Fresnel coefficients with permanent magnetism

Question

I am familiar with the standard derivation of the Fresnel coefficients for linear magneto-dielectrics (for instance https://en.wikipedia.org/wiki/Fresnel_equations#Derivation). However, I was wondering how to modify those derivation for materials with permanent magnetization.

During the derivation they typically work with a magnetizing field of the form: $$ \boldsymbol{H}= \boldsymbol{H}_{\boldsymbol{k}} e^{i(\boldsymbol{k}\cdot\boldsymbol{r}-\omega t)} \tag{1} $$ This follows from $H=YE$. Basically, the magnetizing field depends linearly on the electric field.

However, to account for permanent magnetization ($\boldsymbol{M}_0$), we should replace if for:

$$ \boldsymbol{H}= \boldsymbol{H}_{\boldsymbol{k}} e^{i(\boldsymbol{k}\cdot\boldsymbol{r}-\omega t)}- \boldsymbol{M}_0 \tag{2} $$ right?

The problem is that this form cannot satisfy the interface conditions (for all $\boldsymbol{r}$, and t). Let us assume that only the transmitted medium has permanent magnetization, then:

$$ (\boldsymbol{H}_i+\boldsymbol{H}_r-\boldsymbol{H}_t)\times \hat{n} = 0 \tag{3} $$

$$ (H_{k_i} + H_{k_r} - H_{k_t} + M_0 e^{-i(k_{\parallel} \cdot r - \omega t)} )\times \hat{n} = 0 \tag{4} $$

How to correctly account for permanent magnetization in the Fresnel coefficients? I am aware some magneto-optic effects like MOKE and MOFE, but those modify the permittivity and permeability, not the interface conditions.

Thank you for your time! :)

Answer 1

If $\boldsymbol{M}_0$ represents the polarization of a truly permanent magnet, then $\boldsymbol{M}_0$ must be independent of $\boldsymbol{H_k}$. Now applying the linearity of Maxwell's equations together with the linearity of the boundary (continuity) conditions at the interfaces it follow that you will have separately the standard Fresnel equations to hold for $\boldsymbol{H_k}$ and the magneto-static continuity equations for the fields of $\boldsymbol{M}_0$.