost all the dynamical behaviors. Another method to gain chaos in an erbium-doped fiber-ring laser system is a coupled-ring system. Theoretical study shows that chaos occurs in this dual-ring system. Again a bifurcation scenario is found in the marginal region from self-pulsation to chaos. Interestingly enough, a duplicated ring can produce synchronized chaos.

© 1998 Optical Society of America

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References

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  1. M. Digonnet, ed., Rare Earth Doped Fiber Lasers and Amplifiers (Marcel Dekker, New York, 1993).
  2. M. W. Phillips, H. Gong, A. I. Ferguson, and D. C. Hanna, Opt. Commun. 61, 215 (1987).
    [CrossRef]
  3. S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 2811 (1992).
    [CrossRef] [PubMed]
  4. S. Bielawski and D. Derozier, J. Phys. (France) III 5, 251 (1995).
    [CrossRef]
  5. F. Sanchez, P. Le Boudec, P. L. Francois, and G. Stephan, Phys. Rev. A 48, 2220 (1993); F. Sanchez and G. Stephan, Phys. Rev. E 53, 2110 (1996).
    [CrossRef] [PubMed]
  6. E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1993).
    [CrossRef]
  7. S. Colin, E. Contesse, P. Le Boudec, G. Stephan, and F. Sanchez, Opt. Lett. 21, 1987 (1996).
    [CrossRef] [PubMed]
  8. D. Marcuse, IEEE J. Quantum Electron. 29, 2390 (1993).
    [CrossRef]
  9. L. Luo and P. L. Chu, Opt. Commun. 135, 116 (1997).
    [CrossRef]
  10. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Weinheim, Germany, 1991), Chap. 7, p. 137.
  11. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
    [CrossRef] [PubMed]
  12. L. M. Pecora and T. L. Carroll, Phys. Rev. A 44, 2374 (1991).
    [CrossRef] [PubMed]
  13. L. Luo and P. L. Chu, Opt. Commun. (to be published).
  14. H. Haken, Light (North-Holland, Amsterdam, 1985), Vol. 2, Chap. 5, p. 98.
  15. O. Svelto, Principle of Lasers, 3rd ed. (Plenum, New York, 1989), p. 214.
  16. L. Reekie, I. M. Jauncey, S. B. Poole, and D. N. Payne, Electron. Lett. 23, 1076 (1987).
    [CrossRef]

1997 (1)

L. Luo and P. L. Chu, Opt. Commun. 135, 116 (1997).
[CrossRef]

1996 (1)

1995 (1)

S. Bielawski and D. Derozier, J. Phys. (France) III 5, 251 (1995).
[CrossRef]

1993 (2)

E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1993).
[CrossRef]

D. Marcuse, IEEE J. Quantum Electron. 29, 2390 (1993).
[CrossRef]

1992 (1)

S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

1991 (1)

L. M. Pecora and T. L. Carroll, Phys. Rev. A 44, 2374 (1991).
[CrossRef] [PubMed]

1990 (1)

L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
[CrossRef] [PubMed]

1987 (2)

M. W. Phillips, H. Gong, A. I. Ferguson, and D. C. Hanna, Opt. Commun. 61, 215 (1987).
[CrossRef]

L. Reekie, I. M. Jauncey, S. B. Poole, and D. N. Payne, Electron. Lett. 23, 1076 (1987).
[CrossRef]

Bielawski, S.

S. Bielawski and D. Derozier, J. Phys. (France) III 5, 251 (1995).
[CrossRef]

S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, Phys. Rev. A 44, 2374 (1991).
[CrossRef] [PubMed]

L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
[CrossRef] [PubMed]

Chenevier, M.

E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1993).
[CrossRef]

Chu, P. L.

L. Luo and P. L. Chu, Opt. Commun. 135, 116 (1997).
[CrossRef]

Colin, S.

Contesse, E.

Derozier, D.

S. Bielawski and D. Derozier, J. Phys. (France) III 5, 251 (1995).
[CrossRef]

S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Ferguson, A. I.

