28 November 2012

The Carvallo paradox and femtosecond lasers - part I

Fourier analysis amounts to writing any signal (including those limited in time) as a sum of infinitely extended harmonic functions. One way of framing this apparent contradiction is the Carvallo paradox:

Since a spectrograph only selects one component of the signal to be analyzed (and this component is infinite in time), it should detect that component both before and after receiving the signal.

The standard answer is that spectrographs have a finite resolution: when selecting light with a given wavelength \(\lambda\), the result is in practice a finite interval \( ( \lambda - \Delta \lambda,  \lambda + \Delta \lambda)\), defining the spectral resolution \( R = \lambda / \Delta \lambda \). Let us assume a (respectable) value \( R = 10^5\) in the visible range, at \(\lambda = 500 \, \text{nm}\).

The uncertainty relation connects the resolution to the lenth of the pulse by: \[ \Delta \omega \Delta t \geq \frac{1}{2} \Rightarrow \Delta t = \frac{R}{2 \omega} \geq 8 \; 10^{-11} \, \text{s} \simeq 100 \, \text{ps}\] where in the second step I used \( \left | \Delta \omega / \omega \right | = \left | \Delta \lambda / \lambda \right |\). A pulse with the spectral sharpness \(\Delta \lambda\) must then be at least 100 ps long.

This is a very short interval for classical spectroscopy measurements. However, using modern optical techniques one can create ultrashort pulses, down to \( \Delta t \simeq 10 \, \text{fs}\). When observing such a pulse at a resolution \( R \) we should then see it as spread out to 100 ps. Where is the error?

More about in in the next part, where I'll also try to give a version of the paradox that is not affected by resolution.


  1. >When observing such a pulse at a resolution R we should then see it as >spread out to 100 ps. Where is the error?

    Answer: there is no error. When observing a 10 fsec pulse at that resolution, we DO see it spread out in time. The method of observation involves diffracting the pulse off of a grating. Gratings are dispersive, both spatially and temporally. There is no paradox here.

    1. Thank you very much for your answer! I was wondering whether:
      1) You could give me a reference on that. In particular, I am very interested in how the pulse length depends on the resolution.
      2) You have a comment on a different version of the paradox:
      where no explicit resolution is defined.