Abstract

Solitons in the model of nonlinear photonic crystals with the transverse structure based on two-dimensional (2D) quadratic- or rhombic-shaped Kronig-Penney (KP) lattices are studied by means of numerical methods. The model can also applies to a Bose-Einstein condensate (BEC) trapped in a superposition of linear and nonlinear 2D periodic potentials. The analysis is chiefly presented for the self-repulsive nonlinearity, which gives rise to several species of stable fundamental gap solitons, dipoles, four-peak complexes, and vortices in two finite bandgaps of the underlying spectrum. Stable solitons with complex shapes are found, in particular, in the second bandgap of the KP lattice with the rhombic structure. The stability of the localized modes is analyzed in terms of eigenvalues of small perturbations, and tested in direct simulations. Depending on the value of the KP’s duty cycle (DC, i.e., the ratio of the void’s width to the lattice period), an internal stability boundary for the solitons and vortices may exist inside of the first bandgap. Otherwise, the families of the localized modes are entirely stable or unstable in the bandgaps. With the self-attractive nonlinearity, only unstable solitons and vortices are found in the semi-infinite gap.

©2011 Optical Society of America

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2011 (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011).
[Crossref]

2010 (2)

S. Arismar Cerqueira, “Recent progress and novel applications of photonic crystal fibers,” Rep. Prog. Phys. 73, 024401 (2010).
[Crossref]

Y. Li, B. A. Malomed, M. Feng, and J. Zhou, “Arrayed and checkerboard optical waveguides controlled by the electromagnetically induced transparency,” Phys. Rev. A 82, 633813 (2010).
[Crossref]

2009 (3)

J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, B. B. Baizakov, and M. Salerno, “Matter-wave vortices and solitons in anisotropic optical lattices,” Physica D 238, 1439–1448 (2009).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

2008 (4)

T. Mayteevarunyoo and B. A. Malomed, “Solitons in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 25, 1854–1863 (2008).
[Crossref]

A. S. Rodrigues, P. G. Kevrekidis, M. A. Porter, D. J. Frantzeskakis, P. Schmelcher, and A. R. Bishop, “Matter-wave solitons with a periodic, piecewise-constant scattering length,” Phys. Rev. A 78, 013611 (2008).
[Crossref]

R. Driben and B. A. Malomed, “Stabilization of two-dimensional solitons and vortices against supercritical collapse by lattice potentials,” Eur. Phys. J. D 50, 317–323 (2008).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

2007 (3)

2006 (4)

Y. Kominis, “Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures,” Phys. Rev. E 73, 066619 (2006).
[Crossref]

Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. 31, 2888–2890 (2006).
[Crossref] [PubMed]

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
[Crossref]

S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B 39, 847 (2006).
[Crossref]

2005 (8)

B. Maes, P. Bienstman, and R. Baets, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, 613–619 (2005).
[Crossref]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semiclass. Opt. 7, R53–R72 (2005).
[Crossref]

Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, “Soliton topology versus discrete symmetry in optical lattices,” Phys. Rev. Lett. 95, 123902 (2005).
[Crossref] [PubMed]

I. M. Merhasin, B. V. Gisin, R. Driben, and B. A. Malomed, “Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity,” Phys. Rev. E 71, 016613 (2005).
[Crossref]

G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region ¡ 20 dB/km) around 1550 nm,” Opt. Express 13, 8452–8459 (2005).
[Crossref] [PubMed]

B. T. Seaman, L. D. Carr, and M. J. Holland, “Nonlinear band structure in Bose-Einstein condensates: nonlinear Schrödinger equation with a Kronig-Penney potential,” Phys. Rev. A 71, 033622 (2005).
[Crossref]

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819–821 (2005).
[Crossref]

A. Fuerbach, P. Steinvurzel, J. A. Bolger, A. Nulsen, and B. J. Eggleton, “Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers,” Opt. Lett. 30, 830 (2005).
[Crossref] [PubMed]

2004 (7)

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29, 1530–1532 (2004).
[Crossref] [PubMed]

F. Du, Y. Q. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183 (2004).
[Crossref]

A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004).
[Crossref] [PubMed]

