Four wave mixing and compensating losses in metamaterials

The conditions under which losses can be compensated in the case of propagation of an backward signal wave under parametric four-wave interaction are analyzed. It is shown that, in the presence of a parametric coupling between forward and backward waves, compensation of signal wave losses by losses of direct waves will allow the parametric amplification and generation of the backward wave to be realized at the threshold pump amplitude. An analytical expression is obtained for the threshold pumping amplitude under phase-matching conditions. It is shown that the threshold pump amplitude increases with increasing losses, the input intensity of the idler wave, and with increasing nonlinear coupling coefficients for pump waves. As a consequence, it is possible to control the threshold pump intensity. and it is possible to obtain significant amplification in the metamaterial, as well as parametric generation of the backward wave.