Abstract

We show that the use of a dispersion oscillating fiber in passive cavities significantly extends the modulational instability to novel high-frequency bands, which also destabilizes the branches of the steady response that are stable with homogeneous dispersion. By means of Floquet theory, we obtain the exact explicit expression for the sideband gain, and a simple analytical estimate for the frequencies of the maximum gain. Numerical simulations show that stable stationary trains of pulses can be excited in the cavity.

© 2014 Optical Society of America

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  1. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
    [CrossRef]
  2. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
    [CrossRef]
  3. S. Coen and M. Haelterman, Phys. Rev. Lett. 79, 4139 (1997).
    [CrossRef]
  4. S. Coen and M. Haelterman, Opt. Lett. 26, 39 (2001).
    [CrossRef]
  5. F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
    [CrossRef]
  6. M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
    [CrossRef]
  7. S. Coulibaly, M. Taki, and M. Tlidi, Opt. Express 22, 483 (2014).
    [CrossRef]
  8. S. M. J. Kelly, Electron. Lett. 28, 806 (1992).
    [CrossRef]
  9. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, Opt. Lett. 18, 1499 (1993).
    [CrossRef]
  10. J. Bronski and J. N. Kutz, Opt. Lett. 21, 937 (1996).
    [CrossRef]
  11. N. J. Smith and N. J. Doran, Opt. Lett. 21, 570 (1996).
    [CrossRef]
  12. F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
    [CrossRef]
  13. F. Kh. Abdullaev and J. Garnier, Phys. Rev. E 60, 1042 (1999).
    [CrossRef]
  14. A. Armaroli and F. Biancalana, Opt. Express 20, 25096 (2012).
    [CrossRef]
  15. M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Opt. Lett. 37, 4832 (2012).
    [CrossRef]
  16. M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Phys. Rev. A 87, 013813 (2013).
    [CrossRef]
  17. C. Finot, J. Fatome, A. Sysoliatin, A. Kosolapov, and S. Wabnitz, Opt. Lett. 38, 5361 (2013).
    [CrossRef]
  18. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef]
  19. S. Coen, H. G. Randle, T. Silvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
    [CrossRef]
  20. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).
  21. K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
    [CrossRef]
  22. M. Haelterman, Opt. Lett. 17, 792 (1992).
    [CrossRef]
  23. S. Trillo and S. Wabnitz, Opt. Lett. 16, 986 (1991).
    [CrossRef]
  24. K. Staliunas, C. Hang, and V. V. Konotop, Phys. Rev. A 88, 023846 (2013).
    [CrossRef]

2014 (1)

2013 (6)

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
[CrossRef]

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

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

S. Coen, H. G. Randle, T. Silvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
[CrossRef]

K. Staliunas, C. Hang, and V. V. Konotop, Phys. Rev. A 88, 023846 (2013).
[CrossRef]

2012 (2)

2001 (1)

1999 (1)

F. Kh. Abdullaev and J. Garnier, Phys. Rev. E 60, 1042 (1999).
[CrossRef]

1997 (1)

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

1996 (3)

J. Bronski and J. N. Kutz, Opt. Lett. 21, 937 (1996).
[CrossRef]

N. J. Smith and N. J. Doran, Opt. Lett. 21, 570 (1996).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

1993 (1)

1992 (4)

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

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

S. M. J. Kelly, Electron. Lett. 28, 806 (1992).
[CrossRef]

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

1991 (1)

1987 (1)

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

1982 (1)

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev and J. Garnier, Phys. Rev. E 60, 1042 (1999).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Akimoto, O.

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

Armaroli, A.

Bahloul, L.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Biancalana, F.

Bouwmans, G.

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

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Opt. Lett. 37, 4832 (2012).
[CrossRef]

Bronski, J.

Cherbi, L.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Coen, S.

Coulibaly, S.

S. Coulibaly, M. Taki, and M. Tlidi, Opt. Express 22, 483 (2014).
[CrossRef]

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Doran, N. J.

Droques, M.

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

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Opt. Lett. 37, 4832 (2012).
[CrossRef]

Emplit, P.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Erkintalo, M.

Fatome, J.

Finot, C.

Garnier, J.

F. Kh. Abdullaev and J. Garnier, Phys. Rev. E 60, 1042 (1999).
[CrossRef]

Haelterman, M.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

S. Coen and M. Haelterman, Opt. Lett. 26, 39 (2001).
[CrossRef]

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

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

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

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

Hang, C.

K. Staliunas, C. Hang, and V. V. Konotop, Phys. Rev. A 88, 023846 (2013).
[CrossRef]

Hariz, A.

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Ikeda, K.

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

Kelly, S. M. J.

S. M. J. Kelly, Electron. Lett. 28, 806 (1992).
[CrossRef]

Kobyakov, A.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Kockaert, P.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Konotop, V. V.

K. Staliunas, C. Hang, and V. V. Konotop, Phys. Rev. A 88, 023846 (2013).
[CrossRef]

Kosolapov, A.

Kudlinski, A.

