Nonlinear optical lattices with a void impurity

We examine a one-dimensional nonlinear (Kerr) waveguide array which contains a single "void" waveguide where the nonlinearity is identically zero. We uncover a new family of nonlinear localized modes centered at or near the void, and their stability properties. Unlike a usual impurity problem, here the void acts like a repulsive impurity causing the center of the simplest mode to lie to the side of the void's position. We also compute the stability of extended nonlinear modes showing significant differences from the usual homogeneous nonlinear array. The transmission of a nonlinear pulse across the void shows three main regimes, transmission, reflection and trapping at the void's position, and we identify a critical momentum for the pulse below (above) where the pulse gets reflected (transmitted), or trapped indefinitely at the void's position. For relatively wide pulses, we observe a steep increase from a regime of no transmission to a regime of high transmission, as the amplitude of the soliton increases beyond a critical wavevector value. Finally, we consider the transmission of an extended nonlinear wave across the void impurity numerically, finding a rather complex structure of the transmission as a function of wavevector, with the creation of more and more transmission gaps as nonlinearity increases. The overall transmittance decreases and disappears eventually, where the boundaries separating passing from non-passing regions are complex and fractal-like.