# 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  
Time-resolved attosecond experiments are nowadays well established in the sub-120 eV EUV photon energy range [51; 68; 75]. With ever-improving few-cycle laser development toward higher pulse energies, this regime will be extended into the soft X-ray water window spectral range. Using thin filters for attosecond pulse shaping in this energy regime is limited to fixed opening and absorption edges of the used materials and thus restricts the degree of freedom. This leaves multilayer mirrors as the only key components for tailored spectral filtering and shaping of an attosecond water window pulse. For that reason, a negatively chirped aperiodic Cr/Sc multilayer mirror, optimized for reflecting sub-70 as pulses from HHG at a central photon energy of 326.3 eV (Ar L\({}_{1}\)-edge) with a bandwidth of about 30 eV (FWHM), was developed for future resonant attosecond photoionization experiments. A positively chirped mirror was realized for comparison reasons [67]. The two different aperiodic Cr/Sc multilayer mirrors were optimized by the thin film program Optilayer (version 8.85), which is described in Section 2. While the first multilayer mirror was designed for the introduction of an averaged negative chirp of approximately \(-8000\) as [2] to compensate for a possible positive chirp of the high harmonic plateau, a second mirror with similar parameters was designed to introduce an averaged positive chirp of approximately \(+8000\) as [2]. Both mirrors were designed such that their central energy coincides with the L\({}_{1}\)-edge of Ar (326.3 eV) at an incidence angle of 45\({}^{\circ}\). The optimized stack designs are shown in Figure 8, where the negative (a) and the positive (b) designs consist of around 95 individual layers. The designs were chosen according to their robustness of GDD against small layer thickness errors. Typical layer thicknesses are between 1 and 2 nm, with the thinnest layers as small as 0.5 nm. For both designs, a top oxide layer of 1.4 nm Cr\({}_{2}\)O\({}_{3}\) out of 0.3 nm Cr was included in the model. These designs principally prove the large degree of freedom in customizing water window attosecond pulses, both in space (substrate shape and incidence angle) and time, utilizing aperiodic Cr/Sc multilayer mirrors. Soft X-ray reflectometry measurements were carried out at the beamline 6.3.2 of the ALS [41] (advanced light source) in Berkeley to retrieve the reflectivity profile in the energy range for which the mirror designs target an incidence angle of 45\({}^{\circ}\). A comparison of the measured and simulated soft X-ray reflectivity for both chirped mirrors is shown in Figure 9.  
The reflectivity simulations were performed using the self-written Fresnel code from Section 2 and the start designs retrieved from the Optilayer (version 8.85) optimization procedure [44]. An average Nevot-Croce roughness factor of \(\sigma=0.49\) nm was retrieved from the simulations to account for interfacial imperfections. Both the central energy and the side peaks of the measurements and the designs agree very well and indicate a nearly perfect experimental implementation of the simulated multilayer stack designs of Figure 8. Also shown is the calculated evolution of the GDD within the reflectivity bandwidth of both multilayer systems, indicating an averaged GDD of \(\approx\)\(\pm 8000\) as\({}^{2}\). Both multilayer systems are very similar in terms of peak energy, spectral bandwidth, and peak reflectivity and only differ by the sign of their GDD. Note that the spectral multilayer phase (and thus the GDD) is not accessible by simple reflectivity measurements. Measurements of the spectral phase, using soft X-ray reflectometry, have been previously reported by detecting the standing-wave assisted total electron yield from the multilayer surface as a function of the photon energy around the Bragg peak [57; 76].