Abstract

We study the modulational instability induced by periodic variations of group-velocity dispersion in the proximity of the zero dispersion point. Multiple instability peaks originating from parametric resonance coexist with the conventional modulation instability because of fourth-order dispersion, which in turn is suppressed by the oscillations of dispersion. Moreover, isolated unstable regions appear in the space of parameters because of imperfect phase matching. This confirms the dramatic effect of periodic tapering in the control and shaping of MI sidebands in optical fibers.

© 2014 Optical Society of America

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References

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  1. V. I. Arnold, A. Weinstein, and K. Vogtmann, Mathematical Methods of Classical Mechanics (Springer, 1989).
  2. L. D. Landau and E. Lifshitz, Course of Theoretical Physics, 3rd. ed. (Butterworth-Heinemann, 1976), Vol. 1.
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  9. K. Staliunas, S. Longhi, and G. J. de Valcàrcel, Phys. Rev. Lett. 89, 210406 (2002).
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  10. F. K. Abdullaev, M. Ögren, and M. P. Sørensen, Phys. Rev. A 87, 023616 (2013).
    [CrossRef]
  11. A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
    [CrossRef]
  12. M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Opt. Lett. 37, 4832 (2012).
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    [CrossRef]
  15. M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, A. Mussot, A. Armaroli, and F. Biancalana, Opt. Lett. 38, 3464 (2013).
    [CrossRef]
  16. C. Finot, J. Fatome, A. Sysoliatin, A. Kosolapov, and S. Wabnitz, Opt. Lett. 38, 5361 (2013).
    [CrossRef]
  17. C. Finot, F. Feng, Y. Chembo, and S. Wabnitz, Opt. Fiber Technol. Available online 10July2014. http://hal.archives-ouvertes.fr/hal-00981213 .
  18. A. Armaroli and F. Biancalana, Opt. Express 20, 25096 (2012).
    [CrossRef]
  19. V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, 1988).
  20. J. A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems (Springer, 2010).
  21. A. Armaroli and F. Biancalana, Phys. Rev. A 87, 063848 (2013).
    [CrossRef]

2014 (1)

A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
[CrossRef]

2013 (5)

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Phys. Rev. A 87, 013813 (2013).
[CrossRef]

A. Armaroli and F. Biancalana, Phys. Rev. A 87, 063848 (2013).
[CrossRef]

F. K. Abdullaev, M. Ögren, and M. P. Sørensen, Phys. Rev. A 87, 023616 (2013).
[CrossRef]

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, A. Mussot, A. Armaroli, and F. Biancalana, Opt. Lett. 38, 3464 (2013).
[CrossRef]

C. Finot, J. Fatome, A. Sysoliatin, A. Kosolapov, and S. Wabnitz, Opt. Lett. 38, 5361 (2013).
[CrossRef]

2012 (2)

2008 (1)

2003 (2)

P. St. J. Russell, Science 299, 358 (2003).
[CrossRef]

A. Kumar, A. Labruyere, and P. Tchofo-Dinda, Opt. Commun. 219, 221 (2003).
[CrossRef]

2002 (1)

K. Staliunas, S. Longhi, and G. J. de Valcàrcel, Phys. Rev. Lett. 89, 210406 (2002).
[CrossRef]

1996 (2)

1993 (1)

1967 (1)

V. I. Karpman and E. M. Krushkal, Pis’ma v Zh. Eksp. Teor. Fiz. (JETP Letters) 6, 277 (1967).

Abdullaev, F. K.

F. K. Abdullaev, M. Ögren, and M. P. Sørensen, Phys. Rev. A 87, 023616 (2013).
[CrossRef]

Ambomo, S.

Armaroli, A.

Arnold, V.

V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, 1988).

Arnold, V. I.

V. I. Arnold, A. Weinstein, and K. Vogtmann, Mathematical Methods of Classical Mechanics (Springer, 1989).

Balasubramanian, S.

