A model of plasma formation induced by UV nanosecond pulse-laser interaction with SiO2 thin film based on nanoabsorber is proposed. The model considers the temperature dependence of band gap. The numerical results show that during the process of nanosecond pulsed-laser interaction with SiO2 thin film, foreign inclusion which absorbs a fraction of incident radiation heats the surrounding host material through heat conduction causing the decrease of the band gap and consequently, the transformation of the initial transparent matrix into an absorptive medium around the inclusion, thus facilitates optical damage. Qualitative comparison with experiments is also provided.
©2008 Optical Society of America
Laser induced damage in optical elements is a key issue for high power laser application, and a large number of publications have been devoted to the studies of damage mechanism [1–9]. It is widely acknowledged that nanoabsorbers are a major initiation source of laser damage in the nanosecond ultraviolet (UV) laser pulses, and the progress of the study of damage initiation mechanism has been recently made by first intentionally introducing laser damage initiators and then studying their behaviors under laser irradiation . It is believed that the initial transparent surrounding film matrix was modified, accompanied by plasma formation and high temperature during a damage event . C.W.Carr et al. performed the direct measurement of the plasma and the lattice temperature during a laser damage event for the first time . M.D.Feit and A.M.Rubenchik gave a model of formation of damage crater due to thermal explosion . However, as to the process of conversion of the transparent matrix into an ionized region followed by damage crater formation, a detailed theoretical model is very challenging. Especially for the initial plasma formation process, it is still not well understood. Several possible models for describing the initial stages of the plasma formation of host materials were proposed, including electron injection through thermionic emission , ionization by thermal UV radiation , and so on. These models can explain some important regularities of laser induced damage, but the material parameters such as band gap dependence of temperature were not considered.
In this paper, we report a possible plasma formation model which involves the temperature dependence of band gap of host material and a detailed laser damage process during the laser pulse, with the assumption that a Pt inclusion locates in the SiO2 thin films.
2. Model of plasma formation
We assume that the damage is initiated by laser light absorption in nanoparticles. Considering a spherical inclusion of radius a embedded in an infinite homogeneous film material, under the sphere symmetry coordinate, the temperature of the host material around the inclusion can be calculated as follows 
Where ρ, c, K are the density, heat capacity and thermal conductivity, respectively. The boundary and initial conditions are at t=0, T=T0=constant, where T0 is the ambient temperature and at infinity T=T0. Generally, the thermal conductivity of the particle is much larger than that of the coatings, so we can assume that the temperature inside the particle is homogeneous. At the interface of the particle and the host, the power absorbed by the inclusion must be equal to the power leaving the surface by conduction plus the rate of change of the heat energy of the inclusion
Where a is the radius of Pt inclusion. ρi is the density and ci is the heat capacity. Kh is the heat conductivity of the host material. We further assume that the absorption Qi of the inclusions is uniform volume absorption of the laser light. As reliable absorption coefficient data are absent, Qi was fixed at a value of 0.1. I is the maximum power density of laser pulse and the suffix h denotes host material and i denotes inclusion. The solution of the Eq. (1) with the boundary condition Eq. (2) gives a simple expression for the particle temperature T heating with a flat-top laser pulse shape and duration τ 
For long pulses such as nanosecond pulse, when the laser intensity is not so high that the multiphoton ionization and impaction ionization will not happen for the wide band gap material, so the material is generally considered to be transparent for the incident laser with the photon energy lower than the band gap . However, when we consider absorbing inclusion embedded in the host material matrix which acts as the initial absorbing source and the temperature dependence of the band gap of the surrounding host material, the absorbers will be heated by the laser pulse and then transfer the heat to the surrounding host material through heat conduction, leading to decrease of the band gap of the host material. As for the dependence of the bang gap on the temperature, K.Saito and A.J.Ikushima report that the optical gap is observed to decrease from 8.5eV to 7eV when the temperature of bulk SiO2 is increased from 4K to 1900K . When the optical band gap is lower than the energy of incident photon, the initially transparent host materials will become absorptive. The band gap of the host material as a function of temperature is expressed as 
Where B1 and B2 are material parameters, Eg0 is the initial band gap of the host material. For indirect band gap materials, if we consider that the electron generation is much larger than the electron-hole recombination and do not take the electron-hole recombination into account during the process of electron generation, when ħω>Eg+pEp, the rate of excitation of electrons to the conduction band by single photon absorption can be written as
Where αh is the absorption coefficient, the absorption coefficient has the form as follows 
Where and B 3 are material parameters. Ep is the energy of phonon.
