## Abstract

We develop a model that describes the effect of size distribution of nanoabsorbers on the subsurface of fused silica on laser-damage probability. Using Mie theory and heat equation, we obtain the correlation between the critical fluence and particle radius. Considering a power law distribution of nanoabsorbers, the curves of laser-damage probability are calculated based on experimental results of contents of contaminations and a fit parameter of size distribution of nanoabsorbers. This paper presents the influence of various potential candidates, jointly, on laser-induced damage.

©2012 Optical Society of America

## 1. Introduction

Laser-induced damage on the exit surface of transparent dielectric materials, such as fused silica (SiO_{2}) and potassium dihydrogen phosphate (KDP), remains today a key limitation for large aperture, high-power laser systems. It is inevitable that the nano-absobing particles and nano-defects originated from polishing and grinding processes are formed on the subsurface of fused silica [1,2]. However, these nanodefects near the surface of optic materials, absorbing sub-band gap light, can initiate breakdown at fluences and intensities far below the intrinsic damage threshold (10^{11} W/cm^{2}) [3–5]. Kozlowski et al. tried to correlate the level of contaminants induced by polishing process with the damage threshold of fused silica [1]. Neauport et al. studied the effect of contaminations on laser damage density of fused silica at 351 nm [2]. Such absorbing centers haven’t been detected yet because of a lack of detectivity in existing characterization techniques. To obtain a better understanding of laser-induced breakdown processes, laser interaction experiments on gold nanoparticles embedded in silica have been performed [6–10]. In their work, Papernov et al. showed that the energy absorbed by a defect of gold particle of 5 nm diameter was not enough to melt and evaporate the volume of silica to create the observed crater [6]. Subsequently, Bonneau et al. studied similar samples containing gold particles of 3 nm diameter [7]. Moreover, Papernov et al. presented the damage threshold as a function of particle size (2-19 nm range) [8]. They addressed the fact that even few-nanometer-diameter particles can lead to significant threshold reduction. In order to enhance the resistance to laser damage of fused silica, a deeply understanding of the underlying physical mechanism is required. Hopper and Uhlmann firstly proposed a classical model that linear absorption of laser energy raised the temperature of precursors until mechanical failure took place [11]. Feit and Rubenchik have extended the idea of size selection to calculate damage density as a function of laser fluence [12].

The purpose of the present article is to refine previous study by considering influence of various potential candidates, jointly, on laser-induced damage. In subsection 2.1, for fused silica containing various particles, we calculate the critical fluence as a function of particle size based on Mie theory and heat equation. In subsection 2.2, considering a power law distribution of nanoabsorbers, we present a model to calculate the damage probability on the surface of fused silica. In section 3, firstly, the curves of laser-damage probability are measured by the experimental setup for damage test. Then, the contents of contaminations are determined by inductively coupled plasma optical emission spectrometry (ICP-OES). Lastly, the laser-damage probability taking into account the contents of contaminations has been calculated. Theoretical results have a good agreement with the experimental data by choosing a fit parameter of size distribution of nanoabsorbers.

## 2. Model

#### 2.1. Threshold equation

We consider the metallic or dielectric particles embedded in a transparent material without absorption such as fused silica. The material thermal and optical parameters used in our simulation have been given in Table 1 [13].

For simplicity, we just consider the absorption of spherical particles in our model. Assuming there is only one nanoabsorber under the laser irradiation, the power *Q* absorbed by the particle is *Q* = *π a* ^{2}*αI*, where, *α* is the absorptivity, *I* the intensity and *a* the particle radius. The absorptivity *α* is calculated with Mie theory [14]. The details of the model for calculation are exposed in Ref [15].

We present in Fig. 1
the absorptivity of various particles (Al, Cu, CeO_{2}, Fe, Zr) embedded in fused silica. The particles of few 100 nm can be detectable by classical optical techniques, so we pay our attention to only the left parts of the absorptivity curves. We can see from Fig. 1 that the absorptivity of Cu and Fe particles is larger than the others (Al, CeO_{2}, and Zr), when the particles radius is less than 160 nm.