A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
[CrossRef]

Balaž, A.

A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
[CrossRef]

Biancalana, F.

Bouwmans, G.

Bronski, J. C.

de Valcàrcel, G. J.

K. Staliunas, S. Longhi, and G. J. de Valcàrcel, Phys. Rev. Lett. 89, 210406 (2002).
[CrossRef]

Doran, N.

Droques, M.

Fatome, J.

Finot, C.

Karpman, V. I.

V. I. Karpman and E. M. Krushkal, Pis’ma v Zh. Eksp. Teor. Fiz. (JETP Letters) 6, 277 (1967).

Kosolapov, A.

Krushkal, E. M.

V. I. Karpman and E. M. Krushkal, Pis’ma v Zh. Eksp. Teor. Fiz. (JETP Letters) 6, 277 (1967).

Kudlinski, A.

Kumar, A.

A. Kumar, A. Labruyere, and P. Tchofo-Dinda, Opt. Commun. 219, 221 (2003).
[CrossRef]

Labruyere, A.

A. Kumar, A. Labruyere, and P. Tchofo-Dinda, Opt. Commun. 219, 221 (2003).
[CrossRef]

Landau, L. D.

L. D. Landau and E. Lifshitz, Course of Theoretical Physics, 3rd. ed. (Butterworth-Heinemann, 1976), Vol. 1.

Lifshitz, E.

L. D. Landau and E. Lifshitz, Course of Theoretical Physics, 3rd. ed. (Butterworth-Heinemann, 1976), Vol. 1.

Longhi, S.

K. Staliunas, S. Longhi, and G. J. de Valcàrcel, Phys. Rev. Lett. 89, 210406 (2002).
[CrossRef]

Martinelli, G.

Matera, F.

Mecozzi, A.

Murdock, J.

J. A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems (Springer, 2010).

Mussot, A.

Nathan Kutz, J.

Ngabireng, C. M.

Nicolin, A. I.

A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
[CrossRef]

Ögren, M.

F. K. Abdullaev, M. Ögren, and M. P. Sørensen, Phys. Rev. A 87, 023616 (2013).
[CrossRef]

Paun, R.

A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
[CrossRef]

Ramaswamy, R.

A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
[CrossRef]

Romagnoli, M.

Sanders, J. A.

J. A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems (Springer, 2010).

Settembre, M.

Smith, N. J.

Sørensen, M. P.

F. K. Abdullaev, M. Ögren, and M. P. Sørensen, Phys. Rev. A 87, 023616 (2013).
[CrossRef]

St. J. Russell, P.

P. St. J. Russell, Science 299, 358 (2003).
[CrossRef]

Staliunas, K.

K. Staliunas, S. Longhi, and G. J. de Valcàrcel, Phys. Rev. Lett. 89, 210406 (2002).
[CrossRef]

Sysoliatin, A.

Tchofo-Dinda, P.

S. Ambomo, C. M. Ngabireng, and P. Tchofo-Dinda, J. Opt. Soc. Am. B 25, 425 (2008).
[CrossRef]

A. Kumar, A. Labruyere, and P. Tchofo-Dinda, Opt. Commun. 219, 221 (2003).
[CrossRef]

Verhulst, F.

J. A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems (Springer, 2010).

Vogtmann, K.

V. I. Arnold, A. Weinstein, and K. Vogtmann, Mathematical Methods of Classical Mechanics (Springer, 1989).

Wabnitz, S.

Weinstein, A.

