RelBiPermittivity

Type:2-Tensor, or section
Range:[v_1, …, v_9]
Default:[1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0]
Appearance:simple

This parameter specifies the relative magnetic permittivity (the coupling between the electric displacement \VField{D} field and the magnetic field \VField{H}) \tilde{\TField{\varepsilon}}_{\mathrm{rel}} as follows:

\begin{alignat*}{1}
\tilde{\TField{\varepsilon}} &= \tilde{\TField{\varepsilon}}_{\mathrm{rel}} \sqrt{\varepsilon_0\mu_0}
\end{alignat*}

Warning

Usually the relative magnetic permittivity \tilde{\TField{\varepsilon}}_{\mathrm{rel}} is defined relative to the background permittivity and permeability. We do not follow this convention as it is not clear how this convention is applied for general an-isotropic tensors \TField{\varepsilon} and \TField{\mu}.

For general bi-anisotropic materials the material constitutive relations read as

\begin{alignat*}{2}
\VField{D} \phantom{x} &= \phantom{x} \TField{\varepsilon} \VField{E} & + \tilde{\TField{\varepsilon}} \VField{H}  \\
\VField{B} \phantom{x} &= \phantom{x} \TField{\mu} \VField{H} & + \tilde{\TField{\mu}} \VField{E}
\end{alignat*}

Typically the tensors \tilde{\TField{\varepsilon}} and \tilde{\TField{\mu}} are not independent from each other. To preserve reciprocity of Maxwell’s equation it is required that

\begin{alignat*}{1}
\tilde{\TField{\varepsilon}} & =  -\tilde{\TField{\mu}}^{\mathrm{T}}
\end{alignat*}

For a bi-anisotropic material without damping additional to the Ohmic losses we have

\begin{alignat*}{1}
\tilde{\TField{\varepsilon}} & =  \tilde{\TField{\mu}}^{\mathrm{H}}.
\end{alignat*}

Isotropic materials are typically classified by the reciprocity parameter \chi and the chirality parameter \kappa,

\begin{alignat*}{1}
\tilde{\TField{\varepsilon}} = & \left(\chi-i\kappa\right)\sqrt{\varepsilon_0 \mu_0} \\
\tilde{\TField{\mu}} = & \left(\chi+i\kappa\right)\sqrt{\varepsilon_0 \mu_0}.
\end{alignat*}

This material definition introduces no additional losses and is reciprocal for \chi = 0.