Analyzing second harmonic generation from arrays of cylinders using Dirichlet-to-Neumann maps

We develop an efficient numerical method for analyzing second harmonic generation (SHG) in two-dimensional photonic crystals composed of nonlinear circular cylinders embedded in a linear background medium. Instead of solving the governing inhomogeneous Helmholtz equation for the second harmonic wave in the entire structure directly, we define and solve a locally generated second harmonic field in each cylinder (independent of all other cylinders), then merge the field together using Dirichlet-to-Neumann (DtN) maps of the unit cells. For linear waves in a unit cell without sources, the DtN map is an operator that maps the wave field to its normal derivative on the boundary, and it can be approximated by a small matrix. A highly accurate pseudospectral method is used to solve the locally generated second harmonic wave in the cylinders. The method was applied to analyze enhanced SHG when the linear power reflectivity peaks at both the fundamental and the second harmonic frequencies.