Abstract

We show that finite external excitation can lead to a traveling wave in an excitable passive optical system with one-dimensional space geometry. We have studied the excitable behavior of this system in parallel with that of its diffusive counterpart and show the effects of optical phase on the traveling-wave solution and its velocity. In two-dimensional space we observe numerically rotating optical spiral waves evolving from a truncated planar wave front.

© 1999 Optical Society of America

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References

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  1. See, for example, the special issuesPhysica D 84(1,??2), June (1994) and Chaos 4(3), September (1994).
  2. R. FitzHugh, Biophys. J. 1, 445 (1961).
    [CrossRef] [PubMed]
  3. J. Nagumo, S. Arimoto, and S. Yoshizawa, Proc. IRE 50, 2061 (1962).
    [CrossRef]
  4. A. Zhabotinsky and A. Zaikin, Nature (London) 225, 535 (1970).
  5. P. Coullet, T. Frisch, J. M. Gilli, and S. Rica, Chaos 4, 485 (1994).
    [CrossRef] [PubMed]
  6. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
    [CrossRef]
  7. W. Lu, D. Yu, and R. G. Harrison, Phys. Rev. A 58, R809 (1998).
    [CrossRef]
  8. P. Coullet, D. Daboussy, and J. R. Tredicce, Phys. Rev. E 58, 5347 (1998).
    [CrossRef]
  9. T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
    [CrossRef]

1998 (2)

W. Lu, D. Yu, and R. G. Harrison, Phys. Rev. A 58, R809 (1998).
[CrossRef]

P. Coullet, D. Daboussy, and J. R. Tredicce, Phys. Rev. E 58, 5347 (1998).
[CrossRef]

1997 (1)

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
[CrossRef]

1994 (2)

P. Coullet, T. Frisch, J. M. Gilli, and S. Rica, Chaos 4, 485 (1994).
[CrossRef] [PubMed]

See, for example, the special issuesPhysica D 84(1,??2), June (1994) and Chaos 4(3), September (1994).

1989 (1)

T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
[CrossRef]

1970 (1)

A. Zhabotinsky and A. Zaikin, Nature (London) 225, 535 (1970).

1962 (1)

J. Nagumo, S. Arimoto, and S. Yoshizawa, Proc. IRE 50, 2061 (1962).
[CrossRef]

1961 (1)

R. FitzHugh, Biophys. J. 1, 445 (1961).
[CrossRef] [PubMed]

Arimoto, S.

J. Nagumo, S. Arimoto, and S. Yoshizawa, Proc. IRE 50, 2061 (1962).
[CrossRef]

Coullet, P.

P. Coullet, D. Daboussy, and J. R. Tredicce, Phys. Rev. E 58, 5347 (1998).
[CrossRef]

P. Coullet, T. Frisch, J. M. Gilli, and S. Rica, Chaos 4, 485 (1994).
[CrossRef] [PubMed]

Daboussy, D.

P. Coullet, D. Daboussy, and J. R. Tredicce, Phys. Rev. E 58, 5347 (1998).
[CrossRef]

FitzHugh, R.

R. FitzHugh, Biophys. J. 1, 445 (1961).
[CrossRef] [PubMed]

Frisch, T.

P. Coullet, T. Frisch, J. M. Gilli, and S. Rica, Chaos 4, 485 (1994).
[CrossRef] [PubMed]

Giacomelli, G.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
[CrossRef]

Gilli, J. M.

P. Coullet, T. Frisch, J. M. Gilli, and S. Rica, Chaos 4, 485 (1994).
[CrossRef] [PubMed]

Giudici, M.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
[CrossRef]

Green, C.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
[CrossRef]

Harrison, R. G.

W. Lu, D. Yu, and R. G. Harrison, Phys. Rev. A 58, R809 (1998).
[CrossRef]

Ikeda, H.

T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
[CrossRef]

Kobayashi, R.

T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
[CrossRef]

Lu, W.

W. Lu, D. Yu, and R. G. Harrison, Phys. Rev. A 58, R809 (1998).
[CrossRef]

Mimura, M.

T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
[CrossRef]

Nagai, T.

T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
[CrossRef]

Nagumo, J.