W. Li and A. Smerzi, “Nonlinear Krönig-Penney model,” Phys. Rev. E 70, 016605 (2004).
[Crossref]

F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369–2371 (2004).
[Crossref] [PubMed]

H. Sakaguchi and B. A. Malomed, “Two-dimensional loosely and tightly bound solitons in optical lattices and inverted traps,” J. Phys. B 37, 2225–2239 (2004).
[Crossref]

M. Theis, M., G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[Crossref] [PubMed]

2003 (7)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11, 2589–2596 (2003).
[Crossref] [PubMed]

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67, 026607 (2003).
[Crossref]

A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express 11, 452–459 (2003).
[Crossref] [PubMed]

T. M. Monro and D. J. Richardson, “Holey optical fibres: Fundamental properties and device applications,” C. R. Physique 4, 175–186 (2003).
[Crossref]

J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
[Crossref]

2002 (1)

M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39 (2002).
[Crossref] [PubMed]

2001 (2)

1999 (1)

D. Hennig and G. P. Tsironis, “Wave transmission in nonlinear lattices,” Phys. Rep. 307, 333–432 (1999).
[Crossref]

1998 (2)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic band cap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

1996 (3)

P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, “Influence of nearly resonant light on the scattering length in low-temperature atomic gases,” Phys. Rev. Lett. 77, 2913–2916 (1996).
[Crossref] [PubMed]

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[Crossref]

E. Lidorikis, Q. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[Crossref]

1994 (1)

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[Crossref]

1991 (1)

H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, K. W. West, and J.Spector, atomically precise superlattice potential imposed on a 2-dimensional electron gas,” Appl. Phys. Lett. 58, 726–728 (1991).
[Crossref]

1988 (1)

Alkeskjold, T. T.

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819–821 (2005).
[Crossref]

Arismar Cerqueira, S.

S. Arismar Cerqueira, “Recent progress and novel applications of photonic crystal fibers,” Rep. Prog. Phys. 73, 024401 (2010).
[Crossref]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29, 1530–1532 (2004).
[Crossref] [PubMed]

B. J.,

Baets, R.

Baizakov, B. B.

T. Mayteevarunyoo, B. A. Malomed, B. B. Baizakov, and M. Salerno, “Matter-wave vortices and solitons in anisotropic optical lattices,” Physica D 238, 1439–1448 (2009).
[Crossref]

Baldwin, K. W.

H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, K. W. West, and J.Spector, atomically precise superlattice potential imposed on a 2-dimensional electron gas,” Appl. Phys. Lett. 58, 726–728 (1991).
[Crossref]

Bang, O.

Bennet, F. H.

Bienstman, P.

Bigot, L.

Binosi, D.

Bird, D. M.

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic band cap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Bishop, A. R.

A. S. Rodrigues, P. G. Kevrekidis, M. A. Porter, D. J. Frantzeskakis, P. Schmelcher, and A. R. Bishop, “Matter-wave solitons with a periodic, piecewise-constant scattering length,” Phys. Rev. A 78, 013611 (2008).
[Crossref]

Bjarklev, A.

Bloch, I.

M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39 (2002).
[Crossref] [PubMed]

Bolger, J. A.

Bouwmans, G.

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[Crossref]

Broeng, J.

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11, 2589–2596 (2003).
[Crossref] [PubMed]

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D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Skorobogatiy, M.

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystals Guiding (Cambridge University Press, 2009).

Smerzi, A.

W. Li and A. Smerzi, “Nonlinear Krönig-Penney model,” Phys. Rev. E 70, 016605 (2004).
[Crossref]

Soukoulis, C. M.

E. Lidorikis, Q. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[Crossref]

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[Crossref]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

Steinvurzel, P.

Stormer, H. L.

H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, K. W. West, and J.Spector, atomically precise superlattice potential imposed on a 2-dimensional electron gas,” Appl. Phys. Lett. 58, 726–728 (1991).
[Crossref]

Sukhorukov, A. A.

A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[Crossref] [PubMed]

R. Fischer, D. Trager, D. N. Neshev, A. A. Sukhorukov, W. Królikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett.96, 023905 (2006).
[Crossref] [PubMed]

Thalhammer, G.

