# One-dimensional formulation (1D FDTD)

The **one-dimensional formulation** assumes that electromagnetic fields vary only along the $x$-axis. FDTD OpenMP keeps the electric component $E_z$ and magnetic component $H_y$, with all other components zero. Both retained components are transverse to the direction of propagation.

Including electric and magnetic loss terms, the equations are:

$$
\begin{aligned}
\frac{\partial H_y}{\partial t}
&=\frac{1}{\mu}\frac{\partial E_z}{\partial x}-\frac{\sigma_m}{\mu}H_y \\
\frac{\partial E_z}{\partial t}
&=\frac{1}{\varepsilon}\frac{\partial H_y}{\partial x}-\frac{\sigma_e}{\varepsilon}E_z
\end{aligned}
$$

The positive spatial derivatives follow from the chosen field orientation. Electric conductivity describes electric losses; fictitious magnetic conductivity provides matched damping in the [[fdtd-openmp-pml|PML]]. Both vanish in the default vacuum interior.

This model represents plane waves passing through layers perpendicular to $x$. Material changes can alter wave speed and produce reflection or transmission. It cannot represent diffraction around an obstacle or a transverse beam profile.

The [[fdtd-openmp-yee-grid|Yee grid]] turns these coupled equations into neighboring array accesses. Update coefficients incorporate local material properties and loss terms before stepping begins.