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We present a new numerical method for the analysis of second-harmonic generation (SHG) in one- and two-dimensional (1D, 2D) diffraction gratings containing centrosymmetric quadratically nonlinear materials. Thus, the nonlinear optical properties of a material are determined by its symmetry properties: non-centrosymmetric materials lack inversion symmetry and therefore allow local even-order SHG in the bulk of the material, whereas this process is forbidden in centrosymmetric materials. The inversion symmetry of centrosymmetric materials is broken at their surface whence they allow local surface SHG. Additionally, centrosymmetric materials give rise to nonlocal (bulk) SHG.
Our numerical method extends the linear generalized source method (GSM), which is an efficient numerical method for solving the problem of linear diffraction in periodic structures of arbitrary geometry. The nonlinear GSM is a three-step algorithm: for a given excitation at the fundamental frequency the linear field is computed using the linear GSM. This field gives rise to a nonlinear source polarization at the second harmonic (SH) frequency. This nonlinear polarization comprises surface and bulk polarizations as additional source terms and is subsequently used to compute the nonlinear near- and far-field optical response at the SH.
We study the convergence characteristics of the nonlinear GSM for 1D and 2D periodic structures and emphasize the numerical intricacies caused by the surface SH polarization term specific to centrosymmetric materials. In order to illustrate the practical significance of our numerical method, we apply it to metallic gratings made of Au and Ag as well as dielectric grating structures made of silicon and investigate the relative contribution of the bulk and surface nonlinearity to the nonlinear optical response at the SH. Particular attention is paid to optical effects that have a competing influence to the nonlinear optical response of the grating structures, namely the resonant local field enhancement and optical losses.
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