**3. Mechanism of ionizations**

The changes in surface morphology and deviation from the τ1/2 dependence of the damage threshold on pulse duration are predicted by a rapid ionization mechanism. As shown in **Figure 4**, field-induced multi-photon ionization produces free electrons that are then rapidly accelerated by the laser pulse. This could be further described as avalanche ionization and tunnel ionization. As shown in **Figure 3**, by absorbing the incident photons, the kinetic energy of a free electron becomes sufficiently high and a

### **Figure 2.**

*Threshold dependence on incident energy density on laser pulse duration. A square root dependence is postulated for pulse durations between picoseconds and microseconds. The parameter is tissue-dependent and is expressed in J/cm<sup>2</sup> [13].*

### **Figure 3.**

*Interplay of photo-ionization, inverse bremsstrahlung absorption, and impact ionization in the process of plasma formation. Recurring sequences of inverse bremsstrahlung absorption events and impact ionization lead to an avalanche growth in the number of free electrons [19].*

part of the energy may be transferred to the bound electrons by collisions to overcome the ionization potential, and hence, produce two free electrons. This is referred to as "*collisional impact ionization"* [16, 19–22]. Subsequently, the free electrons absorb photons and produce more free electrons from the bound electrons. Such a series of the impact ionization process is called "*avalanche ionization"* [16, 17]. The avalanche ionization strongly depends on free electron density and is assumed to vary linearly with the laser intensity [23]. Its efficiency is determined by competition between energy gain through *inverse bremsstrahlung absorption* and loss in energy through the phonon emission. Avalanche ionization is responsible for the ablation of wide bandgap materials at the laser intensities below 10<sup>12</sup> W/cm<sup>2</sup> .

When mode-locked fs-laser pulses with intensity higher than 10<sup>13</sup> W/cm<sup>2</sup> interacts with dielectrics, initiation of multi-photon ionization (MI) takes place. As shown in **Figure 4**, several (*n*) photons with the energy *hν* having wavelength (λ) coherently strike the bound electron, acting like a single photon of *nhν* (energy) at the wavelength of λ/*n*. It results in significantly higher photon flux (>10<sup>31</sup> cm<sup>2</sup> s 1 ) that allows a valance band electron to be freed. Meanwhile, absorbing several photons until the total energy would exceed the values of ionization potential [24, 25]. This absorption process is achieved through the meta-stable quantum state(s).

In Q-switched nanosecond laser pulses associated with low intensities, the initial process for the generation of free electrons is supposed to be thermionic emission, that is, the release of electrons due to thermal ionization. Rather, energies and temperature are usually higher in the case of Q-switched laser pulses because of the associated increment in threshold energy of ablations. Thus, laser-induced distortions with ns-pulses are often accompanied by nonionizing side effects [18].

At intensities above 10<sup>12</sup> W/cm<sup>2</sup> , multi-photon absorption becomes considerably strong and even the seed electrons are not required to initialize the ionization process [10, 26]. However, tunnel ionization should be considered when the intensities are higher than 10<sup>15</sup> W/cm<sup>2</sup> . The tunneling ionization is the process in which the strong

**Figure 4.** *Schematic representation of multi-photon ionization [17, 18].*

incident field suppresses the coulomb potential to allow the tunneling of a bound electron to a free state. The coulomb potential that describes the interaction between two-point charges acting along the line connecting the two charges can be expressed by the following expression; *Vcoulomb* ¼ *<sup>q</sup>*1*q*<sup>2</sup>*=*<sup>4</sup>*πεor*, where r is the distance between two ions, q1 and q2 represent electric charge in coulombs carried by 1 and 2, respectively, and ε<sup>o</sup> is the electrical permittivity of the space. In solids, for example, the bound electron in the valence band is excited through either multi-photon or tunneling to the conduction band and becomes quasi-free.

Keldysh developed the theory that describes the ionization of electrons in condensed matter by intense laser fields [27]. Keldysh parameter used to predict the mechanism that plays a significant role in the ionization process is defined as:

$$\chi = \sqrt{\frac{m o\_o^2 c n\_0 e\_o E\_{gap}}{e^2 I}}\tag{1}$$

where ω<sup>o</sup> denotes the incident laser light frequency, *m* is the reduced mass of the electron, and Egap is the band gap of the material. γ can be qualitatively viewed as the ratio of the incident laser frequency and field strength *(I)* [28]. For relatively weak fields with high frequencies (large γ), multi-photon ionization is more important since electrons have less time to tunnel through the only moderately suppressed coulomb potential than in the small γ case (strong field with low frequency).

### **3.1 Laser-induced breakdown and plasma formation**

When obtaining power densities of the incident laser field (>1012 W/cm<sup>2</sup> ) is equivalent to or more than the coulomb field that binds the electron to its ionic core, the atomic coulomb force of similar magnitude was exerted on to the valance band electron and can be excited to a free state. The recurring sequence of *inverse bremsstrahlung* events and impact ionization leads to an avalanche growth in the number of free electrons. Meanwhile, the irradiance should be high enough to compensate for the losses of free electrons through diffusion out from the focus and through recombination [15, 16, 19, 20]. The energy gain at the vicinity of focus was more rapid than the energy loss by collisions with surrounding particles in media occurring without simultaneous absorption of a photon. These "cascade ionizations" in dielectrics followed by plasma formation leads to a phenomenon called *laser-induced optical breakdown* (LIOB), which can result in permanent material modification. Net free electron density is assumed to saturate at the critical density, at which modification/ ablation of material takes place. For femtosecond lasers, critical density is selected as the free electron density at which plasma oscillation frequency equals to the laser frequency, and can be expressed as:

$$m\_{cr} = \frac{\pi m\_e c^2}{e^2 \lambda^2} \tag{2}$$

where *me* is the electron mass, *c* is the speed of light, e is the electron charge, and λ is the laser wavelength. *ncr* is about 10<sup>21</sup> cm�<sup>3</sup> for 800 nm wavelength. At critical electron density, transparent dielectric materials become opaque.
