Summary

based on the combination of both the systematic deviation and the statistical uncertainty, the median of the statistical uncertainty given by the left panel is used as an error bar. In addition, error bars representing a 3σrange (conversion by eq. A.7 in the appendix) are plotted in red. This result shows, that in the gain determination approaches using¹⁰⁹Cd the gain could be determined with a deviation less than 1% (half of the setting granularity) to the simulation truth in more than 99% of all individual histograms.

Fig. 5.44 shows an overview of the results of the analyses ofjk-grids of calibration line spectra with varying intrinsic ADC DNL under the influence of the four described (sec. 5.3.7) binning environments. For this part of the study, the gain determination was based on offset and noise values determined by the analysis (sec. 5.2) of dark frame grids simulated with DNL properties andjk-grid settings identical to the respective grids of calibration line spectra. The plots show the statistical uncertainty of the gain determination, measured by the median absolute deviation (MAD) of the distribution of the relative residuals of the determined gain to the simulation truth.

• The “worst case” is when a calibration to 1 keV photons is approached with ⁵⁵Fe, an intrinsic DNL withσ_DNL=0.3, but ideal bin boundaries are assumed (upper left panel, blue line): In that case, the median absolute deviation of all results is about 1.6% of the simulated gain, which approximates to a 3σ-span of the result distribution of 7.2%. In other words (and assuming that the result distribution is Gaussian in first order): More than 50% of the results show a deviation of at least 1.6% (0.675σ) from the median of all results.

Considering the higher systematic uncertainty of⁵⁵Fe-based calibrations (see above), this would in many cases lead to a selection of a sub-optimal ADC gain setting.

• With decreasing DNL and increasing relative resolution (higher ADC gain setting, higher calibration line energy) the statistical uncertainty decreases in all test cases. The gain determination accuracy is sufficient for the gain calibration in this example: For the combination of¹⁰⁹Cd as source and a gain of 1 kev/LSB as target, a high intrinsic DNL (σ_DNL=0.3) determined with an accuracy of 5% (lower left plot, blue line, second to last data point), the gain can be determined with an accuracy better than 1% in about 99% (3σ) of the individual histograms – taking into account also the smaller systematic uncertainty.

• Similar to the result presented in fig. 5.16 in sec. 5.2.6, in all cases with an instrinsic DNL withσ_DNL>0.1 a determination of the ADC bin boundaries can improve the accuracy as long as the uncertainty on the binning determination is in the order of 5%. If the DNL of the ADC is small (σ_DNL=0.1 or smaller), than the binning must be determined with an uncertainty better than 5% in order to obtain results better than with simply assuming ideal binning.

5. Calibration methods

ideally_known assume_ideal blur 5% blur 10%

0 0.4 0.8 1.2 1.6

2 55Fe, 1 keV, σ_DNL=0.1

55Fe, 1 keV, σ_DNL=0.2 55Fe, 1 keV, σ_DNL=0.3

mad of residual (in % of simulated gain)

ideally_known assume_ideal blur 5% blur 10%

0 0.2 0.4 0.6 0.8

1 55Fe, 0.5 keV, σ_DNL=0.1

55Fe, 0.5 keV, σ_DNL=0.2 55Fe, 0.5 keV, σ_DNL=0.3

mad of residual (in % of simulated gain)

ideally_known assume_ideal blur 5% blur 10%

0 0.1 0.2 0.3 0.4

0.5 109Cd, 1 keV, σ_DNL=0.1

109Cd, 1 keV, σ_DNL=0.2 109Cd, 1 keV, σ_DNL=0.3

mad of residual (in % of simulated gain)

ideally_known assume_ideal blur 5% blur 10%

0 0.05 0.1 0.15 0.2

0.25 109Cd, 0.5 keV, σ_DNL=0.1

109Cd, 0.5 keV, σ_DNL=0.2 109Cd, 0.5 keV, σ_DNL=0.3

mad of residual (in % of simulated gain)

Figure 5.44: Overview of the statistical uncertainty of different combinations of calibration line source, intrinsic (simulated) DNL, knowledge of ADC binning and gain setting. The statistical uncertainty is assessed by the median absolute deviation (MAD). All results are based on offset and system noise values pre-determined by applying the proposed methods (sec. 5.2) using dark frame jk-grids with identical settings and DNL properties.

study was performed under “ideal conditions”, i.e. ideally known ADC binning and offset and system noise values given by the simulation truth. In a second step, the gain determination algorithm was based on offset and system noise values pre-determined by the analysis of dark-frame jk-grids, that were also under the influence of various ADC binning manipulations. Thereby the conditions and the approach for the calibration of the DSSC detector proposed in in sec. 3.3.2 were reproduced and it was demonstrated, that the concept is applicable: The information that can be gathered with dark frame grids can be applied to the system gain characterization. The statistical uncertainty of the results of the gain determination study can be seen as a measure of the overall accuracy of the calibration approach.

