Abstract

We address light propagation in Vogel optical lattices and show that such lattices support a variety of stable soliton solutions in both self-focusing and self-defocusing media, whose propagation constants belong to domains resembling gaps in the spectrum of a truly periodic lattice. The azimuthally rich structure of Vogel lattices allows generation of spiraling soliton motion.

© 2013 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. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
    [CrossRef]
  4. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
    [CrossRef]
  5. F. Chen, M. Stepic, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, Opt. Express 13, 4314 (2005).
    [CrossRef]
  6. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Express 14, 6055 (2006).
    [CrossRef]
  7. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
    [CrossRef]
  8. J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, Nature 422, 147 (2003).
    [CrossRef]
  9. D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
    [CrossRef]
  10. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
    [CrossRef]
  11. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
    [CrossRef]
  12. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
    [CrossRef]
  13. X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
    [CrossRef]
  14. R. Fischer, D. N. Neshev, S. Lopez-Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, Opt. Express 14, 2825 (2006).
    [CrossRef]
  15. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Opt. Lett. 31, 238 (2006).
    [CrossRef]
  16. A. Ruelas, S. Lopez-Aguayo, and J. C. Gutierrez-Vega, Opt. Lett. 33, 2785 (2008).
    [CrossRef]
  17. F. Ye, D. Mihalache, and B. Hu, Phys. Rev. A 79, 053852 (2009).
    [CrossRef]
  18. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 141 (2008).
    [CrossRef]
  19. F. Diebel, P. Rose, M. Boguslawski, and C. Denz, in Nonlinear Photonics, OSA Technical Digest (online) (Optical Society of America, 2012), paper JM5A.31.
  20. P. Rose, M. Boguslawski, and C. Denz, New J. Phys. 14, 033018 (2012).
    [CrossRef]
  21. A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
    [CrossRef]
  22. Y. V. Kartashov, A. Szameit, R. Keil, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3470 (2011).
    [CrossRef]

2012

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

P. Rose, M. Boguslawski, and C. Denz, New J. Phys. 14, 033018 (2012).
[CrossRef]

2011

2010

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

2009

F. Ye, D. Mihalache, and B. Hu, Phys. Rev. A 79, 053852 (2009).
[CrossRef]

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

2008

2006

2005

2004

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

2003

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
[CrossRef]

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
[CrossRef]

1998

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Assanto, G.

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

Boguslawski, M.

P. Rose, M. Boguslawski, and C. Denz, New J. Phys. 14, 033018 (2012).
[CrossRef]

F. Diebel, P. Rose, M. Boguslawski, and C. Denz, in Nonlinear Photonics, OSA Technical Digest (online) (Optical Society of America, 2012), paper JM5A.31.

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Burghoff, J.

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
[CrossRef]

Chen, F.

Chen, Z.

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Christodoulides, D.

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, Nature 422, 147 (2003).
[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]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
[CrossRef]

Denz, C.

P. Rose, M. Boguslawski, and C. Denz, New J. Phys. 14, 033018 (2012).
[CrossRef]

F. Diebel, P. Rose, M. Boguslawski, and C. Denz, in Nonlinear Photonics, OSA Technical Digest (online) (Optical Society of America, 2012), paper JM5A.31.

Desyatnikov, A. S.

Diebel, F.

F. Diebel, P. Rose, M. Boguslawski, and C. Denz, in Nonlinear Photonics, OSA Technical Digest (online) (Optical Society of America, 2012), paper JM5A.31.

Dreisow, F.

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

Efremidis, N.

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
[CrossRef]

Egorov, A. A.

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Eugenieva, E. D.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Fischer, R.

Fleischer, J.

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
[CrossRef]

Garanovich, I. L.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

Gutierrez-Vega, J. C.

Heinrich, M.

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

Hu, B.

F. Ye, D. Mihalache, and B. Hu, Phys. Rev. A 79, 053852 (2009).
[CrossRef]

Kartashov, Y. V.

Y. V. Kartashov, A. Szameit, R. Keil, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3470 (2011).
[CrossRef]

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

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 141 (2008).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Opt. Lett. 31, 238 (2006).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

Keil, R.

Y. V. Kartashov, A. Szameit, R. Keil, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3470 (2011).
[CrossRef]

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

Kevrekidis, P. G.

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef]

Kip, D.

