User:FDominec/spatial dispersion

Review of local medium parameters
Some phenomena observed in the frequency spectrum are in fact consequences of spatial dispersion, such as Doppler broadening of resonance lines in gases.\cite[p. 359]{landau1984electrodynamics} More precisely, these phenomena are primarily dependent on the wave vector $$\mathbf{k}$$, and their expression by means of the temporal spectrum is a mere approximation based on that the dispersion curve usually defines a simple relation between frequency and wave vector.

The above chapter \ref{disp_rel_local_media} presented the customary approximation of local media, where the electric permittivity in Eq. (\ref{eq_eps_loc}) is composed of two terms: In a very similar way, the local magnetic permeability $$\mu^{\rm(Loc)}_r(\omega)$$ also consists of two different components,
 * one caused by the immediate response of vacuum,
 * and another by the electric response of matter.
 * the magnetic response of vacuum,
 * and, if present, the magnetic response of the matter.

The trivial extension of the local medium description to the nonlocal one could consist in redefining both constitutive parameters as functions of frequency and wave vector: $$\varepsilon_r(\omega) \stackrel{?}{\rightarrow} \varepsilon_r(\omega, \mathbf{k})$$, $$\mu_r(\omega) \stackrel{?}{\rightarrow} \mu_r(\omega, \mathbf{k})$$, but this will not be used.

Magnetic effects can be described by electric displacement
Instead, we will show here that the magnetic response can be fully expressed by a certain form of $$\varepsilon_r(\omega, \mathbf{k})$$ dependence on $$\mathbf{k}$$. In the following theory, the spatial-dispersive function $$\varepsilon_r(\omega, \mathbf{k})$$ will consist of The magnetic permeability $$\mu_r$$ becomes a mere constant of
 * the component caused by the immediate electric response of vacuum,
 * the component caused by the electric response of matter,
 * and a new component fully accounting for the \textit{magnetic} response of matter, thanks to a particular shape of its spatial dispersion.
 * the magnetic response of vacuum [as in Eq. (\ref{eq_ce})].

Additionally, the spatial dispersion allows to describe
 * other phenomena that can not be described by the local theory.

Repeating the Maxwell equation that links the magnetic field $$\mathbf{\tilde{H}}$$ with the electric induction $$\mathbf{\tilde{D}}$$,

$$ \nabla \times \mathbf{\tilde{H}} = \frac{\partial \mathbf{\tilde{D}}} {\partial t},\quad\quad\quad\quad\quad\text{(\ref{eq_me4} again)}$$

it is clear that if one defines new pair of vector fields

$$ \mathbf{H} = \mathbf{\tilde{H}} + \frac{\partial\mathbf{X}}{\partial t}, $$

$$ \mathbf{D} = \mathbf{\tilde{D}}  + \nabla\times \mathbf{X}, $$

then Eq. (\ref{eq_me4}) maintains exactly the same form with the new fields, for any differentiable vector field $$\mathbf{X}$$:

$$ \nabla \times \mathbf{H} = \nabla \times \mathbf{\tilde{H}} + \left(\nabla\times \frac{\partial\mathbf{X}}{\partial t}\right) = \frac{\partial \mathbf{\tilde{D}}}{\partial t}+ \frac{\partial(\nabla\times \mathbf{X})}{\partial t} = \frac{\partial \mathbf{D}} {\partial t}, \label{eq_me4sd} $$

because for well behaved functions the temporal and spatial derivatives commute.

With the freedom of choice of $$\mathbf{X}$$, we impose the above mentioned requirement that whole magnetic response of the matter is expressed by the constitutive equation for permittivity. Therefore in spatial-dispersive theory, the constitutive equation for magnetic induction is defined the same as in vacuum:

$$ \mu_0 \mathbf{H} := \mu_0 \mu^{\rm(Loc)}_r \mathbf{\tilde{H}} = \mathbf{B}. \label{eq_mu_sd}$$ When this equation is rearranged into the form similar to \ref{eq_HHsd}, we obtain a prescription for sought $$\mathbf{X}$$:

$$ \mathbf{H} = \mathbf{\tilde{H}} + (\mu^{\rm(Loc)}_r -1)\mathbf{\tilde{H}} = \mathbf{\tilde{H}} + \underbrace{\left(\frac{\mu^{\rm(Loc)}_r-1}{\mu_0\mu^{\rm(Loc)}_r}\right)\mathbf{B}}_{=:\,\partial\mathbf{X}/\partial t}$$

Without loss of generality, we again restrict the discussion to a plane wave (\ref{eq_pw}), thus the time derivative equals to multiplication by $${\mathrm i}\omega$$.

$$ \mathbf{X} = \frac{1}{{\mathrm i}\omega}\left(\frac{\mu^{\rm(Loc)}_r-1}{\mu_0\mu^{\rm(Loc)}_r}\right)\mathbf{B} = \frac{1}{{\mathrm i}\omega\mu_0}\left(1 - \frac{1}{\mu^{\rm(Loc)}_r}\right)\mathbf{B}. \label{eq_Xsd} $$

The new electric displacement $$\mathbf{D}$$ that also accounts for magnetic phenomena is obtained by substitution of Eq. (\ref{eq_Xsd}) into Eq. (\ref{eq_Dsd}):

$$ \mathbf{D} := \mathbf{\tilde{D}} - {\mathrm i}\mathbf{k}\times \mathbf{X} = \mathbf{\tilde{D}} - {\mathrm i}  \frac{1}{{\mathrm i}\omega\mu_0}\left(1 - \frac{1}{\mu^{\rm(Loc)}_r}\right) \mathbf{k}\times \mathbf{B}$$

By means of the other Maxwell equation, the magnetic induction $$\mathbf{B}$$ can be substituted by $$\mathbf{k}\times\mathbf{E} / \omega$$ to obtain an expression that contains only the electric quantities.

$$ \mathbf{D} = \mathbf{\tilde{D}} - \frac{1}{\omega^2 \mu_0}\left(1 - \frac{1}{\mu^{\rm(Loc)}_r}\right) \mathbf{k}\times(\mathbf{k}\times \mathbf{E}) \label{eq_Dsd3}$$