Scalora, Michael; et al.; Dynamics of short pulses and phase matched second harmonic generation in negative index materials; Optics Express Vol. 14, Issue 11, pp. 4746-4756 (2006); https://www.osapublishing.org/oe/abstract.cfm?uri=oe-14-11-4746
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We study pulsed second harmonic generation in metamaterials under conditions of significant absorption. Tuning the pump in the negative index range, a second harmonic signal is generated in the positive index region, such that the respective indices of refraction have the same magnitudes but opposite signs. This insures that a forward-propagating pump is exactly phase matched to the backward-propagating second harmonic signal. Using peak intensities of ~500 MW/cm², assuming χ⁽²⁾~80pm/V, we predict conversion efficiencies of 12% and 0.2% for attenuation lengths of 50 and 5µm, respectively.
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