\left(\begin{array}{c} \Psi(i,t) \\ \Psi(j,t) \end{array} \right)
\label{twobytwo} &=&\left(
\begin{array}{cc}
{\phantom{+}}\cos \alpha & {\phantom{+}}\sin\alpha
\\ -\sin \alpha & {\phantom{+}}\cos\alpha
\end{array}\right)
\left(\begin{array}{c} \Psi(i,t) \\ \Psi(j,t) \end{array} \right).\end{aligned}$$

Two dimensions {#subsec42}
--------------

Assuming translational invariance with respect to the $z$ direction, the system effectively becomes 2D and the TDME separate into two sets of equations [@BornWolf]. For conciseness, in this section we only discuss the set of equations for the TM-modes. The TE-modes can be treated in exactly the same manner.

The relevant EM fields for the TM-modes in 2D are $\Psi(t)=(\bX_x(x,y,t),\bX_y(x,y,t),\bY_z(x,y,t))^T$, in terms of which TDME Eq.(\[eqn:mtxeqn\]) reads $${\frac{\partial}{\partial t}}\Psi(t) = {\mathcal H}\Psi(t)=
\left(\begin{array}{ccc} 0 & 0 &
          -\frac{1}{{\sqrt{\mu}}}{\frac{\partial}{\partial y}}\frac{1}{{\sqrt{\varepsilon}}} \\ 0 & 0 &
{\phantom{+}}\frac{1}{{\sqrt{\mu}}}{\frac{\partial}{\partial x}}\frac{1}{{\sqrt{\varepsilon}}} \\
-\frac{1}{{\sqrt{\varepsilon}}}{\frac{\partial}{\partial y}}\frac{1}{{\sqrt{\mu}}} &
{\phantom{+}}\frac{1}{{\sqrt{\varepsilon}}}{\frac{\partial}{\partial x}}\frac{1}{{\sqrt{\mu}}} & 0 \end{array}\right)
\Psi(t)\,. \label{eqn:2DTM}$$ We discretize continuum space by simply re-using the one-dimensional lattice introduced above, as exemplified in Fig. \[fig:2DTM\] for the case of the TM-modes. This construction automatically takes care of the boundary conditions if $n_x$ and $n_y$ are odd and yields a real skew-symmetric matrix $H$.

(9.15,5.8) (0.5,0)

In analogy with the 1D case the elements of $\Psi(t)$ are defined by $$\Psi(i,j,t) = \left\{ \begin{array}{lll} Y_z(i,j,t)=\sqrt{\varepsilon_{i,j}}E_z(i,j,t), & \mbox{$i$
even and $j$ even} \\ X_y(i,j,t)=\sqrt{\mu_{i,j}}H_y(i,j,t), & 