## 2 Experimental Design and Methodology
### Response Surface Methodology (RSM)  
In order to investigate the causes of changes in the response variable, the input variables were systematically altered in each experiment. The aim was to develop a  
Figure 1: Schematic of the laser cutting process with all the parameters of this study. Figure 2: The focal point of the laser beam can be determined with the following steps: (**a**) analyzing its position before passing through the laser and (**b**) observing the impact of the laser beam on the acrylic sheet, which reveals the precise location of the focal point [35]. mathematical model that accurately represented the relationship between the response and input variables with the least amount of error [25]. There are various Design of Experiments (DOE) methods available, depending on the type of input variable parameters studied [36]. The current study used Response Surface Methodology (RSM) as the design method, which combines statistical methods and applied mathematics to analyze output variables that are influenced by multiple independent input variables. The experiment was designed to determine the optimal values for three key parameters that initiate the cutting process: laser power, laser speed, and focal plane position (with a maximum laser power of 60 W). A test material made of acrylic in a rectangle shape with 17 different lines was used for the cutting test, using a fixed step between the lines. In Table 2, three independent parameters (laser power, cutting speed, and focal plane position) are considered for the laser cutting test, with variable values ranging from \(-2\) to \(2\). For each parameter, five coded values are taken, with a laser power ranging from 40 W to 60 W with a 5 W interval, a laser speed ranging from 4 mm/s to 16 mm/s, and focal plane positions from \(-2\) to 2 mm (Figure 3). To establish the operational boundaries of each parameter for laser cutting, test samples were fabricated by altering one of the process variables while maintaining a narrow kerf width. The experimental design adopted was the matrix scheme of DOE, as shown in Table 3, with the gas pressure set at a fixed value of 3 bars. The geometrical properties of the kerf width at the top and bottom were measured using an optical microscope from Graticules Ltd. (Kent, UK) at a 70\(\times\) magnification, and the acquired images were evaluated using the ImageJ v1.3 software. Figure 4a displays the 17 experiments of laser cutting on a PMMA sheet with a thickness of 5 mm. The effect of the input parameter variations for the minimum and maximum kerf geometry of the chosen experiments mentioned in Table 3 is depicted in Figure 4b-e. The term \"bulges\" in Figure 4b refers to the sputters and materials flow-out from the cut kerf. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Variable** & **Notation** & **Unit** & \(-\)**2** & \(-\)**1** & **0** & **1** & **2** \\ \hline Laser power & LP & W & 40 & 45 & 50 & 55 & 60 \\ Cutting speed & CS & mm/s & 4 & 7 & 10 & 13 & 16 \\ Focal plane position & FPP & mm & \(-2\) & \(-1\) & 0 & +1 & +2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Independent process parameters with design levels. Figure 3: Variation of the focal plane position on the work piece. Figure 4: (**a**) Actual photograph of the top kerf width for all the samples and the top kerf images with a further magnification of (**b**) #2, (**c**) #5, (**d**) #7, and (**e**) #9.