# Attosecond Pulse Shaping by Multilayer Mirrors
## 2 Simulations of Multilayer Systems: Their Theoretical Optical Performance  
A self-written multilayer Fresnel code algorithm (written with the software MATLAB (version 7.8)) is used for reflectivity and transmission calculations. Either the overall reflectivity (mirror) or the transmission (polarizer, filter) is calculated, together with the relevant phase based on the multiple interface reflections and transmissions. The code utilizes the tabulated atomic scattering factor values from [PERSON] and Gullikson [37]. Values above 30 eV are used to calculate the dispersion and the extinction necessary for the Fresnel equations. Realistic imperfect interfaces, top layer oxidation, or interlayer formation like molybdenum silicide [38] are considered to be the stack design demands. We can thus identify various parameters which are used as input for the simulations. These parameters are either tabulated from former characterization measurements or theoretical studies or should be extracted from one's own research studies and different characterization or analysis methods:* **The optical constants**. Tabulated as in [PERSON] and [PERSON] [37], updated self-determined tabulated values at certain absorption edges [39], or directly measured by spectral ellipsometry [40] for performance approximations in other spectral ranges. * **Realistic interface roughness values**. Extracted from soft [41] or hard X-ray reflectometry [42] measurements, as well as cross-section transmission electron microscopy (TEM) [43]. * **Top layer oxidation**. Analysis methods like interface roughness and, additionally, spectral ellipsometry. The thickness loss of the top layer material, as well as the thickness of the native grown oxide layer, should be determined. It is important since those two layers are responsible for a small aperiodicity within the stack, as well as being the first layers the radiation penetrates through--a very important fact in the case of EUV radiation. * **Interlayer formation**. Analysis methods similar to interface roughness. The self-written code is used for simulating the optical performance of periodic multilayer systems or of already calculated aperiodic designs. The calculations of aperiodic designs were previously realized with the software Optilayer (version 8.85). It is based on a Fresnel code coupled to a 'needle optimization' algorithm [44; 45]. This software is mainly used for the optimization of (aperiodic) chirped mirrors in combination with the self-written Fresnel code. A properly chosen start design is the most important prerequisite for a reasonable optimization. Usually, one starts with a periodic multilayer structure which already consists of the material combination best suited for the targeted spectral range. One adjusts the layer thicknesses to the central energy \(E\) of interest at a certain angle of incidence. The period number \(N\) of the start design is chosen according to the required spectral bandwidth \(\Delta E\) and is estimated by:  
\[N=\frac{E}{\Delta E}\,. \tag{3}\]  
Finally, one or more target functions are defined for the algorithm for which the start design should be optimized. Optimizations can, for instance, include a spectrally broader mirror reflectivity, a higher peak reflectivity, or, in case of chirped mirrors, an optimized spectral phase. The optimized spectral phase is being realized by aperiodic layers which introduce a group delay dispersion (GDD), in contrast to a periodic system having a flat phase. The group delay dispersion is the negative second derivative of the spectral phase, (GDD \(=-d^{2}\phi/d\omega^{2}\)). The needle optimization algorithm [44] takes the periodic design and inserts small (needle-like) additional layers at certain positions in the stack. This is defined by the best benefit for the merit function (MF). The thicknesses of these layers are consecutively decreased or increased until their best thickness values are achieved.