# Surface Plasmon Polariton Triggered Generation of 1D-Low Spatial Frequency LIPSS on Fused Silica
## 3 Results and Discussion
For normal incidence, the LSFL periodicity \(\Lambda_{LSFL}\) equals \(\lambda_{SPP}\), and the orientation of the LSFL is expected to be perpendicular to the polarization of the laser due to the Transverse Magnetic (TM) characteristics of the surface plasmons [20]. For dielectric materials, such as fused silica, the complex refractive index is n = 1.45 + i0 [43], and therefore, the dielectric function \(\epsilon_{m}\) = n\({}^{2}\) is not fulfilling \(e_{m}^{\prime}<-1\). In the case of ultrashort pulse laser irradiation, the dielectric function of the fused silica specimen is altered due to photo- and avalanche ionization [26]. These mechanisms excite electrons from the valance into the conduction band of the material. Above a certain carrier density \(N_{e}\), the dielectric material turns into a transient metallic state [17; 20; 21; 22], allowing the generation of SPP. The new complex dielectric function \(\epsilon_{m}^{*}=e_{m}^{*}+ie_{m}^{*\prime}=\epsilon_{m}+\epsilon_{Drude}\) of the irradiated material is calculated with a Drude model described by Equation (3) [44]. Here, \(e\) is the electron charge, \(\epsilon_{0}\) the vacuum dielectric function, \(m_{e}\) the free \begin{table}
\begin{tabular}{c c c} \hline
**v (mm/s)** & \(\Phi\) (J/cm\({}^{2}\)) & \(\Lambda_{LSFL}\) (nm) \\ \hline
80 & 1.72 & 932 \\
100 & 1.77 & 948 \\
120 & 1.79 & 950 \\
140 & 1.86 & 964 \\
160 & 1.91 & 962 \\
180 & 1.96 & 962 \\
200 & 2.03 & 980 \\
220 & 2.10 & 980 \\
240 & 2.27 & 977 \\ \hline \end{tabular}
\end{table} 
Table 1: Periodicities of the one-dimensional LSFL for different fluences and scanning speeds. Periodicities are determined via transferring the SEM pictures in the Fourier spectra. Figure 3: (**a**) AFM image of one-dimensional LSFL for scanning with v = 140 mm/s and \(\Phi\) = 1.86 J/cm\({}^{2}\); (**b**) corresponding modulation profile revealing an average depth of about 260 nm and a modulation profile resembling a \(\left|\sin(\mathrm{x})\right|\) function. electron mass, \(\omega\) = 2\(\pi\)\(\mathrm{rc}\)/\(\lambda\) the optical frequency, \(m_{opt}^{*}\) = 0.49 [38] the optical effective mass and \(\tau_{D}\) = 0.4 fs [38] the Drude damping time. \[\epsilon_{Drude}=-\frac{e^{2}N_{e}}{\epsilon_{0}m_{opt}^{*}m_{e}\omega^{2}(1+ \frac{i}{\omega\tau_{D}})} \tag{3}\]  
Figure 4a shows the evolution of the complex dielectric function of fused silica for different carrier densities. While the imaginary part \(\epsilon_{m}^{*\prime\prime}\) becomes greater for increasing carrier densities, the real part \(\epsilon_{m}^{*\prime}\) decreases. For a carrier density of 4.58 \(\times\) 10\({}^{21}\) cm\({}^{-3}\), the real part becomes smaller than \(-\)1, fulfilling \(\epsilon_{m}^{*\prime}<-1\) and thus enabling the generation of SPP. Figure 4b depicts the periodicity of \(\lambda_{SPP}\) for increasing \(N_{e}\), starting at a value of 4.58 \(\times\) 10\({}^{21}\) cm\({}^{-3}\) (\(\epsilon_{m}^{*\prime}<-1\)). Please note, \(\epsilon_{m}^{\prime}\) in Equation (2) is replaced by the real part of \(\epsilon_{m}^{*}\) for this approximation. Obviously, the periodicity of \(\lambda_{SPP}\) is dependent on the excited carrier density, and \(\lambda_{SPP}\) does not cross the critical value of 1030 nm (laser wavelength). Furthermore, the increasing LSFL periodicities for larger carrier densities are also suggested from Table 1 for increasing fluence levels. The marked area in Figure 4b shows the regime of the LSFL listed in Table 1, with 932 nm \(<\)\(\Lambda_{LSFL}\) > 980 nm. The corresponding carrier densities are between 1.13 \(\times\) 10\({}^{22}\) cm\({}^{-3}\) and 1.87 \(\times\) 10\({}^{22}\) cm\({}^{-3}\). Again, the SPP theory describes the LSFL generation to be perpendicular to the laser polarization having a wavelength smaller than the laser wavelength, both confirmed by the periodic structures shown in Figure 1.