# Recent Advances in Laser-Induced Surface Damage of KH\({}_{2}\)Po\({}_{4}\) [PERSON]
## 3 Exploration of the Initial Energy Deposition
### The Formation and Evolution of Plasma  
#### 3.1.1 Irradiation of Ultra-Short Pulse Laser  
[PERSON] et al. [61] found that when the pulse period is greater than 100 ps, the damage of the dielectric material is mainly caused by the traditional heat transfer, while the damage of the material originates from the formation of laser-induced plasma when the pulse period is very short (\(<\)10 ps). As a result, the formation of plasma formation plays an important role in the initial energy deposition under the irradiation of short (picosecond to a couple of nanoseconds timescale) and ultra-short pulses (femtosecond timescale) [62; 63; 64]. Based on the understanding of ionization, it can be found that the photoionization will be induced inside the material when high power ultra-short laser pulses irradiate transparent dielectric materials. The formed free electrons will be acted as seed electrons to absorb the incident laser energy through various absorption mechanisms in a very small range and a very short time, which will generate avalanche ionization and make the electrons in the action area reach extremely high temperature quickly. Then, the material will be directly excited from the solid state to the form of high-density plasma. The high-density plasma will transfer their kinetic energy to the lattice over a picosecond timescale and perform energy deposition within the skin depth of the material through the reverse bremsstrahlung regime under the subsequent laser irradiation. During this period, the electron density will continue to increase until the plasma frequency approaches the frequency of the incident laser radiation, i.e., the critical plasma density value is reached. Figure 5: Dependence of avalanche ionization rate \(W_{Al}\) on electric intensity \(E\) calculated by AI models of [PERSON], [PERSON], and [PERSON] of KDP [29]. [PERSON] et al. [27] established a rate equation to describe the evolution of free electron density \(n_{e}(t)\) in the dielectric medium KDP under the irradiation of laser pulse below 10 ps. \[\frac{\partial n_{e}(t)}{\partial t}=W_{Pl}(I(t))+W_{AV}(I(t))\cdot n_{e}(t)-W_{ rel}(n(t),t) \tag{13}\]  
where \(W_{rel}(n(t),t)\) is the plasma energy decay term that associated with the diffusion and recombination of electrons. Figure 6a shows the formation of plasma under the irradiation of ultra-short laser pulses. Based on the criterion of critical electron density, the growth of electron density predicted by different ionization models under the irradiation of femtosecond laser was analyzed in references [65; 66]. In addition, there is a sharper drop in transmission at the near-front surface of dielectric materials in Figure 6c, and the greater the light intensity, the weaker the transmission performance and the stronger the absorption performance. Combined with the deposition energy density shown in Figure 6b, it is clear that the energy deposition density inside the material gradually increases with the increase of laser intensity due to the formation of plasma. [PERSON] et al. [68] showed that there is the high concentration near the focal point and the end of the laser source due to the high laser intensity and the reflection of light by the plasma, respectively. Figure 6: The process of energy deposition and evolution under femtosecond laser irradiation. (**a**) The electron density \(n_{e}(t)\) curve of KDP materials under the femtosecond laser irradiation based on Equation (13), \(\lambda=800\) nm, \(\tau=55\) fs, \(I_{0}=68\) TW/cm\({}^{2}\).