## 3 Results and Discussion
### Structural Characterization  
The influence of the surface roughness upon the apparent contact angle can be explained by the Wenzel state and the Cassie-Baxter state. Regarding Wenzel [30], the inherent wettability of a surface will be enhanced by the surface roughness (see also Equation (1)). Increased roughness will decrease the apparent contact angle on an inherently hydrophilic surface and will increase the apparent contact angle on an inherently hydrophobic surface. This is based on the assumption that the entire surface is wetted, i.e., there is no air gap between the surface and the liquid. The increased roughness, which is induced by laser treatment makes a Wenzel-like wetting more likely. This assumption is in accordance with other research groups [21; 26; 29]. In terms of the hydrophobic steady state after 12 days, which wetting state is present--the Wenzel, Cassie-Baxter, or Hybrid Cassie-Wenzel model--is still being debated [34]. \[cos(\Theta_{W})=rcos(\Theta_{Y}) \tag{1}\]  
In the following discussion, we focus on the wetting behavior directly after laser treatment. In the Wenzel equation, r, generally referred to as the roughness ratio, describes the surface roughness defined by the ratio of the true area of the solid surface to its projected area, i.e., r equals 1 for a perfectly flat surface. However, real surfaces exhibit a certain roughness where \(r>1\). According to Equation (1), wetting described by the [PERSON] model will always lead to lower apparent contact angles \(\Theta_{W}\), as described by [PERSON]'s equation (\(\Theta_{Y}\)) for perfectly flat surfaces. Based on this background, our experimental results of a decreased apparent contact angle upon laser irradiation can be assigned to an enlarged surface generated by laser structuring, i.e., the LSFL, u-ripples, and ablation trenches. To estimate r, we measured the topography using AFM and 3D optical microscopy and determined the modulation length and depth of LSFL, u-ripples, and ablation trenches. Based on the measured topography (c.f. Figure 6), we derive that the surface can be described by either \(|\sin(x)|\) (in case of LSFL) or \(sin(y)\) (in case of u-ripples and ablation trenches) type functions. We, therefore, define the following functions to calculate the real surface area using a numerical approach. Please note that the LSFL are oriented perpendicularly to the u-ripples and ablation trenches (see Figure 2), so the real surface L for LSFL is calculated along the \(x\)-axis, whereas for u-ripples and ablation trenches L is calculated along the \(y\)-axis. \[L_{LSFL}=\int_{0}^{\Lambda_{LSFL}}\sqrt{1+\left(\frac{d_{LSFL}\pi\cos\left( \frac{\pi x}{\Lambda_{LSFL}}\right)}{\Lambda_{LSFL}}\right)^{2}}dx \tag{2}\]  
\[L_{u\text{-}ripple}=2\int_{0}^{\frac{\Lambda_{u\text{-}ripple}}{2}}\sqrt{1+ \left(\frac{d_{u\text{-}ripple}\pi\cos\left(\frac{2\pi x}{\Lambda_{u\text{-}ripple }}\right)}{\Lambda_{u\text{-}ripple}}\right)^{2}}dy \tag{3}\]  
\[L_{ablation}=2\int_{0}^{\frac{\Lambda_{ablation}}{2}}\sqrt{1+\left(\frac{d_{ ablation}\pi\cos\left(\frac{2\pi x}{\Lambda_{ablation}}\right)}{\Lambda_{ablation}} \right)^{2}}dy \tag{4}\]  
Here, \(L_{LSFL}\), \(L_{u\text{-}ripple}\), and \(L_{ablation}\) denote the real length of the surface in the direction orthogonal to the structures. Dividing them by \(\Lambda_{LSFL}\), \(\Lambda_{u\text{-}ripple}\), and \(\Lambda_{ablation}\), respectively, gives the relative extension of the surface by the individual structures. In general, we find that these ratios relate to each other according to  
\[\frac{L_{LSFL}}{\Lambda_{LSFL}}>\frac{L_{u\text{-}ripple}}{\Lambda_{u\text{-}ripple }}>\frac{L_{ablation}}{\Lambda_{ablation}}. \tag{5}\]  
For example, for a fluence of 1.96 J/cm\({}^{2}\) and a pulse overlap of 98%, the LSFL leads to an extension of 20.6%, the u-ripple to 9.5%, and the ablation to 1.3%, respectively. The product of these three ratiosgives the ratio of the true area of the solid surface to its projected area, i.e., r as described in the Wenzel equation.