# Gorilla Glass Cutting Using Femtosecond Laser Pulse Filaments
## 2 Fundamental Concepts  
Femtosecond laser filaments are very thin, and a long string of ionized plasma along the laser beam propagation plane resulting from alternate focusing and defocusing of the femtosecond laser pulse until the peak pulse power remains greater than the critical power required to achieve self-focusing [38]. A femtosecond laser pulse filament is formed when an intense femtosecond laser pulse propagates nonlinearly through a Kerr medium, such as transparent glass material. The overall refractive index of a transparent material is obtained by the following equation [39]. \[n=\left(1+\chi^{(1)}+\chi^{(3)}|\epsilon(\omega)|^{2}\right)^{1/2}=\left(n_{0} ^{2}+\chi^{(3)}|\epsilon(\omega)|^{2}\right)^{1/2}=n_{0}\left(1+\frac{\chi^{(3 )}|\epsilon(\omega)|^{2}}{n_{0}^{2}}\right)^{1/2}\approx n_{0}+\frac{\chi^{(3) }|\epsilon(\omega)|^{2}}{2n_{0}}\equiv n_{0}+\frac{n_{2}}{2}|\epsilon(\omega)| ^{2} \tag{1}\]where \(n_{0}\) is the linear refractive index of the sample glass, \(n_{2}\) is the nonlinear refractive index of the sample glass represented in \(\text{cm}^{2}/\text{W}\), \(\chi^{(1)}\equiv\chi(\omega)\) is the linear susceptibility, \(\chi^{(3)}\) is the third-order nonlinear susceptibility, and \(\varepsilon(\omega)\) is the amplitude of the applied field. Because of the Kerr effect, the refractive index of the glass material is increased by \(\Delta n=n_{2}l\), the value of which depends on \(n_{2}\) of the Kerr medium and the intensity of the laser radiation (\(l\)), represented in \(\text{W}/\text{cm}^{2}\). Due to the third-order nonlinear susceptibility, \(n_{2}\) increases positively with the laser energy in glasses. For the Gaussian laser beam, the highest intensity is obvious at the beam center where the refractive index is highest, and this portion acts like a focusing lens causing self-focusing, which squeezes the laser energy into a tiny focal spot with extended focus position. The critical power of self-focusing can be determined using the following equation [35]:  
\[P_{cr}=\frac{3.77\lambda^{2}}{8\pi n_{0}n_{2}} \tag{2}\]  
where \(\lambda\) represents the wavelength of the incident laser beam. The radius of spot size in any material formed by a Gaussian laser beam can be estimated as follows [36]:  
\[w\approx\frac{\lambda F}{\pi w_{0}} \tag{3}\]  
where \(F\) indicates the focal length, and \(w_{0}\) represents the \(1/\text{e}^{2}\) Gaussian beam radius. We know that defocusing is a natural tendency of light. The repetition of focusing and defocusing prolong the Rayleigh length (\(z_{R}\)), which is responsible for femtosecond laser filamentation. The Rayleigh length of any laser beam is determined as follows [36]:  
\[b=2z_{R}=2\frac{\pi w^{2}}{\lambda} \tag{4}\]  
where \(b\) is the confocal parameter. By controlling the size and shape of beam shaper, placed inside the laser propagation path, we can vary the number of filaments and their orientation inside glass materials. Filament fringes were investigated for a femtosecond laser beam operating at 800 nm wavelength with 120 fs pulse width and 1 kHz pulse repetition rate. To control the number of filaments, the laser energy was varied from 10 to 190 \(\upmu\)J. Figure 1 illustrates the cross-sectional view of the filaments formed inside fused silica glass for different kinds of beam shaper [27]. Figure 1: The cross-sectional view of the digital camera images of the filaments, formed inside fused silica glass, for different kinds of beam shapers. (**a**) A circular pinhole with various diameters: 1.5 mm, 3 mm, 6 mm, 7 mm, and 9 mm; (**b**) rectangular slit with various widths: 1.5 mm, 2.1 mm, 2.5 mm, 3 mm, 3.7 mm, and 4.5 mm. (Reprinted with permission from [27] © The Optical Society.). The conventional femtosecond laser pulse filaments assisted glass cutting technology utilizes a thick Kerr medium such as glass at right angles immediately after the objective lens. This glass is responsible for the formation of filaments in the sample glass. The filament length is highly dependent on the glass thickness.