Finite difference time domain (FDTD) allows to solve differential equations for time evolution. For example, we can analyze ultra-short pulses in free space by solving the Maxwell's equations.
The pulse above is plane wave with gaussian shape in the time domain that propagates in $z$ direction. The wave $E_x$ does not have have shape in $x$ direction it is simple plane wave, $E_x (x,t) = \exp(-t^2/\tau^2 - i \omega_0 t)$)
How does one add transverse dimensions (just $x$ direction) to the FDTD method to solve for Gaussian pulse in space ($x$ direction) and Gaussian pulse in time that propagates in $z$ direction?
How to propagate $$E_x (x,t) = \exp(-t^2/\tau^2 - i \omega_0 t) \exp(-x^2/w_o^2)~?$$