M. Theis, M., G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[Crossref] [PubMed]

Theis, M.

M. Theis, M., G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[Crossref] [PubMed]

Torner, L.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, “Soliton topology versus discrete symmetry in optical lattices,” Phys. Rev. Lett. 95, 123902 (2005).
[Crossref] [PubMed]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semiclass. Opt. 7, R53–R72 (2005).
[Crossref]

Trager, D.

R. Fischer, D. Trager, D. N. Neshev, A. A. Sukhorukov, W. Królikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett.96, 023905 (2006).
[Crossref] [PubMed]

Tsironis, G. P.

D. Hennig and G. P. Tsironis, “Wave transmission in nonlinear lattices,” Phys. Rep. 307, 333–432 (1999).
[Crossref]

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Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

Walraven, J. T. M.

P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, “Influence of nearly resonant light on the scattering length in low-temperature atomic gases,” Phys. Rev. Lett. 77, 2913–2916 (1996).
[Crossref] [PubMed]

West, K. W.

H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, K. W. West, and J.Spector, atomically precise superlattice potential imposed on a 2-dimensional electron gas,” Appl. Phys. Lett. 58, 726–728 (1991).
[Crossref]

Westbrook, P. S.

Windeler, R. S.

Winkler, K.

M. Theis, M., G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[Crossref] [PubMed]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semiclass. Opt. 7, R53–R72 (2005).
[Crossref]

Wu, S. T.

F. Du, Y. Q. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183 (2004).
[Crossref]

Xie, P.

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67, 026607 (2003).
[Crossref]

Yang, J.

J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[Crossref]

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
[Crossref]

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystals Guiding (Cambridge University Press, 2009).

Zacarés, M.

Zhang, X.

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67, 026607 (2003).
[Crossref]

Zhang, Z.-Q.

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67, 026607 (2003).
[Crossref]

Zhou, J.

Y. Li, B. A. Malomed, M. Feng, and J. Zhou, “Arrayed and checkerboard optical waveguides controlled by the electromagnetically induced transparency,” Phys. Rev. A 82, 633813 (2010).
[Crossref]

Zoller, P.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

Zyss, J.

R. Driben, B. A. Malomed, A. Gubeskys, and J. Zyss, “Cubic-quintic solitons in the checkerboard potential,” Phys. Rev. E 76, 066604 (2007).
[Crossref]

Appl. Phys. Lett. (2)

F. Du, Y. Q. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183 (2004).
[Crossref]

H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, K. W. West, and J.Spector, atomically precise superlattice potential imposed on a 2-dimensional electron gas,” Appl. Phys. Lett. 58, 726–728 (1991).
[Crossref]

C. R. Physique (1)

T. M. Monro and D. J. Richardson, “Holey optical fibres: Fundamental properties and device applications,” C. R. Physique 4, 175–186 (2003).
[Crossref]

Eur. Phys. J. D (1)

R. Driben and B. A. Malomed, “Stabilization of two-dimensional solitons and vortices against supercritical collapse by lattice potentials,” Eur. Phys. J. D 50, 317–323 (2008).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. W. Haakestad, T. T. Alkeskjold, M. D. Nielsen, L. Scolari, J. Riishede, H. E. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819–821 (2005).
[Crossref]

J. Comput. Phys. (1)

J. Yang, “Newton-conjugate gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[Crossref]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[Crossref]

J. Opt. B: Quant. Semiclass. Opt. (1)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quant. Semiclass. Opt. 7, R53–R72 (2005).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. B (2)

S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B 39, 847 (2006).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Two-dimensional loosely and tightly bound solitons in optical lattices and inverted traps,” J. Phys. B 37, 2225–2239 (2004).
[Crossref]

J. Phys. D (1)

J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
[Crossref]

Nature (3)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39 (2002).
[Crossref] [PubMed]

Opt. Express (7)

Opt. Lett. (5)

Phys. Rep. (2)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

D. Hennig and G. P. Tsironis, “Wave transmission in nonlinear lattices,” Phys. Rep. 307, 333–432 (1999).
[Crossref]

Phys. Rev. A (3)

