Second-harmonic generation in nonlinear grating coupler

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Abstract
Second-harmonic (SH) wave in the grating coupler is investigated by using the singular perturbation technique with multiple space scales. In order to apply the perturbation technique, we first introduce the perturbation parameter concerned with the nonlinear coefficient and the depth of the grating, and the multiple space scales in the propagation direction. Substituting the expansion of the field and the multiple space scales into the nonlinear waveequation, we can get the governing equations to each order of the perturbation. From the solvability conditions to have nontrivial solutions for each perturbationsolution, that coupled equations for the guided wave, incident wave, and radiated wave can be shown. Also, the radiated angle of fundamental and SH waves and the radiated SH wave for input/output couplers are examined numerically.