PHYSICAL REVIEW E VOLUME 53, NUMBER 1 JANUARY 1996 Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays A. B. Aceves Department of Mathematics and Statistics, University of New Mezico, Albuquerque, ¹iv Mezico 87181 C. De Angelis* Dipartimento di Eiettronica e Informatica, Universita di Padova, Via Gradenigo 6/a, 85181 Padova, Italy T. Peschel, R. Muschall, and F. Lederer University of Jena, Faculty of Physics and Astronomy, Maz Wien -Platz -1, 077/8 Jena, Germany S. Trillo and S. Wabnitz Fondazione Ugo Bordoni, Via B. Castiglione 59, 001/2 Rome, Italy (Received 16 June 1995) We investigate the self-trapping phenomenon in one-dimensional nonlinear w'aveguide arrays. We discuss various approximate analytical descriptions of the discrete self-trapped solutions. We analyze the packing, steering, and collision properties of these solutions, by means of a variational approach and soliton perturbation theory. We compare the analytical and numerical results. PACS number(s): 42. 81. +b, 42.65. Tg, 63. 20.Pw, 46. 10. +z I. INTRODUCTION Arrays of passive or active coupled optical waveguides may be employed for several device applications [1 — 5]. In nonlinear waveguide or fiber arrays (NFA's), the charac- teristics of the device may be tuned by the input power of the beam, which permits ultrafast all-optical switching, as it was first proposed for a passive two-guide coupler in Refs. [6,7]. An interesting property of linearly cou- pled nonlinear waveguide systems, which originates &om the coherent nature of the propagating electromagnetic field, is that the coupling process may also be controlled by varying the relative phase of the input beams [8]. In recent years, much research effort has been dedicated to analyzing the extension of the coupled-mode theory to the case of multiple waveguides [9 — 16]. In particular, it has been pointed out that, with three or more cou- pled nonlinear guides, the coupling process is subject to chaotic spatial behavior [9,10, 17 — 19]. On the one hand, this may enhance the sharpness of the switching charac- teristics, but on the other hand. it may introduce unde- sired instabilities that spoil the proper operation of the array at high powers. The origin of the chaotic instabili- ties is the lack of a suKcient number of conservation laws (or Manley-Rowe relations), which prevents the possibil- ity of solving the coupled mode equations exactly. The dynamical nature of these instabilities in discrete non- linear chains has been studied by several authors in a *Also at the Dept. of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131. more general context [20,21]. In fact, the propagation in NFA's is described by a set of coupled ordinary differ- ential equations (ODE's) that is known as the standard discrete nonlinear Schrodinger equation (DNLSE), since it appears as the natural discretization of the nonlinear Schrodinger equation (NLSE) in the continuum (i. e. , par- tial differential equation). The interest in the dynamics ruled by the DNLSE type of equations goes beyond the field of optics since these equations were also derived in other physical contexts such as, e.g. , condensed matter physics, and in particular for polarons [22], or excitons and defects in molecular chains [23] such as, e. g. , poly- acetylene [24, 25]. Recently, the localized modes of mod- ulated waves in a discrete electrical lattice, described by a DNLSE, have been experimentally observed [26]. In this work, we entirely focus our attention on the existence and control of the propagation of stable local- ized wave packets in waveguide arrays. These spatially localized nonlinear modes of the array originate &om the balance between nonlinearity and linear transverse cou- pling [27]. The existence of difFerent stable self-trapped beams in nonlinear chains is a well known phenomenon in physics [28 — 37]. Much less is known, however, on the possibility of using these beams for the stable trans- port of energy across the array [38 — 45]. So far, the only known localized wave of the bright type, which may move (within a finite range of velocities) in a discrete cubic nonlinear system, is the soliton solution of the integrable so-called Ablowitz-Ladik chain [46,47]. However, optical waveguide arrays are described by the DNLSE, which, on the contrary, is a nonintegrable, so-called standard, dis- cretization [46] of the integrable NLSE [48]. Note that, in the context of condensed matter physics, the transport of 1063-651X/96/53(1)/1172(18)/$06. 00 53 1172 1996 The American Physical Society