In the x-ray diffraction, only the intensity the light that hits the detector is captured while the phase of the light is not determined. The energy flux of the light is the Poynting vector. For a harmonic planar wave, the magnitude of the Poynting vector $\propto E_0^2\cos^2(\omega t+k)$ where $E_0$ is the magnitude of the electric field, $\omega$ is the temporal frequency, $t$ the time and $k$ the phase shift. The detector average over time the energy flux and obtains $\propto E_0^2$ and the phase information is averaged out. I suppose that is the reason for the phase problem. One can determined the phase from, say, correlation function of the diffracted beams.
However, one may suggest to measure $E_0^2\cos^2(\omega t+k)$ as a function of time and determine the phase easily.
My question: Why do we not do that by sampling at higher than the Nyquist frequency, and use the Nyquist-Shannon sampling theorem to solve for the phase? Is the sampling frequency limited by the detector, as it has to be higher than the x-ray frequency? Or is it because of some noise? If so, what is the source of the noise, thermal? If so, why can't we just average over the same phase point over many time periods? Or is it because of some uncertainty principle akin to the Heisenberg kind?