Two methods for general laws of reflection and refraction for metasurface with phase discontinuity

Considering the abrupt phase shift at the interface between the two media,the general reflection and refraction laws at the metasurface with the abrupt phase shift are derived by the two methods based on Fermat's principle and the boundary conditions of continuity,respectively.It is found that the tradition reflection and refraction laws are the special cases of the generalized reflection and refraction laws when the phase shift gradient is equal to zero.Based on the generalized laws of reflection and refraction,the occurrence conditions of the total internal reflection and the full transmission are given and the effect of the refraction index and the phase shift gradient are discussed.It is shown that the arbitrary controlling of the electromagnetic waves can be easily achieved.