## 3 Results
### Micromachining with High-Energy Pulses in Air and with a Water Layer Covering the Sample  
As stated above, the micromachining process is more efficient when cutting the samples if the water layer is present, however, it should be noted that this is true only when cutting thick (\(>\)400 \(\upmu\)m) samples. In fact, if the thickness is below a certain value, cutting in ambient air is faster as compared with cutting in water. When cutting the samples, a number of passes (repetitions/number of pulses) along the same path are required in order to reach a certain depth. Figure 4 shows how the depth evolves when the samples are cut with a given set of laser parameters. It is worth noting that the data in Figure 4 was acquired with an average power of 2 W, which is a departure from the settings displayed in the other graphs (typically 10 W). This was done because the glass is prone to cracking or shattering at higher average power settings (only when cutting in air) and a direct comparison between the two cases is otherwise not possible, however the same tendencies hold. It is worth mentioning that  
Figure 3: Experimentally acquired ablation crater profiles (**a**) and numerically simulated crater profiles using Equation 4 (**b**). The pulse energy is 160 \(\upmu\)J (fluence = 29 J/cm\({}^{2}\)) in this case. for an average laser power of 10 W the absolute levels displayed in the graph are different. However, given these specific settings, the depth increases faster for ablation in air for the first \(\sim\)250 \(\upmu\)m, and then slows down from that point, whereas, for the cutting in water the slope of the curve changes at a slower rate. Ablation in air follows a logarithmic dependence and shows a saturating effect after a certain depth has been reached. This saturation is caused by the V-shape formation and a resulting drop in fluence as the area of the surface increases. In fact, cutting through the entire 1-mm thick sample is impossible when cutting in air. Since the impinging pulses do not produce efficient ablation, a portion of the energy is still absorbed and manifests as thermal input into the sample. Transparent materials are brittle, and therefore due to the thermal load and tensile stress emerging around the cut area, the sample cracks or shatters [48]. For cutting in water, a logarithmic dependency is still observed, however the saturation in depth does not occur for the cutting of 1-mm thick samples. Interestingly, from the graph provided in Figure 4, from a depth of 400 \(\upmu\)m to 1 mm, the increase is almost linear for cutting in water. Such a dependency suggests that for cutting materials in water, the fluence projected onto the walls and bottom of the groove does not decrease as fast as it does for cutting in air, or is compensated in some way, i.e., the V-shape formation is not as pronounced or is suppressed due to certain phenomena happening while the beam is propagating within the water layer. However, if the depth increases semilinearly it implies that the Gaussian intensity distribution must have been transformed, since the amount of energy does not increase during micromachining. In addition, the transformed distribution must be sustained along the propagation direction for at least 600 \(\upmu\)m given the studied case. It must be noted, that such an outcome (depicted in Figure 4) cannot occur if the spatial intensity distribution is Gaussian, regardless of the beam dimensions, due to the nonzero gradient of the Gaussian intensity distribution. This departure from the conventional case (ablation in air vs. ablation in water) was understood by modelling the propagation of a high-intensity beam within a non-linearly absorbing medium (water) by means of the non-linear modified Schrodinger Equation (2). In this case, the energy of the pulses was taken as 140 \(\upmu\)J (fluence = 25.4 J/cm\({}^{2}\)) at 260 fs and from this the peak power exceeds the critical power for self-focusing (CPSF) by approximately 100-fold.