## 3 Results
### Micromachining with High-Energy Pulses in Air and with a Water Layer Covering the Sample
The incident peak power, \(P_{in}\), in this case reads 614 MW, the nonlinear index of refraction was taken as \(3\times 10^{16}\) from the recently published data in [49]. The CPSF in this case is approximately 4 MW. For the numerical simulations, the intensity \(A(r,\,L,\,t)^{2}\) and energy density distributions were evaluated at different locations along the beam's propagation direction \(L\). Our calculations were performed for \(P_{in}=100\)\(P_{CR}\) in order to speed up the convergence of the solution with the numerical algorithm used. Accordingly, the input amplitude  
Figure 4: Ablated depth vs. number of repetitions (**a**), for ablation in water and in ambient air. A magnified view from 1 to 700 repetitions is displayed in (**b**). reads as \(E_{0}=100P_{\sigma}/\left(2\pi\rho_{0}^{2}\right)\). The beam waist at the input plane (on top of the layer of water) was taken as 22 \(\upmu\)m.  
As will be shown later, the transformation of the beam, the rate of energy losses (\(dE\)/\(dZ\)), and breakup are largely impacted by the setting of the focal position relative to the surface of the sample (or layer of water). Figure 5 shows how the beam propagates within the water layer given a specific focusing case (on the surface of the sample) and the parameters listed above. However, it is worth pointing out that this case should be regarded as only one of the possible outcomes illustrating the flattening of the Gaussian intensity distribution. For a different set of parameters, the relative lengths of the transformation region of the beam, flat-top/semi flat-top region and beam breakup due to diffraction and self-focusing may be significantly different. In addition, the transformation of the beam is mostly impacted by the initial energy of the pulse as well as the material properties (n\({}_{2}\) and non-linear losses). Moreover, the three different regions displayed in Figure 5 are a result of how fast the energy is lost when the beam enters the non-linear medium (water) and how fast the phase accumulates during propagation. This is not intuitive to predict since this is a non-linear process and every case needs a separate simulation, however if the peak power exceeds P\({}_{CR}\) by less than 10-fold, a multiple filamentation process is likely due to the initial intensity modulation of the beam, in which case the flat-top-semi flat-top intensity distribution does not manifest. This case is not beneficial for the micromachining of deep microchannels. If the peak power is above the stated value, the flattening of the Gaussian distribution may commence and the higher the pulse energy the faster (propagation distance wise) this happens. For the 40 \(P_{CR}\), the flat-top/semi flat-top region starts after approximately 400 \(\upmu\)m of propagation and is sustained for another 350-400 \(\upmu\)m. For the 200 \(P_{CR}\), we predict that the flat-top-semi flat-top intensity distribution would manifest approximately after 25-50 \(\upmu\)m of propagation and would be sustained by another 1000 \(\upmu\)m.  
In the studied case it is important to note that, within a non-linearly absorbing medium, the Gaussian intensity distribution changes very rapidly and resembles a hyper-Gaussian distribution only after approximately 25 \(\upmu\)m within the medium, while the drop in the intensity is approximately 20%. After approximately 100 \(\upmu\)m of propagation, the beam flattens even more and resembles a flat-top distribution but with the peripheral tails present. It is surprising that from this point up to 550 \(\upmu\)m of propagation this intensity distribution is somewhat sustained, while the losses in intensity are only 50%. After this point, the flat-top intensity distribution breaks up due to diffraction and self-focusing (this is confirmed via the numerical model). It was also noticed that the self-focusing term is mostly responsible for the region where the beam breaks up.