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

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Opt. Lett. 37, 4832 (2012).
[CrossRef]

Kutz, J. N.

Lederer, F.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Lefever, R.

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

Leo, F.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Lugiato, L. A.

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

Martinelli, G.

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

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Opt. Lett. 37, 4832 (2012).
[CrossRef]

Matera, F.

Mecozzi, A.

Mook, D. T.

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).

Mussot, A.

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

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Opt. Lett. 37, 4832 (2012).
[CrossRef]

Nayfeh, A. H.

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).

Randle, H. G.

Romagnoli, M.

Settembre, M.

Silvestre, T.

Smith, N. J.

Staliunas, K.

K. Staliunas, C. Hang, and V. V. Konotop, Phys. Rev. A 88, 023846 (2013).
[CrossRef]

Sysoliatin, A.

Taki, M.

S. Coulibaly, M. Taki, and M. Tlidi, Opt. Express 22, 483 (2014).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Tlidi, M.

S. Coulibaly, M. Taki, and M. Tlidi, Opt. Express 22, 483 (2014).
[CrossRef]

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
[CrossRef]

Trillo, S.

Wabnitz, S.

Electron. Lett. (1)

S. M. J. Kelly, Electron. Lett. 28, 806 (1992).
[CrossRef]

Opt. Commun. (1)

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (10)

Phys. Lett. A (1)

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Phys. Rev. A (3)

M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, Phys. Rev. A 88, 035802 (2013).
[CrossRef]

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

K. Staliunas, C. Hang, and V. V. Konotop, Phys. Rev. A 88, 023846 (2013).
[CrossRef]

Phys. Rev. E (1)

F. Kh. Abdullaev and J. Garnier, Phys. Rev. E 60, 1042 (1999).
[CrossRef]

Phys. Rev. Lett. (4)

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

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]

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

Other (1)

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1979).

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Figures (3)

Fig. 1.
Fig. 1.

Level plot of MI gain in the plane (ω,Pu) with (a), (c) normal and (b), (d) anomalous average GVD. (a), (b) Sinusoidal dispersion β(z)=±1+sin(2πz). (c), (d) Piecewise constant dispersion L1=L2=1/2, β1=±1+0.8, β2=±10.8. Dashed–dotted black curves, MI peak gain frequencies from Eq. (10). The horizontal dashed lines stand for Pu± (Pu<P<Pu+ correspond to the negative slope branch of the bistable response). Number 1 denotes the MI branch of the homogeneous cavity. Parameters: δ=π/5, α=θ2/2=0.05. (e) Sketch of periodic dispersion profiles over one period: sinusoidal (blue curve) and piecewise constant (red curve). Average dispersion βav=(β1+β2)/2, modulation depth βm=(β1β2)/2.

Fig. 2.
Fig. 2.

(a), (b) Intracavity power |u|2 at integer z (round trips) from the numerical solution of the LLE: (a) even round trips, (b) odd round trips. (c) Stationary pulse trains at even (blue curves) and odd (red curves) round trips. The dashed curves are the numerical solution of the map [Eqs. (1) and (2)]. (d) Power |u(z,t0)|2 evaluated at t=t0, corresponding to the maximum of the pulse train. Solid blue curve, LLE; dashed red curve, map. The dots correspond to observable field at the output coupler. Parameters: β(z)=1+sin(2πz/Λ) (average anomalous GVD), Λ=1, δ=π/5, α=θ2/2=0.05, Pu=0.15 (lower branch of steady state), which yield ωmax=2.37 [red cross in Fig. 1(b)] with gain g(ωmax)=0.01.

Fig. 3.
Fig. 3.

(a) Level plot of MI gain in the monostable regime with piecewise constant average normal GVD. Dashed–dotted black curves, MI peak gain frequencies from Eq. (10). (b) Section at Pu=0.15 [horizontal black line in panel (a)], showing the first two MI tongues. (c) Intracavity field power at each round trip z=1,2,, calculated from the numerical solution of the LLE [Eq. (3)]. (d) Intracavity field power evolution evaluated at time t0, corresponding to the maximum of the pulse train. Solid blue curve, LLE; dashed red curve, map. The dots correspond to the observable field at the output coupler. Parameters: β1,2=1±0.9, L1=L2=Λ/2=0.5, δ=0, α=θ2/2=0.05, Pu=0.15.

Equations (10)

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

un+1(z=0,t)=θuin(t)+ρeiδun(z=1,t),
iunzβ(z)22unt2+|un|2un=0,
iuzβ(z)22ut2+|u|2u=(δiα)u+iS,
P=Pu[(Puδ)2+α2],
M=[iΩ2(z)αiu02iu02*iΩ2(z)α],
g(ω)=α+4Puδωδω23Pu2,δω=δβ2ω2,
ωmax=2β(δ2Pu),g(ωmax)=Puα.
λ1,2=Δ2±Δ24W,
Δ=eαL[2cos(k1L1)cos(k2L2)σsin(k1L1)sin(k2L2)],
ωmax=±[2βav(mπΛ)2+Pu2]+2βav(δ2Pu),

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