Abstract

We show that the geometrically induced potential existing in undulated slab waveguides dramatically affects the properties of solitons. In particular, whereas solitons residing in the potential maxima do not feature power thresholds and are stable, their counterparts residing in the potential minima are unstable and may exhibit a power threshold for their existence. Additionally, the geometric potential is shown to support stable multipole solitons that cannot be supported by straight waveguides. Finally, the geometric potential results in the appearance of the effective barriers that prevent transverse soliton motion.

© 2011 Optical Society of America

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  1. H. Jensen and H. Koppe, Ann. Phys. 63, 586 (1971).
    [CrossRef]
  2. R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).
    [CrossRef]
  3. H. Aoki, M. Koshino, D. Takeda, and H. Morise, Phys. Rev. B 65, 035102 (2001).
    [CrossRef]
  4. M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
    [CrossRef] [PubMed]
  5. S. Longhi, Laser Photonics Rev. 3, 243 (2009).
    [CrossRef]
  6. S. Longhi, Opt. Lett. 32, 2647 (2007).
    [CrossRef] [PubMed]
  7. A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
    [CrossRef] [PubMed]
  8. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRS Bull. 31, 620 (2006).
    [CrossRef]
  9. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794(1988).
    [CrossRef] [PubMed]
  10. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
    [CrossRef]
  11. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
    [CrossRef]

2010 (1)

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

2009 (2)

S. Longhi, Laser Photonics Rev. 3, 243 (2009).
[CrossRef]

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

2008 (1)

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

2007 (1)

2006 (1)

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRS Bull. 31, 620 (2006).
[CrossRef]

2003 (1)

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

2001 (1)

H. Aoki, M. Koshino, D. Takeda, and H. Morise, Phys. Rev. B 65, 035102 (2001).
[CrossRef]

1988 (1)

1981 (1)

R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).
[CrossRef]

1971 (1)

H. Jensen and H. Koppe, Ann. Phys. 63, 586 (1971).
[CrossRef]

Aoki, H.

H. Aoki, M. Koshino, D. Takeda, and H. Morise, Phys. Rev. B 65, 035102 (2001).
[CrossRef]

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]

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794(1988).
[CrossRef] [PubMed]

da Costa, R. C. T.

R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).
[CrossRef]

Diner, B. A.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Dreisow, F.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

Heinrich, M.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

Itoh, K.

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRS Bull. 31, 620 (2006).
[CrossRef]

Jagota, A.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Jensen, H.

H. Jensen and H. Koppe, Ann. Phys. 63, 586 (1971).
[CrossRef]

Joseph, R. I.

Kartashov, Y. V.

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

Keil, R.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

Koppe, H.

H. Jensen and H. Koppe, Ann. Phys. 63, 586 (1971).
[CrossRef]

Koshino, M.

H. Aoki, M. Koshino, D. Takeda, and H. Morise, Phys. Rev. B 65, 035102 (2001).
[CrossRef]

Lederer, F.

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

Longhi, S.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

S. Longhi, Laser Photonics Rev. 3, 243 (2009).
[CrossRef]

S. Longhi, Opt. Lett. 32, 2647 (2007).
[CrossRef] [PubMed]

Lustig, S. R.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Mclean, R. S.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Morise, H.

H. Aoki, M. Koshino, D. Takeda, and H. Morise, Phys. Rev. B 65, 035102 (2001).
[CrossRef]

Nolte, S.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRS Bull. 31, 620 (2006).
[CrossRef]

Richardson, R. E.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Schaffer, C.

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRS Bull. 31, 620 (2006).
[CrossRef]

Segev, M.

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

Semke, E. D.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Silberberg, Y.

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

Szameit, A.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

Takeda, D.

H. Aoki, M. Koshino, D. Takeda, and H. Morise, Phys. Rev. B 65, 035102 (2001).
[CrossRef]

Tassi, N. G.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Torner, L.

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

Tünnermann, A.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

Vysloukh, V. A.

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

Watanabe, W.

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRS Bull. 31, 620 (2006).
[CrossRef]

Zheng, M.

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Ann. Phys. (1)

H. Jensen and H. Koppe, Ann. Phys. 63, 586 (1971).
[CrossRef]

Laser Photonics Rev. (1)

S. Longhi, Laser Photonics Rev. 3, 243 (2009).
[CrossRef]

MRS Bull. (1)

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, MRS Bull. 31, 620 (2006).
[CrossRef]

Nat. Mater. (1)

M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. S. Mclean, S. R. Lustig, R. E. Richardson, and N. G. Tassi, Nat. Mater. 2, 338 (2003).
[CrossRef] [PubMed]

Opt. Lett. (2)

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)

R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).
[CrossRef]

Phys. Rev. B (1)

H. Aoki, M. Koshino, D. Takeda, and H. Morise, Phys. Rev. B 65, 035102 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[CrossRef] [PubMed]

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

(a) Example of periodically curved quasi-one-dimensional channel. Profiles of fundamental solitons centered on the points with (b),(c) zero and (d) maximal channel curvature. Profiles of dipole solitons supported by cur ved channel are shown in (e) and (f). In all cases, field modulus distributions are shown, while a = 2 .

Fig. 2
Fig. 2

(a) U versus b for fundamental soliton residing in the point with zero curvature at a = 2 (curve 1) and a = 3 (curve 2). Circle corresponds to soliton in Fig. 1b. (b) U versus b for fundamental soliton residing in the point with maximal curvature (curve 1) and for dipole soliton (curve 2) at a = 2 . Circles correspond to solitons from Figs. 1d, 1e, 1f. (c) δ versus b for fundamental soliton in zero curvature point at a = 2 .

Fig. 3
Fig. 3

(a) Decay of unstable fundamental soliton in zero curvature point, and stable propagation of (b) fundamental soliton residing in the point of maximal curvature and (c) dipole soliton. White noise was added into input field distributions. In all cases, b = 4.95 , a = 2 .

Fig. 4
Fig. 4

Critical input tilt (a) versus bending amplitude for solitons with U = 2.8 and (b) versus soliton’s energy flow at a = 2 . (c) Snapshot images showing motion of soliton with U = 2.8 along curved channel with a = 1.5 for θ > θ cr . Labels “ S inp ” and “ S out ” denote input and output soliton positions, respectively.

Equations (2)

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i q ξ = 1 2 ( 2 q η 2 + 2 q ζ 2 ) p R ( η , ζ ) q q | q | 2 ,
δ u = 1 2 ( 2 v η 2 + 2 v ζ 2 ) + b v p R v w 2 v , δ v = + 1 2 ( 2 u η 2 + 2 u ζ 2 ) b u + p R u + 3 w 2 u ,

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