B. T. Seaman, L. D. Carr, and M. J. Holland, “Nonlinear band structure in Bose-Einstein condensates: nonlinear Schrödinger equation with a Kronig-Penney potential,” Phys. Rev. A 71, 033622 (2005).
[Crossref]

A. S. Rodrigues, P. G. Kevrekidis, M. A. Porter, D. J. Frantzeskakis, P. Schmelcher, and A. R. Bishop, “Matter-wave solitons with a periodic, piecewise-constant scattering length,” Phys. Rev. A 78, 013611 (2008).
[Crossref]

Y. Li, B. A. Malomed, M. Feng, and J. Zhou, “Arrayed and checkerboard optical waveguides controlled by the electromagnetically induced transparency,” Phys. Rev. A 82, 633813 (2010).
[Crossref]

Phys. Rev. B (2)

Q. Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Wave propagation in nonlinear photonic band-gap materials,” Phys. Rev. B 53, 15577–15585 (1996).
[Crossref]

E. Lidorikis, Q. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[Crossref]

Phys. Rev. E (5)

P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67, 026607 (2003).
[Crossref]

W. Li and A. Smerzi, “Nonlinear Krönig-Penney model,” Phys. Rev. E 70, 016605 (2004).
[Crossref]

I. M. Merhasin, B. V. Gisin, R. Driben, and B. A. Malomed, “Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity,” Phys. Rev. E 71, 016613 (2005).
[Crossref]

Y. Kominis, “Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures,” Phys. Rev. E 73, 066619 (2006).
[Crossref]

R. Driben, B. A. Malomed, A. Gubeskys, and J. Zyss, “Cubic-quintic solitons in the checkerboard potential,” Phys. Rev. E 76, 066604 (2007).
[Crossref]

Phys. Rev. Lett. (5)

P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, “Influence of nearly resonant light on the scattering length in low-temperature atomic gases,” Phys. Rev. Lett. 77, 2913–2916 (1996).
[Crossref] [PubMed]

M. Theis, M., G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[Crossref] [PubMed]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

Y. V. Kartashov, A. Ferrando, A. A. Egorov, and L. Torner, “Soliton topology versus discrete symmetry in optical lattices,” Phys. Rev. Lett. 95, 123902 (2005).
[Crossref] [PubMed]

A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[Crossref] [PubMed]

Physica D (1)

T. Mayteevarunyoo, B. A. Malomed, B. B. Baizakov, and M. Salerno, “Matter-wave vortices and solitons in anisotropic optical lattices,” Physica D 238, 1439–1448 (2009).
[Crossref]

Prog. Opt. (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

Rep. Prog. Phys. (1)

S. Arismar Cerqueira, “Recent progress and novel applications of photonic crystal fibers,” Rep. Prog. Phys. 73, 024401 (2010).
[Crossref]

Rev. Mod. Phys. (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011).
[Crossref]

Science (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic band cap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Other (4)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystals Guiding (Cambridge University Press, 2009).

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
[Crossref]

R. Fischer, D. Trager, D. N. Neshev, A. A. Sukhorukov, W. Królikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett.96, 023905 (2006).
[Crossref] [PubMed]

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Figures (14)
Fig. 1
Fig. 1 (Color online) (a) The transverse structure of the photonic crystal of the Kronig-Penney type, with the square-shaped modulation pattern corresponding to Eq. (2). Panels (b), (c) and (d) display the linear dispersion relation for this photonic crystal with depth U = 10 in Eq. (2), plotted along the edge Γ → X → M → Γ of the irreducible Brillouin zone, for three values of the duty-cycle parameter: DC = 0.75, DC = 0.5 and DC = 0.25, respectively. Photonic bandgaps are shaded (the lowest one is the semi-infinite gap), while white areas represent the first two Bloch bands.

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Fig. 2
Fig. 2 (Color online) (a) The shape of the rhombic 2D Kronig-Penney structure with duty cycle DC = 0.5. (b) The linear dispersion relation for this structure, formed by the superposition of two modulation functions (2) with U = 10, is plotted along the edge Γ → X → M → Γ of the irreducible Brillouin zone. Shaded and white areas correspond to the gaps (the semi-infinite and two finite ones) and the first two Bloch bands, respectively.

