Geometries for the coherent control of four-wave mixing in graphene multilayers

Deeply sub-wavelength two-dimensional films may exhibit extraordinarily strong nonlinear effects. Here we show that 2D films exhibit the remarkable property of a phase-controllable nonlinearity, i.e., the amplitude of the nonlinear polarisation wave in the medium can be controlled via the pump beam phase and determines whether a probe beam will "feel" or not the nonlinearity. This is in stark contrast to bulk nonlinearities where propagation in the medium averages out any such phase dependence. We perform a series of experiments in multilayer graphene that highlight some of the consequences of the optical nonlinearity phase-dependence, such as the coherent control of nonlinearly diffracted beams, single-pump-beam induced phase-conjugation and the demonstration of a nonlinear mirror characterised by negative reflection. The observed phase sensitivity is not specific to graphene but rather is solely a result of the dimensionality and is therefore expected in all 2D materials.