## 3 3D Micromachining Results  
Given the experimental results shown in Figure 4, it is possible to inversely derive the hole-arrayed mask pattern required to achieve the desired machining depth of any targeted 3D microstructure. More specifically, the required radius of each and every hole in the mask pattern shown schematically in Figure 2a can be determined by cross-checking the assigned machining depth at the location of each hole with the experimentally determined value of the laser machining rate shown in Figure 4.  
The practical feasibility of the proposed excimer 3D laser machining system was evaluated by patterning three different structures on PC substrates, namely conical, trihedral, and pyramidal. In each case, the mask pattern had dimensions of 12 \(\times\) 12 mm\({}^{2}\) (resulting in a 1.2 \(\times\) 1.2 mm\({}^{2}\) machining area). Furthermore, the square element size was set as 10 \(\upmu\)m in every case. Finally, the targeted feature size of the machined microstructures was given as 50 \(\upmu\)m, corresponding to the base diameter of the conical-shaped microstructure and the baseline lengths of the trihedral and pyramidal microstructures, respectively. For each mask, a square array of microstructures was fabricated with an array pitch of 75 \(\upmu\)m. In every case, the structure height and total machining depth were controlled by the number of laser shots fired during the machining process. The laser fluence is 600 mJ/cm\({}^{2}\), and 100 pulses are used. The 3D excimer laser micromachining method proposed here is a subtractive one, so the desired microstructures' surface profiles are achieved by the variation of machine depth at different position of the sample's surface. Let \(h\left(x,y\right)\) be the machining depth at position \(\left(x,y\right)\) and \(h\left(x,y\right)\) is pre-determined for each chosen microstructure. To design the hole-opening mask pattern as schematically shown in Figure 2a, we need to determine the hole-opening-area-ratio of each discrete square element, that is, the \(HR\) (\(x_{i}\), \(y_{j}\)) for the (\(i\)th, \(j\)th) square element in Figure 2a centering at the coordinates of (\(x_{i}\), \(y_{j}\)). The hole-opening-area-ratio of each discrete square element can then be determined by  
\[HR\big{(}x_{i},y_{j}\big{)}=f^{-1}\big{(}h\big{(}x_{i},y_{j}\big{)}/c_{o}\big{)} \tag{1}\]  
Figure 4: Normalized laser machined rate as a function of hole-area opening ratio when using a hole-arrayed mask patterns with three different square element sizes. where \(f(\cdot)\) is the functional relationship between the normalized machining rate and the hole-area-opening-ratio, as being experimentally determined and shown in Figure 4. Since we are using the 10 \(\upmu\)m element size, the solid line with open square symbols in Figure 4 is used. In Equation (1), the \(f^{-1}(\cdot)\) is the inverse function of \(f(\cdot)\), and \(c_{o}\) is an adjustable constant to cope with the laser fluence and the number of laser shots used in the excimer laser micromachining. Figures 5-7 present the experimental results for the conical, trihedral, and pyramidal microstructures, respectively. Figures 5a, 6a and 7a show the microstructure arrays within the 1.2 \(\times\) 1.2 mm\({}^{2}\) machining areas, while Figures 5b, 6b and 7b show the respective designed machined depth contours. Finally, Figures 5c, 6c and 7c show the actual machined contours of the three microstructures, as measured by a confocal microscope. In Figure 5c,d, it can be seen that the main features of the desired 3D profile are obtained in every case. However, it is noted that the sharp profile features, e.g., points, corners, ridges, and edges, are missing. This finding is reasonable since in the present study, the machining process relies on the optical diffraction effect to achieve the required 3D microstructure profile, so only analog surface profiles can be obtained. Nevertheless, the experimentally obtained microstructures are in good general agreement with the designed profiles.