Kivshar, Y. S.

Krolikowski, W.

Lederer, F.

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

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Express 14, 6055 (2006).
[CrossRef]

Longhi, S.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

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

Lopez-Aguayo, S.

Manela, O.

Martin, H.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Mihalache, D.

F. Ye, D. Mihalache, and B. Hu, Phys. Rev. A 79, 053852 (2009).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Neshev, D.

Neshev, D. N.

Nolte, S.

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

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Express 14, 6055 (2006).
[CrossRef]

Ostrovskaya, E. A.

Pertsch, T.

Rose, P.

P. Rose, M. Boguslawski, and C. Denz, New J. Phys. 14, 033018 (2012).
[CrossRef]

F. Diebel, P. Rose, M. Boguslawski, and C. Denz, in Nonlinear Photonics, OSA Technical Digest (online) (Optical Society of America, 2012), paper JM5A.31.

Ruelas, A.

Runde, D.

Rüter, C. E.

Segev, M.

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

F. Chen, M. Stepic, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, Opt. Express 13, 4314 (2005).
[CrossRef]

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
[CrossRef]

Shandarov, V.

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, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[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]

Stepic, M.

Sukhorukov, A. A.

Szameit, A.

Torner, L.

Y. V. Kartashov, A. Szameit, R. Keil, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3470 (2011).
[CrossRef]

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

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 141 (2008).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Opt. Lett. 31, 238 (2006).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[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]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Express 14, 6055 (2006).
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, A. Szameit, R. Keil, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3470 (2011).
[CrossRef]

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

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 141 (2008).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Opt. Lett. 31, 238 (2006).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

Wang, X.

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef]

Ye, F.

F. Ye, D. Mihalache, and B. Hu, Phys. Rev. A 79, 053852 (2009).
[CrossRef]

Nature

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

New J. Phys.

P. Rose, M. Boguslawski, and C. Denz, New J. Phys. 14, 033018 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

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

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
[CrossRef]

Phys. Rev. A

F. Ye, D. Mihalache, and B. Hu, Phys. Rev. A 79, 053852 (2009).
[CrossRef]

Phys. Rev. Lett.

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

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902(2003).
[CrossRef]

Prog. Opt.

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

Other

F. Diebel, P. Rose, M. Boguslawski, and C. Denz, in Nonlinear Photonics, OSA Technical Digest (online) (Optical Society of America, 2012), paper JM5A.31.

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Figures (4)

Fig. 1.
Fig. 1.

(a)–(c) Dynamics of propagation of a single-site excitation in a linear Vogel lattice with p=8. (d) Example of a Vogel lattice with a1=1.7 and a2=π(351/2). Snapshot images showing spiraling soliton motion (e) outward and (f) inward to the center of a lattice with a1=2.5, a2=0.06π, and p=3. In (e) the soliton with U=2.28 was set into motion by multiplying the input field with exp(iαζ), with α=1.4, while in (f) the soliton with U=1.49 was set in motion by means of an input tilt exp(iαη), where α=0.6. Snapshot images are superimposed on the lattice shape. White arrows indicate the direction of motion.

Fig. 2.
Fig. 2.

Field modulus distributions for solitons supported by a Vogel lattice with depth p=8 in focusing media with (a) b=3.01, (b) b=3.05, and (c) b=3.20, and defocusing media with (d) b=0.30, (e) b=0.75, and (f) b=1.98. Solitons from (a) through (c) correspond to circles in Fig. 3(a), while solitons from (d) through (f) correspond to circles in Fig. 3(b).

Fig. 3.
Fig. 3.

Energy flow versus propagation constant for solitons in (a) focusing and (b) defocusing media when p=8. (c) Domains of soliton existence (white regions), stability (regions marked with “s”), and instability (regions marked with “u”) on the plane (p,b). The domains of stability in the semi-infinite gap are shown for focusing media, while in the first finite gap they are shown for defocusing media.

Fig. 4.
Fig. 4.

(a) Orbital angular momentum and (b) angular velocity of spiraling solitons versus ξ. All parameters are as in Fig. 1(e). The dashed line in (a) corresponds to Lξ=0.

Equations (2)

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

iqξ=12(2qη2+2qζ2)+σq|q|2pR(η,ζ)q.
ddξLξ=2pR(η|q|2/ζζ|q|2/η)dηdζ

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