Second harmonic generation in grating-assisted nonlinear directional couplers

The grating-assisted nonlinear directional couplers are investigated by using the singular perturbation technique with multiple space scales. One of the waveguides of the nonlinear directional couplers has second order nonlinear cladding whose surface has a grating profile. The period of the grating is determined to satisfy the quasi-phase matching condition between the fundamental wave and second harmonic (SH) wave. From the solvability condition to have nontrivial solutions for each order, the coupled-mode equations between fundamental and SH waves are derived. In the numerical examples, lithium niobate (LiNbO₃) is chosen as second-order nonlinear medium. After the coupled-mode equations are examined numerically, it is found that the SH wave separates from the fundamental wave due to the difference of the coupling length between the fundamental and SH waves