  We study the second-harmonic generation 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 harmonics. We show that this novel
parametric process can convert a surface of the left-handed metamaterial into
an effective mirror totally reflecting the second harmonics generated by an
incident wave. We derive and analyze theoretically the coupled-mode equations
for a semi-infinite nonlinear metamaterial. We also study numerically the
second-harmonic generation by a metamaterial slab of a finite thickness, and
reveal the existence of multistable nonlinear effects.
