# Pulse dispersion and Fourier transform

According to the theory of Fourier transform, the time domain and the frequency domain contain the same kind of information (if you know the frequency domain, you know the time domain and the other way round).

With respect to the propagation of a short laser pulse in despersive media, we can see that the pulse gets broader in time domain, while the pulse stays constant in frequency domain.

How is it now possible that both domains contain the same kind of information? I mean, how is the broadening in the time domain contained/encoded within the frequency domain?

• It's not clear what material or wave you are talking about, but I interpret those charts to say that some of the signal is taking a longer/shorter path, but there is no frequency change to the signal. – JMLCarter Jan 13 '17 at 17:37
• Sorry, forgot to mention that: I'm talking about a short laser pulse (electromagnetic wave). And the material has a dispersion effect to that pulse, so it has a dispersion coefficient $n$. The form of the wave could for example be Gaussian. – Katie Jan 14 '17 at 0:12

This is your own, most excellent and succinct answer to your own question, "How is it now possible that both domains contain the same kind of information?". The information of a communication or message is knowledge sent that the receiver does not already know. The Fourier transform is a bijective (indeed unitary with respect to the $\mathbf{L}^2$ norm) automorphism on the class of Tempered Distributions, so a tempered distribution (practically, this means anything you're likely to come across in physics including Dirac Deltas and plane waves) wholly defines its Fourier transform (which is also a tempered distribution) and contrariwise. Put simply, there's nothing more you need to know about the function to derive it from its FT and contariwise.