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When an incident plane wave with circular frequency (omega) falls on a grating coated by a layer of nonlinear material, it generates a nonlinear polarization PNL(2(omega) ) which acts as a source term and produces a second harmonic (SH) field called signal. The excitation of an electromagnetic resonance like surface plasmon or a guided wave increases the local field and thus the signal. The problem is to be able to compute and optimize the latter. We have developed a new theory which uses a coordinate transformation mapping the grating profile onto a plane. This simplifies the boundary conditions but complicates the propagation equation. Taking advantage of the psuedoperiodicity of the problem, the Fourier harmonics of the field are solution of a set of first order differential equations with constant coefficients. The resolution of this system via eigenvalue and eigenvector technique avoid numerical instabilities and lead to accurate results which agree perfectly with those found via the Rayleigh method or by the Differential method, when they work. A phenomenological approach is then developed to explain the unusual shape of the resonance lines at 2(omega) , which is based on the poles and zeros of the scattering operator S at (omega) and 2(omega) . It is shown that S(2(omega) ) presents 3 complex poles with 3 associated complex zeros. Their knowledge, plus the nonlinear reflectivity of the plane device allows predicting all the possible shapes of the 2(omega) signal as a function of angle of incidence. The phenomenological study explains an experimental result, found a few years ago, that if 2(omega) lies inside the absorption band of the guiding material instead of the transparent region, the enhanced second harmonic generation (SHG) is changed into a reduced one. It means that in the case phase matching can lead to a minimum instead of maximum. An algorithm is then proposed to maximize the signal intensity; with polyurethane as a guiding material a conversion factor of up to 40% is found when incident power is equal to 40 kW.
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