Self-localization in arrays of defocusing waveguides.

Power-dependent switching between the interacting modes of nonlinear dual-core (two-waveguide) couplers has attracted a great deal of interest during the past several years (see, for example, Synder et al.1 for a review of various physically important situations). In the theoretical papers2 -4 it has been demonstrated that three-waveguide nonlinear directional couplers offer some distinct advantages over the standard two-waveguide coupler proposed by Jensen.5 In fact, two(and even three-) waveguide couplers is a special case of a more general class of n-waveguide systems. Recently Christodoulides and Joseph6 examined the stationary nonlinear solutions for an n-waveguide self-focusing coupler, and Schmidt-Hattenberger et al. 7 have investigated numerically power-controlled switching in multicore (n = 5) couplers. The results are of significant importance in aiding our understanding of how the advantages of the use of n-waveguide couplers (with n > 2) in the cw regime may be useful for pulse (soliton) switching (see, for example, Refs. 8 and 9 and references therein for the soliton switching). Arrays of coupled optical waveguides can exhibit interesting nonlinear phenomena, e.g., self-focusing and stationary localized patterns,6 that are a discrete analog of the effects known in continuum media for a self-focusing nonlinearity. As follows from the results of Ref. 6, an increase of input power in a waveguide array leads to an energy localization in mainly one waveguide, provided that nonlinearity is self-focusing. In this Letter I show that the discrete self-focusing is also possible for arrays of defocusing waveguides, and I give analytical arguments to the physics of self-focusing in discrete waveguide structures with respect to the case of the standard two-waveguide coupler. The model I deal with in this Letter is, in fact, an array of coupled optical waveguides that are lossless, identical, and regularly spaced, with D being the distance between successive waveguides. By use of the formalism of coupled-mode theory and by consideration of only nearest-neighbor interactions, it can be shown that the electric field propagating in the nth waveguide obeys the nonlinear difference-differential equation 6 iCdEn +/3E, + c(En+1 + E-) AIEn1 2 En