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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 4 — Feb. 19, 2007
  • pp: 1706–1711
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Surface solitons supported by Bessel optical potential

Liangwei Dong and Hui Wang  »View Author Affiliations


Optics Express, Vol. 15, Issue 4, pp. 1706-1711 (2007)
http://dx.doi.org/10.1364/OE.15.001706


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Abstract

A simple but new model, an interface separated by 1-D Bessel optical potential with different modulation depth on the opposite side of the interface was proposed. We show this model can support stable surface solitons. Surface solitons form because diffraction and defocusing nonlinearity are balanced by the Bessel potential. We demonstrate numerically such solitons are stable in their entire existence domain even for higher-order Bessel potential.

© 2007 Optical Society of America

1. Introduction

Recently, a new implementation of surface solitary waves was proposed [5

5. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30,2466 (2005). [CrossRef] [PubMed]

] and experimentally created[6

6. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of Discrete Surface Solitons,” Phys. Rev. Lett. 96, 063,901 (2006). [CrossRef]

] in nonlinear optics. Meanwhile, surface gap solitons were predicted[7

7. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface Gap Solitons,” Phys. Rev. Lett. 96, 073,901 (2006). [CrossRef]

] and created in an experiment[8

8. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. 97, 083,901 (2006). [CrossRef]

] at an edge of a waveguiding array built into a self-defocusing continuous medium. Siviloglou et al. reported the experimental creation of discrete surface solitons supported by the quadratic nonlinearity[9

9. G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Observation of discrete quadratic surface solitons,” Opt. Express 14,5508 (2006). [CrossRef] [PubMed]

]. Multipole-mode surface solitons were shown to exist at the interface between two distinct periodic lattices imprinted in Kerr-type nonlinear media[10

10. Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31,2172 (2006). [CrossRef] [PubMed]

]. Polychromatic interface solitons in nonlinear photonic lattices were predicted[11

11. K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett. 31,3125 (2006). [CrossRef] [PubMed]

]. Moreover, two-dimensional vortex soliton can be captured stably by a surface between two optical lattice with different lattice parameters[12

12. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Surface vortex solitons,” .Opt. Express 14,4049 (2006). [CrossRef] [PubMed]

].

Recent studies show that solitons in Bessel optical lattice exhibit some unique features that cannot be found on other schemes. Non-diffracting Bessel beams of different orders can be created experimentally by illuminating through a narrow annular split placed in the focal plane of a lens or axicon[13

13. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177,297 (2000). [CrossRef]

] or by holographic techniques[14

14. N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill, and R. Roy, “Generation of nondiffracting Bessel beams by use of a spatial light modulator,” Opt. Lett. 28,2183 (2003). [CrossRef] [PubMed]

]. Bessel optical lattices have been shown to support stable vortex soliton[15

15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043,902 (2005). [CrossRef]

], spiraling solitons[16

16. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton spiraling in optically induced rotating Bessel lattices,” Opt. Lett. 30,637 (2005). [CrossRef] [PubMed]

] etc. The experimental observation of self-trapping and nonlinear localization of light in modulated Bessel lattices in the form of ring-shaped and single-site states was reported by Fischer et al.[17

17. R. Fischer, D. N. Neshev, S. Lopez-Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of light localization in modulated Bessel optical lattices,” Opt. Express 14,2825 (2006). [CrossRef] [PubMed]

]. Discrete solitons and soliton rotation in Bessel-like photonic lattices were also obsevered[18

18. X. Wang, Z. Chen, and P. G. Kevrekidis, “Observation of Discrete Solitons and Soliton Rotation in Optically Induced Periodic Ring Lattices,” Phys. Rev. Lett. 96, 083,904 (2006).

]. Furthermore, vortex solitons with topological charge larger than unit remain robust in higher-order Bessel lattice under appropriate conditions[19

19. Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, “Soliton Topology versus Discrete Symmetry in Optical Lattices,” Phys. Rev. Lett. 95, 123,902 (2005). [CrossRef]

] and spatiotemporal solitons were predicted to exist stably in Bessel optical lattices[20

20. D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. Kartashov, L.-C. Crasovan, and L. Torner, “Stable Spatiotemporal Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 95, 023,902 (2005). [CrossRef]

].

However, all studies on Bessel lattice soliton in nonlinear optics deal with the two-dimensional problems. So far, surface solitons are only observed in waveguide arrays[6

6. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of Discrete Surface Solitons,” Phys. Rev. Lett. 96, 063,901 (2006). [CrossRef]

, 8

8. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. 97, 083,901 (2006). [CrossRef]

]. Thus, exploring other schemes for implementing surface solitons and investigating the dynamics of such solitons are of important significance. In the present paper, we put forward another way (in an optical modulation slab waveguide) for the realization of surface solitons. An interface separated by 1D Bessel potential with varying potential modulation depth beside the interface is presented. It can be realized easily in experiment by illustrating a Bessel beam with its polarization orthogonal to that of guiding beams into a slab waveguide. The depth of refractive-index modulation beside the center of waveguide can be adjusted by introducing an attenuating plate before the input Bessel beam. Dynamics of solitons supported by this model is discussed in detailed.

