## 5 Results and Discussion  
### Verification and Analysis of Theoretical Model  
According to the five groups of laser scribing parameters in Table 2, the experimental processing is carried out on the laser scribing equipment described in this paper. The depth of different positions of laser scribing under each processing parameter is measured seven times, and the laser experimental data and theoretical model are analyzed according to the measured depth data. The relationship between laser power and scribing depth is as shown in Figure 11a. There is an obvious linear positive correlation between laser power and scribing depth, which conforms to the model of Formula (5). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Frequency/Hz** & **1000** & **2000** & **4000** & **6000** & **10,000** \\ \hline Average power/W & 2.44 & 2.66 & 2.83 & 3.11 & 3.58 \\ Standard deviation/W & 0.013 & 0.013 & 0.0096 & 0.017 & 0.10 \\ \(3\sigma\)/W & 0.040 & 0.040 & 0.029 & 0.050 & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Actual average power under different pulse frequencies (effective measurement time: 2.4 S). Figure 11: Relationship between laser power and speed with scribing depth. (**a**) Relationship between scribing power and scribing depth; (**b**) Relationship between scribing speed and scribing depth. The relationship between laser speed and scribing depth is as shown in Figure 11b. There is an obvious inverse proportionality between laser speed and scribing depth, which also conforms to the model of Formula (5). According to the scribing depth data under different laser power and speed in Figure 11, the corresponding laser power density can be calculated, respectively. The relationship between laser scribing depth and power density can be established, as shown in Figure 12.  
As shown in Figure 12, there is an obvious linear positive correlation between the laser scribing depth and the laser power density, and the calculated trend lines are also very similar. Therefore, the relationship between the laser scribing width and the required laser power density can be expressed by formula:  
\[h\approx 179.49\times\frac{P_{0}}{v_{c}b_{c}}+9.6502\text{ or} \tag{13}\]  
\[h\approx 189.17\times\frac{P_{0}}{v_{c}b_{c}}+7.4170 \tag{14}\]  
This formula is a simplified expression of Equation (5), which is the theoretical model of laser scribing. In the application of laser scribing, when the depth of laser scribing is given, the above formula can be recognized by the NC system. At this time, the laser power can be adjusted adaptively according to the laser speed so as to realize the accurate control of the scribing depth in the NC laser scribing. The data of the testing set were simulated to verify the accuracy of proposed mathematical equations, which is shown in Table 5. The prediction accuracy is defined as follows [35]:  
\[acc=\left(1-\frac{PV-MV}{MV}\right)\times 100\% \tag{15}\]  
where \(PV\) and \(MV\) are the prediction value and measurement value. Figure 12: Relationship between laser power density and scribing depth. (**a**) Relationship between laser power density and scribing depth based on a single factor experimental data of laser power; (**b**) Relationship between laser power density and scribing depth based on a single factor experimental data of laser speed.