## 3 Laser Scribing Mechanism and Theoretical Model
### Theoretical Model of Laser Scribing
#### 3.2.1 Theoretical Model of Laser Scribing  
It is assumed that the transfer of laser energy in the material only exists in the direction perpendicular to the workpiece surface (defined as Z direction), and approximately conforms to the one-dimensional heat transfer equation. According to [PERSON]'s law, the laser intensity attenuates and propagates along the depth z inside the material is as follows:  
\[I(z)=(1-R)I_{0}e^{-Az} \tag{1}\]  
According to the basic theory of thermodynamics, to evaporate or vaporize the material with a thickness of \(h\), it is necessary to heat the material with a depth of \(h\) on the surface to the melting or even vaporization temperature. Since the interaction process between laser and materials conforms to the energy conservation, the energy balance of the scribing process at the scribing line can be described as follows [34]:  
\[(1-R)P_{0}=v_{c}b_{c}hp[c_{p}(T_{m}-T_{\infty})+H_{m}+H_{v}]+P_{HL} \tag{2}\]  
where,  
\(R\) is the reflectivity of the material surface;  
\(P_{0}\) is the laser power radiated on the surface of the material;  
\(\rho\) is the material density;  
\(v_{c}\) is the scribing speed;  
\(b_{c}\) is the scribing width;  
\(h\) is the scribing depth;  
\(c_{p}\) is the specific heat capacity of the material;  
\(T_{p}\) is the melting temperature of the material;  
\(T_{\infty}\) is the environmental temperature of the material;  
\(H_{m}\) is the melting enthalpy of the material;  
\(H_{v}\) is the evaporation enthalpy of the material;  
\(P_{HL}\) is the heat conduction loss in the scribing area.  
The left side of Formula (2) is the laser power absorbed by the material on the surface. On the right side of Formula (2) is the sum of the power required to heat the environment temperature to the evaporation temperature, the power required for melting and evaporation at the incision and the heat conduction loss.  
If all the materials at the scribing area are sublimated, the sublimation enthalpy required for the transformation from a solid state to a gaseous state includes melting enthalpy and evaporation enthalpy. The above energy balance formula is effective for the phenomenon of incision.  
Under the irradiation of a laser beam, if the workpiece material is chemically decomposed, \(H_{m}+H_{v}\) is replaced by the specific enthalpy of decomposition. At this time, the sublimation temperature, evaporation temperature \(T_{v}\), is replaced by the decomposition temperature \(T_{z}\).  
In this theoretical model, it is assumed that the laser scribing line is the same width of the upper and lower scribing seams of the section. In the actual scribing process, the section is generally a conical groove, and the actual cribing width in Formula (2) can be approximately \(0.5\times b_{c}\).  
The following relationship between ablation depth and power density threshold is as follows:  
\[(1-R)P_{th}=\rho v_{c}b_{c}hc_{p}(T_{v}-T_{\infty}) \tag{3}\]  
\(P_{th}\) is the power density threshold.  
When the laser energy irradiates the material surface and is partially absorbed by the material, the energy of the part greater than the ablation threshold makes the material evaporate or even vaporize with thickness \(h\), the energy balance relationship is as follows:  
\[(1-R)(P_{0}-P_{th})=v_{c}b_{c}h\rho(H_{m}+H_{v}) \tag{4}\]  
The above formula can be transformed as follows:  
\[h=\frac{(1-R)(P_{0}-P_{th}-P_{th})}{v_{c}b_{c}\rho(H_{m}+H_{v})} \tag{5}\]  
In general, the laser output power is equal to the power of the material surface, that is \(P_{out}=P_{0}\). \(P_{out}\) is the laser average output power. When the material is assumed to be isotropic, the ablation power density of the material \(P_{th}\), the material density \(\rho\), the melting enthalpy of the material \(H_{m}\) and the evaporation enthalpy of the material \(H_{v}\) are constants. Therefore, the ablation depth \(h\) is determined by the material absorptivity \(R\), laser power \(P_{0}(P_{out})\), scribing speed \(v_{c}\), which is based on Formula (5).