Abstract

A new theoretical approach for the study of the influence of diffraction on Ikeda instability in the Kerr-type nonlinear ring cavity is proposed. The simplicity of this model allows for a complete analytical study of transverse modulational instability in the period-2 regime of the cavity. In particular the emergence of stationary spatial dissipative structures in the profile of the period-doubled transmitted beam is predicted.

© 1992 Optical Society of America

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References

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  1. K. Ikeda, Opt. Commun. 30, 257 (1979).
    [CrossRef]
  2. K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
    [CrossRef]
  3. J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.
  4. D. W. McLaughlin, J. V. Moloney, A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
    [CrossRef] [PubMed]
  5. L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987); L. A. Lugiato, C. Oldano, Phys. Rev. A 37, 3896 (1988).
    [CrossRef] [PubMed]
  6. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
    [CrossRef]
  7. Y. Silberberg, I. Bar-Joseph, J. Opt. Soc. Am. B 1, 662 (1984).
    [CrossRef]
  8. R. Seydel, From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis (Elsevier, New York, 1988).

1987 (1)

L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987); L. A. Lugiato, C. Oldano, Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

1985 (1)

D. W. McLaughlin, J. V. Moloney, A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

1984 (1)

1980 (1)

K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

1979 (1)

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

1976 (1)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[CrossRef]

Akimoto, O.

K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

Bar-Joseph, I.

Daido, H.

K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[CrossRef]

Ikeda, K.

K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

Lefever, R.

L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987); L. A. Lugiato, C. Oldano, Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

Lugiato, L. A.

L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987); L. A. Lugiato, C. Oldano, Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

McCall, S. L.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[CrossRef]

McLaughlin, D. W.

D. W. McLaughlin, J. V. Moloney, A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Moloney, J. V.

D. W. McLaughlin, J. V. Moloney, A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.

Newell, A. C.

D. W. McLaughlin, J. V. Moloney, A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Seydel, R.

R. Seydel, From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis (Elsevier, New York, 1988).

Silberberg, Y.

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[CrossRef]

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

Opt. Commun. (1)

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

Phys. Rev. Lett. (4)

K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

D. W. McLaughlin, J. V. Moloney, A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987); L. A. Lugiato, C. Oldano, Phys. Rev. A 37, 3896 (1988).
[CrossRef] [PubMed]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, Phys. Rev. Lett. 36, 1135 (1976).
[CrossRef]

Other (2)

R. Seydel, From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis (Elsevier, New York, 1988).

J. V. Moloney, in Nonlinear Phenomena and Chaos, S. Sarkar, ed. (Hilger, Bristol, UK, 1986), pp. 214–245.

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Figures (1)

Fig. 1
Fig. 1

Bifurcation diagram in the plane of parameters (Δ, X). The dashed curve represents the plane-wave P2 bifurcation boundary X±(Δ). The regions correspond to η = +1: a, b, the MI-induced P2 regime; c, the plane-wave P2 regime; d, e, the stability of the cw plane-wave solutions; for η = 1: a, e, the stability of the cw plane-wave solutions; b, the plane-wave P2 regime; c, d, the MI-induced P2 regime.

Equations (12)

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E n + 1 ( 0 , x ) = T 1 / 2 E i + R exp ( i ϕ 0 ) E n ( L , x ) , n = 1 , 2 , 3 , ,
E n / z = i γ E n 2 E n + i κ 2 E n / x 2 ,
B n ( 0 , x ) = T 1 / 2 E i + R exp ( i ϕ 0 ) A n ( L , x ) , n = 1 , 2 , 3 , ,
A n + 1 ( 0 , x ) = T 1 / 2 E i + R exp ( i ϕ 0 ) B n ( L , x ) , n = 1 , 2 , 3 , .
A 1 + R exp [ i ( ϕ 0 + α B 2 ) ] = B 1 + R exp [ i ( ϕ 0 + α A 2 ) ] .
A [ T + i ( δ 0 - α A 2 ) ] = B [ T + i ( δ 0 - α B 2 ) ] .
( Y - Z ) [ Y 2 + Z 2 + Y Z - 2 Δ ( Y + Z ) + Δ 2 + 1 ] = 0.
Y ± = Z ± = [ 2 Δ ± ( Δ 2 - 3 ) 1 / 2 ] / 3.
E n ( L , x ) = E n ( 0 , x ) + i γ L E n ( 0 , x ) 2 E n ( 0 , x ) + i κ L 2 E n ( 0 , x ) / x 2 .
2 t R A / t = T 1 / 2 E i ( T + i δ 0 - i α B 2 ) - 2 ( T + i δ 0 ) A + i α ( A 2 + B 2 ) A + 2 i β 2 A / x 2 ,
2 t R B / t = T 1 / 2 E i ( T + i δ 0 - i α A 2 ) - 2 ( T + i δ 0 ) B + i α ( A 2 + B 2 ) B + 2 i β 2 B / x 2 ,
Λ ( K ) = - 1 + [ ( Δ - η K 2 ) X - ( Δ - η K 2 ) 2 - 3 X 2 / 16 ] 1 / 2 .

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