Enhanced second-harmonic generation in media with a weak periodicity

First-order multiple-scale perturbation theory is used to derive a set of coupled-mode equations valid for electromagnetic-wave propagation in a weakly periodic, nonlinear medium with periodicity on the order of a wavelength. We apply this to a problem where the medium has a ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ response and find that the second-harmonic signal generated is enhanced when the fundamental is tuned near the band edge. Results are given for a possible experiment with optical fibers.