# TMz formulation (2D FDTD)

The **TMz (transverse magnetic to $z$) formulation** reduces Maxwell's equations to three field components: $E_z$, $H_x$, and $H_y$. The magnetic field lies in the $xy$ plane, while the electric field points along $z$. All fields vary with $x$, $y$, and time, but have no variation along $z$. This is the two-dimensional model used by [[fdtd-cuda|FDTD CUDA]].

In a linear, isotropic, lossless medium away from sources, the field equations are:

$$
\begin{aligned}
\frac{\partial H_x}{\partial t}&=-\frac{1}{\mu}\frac{\partial E_z}{\partial y}\\
\frac{\partial H_y}{\partial t}&=\frac{1}{\mu}\frac{\partial E_z}{\partial x}\\
\frac{\partial E_z}{\partial t}&=\frac{1}{\varepsilon}\left(\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}\right)
\end{aligned}
$$

Here $\varepsilon$ is the permittivity and $\mu$ is the permeability. The first two equations update the magnetic field from spatial changes in $E_z$. The third updates $E_z$ from the curl of the magnetic field.

The project approximates these derivatives with neighboring samples on a [[fdtd-cuda-yee-grid|Yee grid]]. Conductivity adds damping through precomputed update coefficients. Inside the PML, $E_z$ is split into directional contributions, $E_{zx}$ and $E_{zy}$, whose sum gives the electric field.