Wannier functions and discrete NLS equations for nematicons

José Antonio Vélez-Pérez; Panayotis Panayotaros; José Antonio Vélez-Pérez; Panayotis Panayotaros

doi:10.3934/mine.2019.2.309

Mathematics in Engineering

2019, Volume 1, Issue 2: 309-326. doi: 10.3934/mine.2019.2.309

Previous Article Next Article

Research article Special Issues

Wannier functions and discrete NLS equations for nematicons

Depto. Matemáticas y Mecánica, I.I.M.A.S., Universidad Nacional Autónoma de México, Apdo. Postal 20-726, 01000 Cd. México, México

Received: 12 November 2018 Accepted: 25 January 2019 Published: 12 March 2019

We derive nonlocal discrete nonlinear Schrödinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrödinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrödinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.
- nonlinear optics,
- waveguide arrays,
- nematic liquid crystals,
- periodic media,
- nonlocal media
Citation: José Antonio Vélez-Pérez, Panayotis Panayotaros. Wannier functions and discrete NLS equations for nematicons[J]. Mathematics in Engineering, 2019, 1(2): 309-326. doi: 10.3934/mine.2019.2.309

Related Papers:

Abstract

We derive nonlocal discrete nonlinear Schrödinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrödinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrödinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.

References

[1]	Fratalocchi A, Assanto G, Brzda¸kiewicz KA, et al. (2005) Discrete light propagation and selftrapping in liquid crystals. Opt Express 13: 1808–1815. doi: 10.1364/OPEX.13.001808
[2]	Rutkowska KA, Assanto G, Karpierz MA (2012) Discrete light propagation in arrays of liquid crystalline waveguides, In: Assanto, G., Editor, Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals, Wiley-Blackwell, 255–277.
[3]	Fratalocchi A, Assanto G, Brzda¸kiewicz KA, et al. (2004) Discrete propagation and spatial solitons in nematic liquid crystals. Opt Lett 29: 1530–1532. doi: 10.1364/OL.29.001530
[4]	Fratalocchi A, Assanto G (2005) Discrete light localization in one-dimensional nonlinear lattices with arbitrary nonlocality. Phys Rev E 72: 066608. doi: 10.1103/PhysRevE.72.066608
[5]	Ben RI , Cisneros-Ake L, Minzoni AA, et al. (2015) Localized solutions for a nonlocal discrete NLS equation. Phys Lett A 379: 1705–1714. doi: 10.1016/j.physleta.2015.04.012
[6]	Ben RI, Borgna JP, Panayotaros P (2017) Properties of some breather solutions of a nonlocal discrete NLS equation. Commun Math Sci 15: 2143–2175. doi: 10.4310/CMS.2017.v15.n8.a3
[7]	Kartashov YV, Vysloukh VA, Torner L (2008) Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity. Opt Lett 33: 1747–1749. doi: 10.1364/OL.33.001747
[8]	Alfimov GL, Kevrekidis PG, Konotop VV, et al. (2002) Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential. Phys Rev E 66: 046608. doi: 10.1103/PhysRevE.66.046608
[9]	Pelinovsky D, Schneider G (2010) Bounds on the tight-binding approximation for the Gross- Pitaevskii equation with a periodic potential. J Differ Equations 248: 837–849. doi: 10.1016/j.jde.2009.11.014
[10]	Pelinovsky DE (2011) Localization in Periodic Potentials: From Schrödinger Operators to the Gross-Pitaevskii Equation, Cambridge University Press.
[11]	Fečkan M, Rothos VM (2010) Travelling waves of discrete nonlinear Schrödinger equations with nonlocal interactions Appl Anal 89: 1387–1411.
[12]	Gaididei Y, Flytzanis N, Neuper A, et al. (1995) Effect of nonlocal interactions on soliton dynamics in anharmonic lattices. Phys Rev Lett 75: 2240–2243 doi: 10.1103/PhysRevLett.75.2240
[13]	Johansson M, Gaididei YB, Christiansen PL, et al. (1998) Switching between bistable states in discrete nonlinear model with long-range dispersion. Phys Rev E 57: 4739–4742 doi: 10.1103/PhysRevE.57.4739
[14]	Abdullaev F Kh, Brazhnyi VA (2012) Solitons in dipolar Bose–Einstein condensates with a trap and barrier potential. J Phys B: At Mol Opt Phys 45: 085301. doi: 10.1088/0953-4075/45/8/085301
[15]	Efremidis NK (2008) Nonlocal lattice solitons in thermal media. Phys Rev A 77: 063824. doi: 10.1103/PhysRevA.77.063824
[16]	Peccianti M, De Rossi A, Assanto G, et al. (2000) Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells. Appl Phys Lett 77: 7–9. doi: 10.1063/1.126859
[17]	Peccianti M, Assanto G (2012) Nematicons. Phys Rep 516: 147–208. doi: 10.1016/j.physrep.2012.02.004
[18]	Borgna JP, Panayotaros P, Rial D, et al. (2018) Optical solitons in nematic liquid crystals: model with saturation effects. Nonlinearity 31: 1535. doi: 10.1088/1361-6544/aaa2e2
[19]	Alberucci A, Peccianti M, Assanto G, et al. (2006) Two-color vector solitons in nonlocal media. Phys Rev Lett 97: 153903. doi: 10.1103/PhysRevLett.97.153903
[20]	Izdebskaya Y, Shvedov VG, Assanto G, et al. (2017) Magnetic routing of light-induced waveguides. Nat Commun 8: 14452. doi: 10.1038/ncomms14452
[21]	Kohn W (1959) Analytic properties of Bloch waves and Wannier functions. Phys Rev 115: 809–821. doi: 10.1103/PhysRev.115.809
[22]	Ziman JM (1972) Principles of the Theory of Solids, 2 Eds., Cambridge University Press.
[23]	Bruno-Alfonso A, Nacbar DR (2007) Wannier functions of isolated bands in one-dimensional crystals. Phys Rev B 75: 115428. doi: 10.1103/PhysRevB.75.115428
[24]	Panayotaros P, Marchant TR (2014) Solitary waves in nematic liquid crystals. Phys D 268: 106– 117. doi: 10.1016/j.physd.2013.10.011
[25]	Borgna JP, Panayotaros P, Rial D, et al. (2018) Optical solitons in nematic liquid crystals: large angle model. J Nonlin Sci Preprint.
[26]	Allen G (1953) Band structures of one-dimensional crystals with square-well potentials. Phys Rev 91: 531–533. doi: 10.1103/PhysRev.91.531

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)