A new rigorous electromagnetic theory is developed for studying the second-harmonic generation process that occurs when a high-power laser beam falls on a periodically corrugated waveguide. The waveguide can consist of several layers with corrugated or plane boundaries, with at least one of the layers being filled with a nonlinear lossless or lossy dielectric. The theory uses a nonorthogonal coordinate transformation, which maps the corrugated interfaces onto planes. The Maxwell equations written in covariant form lead to a set of first-order partial differential equations with nonconstant coefficients. Taking into account the periodicity of the system, this set of equations could be transformed into a set of ordinary differential equations with constant coefficients, which can be resolved by an eigenvalue–eigenvector technique, avoiding numerical integration. The theory is developed for both TE and TM polarizations and for any groove geometry and incidence; it can be used, in particular, when surface waves (guided modes or surface plasmon–polaritons) are excited. It is a powerful tool for studying the enhancement of the second-harmonic generation resulting from local field enhancement by electromagnetic resonances. In addition an approach based on the Rayleigh hypothesis is developed. Both theoretical approaches are compared with previously developed differential theory for gratings in nonlinear optics. A spectacular agreement is obtained between the three theories for both TE and TM polarization. The new method is free of limitations concerning the waveguide thickness, the groove depth, and the conductivity of the grating material. It is demonstrated that the limitations of the Rayleigh hypothesis are stronger at the second-harmonic frequency.
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