## 2 Theory of Isofrequency Surfaces in Bianisotropic Materials
\begin{array}{ccc}\overline{\epsilon}_{33}& \overline{X}_{33}\\ \overline{Y}_{33}&\overline{\mu}_{33}\end{array} \right)^{-1}\left( \begin{array} []{ccc}\overline{\epsilon}_{31}&\overline{\epsilon}_{32}&\overline{X}_{31}& \overline{X}_{32}\\ \overline{Y}_{31}&\overline{Y}_{32}&\overline{\mu}_{31}&\overline{\mu}_{32} \end{array} \right)\left( \begin{array}{c}E_{x}\\ E_{y}\\ H_{x}\\ H_{y}\end{array} \right), \tag{7}\]  
while the transverse components satisfy the system of equations  
\[k\left( \begin{array}{cccc}0&0&0&-1\\ 0&0&1&0\\ 0&1&0&0\\ -1&0&0&0\end{array} \right)\left( \begin{array}{c}E_{x}\\ E_{y}\\ H_{x}\\ H_{y}\end{array} \right)=k_{0}\left( \begin{array}{ccc}\overline{\epsilon}_{11}& \overline{\epsilon}_{12}&\overline{X}_{11}&\overline{X}_{12}\\ \overline{\epsilon}_{21}&\overline{\epsilon}_{22}&\overline{X}_{21}& \overline{X}_{22}\\ \overline{Y}_{11}&\overline{Y}_{12}&\overline{\mu}_{11}&\overline{\mu}_{12} \\ \overline{Y}_{21}&\overline{Y}_{22}&\overline{\mu}_{21}&\overline{\mu}_{22} \end{array} \right)\left( \begin{array}{c}E_{x}\\ E_{y}\\ H_{x}\\ H_{y}\end{array} \right)=k_{0}\overline{m}\left( \begin{array}{c}E_{x}\\ E_{y}\\ H_{y}\end{array} \right) \tag{8}\]  
\[\overline{m}=\left( \begin{array}{ccc}\overline{\epsilon}_{11}&\overline{ \epsilon}_{12}&\overline{X}_{11}&\overline{X}_{12}\\ \overline{\epsilon}_{21}&\overline{\epsilon}_{22}&\overline{X}_{21}& \overline{X}_{22}\\ \overline{Y}_{11}&\overline{Y}_{12}&\overline{\mu}_{11}&\overline{\mu}_{12} \\ \overline{Y}_{21}&\overline{Y}_{22}&\overline{\mu}_{21}&\overline{\mu}_{22} \end{array} \right)-\left( \begin{array}{ccc}\overline{\epsilon}_{13}& \overline{X}_{13}\\ \overline{\epsilon}_{23}&\overline{X}_{23}\\ \overline{Y}_{13}&\overline{\mu}_{13}\\ \overline{Y}_{23}&\overline{\mu}_{23}\end{array} \right)\left( \begin{array}{ c}\overline{\epsilon}_{33}&\overline{X}_{33}\\ \overline{Y}_{33}&\overline{\mu}_{33}\end{array} \right)^{-1}\left( \begin{array}{ c}\overline{\epsilon}_{31}&\overline{\epsilon}_{32}&\overline{X}_{31}& \overline{X}_{32}\\ \overline{Y}_{31}&\overline{Y}_{32}&\overline{\mu}_{31}&\overline{\mu}_{32} \end{array} \right).\]We rearrange this and introduce the index of refraction operator \(\hat{N}\):  
\[\hat{N}\left( \begin{array}{c}E_{x}\\ H_{y}\\ H_{x}\\ E_{y}\end{array} \right)=n\left( \begin{array}{c}E_{x}\\ H_{y}\\ H_{x}\\ E_{y}\end{array} \right)\hat{N}=\left( \begin{array}{cccc}-\overline{Y}_{21}&- \overline{\mu}_{22}&-\overline{\mu}_{21}&-\overline{Y}_{22}\\ -\overline{e}_{11}&-\overline{X}_{12}&-\overline{X}_{11}&-\overline{e}_{12}\\ \overline{e}_{21}&\overline{X}_{22}&\overline{X}_{21}&\overline{e}_{22}\\ \overline{Y}_{11}&\overline{\mu}_{12}&\overline{\mu}_{11}&\overline{Y}_{12} \end{array} \right). \tag{9}\]  
The characteristic equation for the eigenvalue problem (Equation (9)) is a quartic equation equivalent to Equation (1), which is valid for a generic bianisotropic medium without an assumption of reciprocity  
\[n^{4}-\text{tr}\Big{(}\hat{N}\big{)}\,n^{3}-\xi n^{2}-\zeta n+\text{det}\Big{(} \hat{N}\big{)}=0, \tag{10}\]  
where  
\[\xi=\frac{1}{6}\Big{(}2\,\,tr\Big{(}\hat{N}^{2}\big{)}-3\,\,tr\Big{(}\hat{N}^{ 2}\Big{)}\,tr\Big{(}\hat{N}\big{)}+tr\Big{(}\hat{N}\big{)}^{3}\Big{)}\]  
Consider reciprocal materials. The material parameters are bound by \(\hat{e}^{T}=\hat{e}\), \(\hat{\mu}^{T}=\hat{\mu}\), \(\hat{X}^{T}=-\hat{Y}\), which is true in the transformed coordinates for the elements of \(\overline{\epsilon}\), \(\overline{\mu}\), \(\overline{X}\), \(\overline{Y}\) and for the elements of \(\overline{m}\) in Equations (8) and (9).