## 3 Results and Discussion
### Influence of Numerical Aperture on Molten Area Characteristics  
It is well known that absorption point moved repeatedly in the laser irradiation direction and molten area was created around the focusing point. Thus, based on the discussion of absorption point movement, molten area characteristics such as the state, size and shape by different numerical apertures were investigated in this section. In order to discuss the state of molten area, considering the results of absorption rate mentioned above, laser intensity \(I_{\text{ab}}\) at the focusing point was calculated using Equation (4). \[I_{\text{ab}}=\frac{4E_{\text{in}}A_{\text{ab}}}{\pi d_{\text{s}}^{2}\tau_{ \text{p}}} \tag{4}\]  
where \(E_{\text{in}}\) is the input laser energy, \(A_{\text{ab}}\) is the absorption rate, \(d_{\text{s}}\) is the diameter of laser beam at focal point, and \(\tau_{\text{p}}\) is the pulse duration [30]. These calculation results were shown in Figure 13a, in which the triangular marks show discontinuous creation of molten area, the circular marks show the continuous creation of molten area without crack, and the cross marks show appearance of crack at the bottom tip. Figure 13b shows the example of these different states observed by the optical microscope, in the case of N.A. of 0.45. The discontinuous creation of molten area was observed from the top view at the pulse energy of 2.2 \(\mu\)J, and the crack appeared at the bottom tip of molten area, as shown in cross sectional view at the pulse energy of 2.5 \(\mu\)J. As can be seen from Figure 13a, N.A. of 0.65 can create a molten area without crack by using the highest laser intensity. This phenomenon shows that N.A. of 0.65 can absorb higher laser energy without cracks, which is expected to result in a large size of molten area compared with other N.A values. Therefore, objective lens with N.A. of 0.65 can achieve stable and efficient formation of molten area. The change in size of molten area under different pulse energies by various numerical apertures will be discussed later. Figure 14 shows the optical micro-photographs of the molten area observed from top and side direction for various numerical apertures. As can be seen from the figure, N.A. of 0.65 can achieve a larger size of molten area without crack generation from cross sectional view. However, the cracks were generated at the bottom tip of molten area, when N.A. of 0.45 was used. In the case of N.A. of 0.85, the size of molten area was small compared with N.A. of 0.45 and 0.65, although a laser beam was tightly focused. Next, influence of numerical aperture on the size of molten area was investigated by measuring the height and width of molten area from cross sectional view under different pulse energies, as shown in Figure 15. The size of molten area varied with change of numerical apertures, which was mainly because the location of absorption point was changed by numerical apertures as mentioned above. On the one hand, the size of molten area increased with increasing pulse energy with all numerical apertures, which is in accordance with the previous results that absorption rate increased with increasing pulse energy for all numerical apertures. On the other hand, lower numerical aperture could create a larger molten area, and the width by N.A. of 0.65 was the largest among all numerical apertures. This phenomenon corresponded to the previous results that N.A. of 0.65 showed the highest absorption rate compared with N.A. of 0.45 and 0.85. In addition, it also can be seen that N.A. of 0.65 could be used in a wider range of pulse energies than other N.A. values, which suggests that N.A. of 0.65 was more appropriate for the micro-welding process. As for N.A. of 0.85, there was almost no difference between the value of the height and width of the molten area. This phenomenon indicated that the molten area with N.A. of 0.85 did not elongate along laser beam axis evidently, which required accurate control of focusing position in the micro-welding of glass.