Propagation properties of Gaussian beam in gradient negative index of refraction material

We apply the ABCD formalism to a gradient negative index medium (NIM) and investigate the propagation and transformation properties of Gaussian beams in this medium. First, we derive the ABCD formalism in a positive gradient NIM and obtain the propagation model. Spatial soliton and the spatial breather propagation in this medium are revealed. Our research suggests that the gradient coefficient has a significant effect on the focusing ability of slab. When the gradient coefficient increases, the quasi-lense effect becomes more prominent and notable. As a result, the focusing ability improves and the beam waist in the focal point shrinks. Second, when Gaussian beams propagate in the negative gradient NIM, the beam waist enlarges as the distance increases. There is neither spatial soliton phenomenon nor breather transmission phenomenon, which is completely different from the propagation characteristics in the positive gradient NIM.