differentiated gaussian pulse

I'm kinda new to FDTD, I would like to know about the differences between Hard Source and Soft Source in FDTD. The book by Dennis Sullivan explained but in a rather confusing approach.

Here is a simple way to look at it

Hard source simply forces the field in a certain region (generally a plane) of the simulation domain to vary with a predefined function of time. The problem with hard source is that waves traveling towards it are reflected off it.

Soft source generally uses a current source as a drive to produce required field. The exact formulation of it is derived from Maxwell's equations. The details can be found in Allen Taflove's book.

A hard source is one that superimposes some wave excitation in the FDTD grid. so you cannot study the reflections or calculate S11 at that point. A soft source is one that caters both the wave excitaion by the mathematical function and the reflection.
In terms of implementation you can set some point in the FDTD grid like:

Ez(i,j)= Eosin(2*pi*f*n*dt);
this would be a hard source. But to implement it in terms of a soft source, you can simply add the reflected component calculated by the FDTD algo.
i.e
Ez(i,j)=Ez(i,j)+Eosin(2*pi*f*n*dt)

This is actually not totally accurate, based on three reasons:
1) The electric field Ez(i,j) is impressed as a current density at the timestep of the magnetic field.
2) The missing operator in front of the sin-function
3) Since the excitation is impressed as a current density, which is coming from the left equations system, we need a minus in front of the amplitude.
this yields for a soft excitation to

Ez(i,j,) = Ez(i,j) - Ehat*sin(2*pi*f*(n+0.5)*dt)
Ehat = (Amplitude*dt)/eps0


Nevertheless, i have some difficulties with the soft excitation, since the impressed excitation function gets differentiated through the curl equations and following time loop, which means, if i superpose a gaussian pulse, i get an differentiated gaussian pulse at the port (see pseudo code). How do i get a "real" gaussian pulse as a soft excitation ?

time loop
[
(1) hx(i,j) = hx(i,j) - (dt/mue0)*[ ez(i,j+1) - ez(i,j) ]
(2) hy(i,j) = hy(i,j) + (dt/mue0)*[ ez(i+1,j) - ez(i,j) ]

(3) ez(i,j) = ez(i,j) + (dt/eps0)*[ hy(i,j) - hy(i-1,j) - hx(i,j) + hx(i,j-1) ]
(4) ez(i,j) = ez(i,j) - (dt/eps0)*exp(-(((n+0.5)*dt-t0)/tw)^2 )
]
After setting the exc-function (line 4), in the next time step the calculation of the magnetic component uses the "new" electric (line 1 and 2) and split the energy to hx(i,j), hx(i,j+1), hy(i,j), hy(i+1,j), the following ez calculation uses this and split it again, the result -> differentiated gaussian impulse :(