Abstract

We introduce a model for two coupled waves propagating in a hollow-core fiber: a linear dispersionless core mode and a dispersive nonlinear surface mode. The linear coupling between them may open a bandgap through the mechanism of avoidance of crossing between dispersion curves. The third-order dispersion of the surface mode is necessary for the existence of the gap. Numerical investigation reveals that the entire bandgap is filled with solitons, and they are stable in direct simulations. The gap-soliton (GS) family includes stable pulses moving relative to the given reference frame up to limit values of the corresponding boost δ, beyond which they do not exist. The limit values are asymmetric for δ0. Recently observed solitons in hollow-core photonic crystal fibers may belong to this GS family.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2004 (5)

2003 (2)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

2002 (1)

C. Conti, G. Assanto, and S. Trillo, J. Nonlinear Opt. Phys. Mater. 11, 239 (2002).
[CrossRef]

1999 (1)

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

1998 (1)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 58, 6708 (1998).
[CrossRef]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

1995 (1)

Yu. S. Kivshar, Phys. Rev. E 51, 1613 (1995).
[CrossRef]

1994 (1)

C. M. de Sterke and J. E. Sipe, Prog. Opt. 33, 203 (1994).
[CrossRef]

1993 (2)

1989 (2)

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

1987 (1)

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

1973 (1)

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Aceves, A. B.

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Aitchison, J. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

Allan, D. C.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

Assanto, G.

C. Conti, G. Assanto, and S. Trillo, J. Nonlinear Opt. Phys. Mater. 11, 239 (2002).
[CrossRef]

Benabid, F.

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, Phys. Rev. Lett. 93, 143907 (2004).
[CrossRef]

Biancalana, F.

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, Phys. Rev. Lett. 93, 143907 (2004).
[CrossRef]

Bird, D. M.

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, Phys. Rev. Lett. 93, 143907 (2004).
[CrossRef]

Birks, T. A.

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

Borrelli, N. F.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Botez, D.

Chen, W.

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 58, 6708 (1998).
[CrossRef]

Conti, C.

C. Conti, G. Assanto, and S. Trillo, J. Nonlinear Opt. Phys. Mater. 11, 239 (2002).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, Prog. Opt. 33, 203 (1994).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

Feng, J.

Fink, Y.

Gallagher, M. T.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Gobel, W.

Gregan, R. F.

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

Helmchen, F.

Ibanescu, Mi.

Joannopoulos, J. D.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

Kivshar, Yu. S.

Yu. S. Kivshar, Phys. Rev. E 51, 1613 (1995).
[CrossRef]

Knight, J. C.

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

Koch, K. W.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Koshiba, M.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Lidorikis, E.

Lim, H.

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 58, 6708 (1998).
[CrossRef]

Malomed, B. A.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 58, 6708 (1998).
[CrossRef]

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

Mangan, B. J.

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Morandotti, R.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

Mortensen, N. A.

Müller, D.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Nabiev, R. F.

Nimmerjahn, A.

Roberts, P. J.

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

Russell, P. St. J.

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

Saitoh, K.

Silberberg, Y.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, Prog. Opt. 33, 203 (1994).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, Phys. Rev. Lett. 93, 143907 (2004).
[CrossRef]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Smith, C. M.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Soljacic, M.

Trillo, S.

C. Conti, G. Assanto, and S. Trillo, J. Nonlinear Opt. Phys. Mater. 11, 239 (2002).
[CrossRef]

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Venkataraman, N.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Wabnitz, S.

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

West, J. A.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Wise, F. W.

Yeh, P.

J. Nonlinear Opt. Phys. Mater. (1)

C. Conti, G. Assanto, and S. Trillo, J. Nonlinear Opt. Phys. Mater. 11, 239 (2002).
[CrossRef]

Nature (London) (1)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, Nature (London) 424, 657 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Lett. A (1)

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Phys. Rev. E (2)

Yu. S. Kivshar, Phys. Rev. E 51, 1613 (1995).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 58, 6708 (1998).
[CrossRef]

Phys. Rev. Lett. (5)

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, Phys. Rev. Lett. 93, 143907 (2004).
[CrossRef]

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

Prog. Opt. (1)

C. M. de Sterke and J. E. Sipe, Prog. Opt. 33, 203 (1994).
[CrossRef]

Radiophys. Quantum Electron. (1)

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Science (1)

R. F. Gregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Solid curves, two branches of dispersion relation (3) for c = 1 and γ = 0.3 . The bandgap is located between the branches. Bold dashed curve inside the bandgap shows dependence E ( k ) for the soliton family found in the gap. The family always fills the entire bandgap and satisfies VK stability condition d E d k > 0 .

Fig. 2
Fig. 2

Generic example of the gap soliton (with k = 0 and δ = 0 ) found inside the bandgap shown in Fig. 1. Note that the absolute values of the fields, (a) U ( τ ) and (b) V ( τ ) , always have an ordinary single-humped shape.

Fig. 3
Fig. 3

Stable evolution of the soliton from Fig. 2, to which an arbitrary initial perturbation was added, with a 2% relative amplitude. The evolution of field v ( z , τ ) is similar to that displayed here for u ( z , τ ) .

Fig. 4
Fig. 4

(a), (b) Generic example of the boosted gap soliton with δ = 0.5 , found at c = 0 and γ = 0.3 . Note that for c = 0 the solitons with δ = 0 do not exist [see (c)]. In terms of u and v the boosted solitons always have a single-humped shape. In the reference frame boosted by velocity shift δ the bandgap exists in the interval δ min < δ < δ max c . (c) Dependences of δ min and δ max on c.

Equations (4)

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

i u z i c u τ + ( 1 2 ) u τ τ + i γ u τ τ τ + u 2 u + v = 0 ,
i v z + i c v τ + u = 0 .
4 k 1 , 2 = ω 2 ( 1 + 2 γ ω ) ± [ ω 4 ( 1 + 2 γ ω ) 2 + 8 c ω 2 ( 2 c + ω + 2 γ ω 2 ) + 16 ] 1 2 .
E ( k ) = + [ U ( τ ; k ) 2 + V ( τ ; k ) 2 ] d τ .

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