Nonlinear topological photonics

For several decades topology has been an indispensable tool in solid state physics. More recently, it became increasingly important in optics and in nonlinear photonics. The chapter focuses on applications of topology to the description of photonic energy bands of nonlinear optical systems. It covers nonlinear optics of topological edge states and corner states in various 1D and 2D platforms including subwavelength resonators, waveguide arrays and integrated silicon photonics. Synergy between topological photonics and nonlinear optics promises novel advanced functionalities such as robust nonlinear generation of parametric waves, topologically protected generation of quantum photon states and all-optical tunability and reconfigurability required in particular for ultrafast optical modulation.