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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?

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Nyquist-Shannon sampling theorem (the same link from the question) states that in order to gather information about a periodic signal using sampling, you must use at least twice the target frequency for sampling in the worst case.

In the case of x-rays, a typical wavelength is $\lambda_x = 0.1$ nm and frequency is $f_x = 3\times{10^{18}}$ Hz. Therefore the sampling frequency should be about $f_s = 6 \times 10^{18}$ Hz $\implies$ $T_s = 1.6 \times 10^{-19}$ s. The fastest available lasers right now are on 30-to-50-femtosecond band in pulsewidth [1] (be aware: this is not even the sampling frequency), and the necessary period for x-ray sampling is about 0.00016 femtoseconds.

So? It is technically impossible to sample x-ray band with today's technology.

Edit: About uncertainty

The uncertainty we would be interested in this system would be the E-t uncertainty, $$σ_E\cdot{}σ_t≥\frac{ℏ}{2}.$$ The energy of these pulses need to be $≥3×10^{−10}$ μJ so that we can significantly distinguish actual pulses than noise, which is already satisfied (pulse energies are about $1-100$ μJ right now).

For x-ray itself (for any light anyway), we always have $σ_E\cdot{}σ_t=h≥\frac{ℏ}{2},$ so no restrictions from uncertainty. (Note that halving the $\sigma_t$ still satisfies the second relation).

These numbers refers to a system at which a specifically manufactured secondary beam is utilized for detection. Now as I said, I currently do not know the capabilities and limits of detectors, so the sampling might be possible with adjusting the detector circuitry, but we cannot even produce controlled light on that frequency using lasers.


References:

[1], Ultrafast fibre lasers, Martin E. Fermann and Ingmar Hartl, Nature Photonics, 2013.

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  • $\begingroup$ As I suspected: writing in my question "Is the sampling frequency limited by the detector, as it has to be higher than the x-ray frequency?". Is that the only reason? $\endgroup$ – Hans May 16 at 4:18
  • $\begingroup$ @Hans I am unaware of limitations on the detector. From E-t uncertainty ($\sigma_E\ \sigma_t\geq\sfrac{\hbar}{2}$), the energy of these pulses need to be $\geq 3\times10^{-10}\ \mu$J so that we can significantly distinguish actual pulses than noise, which is already satisfied (pulse energies are about 1-100 $\mu$J right now. So uncertainty is not a problem so far. The reason we cannot achieve higher frequency lasers is about photonics, and yes, it is because thermal instability (not noise) and noise/aberration from the amplifying media. As I said, detectors nowadays might be unable as well. $\endgroup$ – acarturk May 16 at 10:16

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