Theory of backward second-harmonic localization in nonlinear left-handed media

Recent research on photonic crystals possessing a quadratic nonlinear response has revealed a second-harmonic light localization phenomenon that originates from an all-angle phase matching between counterpropagating Bloch modes at the fundamental and double frequencies [E. Centeno et al., Phys. Rev. Lett. 98, 263903 (2007)]. In this paper, we develop an electromagnetic theory describing the nature of this parametric light localization, which appears in properly design metamaterials or photonic crystals exhibiting nonlinear left-handed behaviors. We demonstrate that interferences between converging phase-matched and diverging anti-phase-matched waves create a localized second-harmonic wave focused on the pump emitter on the scale of half the wavelength. This light trapping is accompanied by the enhancement of the second-harmonic intensity, which linearly increases with the size of the two-dimensional domain. We finally show that the second-harmonic localization effect previously proposed for GaN photonic crystals can also be obtained with LiNbO₃ material.