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Fig. 3
Fig. 3 (Color online) (a) The power-vs.-propagation constant diagram for fundamental solitons (the red curve), diagonal dipoles (the blue curve), and bound complexes of four fundamental solitons (the magenta and black curves pertain to the square-shaped and rhombic complexes, respectively), in the two finite bandgaps of the 2D square-shaped Kronig-Penney structure with duty cycle DC = 0.75 and the self-defocusing nonlinearity. In this and other figures, solid and dashed segments of the curves represent stable an unstable modes, respectively. (b) Typical examples of stable and unstable fundamental solitons for β = 4.4, P = 70.71 and β = 4.0, P = 13.77, respectively. Here and in similar figures below, the left and right panels display, respectively, the distribution of the local amplitude in the stationary solutions, and the respective spectral plane of the stability eigenvalues. (c) Typical examples of stable and unstable square-shaped complexes for β = 4.4, P = 42.31 and β = 4.0, P = 50.79, respectively. (d) A typical example of a stable rhombic complex for β = 4.4, P = 41.98 (an example of unstable rhombuses is not displayed separately, as it does not show anything essentially different from the other modes). (e) Typical examples of stable and unstable diagonal dipoles for β = 4.4 and 4.1, respectively.

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Fig. 4
Fig. 4 (Color online) Examples of stable fundamental solitons, diagonal dipoles, squares, and rhombuses, found in the narrow second bandgap of Fig. 3 at β = 8.35, with the total powers, respectively, P = 1.5013, P = 3.0022, P = 6.0184, and P = 5.9956.

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Fig. 5
Fig. 5 (Color online) Typical examples of the evolution of weakly unstable solitons is displayed by means of the cross section of |Ψ(x,y)|2 drawn through point y = 0: (a) and (b) —a fundamental gap soliton and square-shaped four-soliton complex, both with β = 4.0; (c) —a diagonal dipole with β = 4.1.

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Fig. 6
Fig. 6 (Color online) (a) The red, blue, green, black, magenta and yellow curves show the P(β) dependences for fundamental solitons, straight dipoles, diagonal dipoles, rhombus complexes, square complexes and quadrupoles, respectively, in the single finite bandgap existing at DC = 0.5 in the model with the square-shaped transverse structure and self-defocusing nonlinearity. (b) Generic examples of fundamental solitons and quadrupoles (top and bottom, respectively). (c) Examples of squares and rhombuses. (d) Examples of straight and diagonal dipoles. All the solutions are displayed for β = 2.4. All the solitons and their bound states are stable, in this case.

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Fig. 7
Fig. 7 (Color online) (a) The dotted and solid curves show the integral power, P, as a function of the propagation constant, β, for the square-shaped and rhombic vortices (with topological charge 1), respectively, at DC = 0.75, in the two finite bandgaps of the photonic crystal with the square transverse structure and self-defocusing nonlinearity. (b) A typical example of the amplitude and phase distributions, and the set of perturbation eigenvalues, for the stable square-shaped vortex with β = 4.4 and P = 41.845. (c) The same for a typical unstable one with β = 4.05 and P = 48.949. (d) A typical example of the stable rhombic vortex with β = 4.4 and P = 42.562.

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Fig. 8
Fig. 8 (Color online) (Upper row) and (bottom row): Examples of stable square-shaped and rhombic vortices (the amplitude and phase distributions) found in the narrow second bandgap (9) for DC = 0.75, with β = 8.35 and P = 5.9879 or P = 6.0108, respectively.

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Fig. 9
Fig. 9 (Color online) (a) The P(β) dependence for rhombus-shaped vortices in the finite bandgap at DC = 0.5 in the model with the square-shaped transverse structure and self-defocusing nonlinearity. (b) An example of the amplitude and phase distribution in the stable vortex for β = 2.4 and P = 26.1702.