2. Model

Following Refs. [7

7. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface Gap Solitons,” Phys. Rev. Lett. 96, 073,901 (2006). [CrossRef]

, 15

15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043,902 (2005). [CrossRef]

], we assume that an optical beam propagates along the z-axis in a slab waveguide with defocusing cubic nonlinearity and imprinted transverse refractive index modulation. The generic equation describing the evolution of optical wave packets in this setting is the nonlinear Schrödinger equation:

iAz+122Ax2A2A+R(x)A=0
(1)

where x = X/X 0, z = Z/Ld are the normalized transverse coordinate and propagation distance respectively, X 0 is an arbitrary scale, Ld = k 0 X 2 0 is the diffraction length, corresponding to X 0, k 0 is the wave number, A(x,z) is the dimensionless amplitude of the beam. The function R(x)=prJn2[2βx] at x > 0 and R(x)=plJn2[2β(x)] at x < 0 represents the transverse refractive-index profile, where pl(pr) describes the depth of the refractive-index modulation at the left (right) hand of the interface located at x = 0, β is the radial scale of the n-order Bessel function. Eq. (1) conserves several quantities, including the power P and Hamiltonian H:

P=A2dxH=12[dAdx22R(x)A2+A4]dx
(2)

d2qdx22bq2q3+2Rq=0
(3)

In numerical calculations, we fix the radial scale, β ≡ 2, and vary pl, pr and the propagation constant b. The stability of the soliton can be analyzed by considering the perturbed stationary solution form as A(x,z) = q(x)exp(ibz) + u(x)exp [i(b+λ)z] λ v(x)exp [i(b-λ)z], where the perturbation components u,v could grow with a complex rate λ during propagation. The soliton is stable if the imaginary parts of eigenvalues (λ) equal zero. Substitute the perturbed solution into Eq. (1), we obtained the linear eigen-equations,

Fig. 1. Properties of (surface) soliton with a 0-order Bessel potential. (a) Power of surface soliton with pl = 15 and different pr versus b. (b) Profiles of ground-state soliton with pl = pr = 15. (c) Surface soliton profiles with pl = 25 and pr = 10. (d) Typical bcutoff curve versus pl for different pr.
Fig. 2. Evolution of the solitons with (a) pl = pr = 15,b = 0.2. (b)pl = -23, pr = 16,b = 1.1. (c) pl = 8, pr = 47,b = 2.6. The white noise with variance σ2 noise = 0.01 was added into the initial input.
λu=12d2udx2q2(2u+v)+Rubu
λv=12d2vdx2+q2(2u+v)Rv+bv
(4)

which can be solved numerically.

3. Discussion

First we consider the properties of surface solitons with 0–order Bessel modulation potential [Fig. 1]. The power of such solitons decreases monotonically with the increment of propagation constant [Fig. 1(a)] and vanishes at the upper cutoff bco which determines the existence region of solitons. The soliton with a symmetrical profile corresponds to the ground-state soliton if pl = pr [Fig. 1(b)]. The solitons reside mainly in the center of 0–order Bessel potential if the value of b is large. They will extend and cover several nodes of Bessel function with the decreasing of propagation constant. On the other hand, the soliton profiles are asymmetric when plpr [Fig. 1(c)]. It is the interface that supports such modes in the present scheme, that is, either component of surface soliton beside the interface cannot exist if the other component varnishes or is deformed seriously. We show in Fig. 1(d) those stationary solutions only exist on the upper side of bcutoff curves. The existence region of surface solitons broadens with growth of the modulation parameter pl(pr). The bcutoff curves with different pr converges to a collectively asymptotic line when the value of pl increases. Another very interesting result can be inferred from Fig. 1(d) is that there are families of stationary solutions which are proved to be stable when pl × pr < 0 [e.g. Fig. 2(b)].

Fig. 3. Properties of surfaces soliton with a 1–order Bessel potential. (a) Power of surface soliton with pl = 15 and different pl versus b. (b) Typical bcutoff curves with pl for different pr.

We applied the linear stability analysis [Eq. (4)] on such solutions and found that surface solitons are stable in the entire domain of their existence, i.e., all imaginary parts of eigenvalues (λ) equal zero. To confirm the instability analysis results, we perform extensive numerical simulations using Eq. (1) with the perturbed initial input A(x,z = 0) = q(x)[1 +ρ(x)], where ρ(x) is the white noise with variance σ2 noice. Some examples of evolution of the perturbed solitons are shown in Fig. 2.