M. W. Phillips, H. Gong, A. I. Ferguson, and D. C. Hanna, Opt. Commun. 61, 215 (1987).
[CrossRef]

Glorieux, P.

S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Gong, H.

M. W. Phillips, H. Gong, A. I. Ferguson, and D. C. Hanna, Opt. Commun. 61, 215 (1987).
[CrossRef]

Hanna, D. C.

M. W. Phillips, H. Gong, A. I. Ferguson, and D. C. Hanna, Opt. Commun. 61, 215 (1987).
[CrossRef]

Jauncey, I. M.

L. Reekie, I. M. Jauncey, S. B. Poole, and D. N. Payne, Electron. Lett. 23, 1076 (1987).
[CrossRef]

Lacot, E.

E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1993).
[CrossRef]

Le Boudec, P.

Luo, L.

L. Luo and P. L. Chu, Opt. Commun. 135, 116 (1997).
[CrossRef]

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. 29, 2390 (1993).
[CrossRef]

Payne, D. N.

L. Reekie, I. M. Jauncey, S. B. Poole, and D. N. Payne, Electron. Lett. 23, 1076 (1987).
[CrossRef]

Pecora, L. M.

L. M. Pecora and T. L. Carroll, Phys. Rev. A 44, 2374 (1991).
[CrossRef] [PubMed]

L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
[CrossRef] [PubMed]

Phillips, M. W.

M. W. Phillips, H. Gong, A. I. Ferguson, and D. C. Hanna, Opt. Commun. 61, 215 (1987).
[CrossRef]

Poole, S. B.

L. Reekie, I. M. Jauncey, S. B. Poole, and D. N. Payne, Electron. Lett. 23, 1076 (1987).
[CrossRef]

Reekie, L.

L. Reekie, I. M. Jauncey, S. B. Poole, and D. N. Payne, Electron. Lett. 23, 1076 (1987).
[CrossRef]

Sanchez, F.

Stephan, G.

Stoeckel, F.

E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1993).
[CrossRef]

Electron. Lett. (1)

L. Reekie, I. M. Jauncey, S. B. Poole, and D. N. Payne, Electron. Lett. 23, 1076 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Marcuse, IEEE J. Quantum Electron. 29, 2390 (1993).
[CrossRef]

J. Phys. (France) III (1)

S. Bielawski and D. Derozier, J. Phys. (France) III 5, 251 (1995).
[CrossRef]

Opt. Commun. (2)

M. W. Phillips, H. Gong, A. I. Ferguson, and D. C. Hanna, Opt. Commun. 61, 215 (1987).
[CrossRef]

L. Luo and P. L. Chu, Opt. Commun. 135, 116 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (3)

L. M. Pecora and T. L. Carroll, Phys. Rev. A 44, 2374 (1991).
[CrossRef] [PubMed]

E. Lacot, F. Stoeckel, and M. Chenevier, Phys. Rev. A 49, 3997 (1993).
[CrossRef]

S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990).
[CrossRef] [PubMed]

Other (6)

L. Luo and P. L. Chu, Opt. Commun. (to be published).

H. Haken, Light (North-Holland, Amsterdam, 1985), Vol. 2, Chap. 5, p. 98.

O. Svelto, Principle of Lasers, 3rd ed. (Plenum, New York, 1989), p. 214.

F. Sanchez, P. Le Boudec, P. L. Francois, and G. Stephan, Phys. Rev. A 48, 2220 (1993); F. Sanchez and G. Stephan, Phys. Rev. E 53, 2110 (1996).
[CrossRef] [PubMed]

M. Digonnet, ed., Rare Earth Doped Fiber Lasers and Amplifiers (Marcel Dekker, New York, 1993).

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Weinheim, Germany, 1991), Chap. 7, p. 137.

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Figures (12)

Fig. 1
Fig. 1

Experimental setup of the erbium-doped fiber-ring laser.

Fig. 2
Fig. 2

Spectrum of the lasing light with a cw pump: (a) IP=5.9IPth, (b) IP=11.8IPth.

Fig. 3
Fig. 3

Regions and styles of dynamical behaviors obtained by sweeping the modulation frequency: I¯P=5.9IPth, m=1.

