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 \delta=(2q+0.5)\pi (\delta is the initial phase difference of magnetic components h_x and h_y of incident electromagnetic wave, q is integer), which means that it is difficult to realize negative refraction. However, for \delta=2q\pi, \delta=(2q+1)\pi, or \delta=(2q-0.5)\pi, 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 \delta=2q\pi case is similar to \delta=(2q+1)\pi. The only difference between two cases of \delta=2q\pi and \delta=(2q+1)\pi is that the x-direction for \delta=2q\pi corresponds to the y-direction for \delta=(2q+1)\pi, and the y-direction for \delta=2q\pi corresponds to the x-direction for \delta=(2q+1)\pi. The results are valuable for designing and analysing the complex negative refraction of magnetic materials.