An analytical relation for refraction in media having negative permittivity and permeability

NIM is the material with negative permittivity and permeability. The condition of negative refractive at the interface between a positive isotropic material and a NIM material with different lattice groups is discussed. An analytical theory of TE and TM waves that undergo refraction at this interface based on boundary conditions is presented. This example shows how negative and "positive" refraction can occur with the direction of energy current density backward to wavevector. The refraction for different crystalline system is also analyzed. The theoretical analysis shows that "positive" refraction can occur for some special case in a left-hand material.