V. I. Arnold, A. Weinstein, and K. Vogtmann, Mathematical Methods of Classical Mechanics (Springer, 1989).

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

Opt. Commun. (1)

A. Kumar, A. Labruyere, and P. Tchofo-Dinda, Opt. Commun. 219, 221 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. A (4)

A. Armaroli and F. Biancalana, Phys. Rev. A 87, 063848 (2013).
[CrossRef]

F. K. Abdullaev, M. Ögren, and M. P. Sørensen, Phys. Rev. A 87, 023616 (2013).
[CrossRef]

A. Balaž, R. Paun, A. I. Nicolin, S. Balasubramanian, and R. Ramaswamy, Phys. Rev. A 89, 023609 (2014).
[CrossRef]

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Phys. Rev. A 87, 013813 (2013).
[CrossRef]

Phys. Rev. Lett. (1)

K. Staliunas, S. Longhi, and G. J. de Valcàrcel, Phys. Rev. Lett. 89, 210406 (2002).
[CrossRef]

Pis’ma v Zh. Eksp. Teor. Fiz. (JETP Letters) (1)

V. I. Karpman and E. M. Krushkal, Pis’ma v Zh. Eksp. Teor. Fiz. (JETP Letters) 6, 277 (1967).

Science (1)

P. St. J. Russell, Science 299, 358 (2003).
[CrossRef]

Other (5)

V. I. Arnold, A. Weinstein, and K. Vogtmann, Mathematical Methods of Classical Mechanics (Springer, 1989).

L. D. Landau and E. Lifshitz, Course of Theoretical Physics, 3rd. ed. (Butterworth-Heinemann, 1976), Vol. 1.

C. Finot, F. Feng, Y. Chembo, and S. Wabnitz, Opt. Fiber Technol. Available online 10July2014. http://hal.archives-ouvertes.fr/hal-00981213 .

V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, 1988).

J. A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems (Springer, 2010).

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Figures (4)

Fig. 1.
Fig. 1.

Position of instability peaks as a function of FOD β4 for first (black crosses), second (red circles), and third (green plus) order PR. For comparison, the corresponding conventional MI (blue dashed line) is also reported. The arrow shows the approximate region where the instability occurs despite the fact that the resonance condition (or phase matching) cannot be perfectly achieved.

Fig. 2.
Fig. 2.

(a) Resonance tongues for first order PR, with Λ=10 and β4=0.25. The color scale corresponds to the instability gain. The dotted blue lines denote the predicted positions of PR peaks, while the green dash-dotted lines denote the predicted instability margins. The corresponding maximum gain is shown in the inset (solid lines) as a function of the perturbation strength h and is compared with the analytical predictions (dashed lines). The arrows connect each gain curve with the corresponding resonance tongue. (b) Output of split-step Fourier simulation of Eq. (1) at z14, for h=0.5 (corresponding to the dashed horizontal line in panel (a).

Fig. 3.
Fig. 3.

Same as in Fig. 2, but with β4=0.38 and a larger h range. Despite the fact that phase matching of low detuning sidebands is not achieved, we still observe unstable regions for ω4 (a). The simulations in (b) were performed for h=2.5.

Fig. 4.
Fig. 4.

Same as in Fig. 3, but with β4=0.42. Despite the fact that phase matching of low detuning sidebands is not achieved, we still observe an unstable spot in the parameter space for ω4 and h1.5 (a). The simulations in (b) were performed at that value.

Equations (11)

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izA12β2(z)t2A+124β4t4A+γ(z)|A|2A=0.
iza12β2(z)t2a+124β4t4a+γ(z)P0(a+a*)=0.
i|ψ˙=Hs(z)|ψ,
Hs(z)(β2(z)ω22β4ω424γ(z)P0)σ^ziγ(z)P0σ^y,
β2(z)=β0+β˜(z)=β0+hβ1cosΛz,γ(z)=γ0+γ˜(z)=γ0+hγ1cosΛz,
i|ϕ˙=Hpq(z)|ϕ
Hpq=(0c1(z)c2(z)0)
c10c20=[mΛ2]2,
i|ϕ˙I=hHI(z)|ϕI,withHI=eiH0zH˜(z)eiH0z.
κ=λ0(c˜2c20c˜1c10)
gpQPM=[(γ0P0Jp(hω2Λ))2Δkp2]12,

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