Backward phase-matching for nonlinear optical generation in negative-index materials.

Metamaterials have enabled the realization of unconventional electromagnetic properties not found in nature, which provokes us to rethink the established rules of optics in both the linear and nonlinear regimes. One of the most intriguing phenomena in nonlinear metamaterials is 'backward phase-matching', which describes counter-propagating fundamental and harmonic waves in a negative-index medium. Predicted nearly a decade ago, this process is still awaiting a definitive experimental confirmation at optical frequencies. Here, we report optical measurements showing backward phase-matching by exploiting two distinct modes in a nonlinear plasmonic waveguide, where the real parts of the mode refractive indices are 3.4 and -3.4 for the fundamental and the harmonic waves respectively. The observed peak conversion efficiency at the excitation wavelength of ∼780 nm indicates the fulfilment of the phase-matching condition of k(2ω) = 2k(ω) and n(2ω) = -n(ω), where the coherent harmonic wave emerges along a direction opposite to that of the incoming fundamental light.