For 1–order Bessel potential, the power of surface solitons also decreases monotonically with the growth of propagation constant [Fig. 3(a)]. The existence domain of soliton is smaller than that of 0–order Bessel potential with same parameters pl,pr [Fig. 3(b)]. Another important result is that the bcutoff is determined by the value of max (pl,pr). The bcutoff remains the same when pl < pr but increases almost linearly with b when pr > pl. Some typical soliton profiles are laid out in Fig. 4. One can see clearly the influences of the different potential beside the interface on soliton profiles. Similar to n = 0 cases, the profile of soliton is symmetrical with pl = pr but asymmetrical with different pl and pr. There is an intensity dip on the top of ground-state solitons since the Bessel potential now has a local minimum at x = 0 [Fig. 4(a). The dip is almost invisible if the ratio of pl and pr is relatively large. The left component of soliton profile steepens if pl decreases which even holds for pl < 0 [Fig. 4(b) and 4(c)]. The energy distribution of soliton beside the interface is proportional to the ratio between pl and pr. This can be explained in physics as that the deeper the refractive-index is modulated, the more energy is trapped. We performed again the instability analysis on the stationary solutions with 1–order Bessel potential and found similar results with the case of 0–order Bessel potential, that is, the solitons are stable in their existence domain. Numerical simulations of some examples are shown in Fig. 5. Note that the soliton is stable with different sign of pl and pr [Fig. 5(b) and 5(c)].

Fig. 4. Profiles of surface soliton with 1-order Bessel potential. (a) pl = pr = 15,b = 1.6, (b) pl = -5, pr = 25,b = 0.8, (c) pl = -5, pr = 2,b = 0.05, (d) pl = 2, pr = 35,b = 0.5
Fig. 5. Evolution of the solitons shown in Fig. 4. The white noise with variance σ2 noise = 0.01 was added into the initial input.

To explore the effect of Bessel potential on the stabilization of solitons further, we considered the higher-order Bessel potential (n ≥ 2) and found they also support the stable surface solitons. The properties of these solitons are similar to that of n = 0,1. The only difference is that their existence domain decreases when the n value increases for the same parameters pl and pr. Some examples of stable soltions and their propagation simulations are shown in Fig. 6. Note that for n = 5, surface soliton can still propagate stably. We perform instability analysis and propagation simulations for even higher-order Bessel potential(e.g. n = 15) and prove that the stationary solution is stable in its existence domain.

Fig. 6. Profiles and evolution of the solitons with higher-order Bessel potential. (a,d) pl = pr = 20,b = 0.2, (b,e) pl = 10, pr = 20,b = 0.5 with n = 2. (c,f) pl = 100, pr = 250,b = 1.8 with n = 5. The white noise with σ2 noise = 0.01 was added into the initial input in (d,e,f).

4. Conclusions

In conclusion, we present a new but simple model, an interface separated by different 1–D Bessel potential with a defocusing cubic nonlinearity. Numerical results show that this model supports stable surface solitons. Surface solitons with different-order Bessel potential are shown to be stable in the entire domain of their existence. Our theoretic results enrich the concept of surface soliton and may be generalized to the case of Bose-Einstein condensates trapped in 1D Bessel potential with repulsive interatomic interactions. The model could be realized in experiment based on experimental scheme presented in reference[15

15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043,902 (2005). [CrossRef]

].

Acknowledgments

The authors are indebted to Dr. Fangwei Ye and Yongjin Jiang for useful discussions. The work is supported partly by the National Natural Science Foundation of China (Grant No. 10575087)

References and links

1.

L. Stenflo, “Theory of nonlinear plasma surface waves,” Physica Scripta T63,59 (1996). [CrossRef]

2.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and nonlinear wave equations (Academic Press, London., 1982).

3.

G. Maugin, Nonlinear Waves in Elastic rystals (Oxford University Press: Oxford, 2000).

4.

I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Surface multi-gap vector solitons,” Opt. Express 14,4780 (2006). [CrossRef] [PubMed]

5.

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30,2466 (2005). [CrossRef] [PubMed]

6.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of Discrete Surface Solitons,” Phys. Rev. Lett. 96, 063,901 (2006). [CrossRef]

7.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface Gap Solitons,” Phys. Rev. Lett. 96, 073,901 (2006). [CrossRef]

8.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. 97, 083,901 (2006). [CrossRef]

9.

G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Observation of discrete quadratic surface solitons,” Opt. Express 14,5508 (2006). [CrossRef] [PubMed]

10.

Y. V. Kartashov and L. Torner, “Multipole-mode surface solitons,” Opt. Lett. 31,2172 (2006). [CrossRef] [PubMed]

11.

