# Frequency distribution not determined by the temporal variation of the pulse?

1. ### Jano L.

1,213
Frequency distribution not determined by the temporal variation of the pulse???

In his book Modern Optical Spectroscopy, William Parson says

Light from an incoherent source such as a xenon
flash lamp contains a distribution of frequencies that are unrelated to the length
of the pulse because atoms or ions with many different energies contribute to the
emission.

http://books.google.com/books?id=O3... frequencies in Short pulses of light&f=false

This assertion sounds strange if we recall the well-known fact that the Fourier transform of the pulse is the broader the shorter is the pulse.

Maybe there is a hint in the word "incoherent" which he used. This word is used very often in spectroscopy, but I never found a clear and simple definition. My understanding is that incoherent light is "light whose many spectral components have random phases", i.e. light whose E(t) is chaotic. Is this right?

If so, even a chaotic light in a pulse still can be Fourier transformed and the relation between the length of the pulse and the distribution of frequencies holds.

2. ### Cthugha

1,679
Re: Frequency distribution not determined by the temporal variation of the pulse???

You can roughly transform incoherent as used in classical optics as having short coherence time and therefore the relative phase of the components will randomize quickly. Taken to the extreme this will mean that the phases of the components are random.

Now you can indeed go ahead and Fourier transform what you see, but the resulting emission "pulse" is not Fourier-limited. In other words having some spectral distribution with well defined phases (complete coherence) gives you the shortest possible pulse when Fourier transformed. If you now decrease coherence, the spectral distribution will stay the same, but the pulse duration will become longer and longer until you arrive at CW light for completely incoherent light. So the Fourier transform just gives a lower bound for pulse length.

3. ### Jano L.

1,213
Re: Frequency distribution not determined by the temporal variation of the pulse???

Thank you Ctugha, now I understand. To be sure, I plotted the function
$$E(t) = \sum_k \sin(k \omega_0 t - \phi_k)$$
for 1) $\varphi_k \in (0,2\pi)$ and 2) for $\varphi_k \in (0,\pi)$. In the first case, the function is completely random without any central peak, while the second one has well-defined peak.

So it can be said that the Heisenberg relation gives the width of the pulse made of coherent sine waves, but the wave made of incoherent waves has a peak which is much broader or even unrecognizable.