Introduction to solitons in photonic lattices

In this chapter, we present a review of optical systems that have a periodic variation in their index transverse to the direction of propagation. Such photonic systems include photonic crystal fibers, which have a large index variation that controls frequency dispersion, and coupled waveguide arrays, which have a relative small index variation that controls spatial diffraction. Here, we will focus on the latter case and consider 1+1D and 2+1D dynamics. A photonic lattice has the advantage that the refractive index contrast requirements are low, and thus, for example, a 2D bandgap can be established for index modulations of the order 10^-3. Light excitation is quite simple because the optical wave is launched in a direction that is almost perpendicular to the direction of the index modulations. Also, nonlinearity can quite easily manifest in such systems simply because of low refractive index modulations. As a result, nonlinear self-localized structures or solitons are possible for nonlinear index modulations of the order of 10^-4. An optical periodic system with periodicity along the transverse direction was first theoretically studied in 1965 [1] in the linear regime. In that work the diffraction pattern of an array of identical fibers has been found using coupled mode theory in terms of Bessel functions. Experimentally this behavior was reported in 1973 in an array of optical waveguides [2].