## 3 Optimization with Design of Computer Experiments  
In this study, the aim of design optimization is to achieve homogeneous laser power density distribution \(I\) at focus plane with laser spot radius \(R_{rms}\) at 300 \(\upmu\)m. The proposed design is a constrained optimization problem. It is defined as a maximum value of \(I\), subject to the constraint that \(R_{rms}\) must be limited at 300 \(\upmu\)m within an allowable tolerance \(\varepsilon\), which can be represented by  
\[Maximum:\;I\;(x,z) \tag{7}\]  
\[subject\;to\;R_{rms}(x,z)-300\leq\varepsilon\]  
To achieve this, we used the DoCE and response surface models. The response surface model includes two types of inputs: controllable variables and uncontrollable variables. The controllable variables were inputs that can be specified freely by engineers. The uncontrollable variables were those that vary according to the environmental uncertainty that the engineers have no direct control. Here, two input design variables, laser power (\(P\)), and laser beam focal location (\(F_{z}\)) were considered controllable variables, while water cooling indicator \(h\) (here, convection coefficient was used for simplification) was an uncontrollable variable. It was because the water cooling indicator had a dynamically varying range controlled by manufacturing process environmental conditions. For the response surface models, a set of experiments was conducted to obtain the laser power density distribution and spot size at focal plane positions under different levels of the input design variables. We adopted a modified central composite design (CCD) [11] for the current simulation-based experiment designs. The modified CCD design can include the main effects of the controllable and uncontrollable variables, the quadratic effects of controllable variables, and interactions between the controllable and uncontrollable variables. The design variables and the related coded levels are summarized in Table 2. The performance responses of laser power density \(I\) and RMS spot radius \(R_{rms}\) at laser beam focal plane were considered as output variables. The thirteen experimental designs by RSM and the results of the different response values computed by the STOP simulations are presented in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Original Level** & **Coded Level** & \(\mathbf{-1.732}\) & \(\mathbf{-1}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\mathbf{1.732}\) \\ \hline \(P\) (kW) & \(x_{1}\) & \(  
A quadratic polynomial approximation model for RSM is used in the form given in Equation (8) so that the second-order interactions (cross-product terms) can be modeled as  
\[ \begin{split} Y_{u}(x,z)=\beta_{0}+\beta_{1}x_{u1}+\beta_{2}x_{u2}+ \beta_{11}x_{u1}^{2}+\beta_{22}x_{u2}^{2}+\beta_{12}x_{u1}x_{u2}\\ +\gamma z_{u}+\delta_{1}x_{u1}z_{u}+\delta_{2}x_{u2}z_{u}+ \varepsilon_{u}\;(u=\;1,2\ldots,\;N)\end{split}  \tag{8}\]  
where \(Y_{u}\) represents the response characteristic of \(I\) or \(R_{rms}\) in the \(u\)th experiment; \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\), \(\beta_{11}\), \(\beta_{22}\), \(\beta_{12}\), \(\gamma\), \(\delta_{1}\), and \(\delta_{2}\) are coefficients to be estimated; \(x_{u1}\), \(x_{u2}\), and \(z_{u}\) represent the laser power \(P\), laser beam focus plane position \(R_{z}\), and water cooling indicator \(h\) in the \(u\)th experiment, respectively; and \(\varepsilon_{u}\) is the random error. With \(N\) experiments, the response surface model in Equation (8) can be further rewritten by matrix notation:  
\[Y(x,z)=\;X\beta+\varepsilon \tag{9}\]  
where  
\[X=\left[ \begin{array}{cccccccc}1&x_{11}&x_{12}&x_{11}^{2}&x_{12}^{2}&x_{11}x _{12}&z_{1}&x_{11}z_{1}&x_{12}z_{1}\\ 1&x_{21}&x_{22}&x_{21}^{2}&x_{22}&x_{21}x_{22}&z_{2}&x_{21}z_{2}&x_{22}z_{2}\\ \vdots&&&&&&\vdots\\ \vdots&&&&\vdots\\ 1&x_{N1}&x_{N2}&x_{N1}^{2}&x_{N2}^{2}&x_{N1}x_{N2}&z_{N}&x_{N1}z_{N}&x_{N2}z_{N }\end{array} \right]\]  
\[Y=\left[ \begin{array}{c}Y_{1}\\ Y_{2}\\ \vdots\\ \vdots\\ Y_{N}\end{array} \right]\beta=\left[