Diamagnetism and the dispersion of the magnetic permeability

Abstract

It is well known that the usual Kramers--Kronig relations for the relative permeability function $μ(ω)$ are not compatible with diamagnetism ($μ(0)<1$) and a positive imaginary part ($\text{Im}\,μ(ω)>0$ for $ω>0$). We demonstrate that a certain physical meaning can be attributed to $μ$ for all frequencies, and that in the presence of spatial dispersion, $μ$ does not necessarily tend to 1 for high frequencies $ω$ and fixed wavenumber $\mathbf k$. Taking the asymptotic behavior into account, diamagnetism can be compatible with Kramers--Kronig relations even if the imaginary part of the permeability is positive. We provide several examples of diamagnetic media and metamaterials for which $μ(ω,\mathbf k)\not\to 1$ as $ω\to\infty$.