Fastfourie​rtransform​ation x-axes units

Hi,

I do have question about FFT. Its more of a theoretical question then a coding quest.

Iam computing my own laser spectra using random numbers between Pi and -Pi at sample intervals:

omega_i - omega_i+1 << 2*pi/t ; where t is my theoretical pulse duration of 25 fs.

So i got this spectra of random pi at 700 samplepoints (i need 700) and i do a fft of this signal to get it into the frequency domain i guess(?). After that iam using a gaussianfilter to limit the spectra and afterwards doing a back fft to get it into the time domain(?). I can now use another programm to measure the pulse duration. Now i want to check if the programm is doing it right, by inserting a spectra wich duration i can chose but as you maybe noticed i cant figure out how the x-axes is changing with the fft´s.

so my question: how can i get the x-axes to give me a time domain so i can check the Pulse duration in a plot via FullWidthHalfmaximum. and is it even the right way to do two FFT? Iam thinking about for like 2 days and im not entirely sure anymore.

Thanks again

JT