## 2 Materials and Methods
### Experimental Procedure and Analysis Methods  
Figure 3 shows the welding path and five sectioning locations used to examine the transverse cross-sections. In order to increase the welding length within the limited chamber space, the welding path was designed to be an Archimedean spiral shape by combining linear translational and rotational motion. The laser beam location can be expressed using the following equation:  
\[x=(r_{0}\ -\ vt)\cos\omega t \tag{1}\] \[y=(r_{0}-vt)\sin\omega t\qquad(0\ s\leq t\leq 90\ s)\]  
where \(r_{0}\) is the x-coordinate of the initial location (60 mm), \(v\) is the velocity of the linear translation stage (0.333 mm/s), and \(\omega\) is the angular velocity of the rotational stage (0.349 rad/s). Since the welding speed decreases over time, the laser power linearly decreases over time to maintain the energy per unit length at 23.9 J/mm (Figure 4). The total welding length was 1.41 m, and the total welding time was 90 s.  
Figure 3: Welding shape and sectioning positions (a–e). Figure 2: The two mechanisms used to prevent fume adhesion to the coupling glass: (**a**) the local pressure difference across the aperture and (**b**) the trajectory change of fume particles. In the experiments used to analyze the effect of process parameters on the coupling glass contamination and penetration depth, the inner diameter of the cylinder gas nozzle (D\({}_{\text{nozzle}}\)) and the shielding gas flow rate were set at three and four levels of control parameters, respectively. Each level of control parameter and the names of the conditions are given in Table 1. The maximum shielding gas flow rate was chosen by considering the maximum pumping rate of the vacuum pump. In order to visualize and quantify the contamination during the VLBW process, the transmittance of the coupling glass was measured using a spectrum transmission meter, which has an IR peak wavelength of 940 nm and a resolution of 0.1%. The wavelength selected was the highest wavelength available in the spectrum transmission meter and the wavelength nearest to the laser wavelength (1070 nm). Transmittance was measured within the process laser transmission region, the diameter of which was 10 mm (Figure 5a). A total of 41 transmittance values were recorded at 1 mm intervals in eight radial directions from the center, as shown in Figure 5b. The measured transmittance was converted into a contamination index, as follows:  
\[\text{Contamination Index [\%]}=1-\text{Transmittance [\%]} \tag{2}\]  
Figure 5c indicates the analysis domains that were divided to define the spatial distribution of contamination. If the welding time rises vertically, the center of the coupling glass will be the most contaminated, and the degree of contamination will be lower toward the periphery. Therefore, it was expected that the contamination of the coupling glass by the welding \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **D\({}_{\text{nozzle [mm]}}\)** & **Flow Rate [I/min]** \\ \hline D0\_F0.0 1 & - & 0.0 \\ \hline D4\_F0.5 & & 0.5 \\ D4\_F1.0 & & 1.0 \\ D4\_F2.0 & & 2.0 \\ D4\_F4.0 & & 4.0 \\ \hline D6\_F0.5 & & 0.5 \\ D6\_F1.0 & & 1.0 \\ D6\_F2.0 & & 2.0 \\ D6\_F4.0 & & 4.0 \\ \hline D8\_F0.5 & & 0.5 \\ D8\_F1.0 & & 1.0 \\ D8\_F2.0 & & 2.0 \\ D8\_F4.0 & & 4.0 \\ \hline \hline \end{tabular} 1  
\end{table} 
Table 1: Welding conditions. Figure 4: Variation in laser power over time for constant line energy. standard normal distribution curve. Accordingly, they were divided into three domains, such as sigma zone. A domain was set every 1/3 r\({}_{\text{i}}\), where r\({}_{\text{i}}\) is the radius of the process laser transmission region, 5 mm. The median value of the contamination indexes in each domain was used to represent contamination within the domain. To analyze the correlation between the contamination index and the welding consistency, the penetration depths of the welding specimens were measured after the 1 st welding trial (W1) and the 10 th welding trial (W10), respectively.