In order to be closer to the experimental conditions, we consider a Gaussian temporal profile, *I* = *I*_{0}exp[-4(*t*^{2}/*τ*^{2})]. The temperature evolution in the spherical particle can be obtained by solving the equation of heat conduction

*T*is finite as

_{i}*r*→ 0 and

*T*→ 0 as

_{s}*r*→ ∞. The boundary condition is

*T*,

*C*, and

*D*present the temperature, thermal conductivity and thermal diffusivity, respectively. The suffixes

*i*and

*s*denote nanoparticle and surrounding matrix. Damage is assumed to take place when the temperature of the surrounding matrix reaches a critical value

*T*(2200 K) [16]. We consider that various particles embedded in fused silica are irradiated with energy of 5 J/cm

_{c}^{2}, during pulse duration of 7 ns and numerically calculate the solution. Figure 2 illustrates the evolution of temperature at various particle-silica interfaces with particle radius of 5 nm.

We can see from Fig. 2 that only Cu and Fe particles can induce damage in this condition. By calculating the heating of the particle as a function of the fluence, we can obtain the critical fluence *F _{c}* required to reach the critical temperature

*T*

_{c}_{.}Subsequently,

*F*can be plotted as a function of the particle radius by iterating above calculation for different particle radius.

_{c}As shown in Fig. 3
, Cu and Fe particles require lower fluence to initiate damage compared to the others (Al, CeO_{2}, and Zr), when the particles radius is less than 160 nm.

#### 2.2. Laser-damage probability

It has been described that defects have different size distribution [15,17–20], and we suppose there exists a power law distribution [15,19,20]. We take the number density *n*(*a*) of particles (per area, per size) to have a power law form as a function of size *a* with exponent *γ*. *a* is the particle radius with the range of *a*_{min}≤ *a* ≤ *a*_{max}. Then the number density *n*(*a*) is given by [19]

*N*is the total density of per area. The suffix

*k*presents different types of particles. We can see in subsection 2.1 that particles of few nanometers cannot initiate damage under our experimental condition, so we insert the lower size limit

*a*

_{min}from the results of calculation. The upper size limit

*a*

_{max}is just a convenience in numerical computations, since it has very small effect on the results of the model. Thereby, the total volume

*V*of particles (per area) can be expressed as

_{k}*γ*= 4, the solution can be written as

*γ*≠4, the solution can be written as

*V*can also be expressed as

_{k}*V*/

_{k}= m_{k}*ρ*.

_{k}*m*is the total mass of particles (per area) and

_{k}*ρ*is mass density of the particles. With the knowledge of the critical fluence as mentioned in subsection 2.1 and the defect size distribution, we can obtain the density of damaging defects as a function of critical fluence,

_{k}*g*(

*F*)where,

_{c,k}*g*(

*F*) gives the number of defects per unit area that damage at fluence between

_{c,k}*F*and

_{c,k}*F*+

_{c,k}*dF*. Thus, the number of defects

_{c,k}*N*(

*F*) located under the laser spot area

*S*(

_{Fc,k}*F*) which can induce damage at the fluence

*F*can be expressed as

*S*(

_{Fc,k}*F*) can be written as

*S*(

_{Fc,k}*F*) = (

*πω*

_{0}

^{2}/2) ln(

*F/F*) and

_{c,k}*ω*

_{0}is the spot radius. The damage probability

*P*(

*F*) is the probability of defect receiving more fluence than its critical fluence, which can be expressed as [20]once

*m*is given, the curves of damage probability can be calculated as a function of fluence by choosing a fit parameter

_{k}*γ*.

