Dynamics of solitary waves in nonlinear media with Bragg gratings P. Papagiannis, L. Halastanis, I. Tsopelas, N. Moshonas, Y. Kominis K. Hizanidis Department of Electrical and Computer Engineering, National Technical University of Athens, Athens ABSTRACT We study the dynamics of beams propagating in a planar waveguide with Kerr-type nonlinearity where a Bragg grating is written and diffraction is taken under consideration. The interaction of the forward field with the backscattered one due to the presence of the grating is considered both in the case of planar waves , and in the case of pulse propagation. Our results are demonstrated via numerical simulation of the governing propagation equations. Keywords: Kerr nonlinearity, Bragg grating 1. INTRODUCTION Wave propagation in periodic media exhibits many interesting phenomena. It is characterized by the existence of frequency stop bands or else band gaps near the Bragg frequencies which are associated with the periodicity of the media. 1 In the optical case we are mostly interested in the variation of the refractive index. In order to study the propagation of the optical wave fields we can use the coupled mode theory as long as the periodic variation of the refractive index is weak. 2 In the case of linear media fields with frequencies within the stop band are strongly reflected. Permanent grating structures can be also written in nonlinear media as firstly demonstrated in Ge-doped fibers 3 or more recently in As 2 S 3 based fibers. 4 Outside the frequency gap and at intensities of high transmittivity spatial resonances called gap solitons occur. Phenomena studied in such nonlinear periodic structures are optical bistability, 5 pulse compression, 6 self-pulsations, and chaos. 7 Standing wave optical solitons, 89 have been found to exist at frequencies within the stop band.It has also been reported that slow bragg solitons 10 propagate at frequencies near the Bragg resonance with power spectra falling within the band gap. Finally for large detuning of the carrier frequency the wave propagation can be described in terms of a nonlinear Schrodinger equation. 11 As we will see in the next section the nonlinear coupled mode equations, describe the interaction of two counter propagating beams which couple each other linearly through the presence of the grating and nonlinearly due to the intensity-dependent nonlinear refractive index. In the nonlinear coupling is also included the so-called holographic focusing, 12 a mechanism that produces a secondary longitudinal grating as a consequence of the interference of the counter propagating fields. This phenomena is explained without the presence of an external grating in the case of a Kerr nonlinearity 12 and demonstrated experimentally in media with photorefractive screening nonlinearity 13 where the existence of vector solitons is reported. The influence of holographic focusing is also demonstrated in media with a written grating 14 . 2. MODEL EQUATIONS We consider wave propagation in a medium with weakly periodic index profile. The linearly polarized electric field E(z,t) is taken to be of the form E(z,t)= E f (z,t)e -i(ωt-kz) + E b (z,t)e -i(ωt+kz) + c.c., where E f,b (z,t) the slowly varying amplitudes of the forward-, backward-propagating fields. The nonlinear coupled mode equations that describe the propagation of these fields are, 1516 : i ∂E f ∂z + i V ∂E f ∂t + δE f + 1 2k ∂ 2 E f ∂x 2 + κE b + Γ(|E f | 2 +2|E b | 2 )E f =0 -i ∂E b ∂z + i V ∂E f ∂t + δE f + 1 2k ∂ 2 E b ∂x 2 + κE f + Γ(|E b | 2 +2|E f | 2 )E b =0 (1) Nonlinear Optics and Applications II, edited by Mario Bertolotti, Proc. of SPIE Vol. 6582, 65821E, (2007) 0277-786X/07/$18 · doi: 10.1117/12.722748 Proc. of SPIE Vol. 6582 65821E-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 08/13/2013 Terms of Use: http://spiedl.org/terms