# Recent Advances in Laser-Induced Surface Damage of KH\({}_{2}\)Po\({}_{4}\) [PERSON]
## 2 Mechanism of Pulsed Laser-Induced Surface Damage
### Ionization Damage  
#### 2.2.1 Photoionization  
With the improvement of the processing technology of KDP crystal materials, the impurities in KDP crystal materials can be avoided gradually. The study of intrinsic damage gradually became clearer. As a transparent crystal material, KDP crystals without any impurities have relatively low absorption of incident laser energy. Therefore, the impurity-based TE model is no longer applicable [46]. For the laser-induced damage without impurities, [PERSON] et al. [47; 48; 49] proposed that the laser damage of transparent optical crystal materials was caused by the change of energy level in multiphoton absorption, and a locally dense plasma area will be formed in the process of damage. In addition, the birth of high-power density ultrashort pulse laser made rapid ignition become the key for laser driven fusion engineering. The appearance of ultra-short pulse laser was helpful to understand the interaction between laser and transparent optical materials. The intrinsic damage of KDP crystal materials under laser irradiation has gradually become the key point of research. Photoionization refers to the process in which the electrons in the equilibrium state inside the material absorb photons directly and are excited to the excitation state when the laser interacts with optical crystal materials [50]. Actually, the excitation condition for electrons is to ensure that the energy of the incident photon is equal to the band gap between the excited state and the ground state. In general, the electron absorbs only one photon from ground state to excited state. However, if the energy of a single photon is less than this bandgap, the electron cannot be excited [51]. [PERSON] et al. [25] noted the laser intensity and frequency, and proposed a comprehensive theory of photoionization. It is possible that an electron will absorb two or more photons and transition if the width of this band gap is equivalent to the total energy of two or more photons under the intense electric field. This kind of mechanism is called multiphoton ionization regime. Nowadays, the laser electric field has been able to reach a certain level, far exceeding the perturbation limit of crystal materials [52; 53]. This situation means that the external reinforcement field will break the Coulomb well of the bound electrons and excite the free electrons through short tunnel ionization. As a result, there are two different regimes of photoionization, the multiphoton ionization (MPI) and the tunneling ionization (TI). [PERSON] used the following formula to describe the photoionization rate based on the understanding of photoionization. \[W_{PI}(I(t))=\frac{2\omega}{9\pi}\Bigg{(}\frac{\omega m_{e}{}^{\star}}{\hbar \sqrt{\gamma_{1}}}\Bigg{)}Q(r,x)\exp\Bigg{[}-\pi\langle x+1\rangle\frac{K( \gamma_{1})-E(\gamma_{1})}{E(\gamma_{2})}\Bigg{]} \tag{6}\]  
where \(\hbar\) is the reduced Planck constant. \(m_{e}\) represents effective electronic mass. \(\omega\) is the angular frequency of the incident laser. \(K(\gamma)\), \(E(\gamma)\) represent the first and second complete elliptic integral, respectively. \(\gamma\) is the adiabatic parameter of the solid material, also known as the Keldysh parameter, which can be shown as  
\[\gamma=\frac{\
u}{e}\Bigg{[}\frac{m_{e}cn\varepsilon_{0}E_{g}}{I}\Bigg{]}^{1/2} \tag{7}\]  
where \(\
u\) is the laser frequency, \(m_{e}\), \(e\), and \(n\) are the reduced mass, electronic charge, and the index of refraction of materials, respectively. \(E_{g}\) is the band gap, which is 7.7 eV. Figure 4[29; 53; 52] shows the photoionization rate as a function of laser intensity for 800 nm laser in KDP. The effect of multiphoton ionization and tunneling ionization on photoionization can be accurately characterized by the Keldysh coefficient \(\gamma\), as shown in Figure 4.  
The process of photoionization is quite different under the strong field and the weak field. It can be seen that the photoionization is a mix between tunneling and multiphoton ionization when the Keldysh coefficient is around 1.5.