Periodic nanostructures in photonics facilitate a far-reaching control of light propagation and light-matter interaction. This article reviews the current status of this subject, including both recent progress and well-established results. The primary focus is on the basic physical principles and potential applications associated with the existence of Bragg scattering, photonic band structures, and engineered effective-medium properties in periodic dielectric and metallo-dielectric systems. In addition, we discuss advantages and limitations of various theoretical and numerical approaches as well as of those fabrication techniques that have specifically been developed for this field.

We consider a discrete model that describes a linear chain of particles coupled to a single-site defect with instantaneous Kerr nonlinearity. We show that this model can be regarded as a nonlinear generalization of the familiar Fano-Anderson model and it can generate amplitude-dependent bistable resonant transmission or reflection. We identify these effects as the nonlinear Fano resonance and study its properties for continuous waves and pulses.

We study the light transmission in two-dimensional photonic crystal waveguides with embedded nonlinear defects. First, we derive the effective discrete equations with long-range interaction for describing the waveguide modes, and demonstrate that they provide a highly accurate generalization of the familiar tight-binding models which are employed, e.g., for the study of the coupledresonator optical waveguides. Using these equations, we investigate the properties of straight waveguides and waveguide bends with embedded nonlinear defects and demonstrate the possibility of the nonlinearity-induced bistable transmission. Additionally, we study localized modes in the waveguide bends and (linear and nonlinear) transmission of the bent waveguides and emphasize the role of evanescent modes in these phenomena.