## 3 Analysis of Processing Efficiency
### Laser Power Input (\(P_{in}\))  
Laser intensity distribution can be expressed by a combination of a Gaussian beam profile and a Laguerre polynomial [15]. This expression can be written as  
\[I_{pl}(\rho,\theta)=I_{0}\rho^{l}[L_{p}^{l}(\rho)]^{2}(\cos l\theta)^{2}e^{-\rho} \tag{4}\]  
where \(I_{0}\) is the laser intensity, \(\rho=\frac{2r^{2}}{w^{2}}\) is a parameter related to the laser beam radius (\(w\)) and distance (\(r\)) from the heat source, and \(\theta=\tan^{-1}\frac{y}{x}\) is the angle between the reference direction on the chosen plane. The laser beam radius, which is one half of the beam diameter, was set to 11.5 \(\upmu\)m in this study. The line from the origin, \(r\)\(\left(=\sqrt{x^{2}+y^{2}}\right)\), is the radial distance, and \(L_{p}^{l}\) is the associated Laguerre polynomial of order \(p\) and index \(l\). The Laguerre polynomial is expressed as  
\[L_{p}^{l}(\rho)=\frac{\rho^{-l}e^{\rho}}{p!}\frac{d^{p}}{d\rho^{p}}(e^{-\rho} \rho^{p+1}) \tag{5}\]  
From Equations (4) and (5), the laser power can be obtained and expressed as  
\[P_{laser}=\int_{0}^{2\pi}\int_{0}^{\infty}I_{pl}(\rho,\varphi)rdrd\varphi \tag{6}\]  
From Equation (4), when \(p=0\) and \(l=0\), the Gaussian distribution can be expressed as  
\[I(r)=\frac{2P_{0}}{\pi w^{2}}e^{-\frac{2r^{2}}{w^{2}}} \tag{7}\]  
where \(P_{0}\) is a given laser power by a laser cutting system. Even though the laser power is given by the system, the real laser energy irradiated on the material surface could be different. Therefore, this study calculated a more precise laser power considering its approaching and leaving time. Laser power irradiation is shown in Figure 3. First, at the beginning of the laser cutting process, the laser is approaching the given area (\(2 wl_{g}\)) and the laser power is increased to the certain laser power (\(P_{0}\)). After the laser is moving inside the given area with the given length (\(l_{g}\)), a constant laser power is maintained; then the laser power decreases as the laser is moving further away from the given area.  
Figure 2: Definition of areas for cutting efficiency.  
The given length (\(l_{g}\)) was set to 1 mm in this study. The length is cancelled out when calculating the efficiency. Based on this, the function of the laser power can be expressed as follows:  
\[\int_{w-vt}^{\infty}{\int_{-\infty}^{\infty}{I(x,y)dydx}}=\frac{{P_{0}}}{2}(1- \text{erf}(\sqrt{\frac{2}{w^{2}}}(w-vt))),0<t<t_{a} \tag{8}\]  
\[\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{I(x,y)dydx}}={P_{0}},{t_{a}} \leq t\leq{t_{l}} \tag{9}\]  
\[\int_{-\infty}^{w-vt+z}{\int_{-\infty}^{\infty}{I(x,y)dydx}}=\frac{{P_{0}}}{2} (1-\text{erf}(\sqrt{\frac{2}{w^{2}}}(w-vt+l_{g}))),{t_{l}}<t<t_{f} \tag{10}\]  
where \(t_{a}\big{(}=\frac{2w}{v}\big{)}\) is the approaching time, \(t_{l}\Big{(}=\frac{l_{g}}{v}\Big{)}\) is the leaving time, and \(t_{f}\Big{(}=\frac{l_{g}}{v}+\frac{2w}{v}\Big{)}\) is the final time. The detailed derivation is given in Appendix A. The total laser energy input irradiated in the given area can be calculated by integrating the graph shown in Figure 3 and its expression is shown as follows:  
\[E_{in}=E_{a}+E_{c}+E_{l} \tag{11}\]  
where \(E_{in}\) is the total laser energy, \(E_{a}\) is the laser energy irradiated from the beginning to the approaching time, \(E_{c}\) is the laser energy where the laser is fully irradiating the given area, and \(E_{l}\) is the laser energy irradiated from the leaving time to the final time. Finally, the laser power input can be calculated by dividing the total laser energy by the working time.  
\[P_{i}=\frac{E_{in}}{t_{f}} \tag{12}\]  
Figure 3: Description of laser power irradiation with a laser power of 250 W, speed of 5000 mm/s, and distance of 100 μm.