## 3 Results and Discussion  
### Analysis of Inherent Errors in Helical Drilling Systems  
The error of the helical drilling system is mainly composed of light spot shape error and mechanical rotation error. The laser light spot is not circular after being focused by the objective lens but is an approximate ellipse, as shown in Figure 5a. As the Dove prism \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Average power \(P\) (W) & \(\sim\)20 \\ Repetition frequency \(f_{rep}\) (Khz) & 50–400 \\ Wavelength \(\lambda\) (nm) & 1030 \\ Pulse width \(\tau\) (fs) & 400 \\ Spot diameter \(d\) (mm) & 4 \\ Beam quality factor \(M^{2}\) & \(\leq\)1.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of femtosecond laser. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**C\(\leq\)** & **Si\(\leq\)** & **Mn\(\leq\)** & **P\(\leq\)** & **S\(\leq\)** & **Cr\(\geq\)** & **Ni\(\geq\)** \\ \hline
0.05–0.15 & 1.00 & 1.00 & 0.025 & 0.015 & 20.5–23.5 & Balance \\ Mo\(\geq\) & Cu\(\leq\) & N\(\leq\) & Al\(\leq\) & Ti\(\leq\) & Fe\(\leq\) & Co\(\leq\) \\
8.0–10.0 & 0.50 & - & 0.50 & 0.15 & 17.0–20.0 & 0.50–250 \\ V\(\leq\) & W\(\leq\) & B\(\leq\) & & & & \\ - & 0.20–1.00 & 0.01 & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: Chemical composition of GH3536 nickel-based superalloy (wt %). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Angle of** & & & **Laser** & & **Gas** & **Cleaning** \\
**Wedge-** & **Rotation Rate** & **Output Laser** & **Repetition** & & **Protetive** & **Pressure** & **Fluid** \\
**Shaped Prism** & **(\(\prime\)/min)** & **Power (w)** & **(kHz)** & **Gas** & **(MPa)** & **Fluid** \\ (\(\prime\)) & & & & **(kHz)** & **Gas** & **(MPa)** & **Fluid** \\ \hline
7 & 2000 & 7 & & & & \\
8 & 2500 & 8 & 100 & Compressed & & \\
9 & 3000 & 9 & 200 & air & 0.4 & Ethyl alcohol \\
10 & 3500 & 10 & 400 & air & & \\ \hline \end{tabular}
\end{table} 
Table 3: Drilling parameters of film cooling holes. rotates, the light spot also rotates, at twice the speed of the prism. The pore wall of the processed hole is not always cut by the same position of the light spot circumference. Therefore, the roundness error of the light spot shape will be reflected on the roundness error of the processed hole. The spot shape error is mainly caused by the quality of the laser beam and the lens light. At the same time, the light spot shape is also not consistent at different positions along the cutting trajectory, and this difference can also affect the roundness error of the processed hole. As shown in Figure 5a, there is an obvious difference in the shape of the light spot at 0\({}^{\circ}\) and 180\({}^{\circ}\) positions. The latter is significantly larger than the former. This will cause a change in the width of the cutting path when the light spot rotates, as shown in Figure 5b. The left cutting path is significantly wider than the right cutting path. The more irregular the shape of the spot, the greater the roundness error. The errors above, combined with the rotational errors of the mechanical system, will ultimately affect the roundness error of the processed hole. According to the width variation of the cutting path in Figure 5b, theoretically, if all materials on the cutting path are removed, the roundness error is about 7 \(\upmu\)m. However, because the size of the spot in Figure 5 includes a large heat-affected zone, the actual roundness error will be less than 7 \(\upmu\)m, if only the focus spot size or Rayleigh length range is considered. The size of the heat-affected zone is around 50 \(\upmu\)m. Observing the completely burned-through area of the material in a single spot in Figure 5a, the diameter of the focus spot in this system is about 20-25 \(\upmu\)m; this is consistent with the theoretical value mentioned in Section 2. If the ablation threshold of the material is considered, and the material is strictly removed just at the focal point, the final hole roundness error is about 3-5 \(\upmu\)m in our helical drilling system.