Modulational instability and solitary waves in one-dimensional lattices with intensity-resonant nonlinearity Milutin Stepić * Vinča Institute of Nuclear Sciences, P. O. Box 522, 11001 Belgrade, Serbia Aleksandra Maluckov and Marija Stojanović Faculty of Mathematics and Sciences, University of Niš, P. O. Box 224, 18001 Niš, Serbia Feng Chen School of Physics, Shandong University, 250100 Jinan, China Detlef Kip Institute of Physics and Physical Technologies, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany Received 28 August 2008; published 17 October 2008 We study theoretically light beam propagation in one-dimensional periodic media with intensity-resonant nonlinearity. The phenomenon of discrete modulational instability is investigated in detail as well as the conditions for the existence and stability of fundamental lattice and surface soliton modes. According to the linear stability analysis, only on-site solitons are stable. The mobility of lattice solitons is analyzed by both free energy and mapping concepts. Only broad solitons may freely traverse the lattice. DOI: 10.1103/PhysRevA.78.043819 PACS numbers: 42.65.Tg, 63.20.Pw, 42.25.Gy I. INTRODUCTION Discrete spatial solitons are robust nonlinear structures capable of maintaining their shape during propagation. In lossless media, they are conditioned by an exact balance be- tween diffraction and nonlinearity. Such stable structures naturally emerge as convenient energy carriers in various settings including Bose-Einstein condensates 1, biomol- ecules 2, and nonlinear transmission lines 3. Experimen- tally, research into discrete solitons in nonlinear optics has proved to be particularly successful and there are many re- search reports that witness in favor of the feasibility of an all-optical concept. Uniform nonlinear waveguide arrays NWAs represent an arrangement of mutually parallel chan- nel waveguides that are weakly linearly coupled. Discrete solitons in NWAs were proposed two decades ago 4 and observed thereafter in media exhibiting cubic 5, saturable 6, quadratic 7, and nonlocal 8 nonlinear responses, to cite only a few. There is an ongoing demand for reliable, low-cost, and environment-friendly optical materials with fast response at low power level 9. Indium phosphide InP is a binary semiconductor with zinc-blende crystal structure and F4 ¯ 3m group symmetry. This material is extensively used in high- power and high-frequency electronics because of its superior electron mobility with respect to the more common semicon- ductors silicon and gallium arsenide. It possesses a direct band gap, making it useful for optoelectronic devices like laser diodes. Moreover, InP has one of the longest lifetimes of optical phonons of any compound with the zinc-blende crystal structure. This photorefractive semiconductor has a tiny electro-optic coefficient which, in turn, would require high applied fields for external biasing. On the other hand, it has been shown that InP doped with iron InP:Fe exhibits a resonant enhancement of both light-induced space-charge fields and two-wave mixing gain with a measured microsec- ond response at microwatt power level and telecommunica- tion wavelengths 10–12. Self-deflection, self-focusing, and spatial solitons in this material have been investigated re- cently in the bulk 13–15. In this paper, we present a theoretical model for light propagation in periodic media with intensity-resonant non- linearity Sec. II, briefly discuss the phenomenon of discrete modulational instability Sec. III, and investigate the exis- tence and stability of lattice and surface solitons Sec. IV. One part of Sec. IV is devoted to soliton mobility. Finally, conclusions are given in Sec. V. II. MATHEMATICAL MODEL Paraxial optical beam propagation in linear one- dimensional 1D periodic media may be described by the following partial differential equation 16: i k z k U z + 1 2k  2 U x 2 - k z 2 2k U + kn r x n 0 U = U , 1 where x stands for the transverse coordinate, U is the ampli- tude of the electric field,  is the linear propagation constant, k x denotes the transverse component Bloch momentum of wave number k =2n 0  0 -1 , and  0 represents the vacuum wavelength of light. To a good extent, the periodically modu- lated refractive index, which defines the periodic lattice, may be described by n r x = n 0 +  cos 2 x /  where n 0 is the re- fractive index of the light in the substrate n 0 = 3.205 for InP:Fe at  0 = 1.3159 m,  is the modulation amplitude, and  is the period of the lattice. This equation can be solved * mstepic@vinca.rs PHYSICAL REVIEW A 78, 043819 2008 1050-2947/2008/784/0438197 ©2008 The American Physical Society 043819-1