Exact solution for permeability of a ferromagnet under parametric bichromatic irradiation

If two parametrically coupled electromagnetic fields are applied to a ferromagnet, a non-linearity of the equation of precession of magnetic moment strongly affects the permeability. If the coupling constant $\vert \beta \vert <1$ between the probe (p) wave and the support (s) wave is small, the permeability $\hat {\mu }(\omega)$, at the frequency of the probe wave $\omega =\omega _p =2\omega _s$, is still monochromatic-like, with the re-normalized resonance spin waves frequency $\Omega _r \to \Omega (1+\beta )$. If a coupling is strong, $\beta \;\,\le -\,\,1$, unusual response effects are possible (the magnetic transparency and profoundly non-monochromatic permeability $\hat{\mu}(\omega)$. Possible optical applications to the homogeneous negative refractive index materials are discussed.