It has to be noted, that this study only assesses the accuracy of the gain determination algorithm on a very broad range of gain settings (sec. 3.2.2) and does not compare the final selection of the gain setting (i.e. the gain calibration) with the best possible gain setting given by the simulation truth, as was done so in sec. 5.2.7 for the offset calibration. Also, the study is not based on actual results of the ADC binning determination algorithm ([31, 32]) – the uncertainty of the binning determination has been simulated by Gaussian blurring as described in sec. 5.2.5.

As soon as the DSSC calibration algorithms will be integrated in the framework of the XFEL detector operation software ([63]), these studies will be revisited and also applied to experimental data.

Examples with¹⁰⁹Cd as a calibration line source show a low systematic deviance (< 0.3%) and in many cases the achievable statistical accuracy is sufficient for the calibration of the gain

setting that has a granularity of approximately 1%.

The results obtained with⁵⁵Fe are heavily impaired by the lower relative ADC resolution of the calibration line peak causing significantly higher statistical uncertainties. It also noteworthy, that the determined systematic deviation remains very stable for each combination of calibration line source, detector gain setting and knowledge of the ADC binning. It is possible, that the higher systematic deviation of the gain determination approaches based on the⁵⁵Fe is not only due to the lower relative ADC resolution of the⁵⁵Fe-lines, but due to a worse adaption of the global fit function to the spectral response. If calibration line energies this low (< 10 keV) should be used for the calibration of the DSSC, a refinement of the composition of the global fit shape (sec. 5.3.2) taking into account the signal pile-up and improving the model of the low-energy trough between calibration line and noise peak as well as a further revision of the start value algorithm could lead to an improvement.

In total (16+48)·100·1024 ∼6.5·10⁶ individual histograms have been analyzed in this study. For each individual histogram, a likelihood-fit together with numerical integration over the fit-function was performed. Due to the low number of parameters in the fit function the pure processing time for the entire test field was not more than 36 hours on 32 CPUs⁷. From current experimental results it can be estimated that after the determination of offset and system noise with a dark frame ‘I_ramp– pixel delay’ grid, about 100 different gain and offset setting combinations will remain as "candidates" for the final calibration and will have to be characterized with the proposed method. For the 10⁶ pixels of the DSSC the total computing time of the calibration would therefore be about 23 days. Of course, the process can be sped up easily by parallelization. Using the synchrotron radiation of the PETRA III facility (Hamburg, Germany) or the even higher energy of the EU.XFEL FXE beamline and an X-ray fluorescense target, an individual spectrum with every pixel illuminated can be recorded in about 1-2 minutes [62].

The fit-based method for the gain determination allows for two different approaches:

• The results of the offset and noise characterization be taken into account by fixing the respective fit parameters to the determined values. This approach has been used in the study based on the analysis of simulated spectra in this chapter. By reducing the number of free fit parameters in this way, a high fit accuracy and stability can be achieved.

• The fitting of the calibration line spectra measured with the DSSC can also be performed without prior knowledge of noise or offset. Examples for this approach can be found in the experimental part of this work in ch. 6.

The typical spectral response of DSSC sensor pixels to individual calibration line sources has been measured with high ADC resolution so far with the SPIX setup (e.g. fig. 5.32), operating prototype DSSC DEPFET pixels. The MiniSDD sensor pixels foreseen for Day0-operation have similar properties regarding their spectral response in the lower energy range that is relevant for the gain calibration. The reason is that the characteristic shape of the low-energy trough between noise peak and calibration lines is mainly dominated by charge sharing due to the size and shape of the pixel and its internal electrical potential structure. The separation of the signal electron

7Intel Xeon Processor E5-2630 v3

5. Calibration methods charge cloud between the pixels is dominated by the potential due to the drift rings (sec. 3.1.2).

These structures are implemented almost identical on the two different DSSC sensor concepts.

Currently, the possibility to record calibration line spectra with high ADC resolution with DSSC prototypes and the final DSSC detector ladder systems (sec. 3.1) is being investigated.

With this approach, the characteristic spectral response of the DSSC sensor pixels could be characterized without the need for additional, isolated sensor measurements.