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Fig. 10
Fig. 10 (Color online) The power-vs.-propagation constant curves for the fundamental solitons in two finite bandgaps of the photonic crystal with the rhombic transverse structure (see Fig. 2) and self-defocusing nonlinearity. In the second bandgap, the top branch is unstable, while the two lower ones are stable. (b) The largest instability growth rate of the gap solitons as functions of the propagation constant, β. The solid and dotted curves in the first bandgap in (b) correspond, respectively, to the bottom and top portions of the P(β) curve in (a). The curve in the second bandgap in (b) corresponds to the top P(β) branch in the same bandgap in (a). A typical example of stable solitons in the first bandgap is shown in panel (c) for β = 3.52 and P = 60.00. (d) Examples of unstable solitons with β = 3.9 in the first bandgap for P = 8.5687 and P = 26.6719 (top and bottom rows, respectively).

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Fig. 11
Fig. 11 (Color online) (a) and (b): The evolution of unstable solitons corresponding to the top and bottom rows in Fig. 10 is shown in the cross section of |Ψ(x,y)|2 drawn through y = 0. (c): The same for the unstable pattern from Fig. 12(b).

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Fig. 12
Fig. 12 (Color online) Typical examples of solitons, found in the second finite bandgap of the photonic crystal with the rhombic structure and self-defocusing nonlinearity, for β = 2.1. In (a), the top and bottom rows display examples of stable solitons, with P = 5.9460 and P = 13.4476, which correspond, respectively, to the lower and middle P(β) curves in the second bandgap in Fig. 10. (b) A typical example of an unstable soliton with P = 53.0894, which corresponds to the upper curve in the second bandgap in Fig. 10.

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Fig. 13
Fig. 13 (Color online) Examples of unstable vortices found, respectively, in the first and second finite bandgaps of the model with the rhombic structure, with β = −4.25, P = 30.4432 (a), and β = 2.1, P = 64.2146 (b).

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Fig. 14
Fig. 14 (Color online) (a) The power-vs.-propagation-constant dependence for solitons supported by the self-focusing nonlinearity in the model with the rhombic transverse structure. (b) A typical example of an unstable soliton, with β = 8.0 and P = 48.0398. (c) Decay of this soliton in direct simulations.

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Equations (12)

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i Ψ z   +   Ψ x x   +   Ψ y y   +   W ( x , y )   ( 1   +   σ | Ψ | 2 )   Ψ   =   0 .
W ( x , y )   =   { U , L n   <   x   <   D   +   L n   and   L n   <   y   <   D   +   L n , 0 , D   +   L n   <   x   <   L   ( 1   +   n )   and / or   D   +   L n   <   y   <   L ( 1   +   n ) ,
− β Φ   +   Φ x x   +   Φ y y   +   W ( x , y )   ( Φ   +   σ Φ 3 )   =   0 .
P ( β )   ≡   ∫ − ∞ + ∞ ∫ − ∞ + ∞ | Ψ ( x , y ) | 2 d x d y ,
[ ( ∂ x   +   i k x ) 2   +   ( ∂ y   +   i k y ) 2   +   W ( x , y ) ] φ ( x , y )   =   β φ ( x , y )
Ψ ( x , y , z )   =   e i β z   { Φ 0   ( x , y )   +   [ v ( x , y )   −   w ( x , y ) ] e λ z   +   [ v *   ( x , y )   −   w *   ( x , y ) ]   e λ * z }
L ψ   =   λ ψ ,
L = i   ( G 0 ∇ 2   +   G 1 ∇ 2   +   G 2 − G 0 ) ,   ψ   =   ( v w ) G 0 = 1 2   ( Φ 0 2   −   Φ 0 * 2 ) σ W ( x , y ) G 1 = − β   +   W ( x , y )   +   2 σ W ( x , y ) | Φ 0 | 2   −   1 2 σ W ( x , y )   ( Φ 0 2   +   Φ 0 * 2 ) G 2 = − β   +   W ( x , y )   +   2 σ W ( x , y ) | Φ 0 | 2   +   1 2 σ W ( x , y )   ( Φ 0 2   +   Φ 0 * 2 )
8.315   <   β   <   8.398 ,
−   4.15   <   β   <   − 4.00 ,   24.14   <   P   <   25.69 ,
−   3.70   <   β   <   − 3.50 ,   30.52   <   P   <   75.20 ,
−   4.30   <   β   <   − 4.27 ,   10.35   <   P   <   10.63.

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