Assuming a material well-approximated by a real permittivity and permeability, the relationship between permittivity ε ε r , permeability μ μ r and refractive index n is given by n = ± ± ε ε r μ μ r . All known non-metamaterial transparent materials (glass, water,...) possess positive ε ε r and μ μ r . By convention the positive square root is used for n. However, some engineered metamaterials have ε ε r and μ μ r < 0 . Because the product ε ε r μ μ r is positive, n is real. Under such circumstances, it is necessary to take the negative square root for n. When both ε ε r and μ μ r are positive (negative), waves travel in the forward (backward) direction. Electromagnetic waves cannot propagate in materials with ε ε r and μ μ r of opposite sign as the refractive index becomes imaginary. Such materials are opaque for electromagnetic radiation and examples include plasmonic materials such as metals (gold, silver,...).