Abstract

We investigate the modulation instability of multiple four-wave mixing arising from a dual-frequency pump in a single-mode fiber or waveguide. By applying the Floquet theory on account of the periodic nature of four-wave mixing, we reveal a collective type of instability occurring in the anomalous dispersion regime. Our interpretation of the linear stability analysis is validated by the numerical solution of the nonlinear Schrödinger equation.

© 2011 Optical Society of America

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  1. J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
    [Crossref] [PubMed]
  2. D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
    [Crossref] [PubMed]
  3. S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
    [Crossref] [PubMed]
  4. D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
    [Crossref]
  5. V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. G. Someda, and G. Vadalà, Opt. Lett. 35, 1440 (2010).
    [Crossref] [PubMed]
  6. S. Trillo and A. Valiani, Opt. Lett. 35, 3967 (2010).
    [Crossref] [PubMed]
  7. E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
    [Crossref]
  8. C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
    [Crossref]
  9. J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
    [Crossref]
  10. G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [Crossref] [PubMed]
  11. G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
    [Crossref] [PubMed]
  12. J. E. Rothenberg, Phys. Rev. A 42, R682 (1990).
    [Crossref]
  13. G. Millot, S. Pitois, and P. Tchofo-Dinda, J. Opt. Soc. Am. B 19, 454 (2002).
    [Crossref]
  14. J. E. Rothenberg, Phys. Rev. Lett. 64, 813 (1990).
    [Crossref] [PubMed]
  15. S. Trillo and S. Wabnitz, Phys. Rev. E 55, R4897 (1997).
    [Crossref]
  16. S. Trillo and S. Wabnitz, Phys. Rev. E 56, 1048 (1997).
    [Crossref]

2010 (2)

2006 (1)

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[Crossref]

2002 (2)

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[Crossref]

G. Millot, S. Pitois, and P. Tchofo-Dinda, J. Opt. Soc. Am. B 19, 454 (2002).
[Crossref]

2001 (1)

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[Crossref]

1998 (1)

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[Crossref]

1997 (2)

S. Trillo and S. Wabnitz, Phys. Rev. E 55, R4897 (1997).
[Crossref]

S. Trillo and S. Wabnitz, Phys. Rev. E 56, 1048 (1997).
[Crossref]

1994 (2)

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
[Crossref] [PubMed]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[Crossref] [PubMed]

1991 (1)

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[Crossref] [PubMed]

1990 (2)

J. E. Rothenberg, Phys. Rev. A 42, R682 (1990).
[Crossref]

J. E. Rothenberg, Phys. Rev. Lett. 64, 813 (1990).
[Crossref] [PubMed]

1989 (1)

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[Crossref] [PubMed]

1987 (1)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[Crossref] [PubMed]

Agrawal, G. P.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[Crossref] [PubMed]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[Crossref] [PubMed]

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[Crossref] [PubMed]

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[Crossref] [PubMed]

Beletic, J. W.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[Crossref]

Cestier, I.

Chraplyvy, A. R.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[Crossref]

Ciaramella, E.

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[Crossref]

Colman, P.

Combrié, S.

Curti, F.

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[Crossref]

De Rossi, A.

Eckhouse, V.

Eisenstein, G.

Fatome, J.

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[Crossref]

Hart, D. L.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[Crossref]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
[Crossref] [PubMed]

Judy, A. F.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[Crossref]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
[Crossref] [PubMed]

Kennedy, T. A. B.

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
[Crossref] [PubMed]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[Crossref] [PubMed]

McKinstrie, C. J.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[Crossref]

Millot, G.

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[Crossref]

G. Millot, S. Pitois, and P. Tchofo-Dinda, J. Opt. Soc. Am. B 19, 454 (2002).
[Crossref]

Pitois, S.

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[Crossref]

G. Millot, S. Pitois, and P. Tchofo-Dinda, J. Opt. Soc. Am. B 19, 454 (2002).
[Crossref]

Radic, S.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[Crossref]

Rothenberg, J. E.

J. E. Rothenberg, Phys. Rev. A 42, R682 (1990).
[Crossref]

J. E. Rothenberg, Phys. Rev. Lett. 64, 813 (1990).
[Crossref] [PubMed]

Roy, R.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[Crossref]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
[Crossref] [PubMed]

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[Crossref] [PubMed]

Santagiustina, M.

