## 3 Results and Discussion
### Top Kerf Width  
Table 4 presents the results of the analysis of variance for the top kerf width, which revealed that the laser's focal plane position (FPP) has a significant linear and quadratic effect on the response variable. Additionally, all parameter interactions are significant, as shown in Figure 5. The regression equation obtained is significant, while the lack of fit is not significant. A regression equation is considered good if it is significant and the lack of fit is insignificant. Hence, based on the analysis, the final regression equation in terms of the coded parameter values can be expressed as Equation (1). \[\left(Top\ kerf\ width\right)^{1.19}=-182.25+7.06\times LP+4.49\times CS+0.89 \times FPP-0.07\times LP^{2}-0.19\times CS^{2}-2.56\times FPP^{2} \tag{1}\]  
In Figure 6, the response surface plot for the top kerf width is shown with respect to the input variables. In Figure 6, it can be observed that decreasing the laser focal plane position (FPP) leads to a decrease in the top kerf width. This indicates that, when the laser spot point is located deeper in the specimen, the upper kerf width is reduced. On the other hand, when the focused laser beam is used at the zero level of the FPP, the laser power spreads over a wider surface area, resulting in more heat energy interacting with the specimen and increasing the top kerf width. The figure clearly shows that the maximum top kerf width occurs in the middle (level 0) of the FPP and in the middle level of the cutting speed. The observation of the laser power and cutting speed can be explained with the heat input value mentioned in Equation (4) and Table 3, as follows:  
\[\text{Heat input}=\text{Laser power/Cutting speed} \tag{2}\]  
The increase in laser power and decrease in cutting speed lead to an increase in the heat input and a wider kerf width due to the larger area of material being melted and, also, more bulges. It is important to note that in samples NO. 1, 11, 7, and 17 complete cutting was not achieved, and, therefore, none of the responses could be measured (Table 3). This can be explained using the heat input concept in Equation (2). In these samples, the laser power was too low, the cutting speed too high, and the focal plane position was on or near the top surface, which resulted in insufficient energy from the laser beam for effective cutting. A comparison of the samples No. 11 and No. 14 indicates that, although the laser \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source of Variation** & **Degree of Freedom** & **Sum of Squares** & **Mean Squares** & **F Value** & \(p\) **Value** \\ \hline Model & 200.63 & 6 & 33.44 & 3.62 & 0.0357 \\ A-LP & 0.071 & 1 & 0.071 & \(7.667\times 10^{-3}\) & 0.9320 \\ B-CS & 39.06 & 1 & 39.06 & 4.23 & 0.0669 \\ C-FPP & 12.78 & 1 & 12.78 & 1.38 & 0.2668 \\ AB & 60.73 & 1 & 60.73 & 6.57 & 0.0282 \\ BC & 61.93 & 1 & 61.93 & 6.70 & 0.0270 \\ C\({}^{2}\) & 127.75 & 1 & 127.75 & 13.82 & 0.0040 \\ Residual & 92.43 & 10 & 9.24 & & \\ Lack of Fit & 72.00 & 8 & 9.00 & 0.88 & 0.6318 \\ Pure Error & 20.43 & 2 & 10.22 & & \\ Total & 293.06 & 16 & & & \\ \hline R-Sq = \% 68.46 & & & R-Sq (adj) = \% 49.54 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Revised analysis of variance of the top kerf width. Figure 7 depicts the residual plot for the top kerf width, where the normal probability diagram shows that the response of the top kerf width is randomly scattered around the diagonal line and follows a normal distribution in comparison to the other responses. As a result, the final regression model obtained is a suitable model for predicting and investigating the effects of the parameters in proportion to the other responses. Therefore, the mathematical equation derived from the model is a desirable tool for predicting and investigating the effects of the experimental parameters. From the residuals plot, it appears that there is a moderate negative correlation between the heat input and the top kerf width, as the points tend to cluster in a downward-sloping pattern.