Abstract

Modulation instability in a passive fiber cavity is revisited. We address the problem in the statement with a continuous-time Ikeda map, rather than in the mean-field limit. It is found that plane wave solutions are unstable for both normal and anomalous dispersion regimes of an optical fiber. The origin of the instability in the continuous-time Ikeda map is in the mode mixing introduced by the beam splitter. The obtained conditions for the instability were compared with ones known for the discrete-time Ikeda map, showing appreciable difference, which, however reduces in the mean-field limit.

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References

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  1. K. Ikeda, Opt. Commun. 30, 257 (1979).
    [CrossRef]
  2. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).
  3. K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709(1980).
    [CrossRef]
  4. M. Haelterman, Opt. Lett. 17, 792 (1992).
    [CrossRef] [PubMed]
  5. S. Coen and M. Haelterman, Phys. Rev. Lett. 79, 4139(1997).
    [CrossRef]
  6. D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
    [CrossRef] [PubMed]
  7. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209(1987).
    [CrossRef] [PubMed]
  8. M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, Opt. Lett. 32, 662 (2007).
    [CrossRef] [PubMed]
  9. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745(1992).
    [CrossRef] [PubMed]
  10. G. Kozyreff, T. Erneux, M. Haelterman, and P. Kockaert, Phys. Rev. A 73, 063815 (2006).
    [CrossRef]
  11. G. Agez, M. G. Clerc, and E. Louvergneaux, Phys. Rev. E 77, 026218 (2008).
    [CrossRef]

2008

G. Agez, M. G. Clerc, and E. Louvergneaux, Phys. Rev. E 77, 026218 (2008).
[CrossRef]

2007

2006

G. Kozyreff, T. Erneux, M. Haelterman, and P. Kockaert, Phys. Rev. A 73, 063815 (2006).
[CrossRef]

1997

S. Coen and M. Haelterman, Phys. Rev. Lett. 79, 4139(1997).
[CrossRef]

1992

1987

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209(1987).
[CrossRef] [PubMed]

1985

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

1980

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

1979

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

Agez, G.

G. Agez, M. G. Clerc, and E. Louvergneaux, Phys. Rev. E 77, 026218 (2008).
[CrossRef]

Akimoto, O.

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

Clerc, M. G.

G. Agez, M. G. Clerc, and E. Louvergneaux, Phys. Rev. E 77, 026218 (2008).
[CrossRef]

Coen, S.

S. Coen and M. Haelterman, Phys. Rev. Lett. 79, 4139(1997).
[CrossRef]

Daido, H.

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

Erneux, T.

G. Kozyreff, T. Erneux, M. Haelterman, and P. Kockaert, Phys. Rev. A 73, 063815 (2006).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).

Haelterman, M.

G. Kozyreff, T. Erneux, M. Haelterman, and P. Kockaert, Phys. Rev. A 73, 063815 (2006).
[CrossRef]

S. Coen and M. Haelterman, Phys. Rev. Lett. 79, 4139(1997).
[CrossRef]

M. Haelterman, Opt. Lett. 17, 792 (1992).
[CrossRef] [PubMed]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745(1992).
[CrossRef] [PubMed]

Ikeda, K.

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

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

Kockaert, P.

G. Kozyreff, T. Erneux, M. Haelterman, and P. Kockaert, Phys. Rev. A 73, 063815 (2006).
[CrossRef]

Kozyreff, G.

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209(1987).
[CrossRef] [PubMed]

Louvergneaux, E.

Lugiato, L. A.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209(1987).
[CrossRef] [PubMed]

McLaughlin, D. W.

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

Moloney, J. V.

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

Mussot, A.

Newell, A. C.

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

Taki, M.

Tlidi, M.

Trillo, S.

Vladimirov, A. G.

Wabnitz, S.

Opt. Commun.

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

Opt. Lett.

Phys. Rev. A

G. Kozyreff, T. Erneux, M. Haelterman, and P. Kockaert, Phys. Rev. A 73, 063815 (2006).
[CrossRef]

Phys. Rev. E

G. Agez, M. G. Clerc, and E. Louvergneaux, Phys. Rev. E 77, 026218 (2008).
[CrossRef]

Phys. Rev. Lett.

S. Coen and M. Haelterman, Phys. Rev. Lett. 79, 4139(1997).
[CrossRef]

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

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209(1987).
[CrossRef] [PubMed]

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

Other

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).

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Figures (1)

Fig. 1
Fig. 1

MI gain G discr and G cont for the discrete-time model [Eqs. (2)] and for the continuous-time one [Eqs. (1)]. The normal dispersion regime is illustrated in panels (a) and (b), while the anomalous one is shown in panels (c) and (d).

Equations (11)

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

A ( 0 , t + t R ) = T E + R e i ϕ A ( 1 , t ) ,
A x = i A t t + i σ | A | 2 A .
A n + 1 ( 0 , t ) = T E + R e i ϕ A n ( 1 , t ) ,
A x n = i A t t n + i σ | A n | 2 A n .
( u ( 0 , t + t R ) v ( 0 , t + t R ) ) = R ( cos ϕ A sin ϕ A sin ϕ A cos ϕ A ) ( u ( 1 , t ) v ( 1 , t ) )
u ˜ ( s , x ) = s 2 ( α 1 ( s ) e λ ( s ) x + α 2 ( s ) e λ ( s ) x ) ,
v ˜ ( s , x ) = λ ( s ) ( α 1 ( s ) e λ ( s ) x α 2 ( s ) e λ ( s ) x ) ,
( u ˜ ( 0 , s ) v ˜ ( 0 , s ) ) = e i s t R R ( cos ϕ A sin ϕ A sin ϕ A cos ϕ A ) ( u ˜ ( 1 , s ) v ˜ ( 1 , s ) ) .
e i s t R = R [ b ( s ) ± b 2 ( s ) 1 ] ,
c 2 s 4 [ 1 2 σ c 2 A 0 2 ] s 2 + 2 i θ s + θ 2 = 0 ,
A m = 1 c 2 ( 0 c 2 0 0 2 σ A 0 2 K 2 2 K c 2 π i m 1 0 0 0 c 2 2 π i m 1 K 2 2 K c ) .

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