Negative refraction in ferromagnetic materials under external magnetic field： a theoretical analysis

We present a detailed theoretical analysis on the possibilities and conditions for negative permeability and negative refraction occuring in the magnetic materials with both pronounced magnetic and dielectric responses to electromagnetic waves. The results indicate that the permeability is always positive for 5 = （2q ＋0.5）π （5 is the initial phase difference of magnetic components hx and hy of incident electromagnetic wave, q is integer）, which means that it is difficult to realize negative refraction. However, for 5 = 2qπ, 5 = （2q ＋ 1）π, or 5 = （2q - 0.5）π, the negative permeability occurs at some range of free procession frequency, which means that the refraction can become negative under certain conditions. Further analysis reveals that for general positive permittivity there are various opportunities for realizing the negative refraction provided that some requirements are met. One concludes also that the refractive index for δ = 2qπ case is similar to 5 = （2q ＋ 1）π The only difference between two cases of δ = 2qπ and 5 = （2q ＋ 1）π is that the x-direction for δ = 2qπ corresponds to the y-direction for 5 = （2q ＋ 1）π and the y-direction for 5 = 2qπ corresponds to the x-direction for δ = （2q ＋ 1）π. The results are valuable for designing and analysing the complex negative refraction of magnetic materials.