Induced Focusing of Optical Wave From Cross-Phase Modulation in Nonlinear Metamaterials

In this paper, we investigate the novel behavior of induced focusing in nonlinear metamaterials (MMs) through designing probe-pump configuration, where the coupled waves may simultaneously experience negative refractive index or positive refractive index, or one experiences positive refractive index and another experiences negative refractive index both analytically and numerically. It is shown that cross-phase mod- ulation (XPM) in MM not only brings some new features to the behavior of induced focusing, but also makes it possible in the ordinary positive-index materials (OMs), where it is otherwise impossible. First, we reveal that weak probe beam can be focused by strong pump beam even when their centers are completely overlapping each other. Moreover, it is found that the separated distance between coupled waves has an optimal value, which results in the strongest focusing of probe beam. Finally, another most notable property is that a spatial induced focusing is found to occur for different combinations of the signs of refractive index decided by the probe and pump beams, respectively, implying that a spatial XPM in MMs can be used as a more powerful tool for controlling the inducing focusing, when compared with the corresponding case in OMs. Our findings demonstrate that MMs can provide us unique opportunities unattainable in OMs to manipulate the all-optical control of optical focusing behavior through the new physical mechanism of XPM.