Second-order nonlinear generation for electric fields propagating in two spatial dimensions

Abstract A derivation is conducted in two spatial dimensions to obtain the electric field pulse generated through second-order nonlinear processes, given electric fields incident onto a non-centrosymmetric crystal at a non-normal angle. Within this detailed derivation, it is necessary to perform a two-dimensional evaluation of the second-order nonlinear wave equation and allow the electric fields (along with their varying spectral components) to propagate through the crystal at different angles. The derived equations are employed within a representative example to depict electric field generation from a GaSe crystal, whereby we consider various incident angles and provide a comparison between theory and experiment. The developed model has the ability to evaluate sum frequency generation and difference frequency generation in any non-centrosymmetric crystal for electric fields at any incident angle.