P=±1: Positive value for emitting phonon and negative value for absorbing phonon.
Based on the model discussed above, we have studied the evolution of the free electrons in host material. The main parameters used in calculation are listed in table 1. As the values of reliable material parameter are absent, the values of B1, B2, and B3 quoted are roughly estimated according to the semiconductor material of Si . The other main parameters are chosen from Ref. 7.
Figure 1 shows the evolution of electron density for the single photon absorption at different laser intensity.
From Fig.1, we can see that the higher the laser intensity, the shorter the duration of the single photon ionization. The electron density is almost the same at different laser intensity at the collapse of band gap.
Figure 2 shows the electron densities at different band gaps. From Fig. 2, we can see that it takes longer time to cause the collapse of the band gap for the host material with the wider band gap. If we take the collapse of the band gap as the damage standard, the host material with the wider band gap has the higher damage threshold. From Fig. 1 and Fig. 2, we can also see that the critical density (~8×1021cm-3) for 355 nm wavelength is not reached during the process of the collapse of band gap . For the remainder laser pulse, due to the collapse of the band gap, the electron concentration can easily increase up to the critical density. To estimate the ionization time, we take the free electrons created by single photon ionization as the initial electrons of impact ionization. The ionization rate by electron impact is given by 
Where n is the free electron density and I is the laser intensity. The impact ionization coefficient is set as β~ 10cm2/ns GW . Using the parameters in Table 1, the ionization time is about 0.3 ns.
Because B1, B2 and B3 are three important parameters in our calculation, we also investigate the influence of the parameters B1, B2, B3 on the calculated electron density value. We found that B2 had little influence on the calculated electron density. Figure 3 and Fig. 4 show the electron density values at different values of B1 and B3 with the other parameters remained unchange, respectively.
It can be seen from Fig. 3 that B1 affects the time for the collapse of band gap, and the time becomes shorter with larger B1. From Fig. 4, we can see that the electron density increase with increasing B3. So it can be inferred that the host materials with larger B1and B3 can be damaged more easily during the laser irradiation.
From above calculations, we can know that it takes approximately less than one half of laser pulse duration to cause the band gap collapse of host material and induce its ionization. The ionized region of the inclusion-surrounding material matrix will serve as a new absorption source responsible for the subsequent ionization of the new region of the host material during the remaining pulse duration, leading to the growth of absorbing-volume (plasma ball) of the host material. In order to give qualitative comparison with experiments, we assume that the growth of the ionized radius r can be obtained by the model proposed in Ref. 6
Where the growth factor γ scales linearly with F.
The growth of the plasma sphere may tend to saturate and its size would be comparable to the laser radiation wavelength at the end of the pulse. The energy of the plasma sphere can be estimated as
At high fluence, the energy density in the plasma sphere is much larger than the typical evaporation energy density and will lead to microexplosion of the host materials. The radius R of a produced damage craters due to microexplosion is related to the E. According to Ref. 18, the maximum crater radius is given by
Where with is the maximum lodging depth at which a crater can still be formed. Where c is the critical velocity estimated by ρc 2=G and G is the characteristic strength of host material G, ρ is the host material density, α is the fraction of the deposited energy going into hydro motion, its typical value is about 10%.
Taking the fraction of the absorbed energy that appears in mechanical wave as α=0.1, material density ρ=2210 Kg/cm3, and the strength of SiO2 coatings G=300MPa, the maximum crater radius is about 1.2 µm. SiO2 coating was deposited by electron beam evaporation. Under a laser fluence of 10 J/cm2, the experimental damage crater measured by Veeco optical profiling system was shown in Fig. 5.
We can see from Fig. 5 that the sizes of the experimental and theoretical damage craters agree within an order of magnitude. However, it should be noted that the theoretical model is rough and just a qualitative comparison with experiment. The assumption of Pt inclusion embed in the host materials may not be justified. The reason that we take the Pt inclusion as the initial absorption source is that the true absorbers in the thin film are difficult to be identified and the Pt inclusion has known physical parameters which may be similar to the true absorbers. On the other hand, the physical parameters used in our calculation are roughly estimated.
In conclusion, we have given a damage model for the reaction of UV nanosecond pulse with SiO2 thin film, assuming that the damage initiators are Pt nanoabsorbers. In this model, a possible plasma formation process which takes the temperature dependence of band gap of host material into account was proposed. The critical temperature of the collapse of band gap of host materials sets the damage threshold. Qualitative comparison with experiments is provided.
This work is supported by National Natural Science Foundation of China under Grant No.60678004. I would like to thank Prof. Ya Cheng for language editing.
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