## 4 Discussion  
### Machining Thresholds  
As the Rayleigh length of the laser is 10 mm and far greater than the thickness of the sample (\(\sim\)1 mm), the beam radius is approximately constant throughout the sample; it is 42.5 um on the upper surface and 42.7 um on the lower surface. Thus, the laser fluence is approximately constant on both surfaces of a given sample. The grooves are thought to have formed on the lower surface due to constructive interference on said surface. To describe this phenomenon in detail, it is assumed that laser absorption within the sample is extremely small and only first-order reflection is considered. Single-crystal diamond is known to be quite transparent at the wavelength used. However, some local  
Figure 6: SEM image of the dark cloud found around the groove on the upper surface of the HPHT sample, machined by a fluence of 10.3 J/cm\({}^{2}\); (**a**) the entire cloud and (**b**) in detail. Figure 7: Raman spectra of the point 10 μm from the edge of the groove. absorption occurs due to the presence of crystal defects and impurities. Absorption \(\alpha\) is calculated as follows:  
\[\alpha\ =1-r-t \tag{1}\]  
where \(t\) is the transmission rate and \(r\) is the reflection rate. The transmission rate of diamond is known to be \(\sim\)70% and the reflection rate is \(\sim\)17% [18]. The assumption of a low absorption rate is shown to be applicable as it is roughly 13%. The light passes through the diamond sample as shown in Figure 8. The transmission rates for the samples used, through the 2 surfaces, were 75.7% for HPHT diamond, and 83.9% for CVD diamond respectively. Figure 8: Schematic of the mechanism of interference at both surfaces (light ray is shown at an angle for illustration) showing (**a**) laser interaction with the sample; (**b**) destructive interference at the upper surface and (**c**) constructive interference at the lower surface. Using the wave theory of light, the laser intensity \(I\) can be described by the following equation. \[I(x,t)=K^{2}=\ E(x,t)\times E^{*}(x,t) \tag{2}\]  
where \(K\) is wave amplitude, \(E\) is the electric field strength with \(E^{*}\) as its complex conjugate, \(x\) is displacement and \(t\) is time so that  
\[I=\ E^{2} \tag{3}\]  
The initial laser intensity is \(I_{0}=E_{0}^{2}\), where \(E_{0}\) is the initial electric field strength. Additionally, laser light differs from ordinary light in that it produces photons of the same frequency, wavelength and phase [19]. Therefore, it can be assumed that the phase is generally constant. In a Gaussian beam, a phase shift (Guoy phase shift) is present near the beam waist, resulting in an increase in the wavelength. However, it is not observable in most experiments using a fundamental Gaussian beam (which was used in the present study) and only becomes significant for higher-order Gaussian modes [20]. Assuming that the light ray reaches the diamond at normal incidence, the following equations stand true for the upper surface. \[I_{\text{ref}}=RE_{0}^{2} \tag{4}\]  
\[I_{\text{in}}=(1-R)E_{0}^{2} \tag{5}\]  
where \(I_{\text{ref}}\) is intensity reflected from the upper surface, \(I_{\text{in}}\) is the intensity within the diamond and given that \(n_{\text{m}}\) is the refractive index of the diamond and \(n_{0}\) is that of air, the reflectance \(R\) can be written as:  
\[R\ =\ \left(\frac{n_{\text{m}}-n_{0}}{n_{\text{m}}\ +\ n_{0}}\right)^{2} \tag{6}\]  
At this surface, destructive interference occurs between the incident ray and the reflected ray. This is because the refractive index of air is less than the refractive index of diamond, so that a phase change of \(\pi\) accompanies the reflection [21]. By the principle of superposition, the incident ray and the reflected ray can be combined to give the intensity \(I_{\text{upper}}=I_{\text{in}}\) at the upper surface. Figure 7:0  
After crossing the upper interface, \(I_{\text{in}}\) passes through the diamond to reach the bottom surface. Here, the following equations are used.