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

We present a detailed theoretical analysis on thepossibilities and conditions for negative permeability and negativerefraction occuring in the magnetic materials with both pronouncedmagnetic and dielectric responses to electromagnetic waves. Theresults indicate that the permeability is always positive for$de=(2q+0.5)pi$ ($de$ is the initial phase difference of magneticcomponents $h_{x}$ and $h_{y }$ of incident electromagnetic wave,$q$ is integer), which means that it is difficult to realizenegative refraction. However, for $de=2qpi, de=(2q+1)pi$, or$de=(2q-0.5)pi$, the negative permeability occurs at some range offree procession frequency, which means that the refraction canbecome negative under certain conditions. Further analysis revealsthat for general positive permittivity there are variousopportunities for realizing the negative refraction provided thatsome requirements are met. One concludes also that the refractiveindex for $de=2qpi$ case is similar to $de=(2q+1)pi$. The onlydifference between two cases of $de=2qpi$ and $de=(2q+1)pi$ isthat the $x$-direction for $de=2qpi$ corresponds to the$y$-direction for $de=(2q+1)pi$, and the $y$-direction for$de=2qpi$ corresponds to the $x$-direction for $de=(2q+1)pi$.The results are valuable for designing and analysing the complexnegative refraction of magnetic materials.