Electric and magnetic losses and gains in determining the sign of refractive index

Within the framework of classic electromagnetic theories, we have studied the sign of refractive index of optical medias with the emphases on the roles of the electric and magnetic losses and gains. Starting from the Maxwell equations for an isotropic and homogeneous media, we have derived the general form of the complex refractive index and its relation with the complex electric permittivity and magnetic permeability, i.e. n=ϵμ, in which the intrinsic electric and magnetic losses and gains are included as the imaginary parts of the complex permittivity and permeability, respectively, as ϵ = ϵr + iϵi and μ = μr + iμi. The electric and magnetic losses are present in all passive materials, which correspond, respectively, to the positive imaginary permittivity and permeability ϵi > 0 and μi > 0. The electric and magnetic gains are present in materials where external pumping sources enable the light to be amplified instead of attenuated, which correspond, respectively, to the negative imaginary permittivity and permeability ϵi < 0 and μi < 0. We have analyzed and determined uniquely the sign of the refractive index, for all possible combinations of the four parameters ϵr, μr, ϵi, and μi, in light of the relativistic causality. A causal solution requires that the wave impedance be positive Re{Z} > 0. We illustrate the results for all cases in tables of the sign of refractive index. One of the most important messages from the sign tables is that, apart from the well-known case where simultaneously ϵ < 0 and μ < 0, there are other possibilities for the refractive index to be negative n < 0, for example, for ϵr < 0, μr > 0, ϵi > 0, and μi > 0, the refractive index is negative n < 0 provided μi/ϵi > μr/∣ϵr∣.