K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, “Polychromatic interface solitons in nonlinear photonic lattices,” Opt. Lett. 31,3125 (2006). [CrossRef] [PubMed]

12.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Surface vortex solitons,” .Opt. Express 14,4049 (2006). [CrossRef] [PubMed]

13.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177,297 (2000). [CrossRef]

14.

N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill, and R. Roy, “Generation of nondiffracting Bessel beams by use of a spatial light modulator,” Opt. Lett. 28,2183 (2003). [CrossRef] [PubMed]

15.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043,902 (2005). [CrossRef]

16.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton spiraling in optically induced rotating Bessel lattices,” Opt. Lett. 30,637 (2005). [CrossRef] [PubMed]

17.

R. Fischer, D. N. Neshev, S. Lopez-Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of light localization in modulated Bessel optical lattices,” Opt. Express 14,2825 (2006). [CrossRef] [PubMed]

18.

X. Wang, Z. Chen, and P. G. Kevrekidis, “Observation of Discrete Solitons and Soliton Rotation in Optically Induced Periodic Ring Lattices,” Phys. Rev. Lett. 96, 083,904 (2006).

19.

Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, “Soliton Topology versus Discrete Symmetry in Optical Lattices,” Phys. Rev. Lett. 95, 123,902 (2005). [CrossRef]

20.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. Kartashov, L.-C. Crasovan, and L. Torner, “Stable Spatiotemporal Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 95, 023,902 (2005). [CrossRef]

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 18, 2006
Revised Manuscript: January 18, 2007
Manuscript Accepted: January 24, 2007
Published: February 19, 2007

Citation
Liangwei Dong and Hui Wang, "Surface solitons supported by Bessel optical potential," Opt. Express 15, 1706-1711 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1706


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References

  1. L. Stenflo, "Theory of nonlinear plasma surface waves," Physica Scripta T63, 59 (1996). [CrossRef]
  2. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and nonlinear wave equations (Academic Press, London, 1982).
  3. G. Maugin, Nonlinear Waves in Elastic Crystals (Oxford University Press: Oxford, 2000).
  4. I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, "Surface multi-gap vector solitons," Opt. Express 14, 4780 (2006). [CrossRef] [PubMed]
  5. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, "Discrete Surface Solitons," Opt. Lett. 30, 2466 (2005). [CrossRef] [PubMed]
  6. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of Discrete Surface Solitons," Phys. Rev. Lett. 96, 063,901 (2006). [CrossRef]
  7. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Surface Gap Solitons," Phys. Rev. Lett. 96, 073,901 (2006). [CrossRef]
  8. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite waveguide arrays," Phys. Rev. Lett. 97, 083,901 (2006). [CrossRef]
  9. G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Observation of discrete quadratic surface solitons," Opt. Express 14, 5508 (2006). [CrossRef] [PubMed]
  10. Y. V. Kartashov and L. Torner, "Multipole-mode surface solitons," Opt. Lett. 31, 2172 (2006). [CrossRef] [PubMed]
  11. K. Motzek, A. A. Sukhorukov, and Y. S. Kivshar, "Polychromatic interface solitons in nonlinear photonic lattices," Opt. Lett. 31, 3125 (2006). [CrossRef] [PubMed]
  12. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Surface vortex solitons," Opt. Express 14, 4049 (2006). [CrossRef] [PubMed]
  13. J. Arlt and K. Dholakia, "Generation of high-order Bessel beams by use of an axicon," Opt. Commun. 177, 297 (2000). [CrossRef]
  14. N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill, and R. Roy, "Generation of non diffracting Bessel beams by use of a spatial light modulator," Opt. Lett. 28, 2183 (2003). [CrossRef] [PubMed]
  15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stable ring-profile Vortex Solitons in Bessel Optical Lattices," Phys. Rev. Lett. 94, 043,902 (2005). [CrossRef]
  16. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Soliton spiraling in optically induced rotating Bessel lattices," Opt. Lett. 30, 637 (2005). [CrossRef] [PubMed]
  17. R. Fischer, D. N. Neshev, S. Lopez-Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, "Observation of light localization in modulated Bessel optical lattices," Opt. Express 14, 2825 (2006). [CrossRef] [PubMed]
  18. X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of Discrete Solitons and Soliton Rotation in Optically induced Periodic Ring Lattices," Phys. Rev. Lett. 96, 083,904 (2006).
  19. Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, "Soliton Topology versus Discrete Symmetry in Optical Lattices," Phys. Rev. Lett. 95, 123,902 (2005). [CrossRef]
  20. D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. Kartashov, L.-C. Crasovan, and L. Torner, "Stable Spatiotemporal Solitons in Bessel Optical Lattices," Phys. Rev. Lett. 95, 023,902 (2005). [CrossRef]

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