Fig. 4
Fig. 4

Period-doubling route to chaos. The modulation frequencies are the following: (a) 9.3 kHz, repeating every two pulses; (b) 9.7 kHz, repeating every four pulses; (c) 9.8 kHz, repeating every eight pulses; (d) 10.3 kHz, chaos. The x axis has 100 μs/unit, 1 ms in total; the y axis is in arbitrary units.

Fig. 5
Fig. 5

Intermittently chaotic route to chaos. The modulation frequencies are the following: (a) 12.3 kHz, chaos; (b) 12.7 kHz, intermittency; (c) 13.2 kHz, slight intermittency; (d) 13.6 kHz, resonantly enhanced pulses. The x axis has 100 μs/unit, 1 ms in total; the y axis is in arbitrary units.

Fig. 6
Fig. 6

Simulation of the period-doubling route. I¯P=5.9IPth, m=1, k=64 600, g=89 200. The modulation frequencies are the following: (a) 9 kHz, repeating every two pulses; (b) 9.4 kHz, repeating every four pulses; (c) 9.47 kHz, repeating every eight pulses; (d) 9.62 kHz, chaos. The y axis is in arbitrary units.

Fig. 7
Fig. 7

Simulation of the intermittency route. The modulation frequencies are the following: (a) 12.5 kHz, chaos; (b) 12.9 kHz, intermittency; (c) 13.3 kHz, slight intermittency; (d) 13.7 kHz, resonantly enhanced pulses. The y axis is in arbitrary units.

Fig. 8
Fig. 8

Simulation of sweeping the modulation frequency.

Fig. 9
Fig. 9

Erbium-doped fiber dual-ring laser system. The fundamental system consists of ring a and ring b. Ring b is a slave subsystem driven by the field from ring a.

Fig. 10
Fig. 10

Stationary states p1, p2, p3, and p4. The crossing points by the dashed lines are unstable. The x and y axes are in arbitrary units. ka=1000, kb=1000, ga=10 500, gb=5200, IPa=4, IPb=4.

Fig. 11
Fig. 11

Period-doubling route and strange attractor projected on the plane of Ea and Eb. ka=1000, kb=1000, ga=10 500, C0=0.2, and gb as follows: (a) 4370, single period; (b) 4470, two periods; (c) 4515, four periods; (d) 4524, eight periods; (e) 4535, chaos; (f) 5200, developed chaos. The x axis shows Ea in arbitrary units; the y axis shows Eb in arbitrary units.

Fig. 12
Fig. 12

Chaotic and synchronizedly chaotic behaviors: (a) Ea, (b) Eb, (c) Ea-Eb, (d) Eb, (e) Eb-Eb; Eb(τ=0)=0.1, Eb(τ=0)=3.5. The y axis is in arbitrary units; the x axis shows time in units of 10 ms.

Equations (17)

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

I˙L=-kIL+gILD,
D˙=-(1+IP+IL)D+IP-1,
Ip=I¯p(1+mcos ωPτ),
ILs=IPgk-1-gk+1.
E˙a=-ka(Ea+C0Eb)+gaEaDa,
E˙b=-kb(Eb-C0Ea)+gbEbDb,
D˙a=-(1+IPa+Ea2)Da+IPa-1,
D˙b=-(1+IPb+Eb2)Db+IPb-1,
C0Eb=-Ea+gaka Ea(IPa-1)1+IPa+Ea2,
-C0Ea=-Eb+gbkb Eb(IPb-1)1+IPb+Eb2.
E˙b=-kb(Eb-C0Ea)+gbEbDb,
D˙b=-(1+IPb+Eb2)Db+IPb-1,
dqdt=[VaBN-(1/τc)]q,
dNdt=Wp(Nt-N)-2BqN-Nt+Nτ2,
d(N/Nt)d(t/τ2)=-(1+τ2WP+2τ2Bq)(N/Nt)+τ2WP-1.
D˙=-(1+IP+IL)D+IP-1.
I˙L=-kIL+gILD.

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