- #1

Jufa

- 101

- 15

The common answer is that you can decompose this finite signal into an infinite set of infinite harmonic waves with a defined frequency and you then find the actual spectrum of the signal, the same you would find experimentally.

I will now point out what I find that is not consistent about this explanation:

1- One of the conclusions you can reach following this reasoning is that measuring the frequency spectrum of a finite pulse of light actually means finding its Fourier decomposition not within the space covered by the wave, but within all the space (from minus infinite to plus infinite). I find this correspondence between measurement and decomposition in a set of space where the wave does not actually "live" quite counterintuitive.

By "Measuring the spectrum of frequencies of a pulse of light (that must include, at least, one period) " I understand "Measuring the frequency of oscillation of the electric field defined all along the distance covered by the pulse in a certain moment of time." Suppose you create an electromagnetic wave by making an electron oscillate with a certain frequency within an antenna. The wave you get oscillates with the same frequency as the electron does. And if you perform the sort of measurement I described you should get a single frequency (Dirac's Delta). No matter if the antenna has been working for a minute or for decades. I'm not 100% sure of the latter mentioned and it might be the main point of the question. Maybe you cannot know the frequency of the electron oscillation with arbitrary precision (due to quantum phenomena) but, if so, how can the accuracy of this frequency depend on the time the antenna has been working?

2- Let's assume there is not such a quantum reason (I have not found it anywhere). Then what we have is two ways of measuring that provide different results: The diffraction will tell us that the pulse is made of a spectrum of frequencies and the simple measurement of the field oscillation experiment will tell us that the pulse has a single frequency. One would say: "When performing the experiment I described you get a single frequency because the other frequencies are "hided" due to destructive interferences. So the most general experiment you can perform in order to measure all the frequencies of a pulse (even the hided ones) is the one where, for instance, the pulse passes through a diffraction prism. That way waves with different frequency will separate and will not interfere destructively anymore. Then you compute the frequencies of the waves that come out of the prism." And, as far as I know, that is what does happen experimentally. But this suggests me two inconsistencies:

-How do we measure the frequencies that come out of the prism? These waves are finite as well and that means that they must "hide" some sort of spectrum. So the measurement would never end.

-This is the inconsistency that worries me the most: Fourier Theorem and Superposition Theorem all together tell us that the electric field defined by the finite pulse and the infinite sum of infinite harmonic waves are equivalent. That would predict the "prism experiment" already mentioned but it also predicts that you could perform this experiment anywhere, since the extent of the harmonic functions is infinite. That leads to an information transfer faster than c (light speed) which makes no sense. Obviously, the latter does not happen experimentally, which makes me think that there's something about Superposition Theorem or Fourier's Theorem (or even both) that I misunderstand.

Thank you for reading.