## 3 Results and Discussion  
### Influence of Numerical Aperture on Focusing Situation in Glass  
Optical microscope was used to observe the side view of molten area in the scanning direction of laser beam, as shown in Figure 7. The laser beam was tightly focused inside glass by objective lens of N.A. of 0.65 with spherical aberration correction. The pulse repetition rate, scanning velocity, and pulse energy were 1.0 MHz, 100 mm/s, and 1.5 \(\upmu\)J, respectively. As can be seen from the figure, the starting point of laser absorption was created when an ultrashort pulsed laser beam was tightly focused inside the glass, and the absorption area was expanded by the upward movement of the absorption point. Then, this absorption point was dropped down and shifted to the right side to complete the first cycle of molten area formation, and new starting point of laser absorption was established. The process was repeated periodically, and the welding lines were created. Therefore, molten area was not created by only one laser shot, but by the sum of multiple laser shots [19]. It is considered that the focusing condition such as numerical aperture has a great influence on this phenomenon, which is strongly related to the absorbed energy of the laser beam. The results of ray tracking are shown in Figure 8a,b. The most centralized part in the path of ray tracking was defined as the spot, and the distance between upper intersection point and bottom intersection point in laser beam axis was determined as the elongation distance of spot \(d_{1}+d_{2}\). It can be seen from Figure 8a that \(d_{1}+d_{2}\) was equal to 0 for each numerical aperture in air, and higher numerical aperture resulted in diffusion of laser beam. Figure 8b shows that rays of laser beam of each numerical aperture passed through the interface between air and glass. Then, the angle of ray changed because of a different refractive index at the interface between air and glass, according to the Snell's law. It can be seen that rays of laser beam did not intersect at one point, which indicated that the laser beam could not be focused in glass the same way as that in air. The value of \(d_{1}+d_{2}\) was smaller, when numerical aperture was lower, such as 0.45. This result showed that the distance of high power density in laser beam axis was shorter, and it was difficult to maintain absorption of laser energy. However, the rays of laser beam in laser irradiation direction diffused evidently, when the numerical aperture was higher. Therefore, higher numerical aperture indicated that the focusing situation was more likely to be disturbed. This is because higher numerical aperture such as 0.85 resulted in larger angle of incidence, and thus larger angle of refraction. This phenomenon caused loose focusing situation in glass and thus the reduction of power density. Thus, it can be considered that the spot diameter in glass would be decreased with increasing numerical aperture. N.A. of 0.65 showed superior focusing characteristics when compared with N.A. of 0.45 and 0.85, which enables achievement of stable welding of glass. The measurement results of spot diameter and absorption rate for various numerical apertures are discussed in subsequent sections. Figure 7: Schematic illustration of observation of molten area from side view direction. Figure 9 shows the measurement results of laser spot focused in glass. As shown in Figure 9a, the glass plate of 0.3 mm thickness and monocrystalline silicon wafer of 1.0 mm thickness were placed together by optical contact, and then laser beam was focused and scanned around the interface of these two materials from the glass plate side. The top view of the formed weld bead in the case of N.A. of 0.65 and the definition of spot diameter in glass \(D_{\text{in}}\) are shown in Figure 9b,c shows the measurement results of spot diameter in glass \(D_{\text{in}}\), spot diameter in air \(D_{\text{s}}\) and spot expansion ratio \(D_{\text{in}}/D_{\text{s}}\) with various numerical apertures.