Someda, C. G.

Stoev, K.

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
[Crossref] [PubMed]

Tchofo-Dinda, P.

Thompson, J. R.

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[Crossref] [PubMed]

Trillo, S.

S. Trillo and A. Valiani, Opt. Lett. 35, 3967 (2010).
[Crossref] [PubMed]

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[Crossref]

S. Trillo and S. Wabnitz, Phys. Rev. E 56, 1048 (1997).
[Crossref]

S. Trillo and S. Wabnitz, Phys. Rev. E 55, R4897 (1997).
[Crossref]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[Crossref] [PubMed]

Vadalà, G.

Valiani, A.

Wabnitz, S.

S. Trillo and S. Wabnitz, Phys. Rev. E 55, R4897 (1997).
[Crossref]

S. Trillo and S. Wabnitz, Phys. Rev. E 56, 1048 (1997).
[Crossref]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, IEEE J. Sel. Top. Quantum Electron. 8, 538 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (1)

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[Crossref]

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

Opt. Lett. (2)

Phys. Rev. A (5)

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[Crossref] [PubMed]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, Phys. Rev. A 50, 1807 (1994).
[Crossref] [PubMed]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[Crossref] [PubMed]

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[Crossref] [PubMed]

J. E. Rothenberg, Phys. Rev. A 42, R682 (1990).
[Crossref]

Phys. Rev. E (3)

S. Trillo and S. Wabnitz, Phys. Rev. E 55, R4897 (1997).
[Crossref]

S. Trillo and S. Wabnitz, Phys. Rev. E 56, 1048 (1997).
[Crossref]

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[Crossref]

Phys. Rev. Lett. (2)

J. E. Rothenberg, Phys. Rev. Lett. 64, 813 (1990).
[Crossref] [PubMed]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[Crossref] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

(a) Gain curves from Floquet analysis (solid curve) and IC-NLS (dashed curve) versus MI frequency δ ω , for pump detuning Ω = 1.5 and anomalous GVD ( β = 1 ). (b), (c), (d) Structure of the unstable eigenvector at peak of bands (b) 1, (c) 2, and (d) 3. The dashed red lines locate the FWM modes. The arrows connect the dominant MI sidebands to the A k component they pertain to.

Fig. 2
Fig. 2

(a), (c), and (d) Output spectra from Eq. (1): anomalous GVD, (a)  Ω = 1.5 and (c)  Ω = 4 . In (a), the gain band 1 around pumps are reported in thin solid red curve for comparison; (d) normal GVD, Ω = 1.5 . (b) Periodic evolution ( z p = 0.71 ) of pump power fraction relative to (a).

Fig. 3
Fig. 3

(a) Output spectrum with additional input seed at δ ω = 3.3 along the unstable eigenvector of band 2. (b) Power fraction evolution (in log scale) of seed (triangles), idlers (circles), and peak of spontaneous MI growing from noise around the pump (thick solid curve) and the n = 3 FWM order (thin solid curve). The dashed curves give the predicted growth rates associated with peak gain of bands 1 and 2 in Fig. 1.

Equations (5)

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i u z β 2 2 u t 2 + | u | 2 u = 0 ,
u ( z , t ) = 1 2 [ A p 1 ( z ) exp ( i Ω t ) + A p 2 ( z ) exp ( i Ω t ) + A s 1 ( z ) exp ( i 3 Ω t ) + A s 2 ( z ) exp ( i 3 Ω t ) ] ,
i d A p 1 d z = ( | A p 1 | 2 2 + | A p 2 | 2 + | A s 1 | 2 + | A s 2 | 2 ) A p 1 + + A s 1 A s 2 A p 2 * + A s 1 A p 1 * A p 2 + A p 2 2 A s 2 * 2 + β Ω 2 2 A p 1 , i d A s 1 d z = ( | A p 1 | 2 + | A p 2 | 2 + | A s 1 | 2 2 + | A s 2 | 2 ) A s 1 + A p 1 2 A p 2 * 2 + A p 1 A p 2 A s 2 * + 9 β Ω 2 2 A s 1 ,
A k ( z , t ) = [ η k ( z ) + ε k ( z , t ) ] exp ( i ϕ k ( z ) ) ,
d X d z = M p ( δ ω ) X ,

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