## 2 Materials and Methods
### Modelling of Material Removal Rates and Research on Material Removal Methods  
#### 2.4.1 Investigation of the Material Removal Process  
The energy of a single femtosecond laser pulse follows a Gaussian distribution (as depicted in Figure 4), with a central region where it is concentrated and an edge region where it is less evenly distributed. When a coating material is etched with a femtosecond laser, some of the thermal radiation will be reflected by the material's surface, while the remainder will pass through the material's surface into its interior and gradually absorb. The thermal radiation that is not completely absorbed will penetrate the material and create a transmission. The propagation of heat radiation in a material has a defined penetration depth, or thermal penetration depth, for materials with a particular volume and form [35]. The femtosecond laser can be used to selectively fine-process the coating without affecting the coating substrate if the diamond coating thickness is more than its own thermal penetration depth. Without affecting the coating substrate, the femtosecond laser can be utilized to selectively refine the coating. Due to the narrow pulse width of the femtosecond laser, the diamond coating can be assumed not to transfer energy to the lattice and the lattice temperature remains constant under the action of the femtosecond laser [36]. However, temperature impacts cannot be avoided, and the coating's inside component grows. Inside the coating, multiphoton ionization and free electron heating take place. On the surface and bottom layers, laser pulses excite electrons by multiphoton ionization, the initial formation of electrons in a few femtoseconds to obtain a high kinetic energy, with a certain kinetic energy of free electrons can diffuse in the material to a certain depth, causing strong ionization, which turns the substance into a dense plasma [37]. The material can quickly acquire a significant number of high\(-\)temperature free electrons due to multiphoton ionization. Some surface\(-\)ejected  
Figure 3: Schematic of the femtosecond laser ablation of the nanocrystalline diamond coating: (**a**) femtosecond processing system and (**b**) laser scanning of the nano-crystalline diamond coating. material collects in the vicinity as a result of a shock wave or high pressure produced by a fast rise in temperature in a confined volume. The area of the boiling zone and the volume of material ejected both grow as the incident laser energy rises, along with the production of plasma. The interaction between the laser and plasma causes a phase transition, which results in the removal of the material. The ablation threshold for the femtosecond laser etching of diamond coatings is clearly defined, and local energy deposition can be carried out fast and effectively while the material is being treated. When the actual energy\(-\)flux density exceeds the threshold energy density and the energy deposition reaches a particular level, ablation takes place, irreversibly altering the structure of the material. Therefore, it is crucial to evaluate the correlation between the femtosecond laser energy density and ablation threshold. A femtosecond laser pulse's energy has a Gaussian distribution, and the energy\(-\)flux F at various points inside a pulse can be written as  
\[\mathrm{F}=\mathrm{F}_{0}\mathrm{exp}\left(-\frac{8\mathrm{r}^{2}}{\omega_{ \mathrm{d}}^{2}}\right) \tag{1}\]  
where r is the distance from the measured position to the center of the light spot and F\({}_{0}\) is the peak energy\(-\)flux. Compared to single-pulse ablation [38], the energy density F\({}_{\mathrm{th}}\) (N) of multi-pulse ablation has a cumulative impact and is denoted by  
\[\mathrm{F}_{\mathrm{th}}(\mathrm{N})=\mathrm{F}\cdot\mathrm{N}^{\mathrm{S}-1} \tag{2}\]  
where S is the accumulation factor and N is the number of pulses.