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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology

Xiang Gao
• Corresponding author
• Joint Laboratory for Extreme Conditions Matter Properties, Southwest University of Science and Technology and Research Center of Laser Fusion, CAEP, Mianyang 621010, China
• Fundamental science on Nuclear wastes and Environmental Safety Laboratory, Southwest University of Science and Technology, Mianyang 621010, China
• Email
• Other articles by this author:
/ Rong Qiu
• Joint Laboratory for Extreme Conditions Matter Properties, Southwest University of Science and Technology and Research Center of Laser Fusion, CAEP, Mianyang 621010, China
• Other articles by this author:
/ Kunpeng Wang
• Fundamental science on Nuclear wastes and Environmental Safety Laboratory, Southwest University of Science and Technology, Mianyang 621010, China
• Other articles by this author:
/ Jiangmei Zhang
• Fundamental science on Nuclear wastes and Environmental Safety Laboratory, Southwest University of Science and Technology, Mianyang 621010, China
• Other articles by this author:
/ Guorui Zhou
/ Ke Yao
/ Yong Jiang
• Joint Laboratory for Extreme Conditions Matter Properties, Southwest University of Science and Technology and Research Center of Laser Fusion, CAEP, Mianyang 621010, China
• Other articles by this author:
/ Qiang Zhou
• Joint Laboratory for Extreme Conditions Matter Properties, Southwest University of Science and Technology and Research Center of Laser Fusion, CAEP, Mianyang 621010, China
• Other articles by this author:
Published Online: 2017-04-30 | DOI: https://doi.org/10.1515/phys-2017-0025

## Abstract

A model for predicting the size ranges of different potential inclusions initiating damage on the surface of fused silica has been presented. This accounts for the heating of nanometric inclusions whose absorptivity is described based on Mie Theory. The depth profile of impurities has been measured by ICP-OES. By the measured temporal pulse profile on the surface of fused silica, the temperature and thermal stress has been calculated. Furthermore, considering the limit conditions of temperature and thermal stress strength for different damage morphologies, the size range of potential inclusions for fused silica is discussed.

PACS: 79.20.Ds; 42.62.-b; 42.70.Ce

## 1 Introduction

The achievement of the highest output energy from large aperture laser facilities [14] is limited by the obscuration of damage regions induced by high laser fluence [5, 6]. Even optical components with the highest quality such as fused silica can suffer material breakdowns near the surface at laser intensities far below the intrinsic damage threshold (∼100 GW/cm2) [7]. The different damage precursors include submicroscopic defects and inclusions induced by polishing or grinding processes. The presence of submicroscopic cracks, pores and indentations will cause enhance absorption due to local electric field intensification, which can lower the surface breakdown threshold [810]. Nano-absorbing centers such as impurities, contaminants and void fillers will also be embedded on the subsurface of fused silica [2, 11, 13]. These inclusions can absorb laser energy significantly, raising the temperature around the inclusions high enough to cause damage [14]. A statistical trend has been observed in damage tests due to size distributions [1519] (Dirac, power law, Gaussian law) of the inclusions. The damage density and damage probability dependend on laser parameters (fluence, pulse length, spot size, or wavelength) and this has been observed in the experiments [2022].

The purpose of this paper is to describe the method for identifying the size of various inclusions which creat different damage morphologies. In section 2, the thermal and mechanical model is presented to evaluate the temperature and stress based on the light absorption in nanoabsorbers. In section 3, the temporal profile of pulse and the depth profile of impurities have been detected. The size range of various potential impurities which created different types of damage morphologies can be obtained.

## 2 Temperature and thermal stress model of absorbing inclusion

It is assumed that all characteristic dimensions of precursors are smaller than 200 nm since larger precursors can readily be detected by classical optical techniques. It is believed that the presence of submicroscopic defects will cause local enhancement of electric field strength. For three representative geometries, it has been shown that a crack has a larger electric field enhancement factor γ than a cylindrical groove and spherical pore; this can be expressed as γ = m2 [8], where m is the optical index of the host material. For fused silica, m = 1.48, the threshold intensity induced by surface cracks is a factor of 4.8 lower than that of the bulk. However, in practical optical components the surface damage threshold irradiated by nanosecond pulses is two orders of magnitude lower than that of the bulk. Therefore, the inclusion produced by polishing or grinding processes will be the dominant mechanism, giving the lowest surface damage threshold.

In accordance with experimental data, we assume the absorbing inclusions are embedded in the transparent matrix. For simplification, the shape of inclusions is taken to be spherical and they are nanoscale size. The large variations in the thermal parameters of the inclusion only weakly affect temperature evolution in the case of an absorbing inclusion driven mechanism [23]. Thus, the thermal properties of the surrounding matrix are of critical importance for the temperature rise [15], so the calculation only considers the dependence of thermal properties of the host material on the temperature. In this manuscript, considering the real temporal shape of the laser pulse by fitting a measured Gaussian temporal profile, we refined previous model [2022] to calculate the particle temperature.

