Abstract

We consider the propagation of solitons in media with an imprinted transverse periodic or parabolic refractive index modulation when the modulation depth fluctuates slightly along the propagation direction. We find that, under suitable spectral resonance conditions, small transverse soliton oscillations may get parametrically amplified.

© 2009 Optical Society of America

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  1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  2. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
    [CrossRef]
  3. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
    [CrossRef] [PubMed]
  4. M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
    [CrossRef] [PubMed]
  5. K. Staliunas and R. Herrero, Phys. Rev. E 73, 016601 (2006).
    [CrossRef]
  6. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
    [CrossRef] [PubMed]
  7. R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, Opt. Express 15, 3212 (2007).
    [CrossRef] [PubMed]
  8. A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
    [CrossRef]
  9. Y. V. Kartashov, L. Torner, and D. N. Christodoulides, Opt. Lett. 30, 1378 (2005).
    [CrossRef] [PubMed]
  10. C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 31, 1498 (2006).
    [CrossRef] [PubMed]
  11. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 99, 233903 (2007).
    [CrossRef]
  12. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1102 (2004).
    [CrossRef] [PubMed]
  13. F. Kh. Abdullaev, S. A. Darmanyan, and M. R. Djumaev, Phys. Lett. A 141, 423 (1989).
    [CrossRef]
  14. J. J. Garcia Ripoll, V. M. Perez Garcia, and P. Torres, Phys. Rev. Lett. 83, 1715 (1999).
    [CrossRef]

2009 (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
[CrossRef]

2008 (2)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

2007 (2)

2006 (3)

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 31, 1498 (2006).
[CrossRef] [PubMed]

K. Staliunas and R. Herrero, Phys. Rev. E 73, 016601 (2006).
[CrossRef]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

2001 (1)

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

1999 (1)

J. J. Garcia Ripoll, V. M. Perez Garcia, and P. Torres, Phys. Rev. Lett. 83, 1715 (1999).
[CrossRef]

1989 (1)

F. Kh. Abdullaev, S. A. Darmanyan, and M. R. Djumaev, Phys. Lett. A 141, 423 (1989).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, S. A. Darmanyan, and M. R. Djumaev, Phys. Lett. A 141, 423 (1989).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Aitchison, J. S.

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, Opt. Express 15, 3212 (2007).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Y. V. Kartashov, L. Torner, and D. N. Christodoulides, Opt. Lett. 30, 1378 (2005).
[CrossRef] [PubMed]

Cianci, E.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, and M. R. Djumaev, Phys. Lett. A 141, 423 (1989).
[CrossRef]

de Sterke, C. M.

Dignam, M. M.

Djumaev, M. R.

F. Kh. Abdullaev, S. A. Darmanyan, and M. R. Djumaev, Phys. Lett. A 141, 423 (1989).
[CrossRef]

Dreisow, F.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Foglietti, V.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Garanovich, I. L.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 31, 1498 (2006).
[CrossRef] [PubMed]

Garcia Ripoll, J. J.

J. J. Garcia Ripoll, V. M. Perez Garcia, and P. Torres, Phys. Rev. Lett. 83, 1715 (1999).
[CrossRef]

Heinrich, M.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Herrero, R.

K. Staliunas and R. Herrero, Phys. Rev. E 73, 016601 (2006).
[CrossRef]

Iyer, R.

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 99, 233903 (2007).
[CrossRef]

Y. V. Kartashov, L. Torner, and D. N. Christodoulides, Opt. Lett. 30, 1378 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1102 (2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 31, 1498 (2006).
[CrossRef] [PubMed]

Krolikowski, W.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 31, 1498 (2006).
[CrossRef] [PubMed]

Laporta, P.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Lobino, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Longhi, S.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Marangoni, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Minovich, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Mitchell, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Musslimani, Z. H.

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Neshev, D. N.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 31, 1498 (2006).
[CrossRef] [PubMed]

Nolte, S.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Perez Garcia, V. M.

J. J. Garcia Ripoll, V. M. Perez Garcia, and P. Torres, Phys. Rev. Lett. 83, 1715 (1999).
[CrossRef]

Pertsch, T.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Ramponi, R.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Rosberg, C. R.

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Staliunas, K.

K. Staliunas and R. Herrero, Phys. Rev. E 73, 016601 (2006).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Sukhorukov, A. A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 31, 1498 (2006).
[CrossRef] [PubMed]

Szameit, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 99, 233903 (2007).
[CrossRef]

Y. V. Kartashov, L. Torner, and D. N. Christodoulides, Opt. Lett. 30, 1378 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1102 (2004).
[CrossRef] [PubMed]

Torres, P.

J. J. Garcia Ripoll, V. M. Perez Garcia, and P. Torres, Phys. Rev. Lett. 83, 1715 (1999).
[CrossRef]

Tünnermann, A.

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 99, 233903 (2007).
[CrossRef]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1102 (2004).
[CrossRef] [PubMed]

Wan, J.

Opt. Express (1)

Opt. Lett. (3)

Phys. Lett. A (1)

F. Kh. Abdullaev, S. A. Darmanyan, and M. R. Djumaev, Phys. Lett. A 141, 423 (1989).
[CrossRef]

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Phys. Rev. A (1)

A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, Phys. Rev. A 78, 031801 (2008).
[CrossRef]

Phys. Rev. E (1)

K. Staliunas and R. Herrero, Phys. Rev. E 73, 016601 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

J. J. Garcia Ripoll, V. M. Perez Garcia, and P. Torres, Phys. Rev. Lett. 83, 1715 (1999).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 99, 233903 (2007).
[CrossRef]

Prog. Opt. (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Random soliton swinging in parabolic potential for two different samples of colored noise at S = 0.0153 . In (a) random phase is close to ϕ 0 π 2 , while in (b) it is close to ϕ 0 π 2 . (c) Growth of accumulated kinetic energy versus propagation distance for different realizations at S = 0.0510 . In all cases Ω n = 2 Ω 0 , μ = 0.2 , α 0 = 0.02 , and p = 0.2 .

Fig. 2
Fig. 2

Accumulated kinetic energy: (a) in parabolic potential, versus Ω n at S = 0.0153 ; (b) in parabolic potential, versus S at Ω n = 2 Ω 0 ; (c) in harmonic potential, versus Ω n at S = 0.0153 ; (d) in harmonic potential, versus S at Ω n = 1.72 Ω 0 . In all cases μ = 0.2 , α 0 = 0.02 , p = 0.2 , L = 256 , and M = 300 .

Fig. 3
Fig. 3

Histograms of E ( L ) for parabolic potential at (a) S = 0.0077 and (b) S = 0.0153 . In both cases Ω n = 2 Ω 0 , p = 5.2 , μ = 0.2 , and α 0 = 0.02 . (c) Histogram for the case of monochromatic noise ( S 0 ) with fluctuating initial phase ϕ 0 uniformly distributed on the segment [ π , π ] .

Fig. 4
Fig. 4

Random swinging under the influence of broadband white noise at Ω n = 0 , p = 0.2 . Histograms of E ( L ) in parabolic potential at (a) μ = 0.2 and (b) μ = 0.25 . (c) Accumulated kinetic energy K ( L ) versus white-noise bandwidth S for harmonic potential (curve 1) and for parabolic potential (curve 2) at μ = 0.2 , M = 300 .

Equations (1)

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i q ξ = 1 2 2 q η 2 q | q | 2 p Q ( ξ ) R ( η ) q .

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