Second-harmonic generation in nonlinear left-handed metamaterials

We study the second-harmonic generation (SHG) in left-handed metamaterials with a quadratic nonlinear response. We demonstrate a novel type of the exact phase matching between the backward-propagating wave of the fundamental frequency and the forward-propagating wave of the second harmonic (SH). We show that this novel parametric process can convert a surface of the left-handed metamaterial into an effective mirror that totally reflects the SH generated by an incident wave. We derive and analyze theoretically the coupled-mode equations for a semi-infinite nonlinear metamaterial. We also numerically study the SHG by a metamaterial slab of a finite thickness and reveal the existence of multistable nonlinear effects.