Exposure Time and Random Noise

In this section, we are looking at the effects of different CCD exposure times on the reconstructed signal. Furthermore, the effects of random noise sources (shot noise and read out noise) will be investigated.

For the calculation of the exposure times, the photon flux of the XUV source was set to 5×10¹¹photons/s which is comparable to the experiments with the synchrotron radiation. In general, the exposure time influences the maximum measurable scattering angle and consequently the shape of the plateaus of the cross-corrections.

Figure 5.5c shows the relative error of the normalized amplitude and the relative error of the phase of the cross-correlation c12. In Fig. 5.5a, two selected scattering patterns (for 15 ms and 155 ms) are shown, with the corresponding cross-correlation displayed in

Fig. 5.5b.

As is expected and can be seen see from Fig. 5.5a, shorter exposure times mean less intensity detected at higher diffraction angles. If we examine the example of low exposure in Fig. 5.5b (15 ms), the noise on the phase plateau is stronger than on the amplitude plateau. The noise stems from the stronger intensity cut-off⁵ in the scattering pattern, increasing the areas of zero intensity around the destructive interference. The relative error of the phase in Fig. 5.5c is less influenced by the noise on the plateau because the values of the plateau gets averaged for the reconstruction. As in the last section, the normalized amplitude’s relative error is more affected by the noise due to the noise contributions of both the material-reference cross-correlation and the normalization cross-correlation.

5Only simulated photon events ≥1 are counted in this simulation

1 ms

Figure 5.6: Simulation of triple slit scattering with shot noise and varying exposure times.

The data points are averaged over 10 different simulations (photonflux 5×10¹¹photons/s, sample-CCD distance 0.125 m). a Zoomed-in (2x) scattering images of triple slits for 1 ms and 136 ms exposure time plotted in logarithmic scale. b Center slice thorough cross-correlations of the three slits. The cross-correlations are described in the caption of Fig. 5.4. The inset in the lower left (upper right) displays a 20 times scaled region of the left (middle) plateau. The upper (lower) panel corresponds to the 1 ms (136 ms) scattering pattern on the left in a. c Relative error of reconstructed phase shift and transmission amplitude to ground truth as function of exposure times.

For exposure times >140 ms, the relative error diverges. Here, the CCD begins to saturate. The reconstruction is very sensitive to this error. Just a few oversaturated pixels in the center of the diffraction pattern will lead to a disturbed reconstruction. The multiplicative function, which degenerates the true diffraction pattern into the saturated one, will lead to a convolution with an ariy-like pattern in real space. This error can be corrected in Fourier space through inpainting of the oversaturated areas (not shown).

For a XUV source with 5×10¹¹photons/s, even an exposure time of 15 ms leads to acceptable values with only 2 % relative error. During our HHG experiment, the photon flux behind the sample was more likely to be in the order of 1×10⁸photons/s (see Ch. 3.2).

An acceptable exposure time would thus be in the order of 10 s.

For the last simulation, the probability that a photon was detected was equal to one within the intensity cut-off determined by the exposure time. Next, we are investigating the effect of shot noise, following the Poisson distribution, on reconstructions with different exposure times. For the shot noise simulation, the shot noise generator from [75] was used.

The relative errors in Fig. 5.6c were averaged over 10 runs each. The error bars indicate the standard deviation of this average. As before with lower exposure times, when the destructive interference minima were not measured smoothly, the relative error diverges.

The influence of the Poisson noise can be seen from the noise on the cross-correlation plateaus. However, compared to the the noise-free data from Fig. 5.5, the average of the values of the correlation plateaus is only affected slightly by the Poisson noise distribution in the Fourier plane.

We now introduce Gaussian noise to the background to simulate the readout noise of

19 ms

Figure 5.7: Simulation of triple slit scattering with readout noise and varying expo-sure times. The data points are averaged over 10 different simulations (photonflux 5×10¹¹photons/s, sample-CCD distance 0.125 m). aZoomed-in (2x) scattering images of triple slits for 1 ms and 19 ms exposure time plotted in logarithmic scale. b Center slice thorough cross-correlations of the three slits. The cross-correlations are described in the caption of Fig. 5.4. The inset in the lower left (upper right) displays a 20 times scaled region of the left (middle) plateau. The upper (lower) panel corresponds to the 1 ms (19 ms) scattering pattern on the left in a. c Relative error of reconstructed phase shift and transmission amplitude to ground truth as function of exposure times.

the CCD (σ = 15 counts), which is comparable to the Gaussian noise from the experiment.

The data points in Fig. 5.7c are averaged over 10 different runs. The error bars indicate the standard deviation of this average. Even though the noise on the correlation plateaus appears stronger than before (Fig. 5.7b), the evenly distributed Gaussian noise can be mostly averaged out in the reconstruction. Compared to the shot noise simulation at 1 ms, the effect of the read out noise has a stronger impact on the relative error. We attribute this to the disturbance of the destructive interference minima by the read-out noise.

CHAPTER 6