A transient analysis of negative refraction at the interface between two transmission-line grids

In response to a few recent publications that deny the existence of negative refraction at a single interface on the grounds that it violates causality, an investigation is performed using two-dimensional transmission-line grids. Containing only passive elements and simulated using a standard circuit simulator, the transient response of an obliquely incident Gaussian beam on a negative-refractive-index material is examined. Phase velocity and group velocity (power) are both shown to refract negatively. The refracted beam front, however, is no longer parallel to the phase fronts. In order to remain causal, it must always maintain a positive angle with respect to the interface. This front is called the group front, and despite facing in a different direction, it propagates in the direction of the group velocity. The effect of adding a second interface to the simulation is also shown. In this case, the wave emerging from the negative-refractive-index slab is parallel to the incident wave.