J. Nagumo, S. Arimoto, and S. Yoshizawa, Proc. IRE 50, 2061 (1962).
[CrossRef]

Nespolo, U.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
[CrossRef]

Rica, S.

P. Coullet, T. Frisch, J. M. Gilli, and S. Rica, Chaos 4, 485 (1994).
[CrossRef] [PubMed]

Tedicce, J. R.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
[CrossRef]

Tredicce, J. R.

P. Coullet, D. Daboussy, and J. R. Tredicce, Phys. Rev. E 58, 5347 (1998).
[CrossRef]

Tsujikawa, T.

T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
[CrossRef]

Yoshizawa, S.

J. Nagumo, S. Arimoto, and S. Yoshizawa, Proc. IRE 50, 2061 (1962).
[CrossRef]

Yu, D.

W. Lu, D. Yu, and R. G. Harrison, Phys. Rev. A 58, R809 (1998).
[CrossRef]

Zaikin, A.

A. Zhabotinsky and A. Zaikin, Nature (London) 225, 535 (1970).

Zhabotinsky, A.

A. Zhabotinsky and A. Zaikin, Nature (London) 225, 535 (1970).

Biophys. J. (1)

R. FitzHugh, Biophys. J. 1, 445 (1961).
[CrossRef] [PubMed]

Chaos (1)

P. Coullet, T. Frisch, J. M. Gilli, and S. Rica, Chaos 4, 485 (1994).
[CrossRef] [PubMed]

Jpn. J. Appl. Math. (1)

T. Tsujikawa, T. Nagai, M. Mimura, R. Kobayashi, and H. Ikeda, Jpn. J. Appl. Math. 6, 341 (1989).
[CrossRef]

Nature (London) (1)

A. Zhabotinsky and A. Zaikin, Nature (London) 225, 535 (1970).

Phys. Rev. A (1)

W. Lu, D. Yu, and R. G. Harrison, Phys. Rev. A 58, R809 (1998).
[CrossRef]

Phys. Rev. E (2)

P. Coullet, D. Daboussy, and J. R. Tredicce, Phys. Rev. E 58, 5347 (1998).
[CrossRef]

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tedicce, Phys. Rev. E 79, 6414 (1997).
[CrossRef]

Physica D (1)

See, for example, the special issuesPhysica D 84(1,??2), June (1994) and Chaos 4(3), September (1994).

Proc. IRE (1)

J. Nagumo, S. Arimoto, and S. Yoshizawa, Proc. IRE 50, 2061 (1962).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Phase-space illustration of excitability for E0=10. The two curves are the nullclines of the two elements in Eqs.  (1). (b) Steady-state relation of the input E0 to the output E. The solid and the dashed curves correspond to stable and unstable steady-state solutions for ϵ=0.001, respectively. They are separated by a Hopf bifurcation, as marked by a filled square at E0,E=10.80,1.13. The other parameters are fixed at C0=9 and η=0.06.

Fig. 2
Fig. 2

Traveling wave-front solutions (solid curves) for Eq.  (4) and (dotted curves) for its diffusive counterpart. For Eq.  (4), the transition from uh to ul of the field amplitude in (a) coincides with an abrupt pulse in the phase of the field in (b). (c) Functional dependence of the velocities of the wave front on the incident amplitude E0 for both equations. The parameters are chosen to be C0=9, η=0.06, and E0=10.

Fig. 3
Fig. 3

Excitable pulse of Eqs.  (1). (a), (b), and (c) correspond to the solutions of the amplitude and phase of the electric field and the temperature variations. The parameters are the same as in Fig.  2, except that ϵ=0.01.

Fig. 4
Fig. 4

Evolution and formation of optical spirals (field amplitude) from a truncated planar wave front in two-dimensional space. In (d) the inner waves are rotating spirals and the outer ones are expanding targets.

Equations (4)

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

k1-1dEdt=-E-2CD1+E2+E0+i2E,k2-1dCdt=-C+ηCE2+C0,
Ex,t=uz,z=x-cτ,τ=k1t,
u-=ul,u=uh,ϕ-=ϕ=0,
iu+cu-u-2Csu1+u2+E0=0,

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