# Attosecond Pulse Shaping by Multilayer Mirrors
## 4 Achieved Results and Examples of Multilayer Mirrors for Attosecond Pulses
### Multilayer Mirrors for Attosecond Pulses in the Water Window Soft X-Ray Range
While this method is very suitable for periodic multilayer systems, its accuracy is limited when applied to aperiodic systems with a weak standing wave. On the other hand, one can characterize the spectral phase with two attosecond methods--the RABBITT technique [50; 77] or by attosecond photoelectron streaking spectroscopy--providing access to a full characterization of the reflected attosecond pulse both in amplitude and phase [55; 59]. However, for the water window, the implementation of both attosecond techniques has not been established yet due to a lack of sufficient photon flux from HHG attosecond sources in this spectral range. Here, we analyze and estimate the influence of the layer errors on the reflectivity and the GDD via simulations. The almost perfect  
Figure 8: Design of negatively and positively chirped Cr/Sc mirrors. Multilayer stack designs of the aperiodic Cr (red) and Sc (blue) layers for the introduced (**a**) negative, and (**b**) positive GDD. An arbitrary line with a layer thickness of \(d=1.5\) nm (dashed black) is depicted for comparison reasons. Figure 9: Soft X-ray measurement of aperiodic Cr/Sc mirrors. Soft X-ray reflectivity measurement (brown) and simulation (blue) for the (**a**) negatively and (**b**) positively chirped Cr/Sc multilayer together with the corresponding simulated GDD (dashed red). match between the designed and the measured reflectivity in Figure 9 proves the correct elimination of systematic deposition errors and leaves only random errors to be analyzed. Both the simulated reflectivity and the GDD of the original design of the negatively chirped multilayer were compared with slightly modified designs. The designs were chosen randomly by joggling each layer thickness within predefined limits, and the merit function (MF) of the reflectivity, as well as the corresponding averaged GDD value, was calculated to analyze the quality of reflectivity and GDD simulations:  
\[\text{MF}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(R_{(\text{sim}),i}-R_{(\text{soft x-ray}),i})^{2}}\,, \tag{5}\]  
where \(N\) is the total number of wavelength samplings with integer \(i\) representing the position of the sample, with samples being equally spaced. One hundred designs were averaged per allowed error, and both the reflectivity and the GDD deviations were calculated. Designs with an overall stack height deviation of more than 0.8% were left out due to a discrepancy with parallel profilometry investigations. From Figure 10a, it can be deduced that the average random thickness error is less than 0.5%, corresponding to an average layer thickness error of approximately 0.05 angstrom. Figure 10b shows that the GDD mean value stays nearly constant, independent of the layer errors. The upper limit of the layer errors of 0.5% corresponds to a maximum mean GDD error of approximately \(\pm\)150 as [2]. To summarize the results, we have a standard deviation of \(\approx\)0.024% in the soft X-ray reflectivity and a mean GDD of \(-8090\pm 150\) as [2] for the aperiodic (negatively chirped) Cr/Sc multilayer mirror.