## 3. Experiment

The experimental setup [15] involves a single mode YAG laser beam with 355 nm wavelength and 7 ns pulse duration. The size of our samples is 35 mm × 35 mm × 3 mm. Single-shot damage measurements are made using the 1-on-1 procedure. In order to have a good accuracy of experimental data, we observe the 50 different regions under the laser irradiation at each fluence *F*. By counting the number of damage regions at each fluence *F*, we can plot the probability curve *P*(*F*). As mentioned in subsection 2.2, laser-damage probability has a correlation with the spot radius *ω*_{0}, so two different kinds of spot radii (155 μm and 360 μm at 1/*e*^{2}) have been applied to check the validity of our model.

Figure 4 shows experimental curves of laser-damage probability measured on the front surface of fused silica irradiated by 355 nm laser pulses with two kinds of spot radii (155 μm and 360 μm).

The inductively coupled plasma optical emission spectrometer (ICP-OES) is used to determine the contents of the contaminations on the subsurface of fused silica, which has a detailed description in Ref [2], and only a brief description is given here. After accurate weighing and thickness measurement, each fused silica sample is etched during 10 minutes in ultra pure grade HF solution. For all the considered samples, this dissolved depth is in the range of 3 μm to 5 μm. The contents of impurities can be obtained by suitable spectral analysis. Due to the use of cerium oxide slurry in polishing process, we can calculate the contents of cerium oxide based on the contents of cerium measured by ICP-OES. Table 2
gives the contents of main impurities (Al, Cu, CeO_{2}, Fe, Zr) on the subsurface (3~5 μm) of our samples.

We assume that the impurities are spherical particles and their mass *m _{k}* (per area) have a homogeneous distribution on the subsurface of fused silica. Thereby,

*m*can be obtained according to the measured results in Table 2. Considering our experimental condition, the lower size limit

_{k}*a*min for various particles can be obtained by calculating the critical fluence as a function of particle radius as mentioned in subsection 2.1 with fluence ranging from 0 J/cm2 to 25 J/cm2. From the result of the calculation, the values of

*a*

_{min}for Cu and Fe particles are lower than others (Al, CeO

_{2}, and Zr). The particle sizes most susceptible to create damage are chosen as the upper size limit

*a*

_{max}for various particles as seen in Fig. 3

_{.}The values of

*a*

_{max}are set to 30 nm for convenience in numerical computations, since they have very small effect on the results of the model. The theoretical laser-damage probability can be calculated with different spot sizes by choosing a fit parameter

*γ*. We now apply the calculated damage probability to fit the experimental data with 155 µm and 360 µm spot radii. By estimating the average dissolved depth is 4 µm, the error for the calculated damage probability is about 0.04. Figure 5 shows the experimental curves of laser-damage probability measured on the surface of fused silica and theoretical curves calculated with the parameter

*γ*= 34.

We can see from Fig. 5 that there is a good agreement between experimental and theoretical results at the set of parameter *γ* = 34. The calculated damage threshold with spot size dependence can be obtained with good agreement to the experimental result. The interesting point is that the model can describe observed spot size dependence since with the same parameter *γ* = 34 the data can be fit with the two different spot sizes. Conversely, once the parameter *γ* is obtained by fitting the experimental data, the defect distribution for various particles embedded in the subsurface of fused silica can be identified.

## 4. Conclusion

We have presented a damage model for the reaction of 355 nm laser pulse with the front surface of silica, assuming that the damage initiators are contaminants induced by polishing process. In this model, laser-damage probability on the front surface fused silica has been calculated. Theoretical curves of laser-damage probability have a good agreement with experimental results measured with two kinds of spot radii (155 μm and 360 μm at 1/*e*^{2}), when the parameter *γ* = 34 is considered in our simulation. Consequently, the size distribution *γ* of nanoabsorbers on the surface of optical materials can be obtained according to the method. Of course the result must be taken with caution since different assumptions have been made for the calculations, but it is noteworthy that the curves of laser-damage probability obtained with different spot sizes have been explained with same size distribution. This model is of interest for the study of polishing process since it improves the knowledge on the properties of metallic or dielectric nanoabsorbers on the subsurface of fused silica.

## Acknowledgments

This work was supported by the Major Program of National Natural Science Foundation of China under Grant No. 60890200.

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