Abstract

Taking up to fourth-order dispersion effects into account, we show that fiber resonators become stable for a large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for an arbitrary small but nonzero fourth-order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and open the way to experimental implementation.

© 2007 Optical Society of America

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  1. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
    [CrossRef]
  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
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    [CrossRef] [PubMed]
  4. A. M. Turing, Philos. Trans. R. Soc. London Ser. B 237, 37 (1952).
    [CrossRef]
  5. J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
    [CrossRef] [PubMed]
  6. P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75, 2873 (1999).
    [CrossRef]
  7. G. Millot, S. Pitois, P. Tchofo Dinda, and M. Haelterman, Opt. Lett. 22, 1686 (1997).
    [CrossRef]
  8. S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys. Rev. A 43, 6162 (1991).
    [CrossRef] [PubMed]
  9. S. Pitois and G. Millot, Opt. Commun. 226, 415 (2003).
    [CrossRef]
  10. J. D. Harvey, R. Leonhardt, S. Coen, G. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, Opt. Lett. 28, 2225 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. G. Kozyreff, S. J. Chapman, and M. Tlidi, Phys. Rev. E 68, 015201(R) (2003).
    [CrossRef]
  15. G. Kozyreff and M. Tlidi, Phys. Rev. E 69, 066202 (2004).
    [CrossRef]

2004

G. Kozyreff and M. Tlidi, Phys. Rev. E 69, 066202 (2004).
[CrossRef]

2003

1999

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75, 2873 (1999).
[CrossRef]

1997

1993

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

1991

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

1990

J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
[CrossRef] [PubMed]

1987

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

1986

K. Taï, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

1952

A. M. Turing, Philos. Trans. R. Soc. London Ser. B 237, 37 (1952).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

Cavalcanti, S. B.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Chapman, S. J.

G. Kozyreff, S. J. Chapman, and M. Tlidi, Phys. Rev. E 68, 015201(R) (2003).
[CrossRef]

Coen, S.

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

da Cruz, H. R.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Gouveira-Neto, A. S.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Haelterman, M.

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75, 2873 (1999).
[CrossRef]

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

G. Millot, S. Pitois, P. Tchofo Dinda, and M. Haelterman, Opt. Lett. 22, 1686 (1997).
[CrossRef]

Hansen, K.

Harvey, J. D.

Hasegawa, A.

K. Taï, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Knight, J. C.

Kockaert, P.

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75, 2873 (1999).
[CrossRef]

Kozyreff, G.

G. Kozyreff and M. Tlidi, Phys. Rev. E 69, 066202 (2004).
[CrossRef]

G. Kozyreff, S. J. Chapman, and M. Tlidi, Phys. Rev. E 68, 015201(R) (2003).
[CrossRef]

Lefever, R.

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

Leonhardt, R.

Lugiato, L. A.

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

Millot, G.

S. Pitois and G. Millot, Opt. Commun. 226, 415 (2003).
[CrossRef]

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75, 2873 (1999).
[CrossRef]

G. Millot, S. Pitois, P. Tchofo Dinda, and M. Haelterman, Opt. Lett. 22, 1686 (1997).
[CrossRef]

Pitois, S.

S. Pitois and G. Millot, Opt. Commun. 226, 415 (2003).
[CrossRef]

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75, 2873 (1999).
[CrossRef]

G. Millot, S. Pitois, P. Tchofo Dinda, and M. Haelterman, Opt. Lett. 22, 1686 (1997).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
[CrossRef] [PubMed]

Russell, P. St. J.

Taï, K.

K. Taï, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Tchofo Dinda, P.

Tlidi, M.

G. Kozyreff and M. Tlidi, Phys. Rev. E 69, 066202 (2004).
[CrossRef]

G. Kozyreff, S. J. Chapman, and M. Tlidi, Phys. Rev. E 68, 015201(R) (2003).
[CrossRef]

Tomita, A.

K. Taï, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Turing, A. M.

A. M. Turing, Philos. Trans. R. Soc. London Ser. B 237, 37 (1952).
[CrossRef]

Wadsworth, W. J.

Wong, G.

Appl. Phys. Lett.

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75, 2873 (1999).
[CrossRef]

Opt. Commun.

S. Pitois and G. Millot, Opt. Commun. 226, 415 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Philos. Trans. R. Soc. London Ser. B

A. M. Turing, Philos. Trans. R. Soc. London Ser. B 237, 37 (1952).
[CrossRef]

Phys. Rev. A

J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
[CrossRef] [PubMed]

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Phys. Rev. E

G. Kozyreff, S. J. Chapman, and M. Tlidi, Phys. Rev. E 68, 015201(R) (2003).
[CrossRef]

G. Kozyreff and M. Tlidi, Phys. Rev. E 69, 066202 (2004).
[CrossRef]

Phys. Rev. Lett.

K. Taï, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

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

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

Rev. Mod. Phys.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. BS, beam splitter.

Fig. 2
Fig. 2

(a) Marginal stability curve for the steady state solution versus MI. Black curve, β 4 0 ; gray curve, β 4 = 0 . (b) Evolution of the cavity intensity stationary state I = ψ S 2 versus the input intensity P = S 2 (the dashed curve corresponds to the unstable case). γ = 10 W 1 km 1 , Φ 0 = 1.98 π , T = 0.35 , L = 10 m , β 2 = 3 × 10 28 s 2 m , β 3 = 0 , β 4 = 6.4 × 10 54 s 4 m .

Fig. 3
Fig. 3

Evolution of the maximum temporal gains (solid black and gray lines) versus (a) the frequency Ω and (b) the output intensity I.

Fig. 4
Fig. 4

(a) Evolution of the frequency of instability versus the pump power with the same parameters as in Fig. 2, except for β 3 = 2 × 10 42 s 3 m . Circles, numerical simulations; solid curves, analytical results. (b), (c), and (d) power spectra for 30, 400, and 900 mW of pump power, respectively.

Equations (2)

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ψ t = S ( 1 + i Δ ) ψ + i ψ 2 ψ i β 2 2 ψ τ 2 + B 3 3 ψ τ 3 + i B 4 4 ψ τ 4 ,
Ω L , U 2 = β 2 ± β 2 2 + 4 ( Δ 2 ) B 4 2 B 4 ,

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