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Consider you have a laser that has a given emission spectrum.

Does this emission spectrum physically represents the electromagnetic field it would emit if it was turned on for an infinite amount of time ?

Indeed, if you turn on and off the laser for a time $T1$ or $T2$ the frequencies in the signal you would see would depend on those times because of Fourier transform properties.

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If a peak in an emission spectrum was infinitely narrow then it would mean then a laser made from that transition would be truly monochromatic if the laser was on for infinite time. However no transition in the world is actually infinitely narrow. Every transition in fact has some linewidth so every laser has a finite linewidth. The narrowest lasers have linewidths on the order of $mHz$ so the laser must be on for a half an hour or so to ensure the measured linewidth is not Fourier limited. (A Fourier limited signal means that the width of your signal in Fourier space is “artificially” broadened due to the fact that you haven’t spent enough time measuring it to see the true spectrum).

If a laser had an infinitely narrow linewidth then any measurement of the linewidth of that laser would be Fourier limited because it is impossible to perform a measurement that takes infinite time.

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  • $\begingroup$ Would you say that then it is more a measuring problem than a fundamental one. In the sense if I can only measure my signal for a period $T$, I will be limited in frequency resolution. To find the spectrum of the laser I would have to measure its output for an infinite amount of time. $\endgroup$ – StarBucK Jan 12 '20 at 16:09
  • $\begingroup$ But fundamentally if I solved all the possible equation of motion inside the cavity of the laser I could predict the spectrum $E(\omega)$ it emits. So that this quantity is well defined, it is just that in practice it will be hard for me to check it in a finite time measurement. $\endgroup$ – StarBucK Jan 12 '20 at 16:11
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    $\begingroup$ Yes, you’re first comment is correct. If you can only measure your signal for a time $T$ your measurement of the spectrum will be Fourier limited. Regarding your second comment, it is possible to know (from theory or otherwise) that a signal is very narrow but not be able to confirm that experimentally due to the (Fourier or otherwise) limited spectral resolution of your detector. $\endgroup$ – Jagerber48 Jan 12 '20 at 16:15
  • $\begingroup$ Thank you very much for your answer. $\endgroup$ – StarBucK Jan 12 '20 at 16:18

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