The temperature of inclusion and host matrix heating by nanosecond pulses is described as $Ckρk∂θk∂t=∇χk∇θk+Qϕ(a−r)$(1)

where θ = TT0 is the increase in temperature T at the heated location over the initial temperature T0 and a is the radius of inclusion. ρk, Ck and χk, respectively, present density, thermal capacity and thermal conductivity. k has two values: 1 corresponds to the inclusion and 2 to the host material. Q is the light intensities absorbed by the inclusion, which could be written as Q = (1 − R)ηI0 exp(−At2/τ2), where η is the absorption cross-section per unit particle volume, R is the reflectivity of the surrounding medium and I0 is the maximum intensity. τ is the pulse duration (at 1/e) and A is a fit parameter about the pulse temporal profile. The function ϕ(r) is expressed as ϕ(r) = 0 at r < 0 and ϕ(r) = 1 at r ≥ 0. The absorption cross-section per unit particle volume η is calculated with Mie theory [24]. $η=2πk2V∑n=1Nstop2n+1Rean+bn−an2+bn2$(2)

an and bn are the coefficients determined with continuity relations [22]. k = 2πma/λ, m is the refractive index of host material and λ is the wavelength of incident light. V is the volume of inclusion. Nstop is a cutoff parameter in the calculation, which can be set to 10. The absorption cross-section per unit volume of various spherical particles in a fused silica matrix has been plotted in Figure 1.

Figure 1

Absorption cross section per unit particle volume as a function of particle diameter

Figure 1 shows that the absorptivity of CeO2 inclusions is lower compared to others (Cu, Fe and Hf). By using the method of Fourier transform, the heat equation can be readily solved numerically to express T [22] $T1=T0+12π∑ω=−∞ω=+∞A1rexpiβ1r−A1rexp−iβ1r−Sexp−iωtΔω$(3) $T2=T0+12π∑ω=−∞ω=+∞A2rexpiβ2r−iωtΔω$(4)

where $S=3ηI08aβ12χ1πτexp−τ2ω216$(5) $A1=χ2aS1−iβ2aχ2−χ1−iχ1β2an1+iaχ1β1n2$(6) $A2=A1n1−aSexp−iβ2a$(7)

where, ${\beta }_{k}=\omega {\rho }_{k}{C}_{k}\cdot \left(1+i\right)/2{\chi }_{k}^{1/2}$, ${n}_{k}=\mathrm{exp}\left(i{\beta }_{k}a\right)+\left(-1{\right)}^{k}\mathrm{exp}\left(-i{\beta }_{k}a\right)$. The temperature rise of local regions in the host material brings about thermal stress. The processes of crack formation, melting or ablation of host material are achieved in the final stages of the LID. To investigate these situations, the thermal stress has to be considered based on the solutions for temperature Eq. (3) and Eq. (4). The thermoelasticity equation can be expressed as [25] $∂∂rr−2∂∂rr2urk=αk1+νk1−νk∂θkr,t∂r$(8)

where ur is the radial displacement vector. αk is the linear expansion coefficient and νk is the Poisson’s ratio. As the limit condition (ur1 finite as r → 0 and ur2 = 0 as r → ∞), the general solution of the equation is $ur1=1+ν11−ν1α1r2∫0rθ1(r)r2dr+B1r$(9) $ur2=1+ν21−ν2α2r2∫arθ2(r)r2dr+B2r2$(10)

According to the deformation tensor definition and Duamel-Naiman relationships, the thermal stress can be described as $εrk=∂urk/∂r,εθk=εφk=urk/r$(11) $σrk=Ek1+νk1−2νk1−νkεrk+2νkεϕk−1+νkαkθk$(12) $σθk=σφk=Ek1+νk1−2νkεφk+νkεrk−1+νkαkθk$(13)

where E is the Young’s modulus. σr and σθ are radial stress and hoop stress, respectively. The thermal stress can be described as $σr1=−2α1E11−ν11r3∫0rθ1r2dr+E1B11−2ν1$(14) $σθ1=α1E11−ν11r3∫0rθ1r2dr+E1B11−2ν1−α1E1θ11−ν1$(15) $σr2=−2α2E21−ν21r3∫arθ2r2dr−1r32E2B21+ν2$(16) $σθ2=2α2E21−ν21r3∫arθ2r2dr+1r3E2B21+ν2−α2E2θ21−ν2$(17)

where, B1 and B2 are obtained by the boundary conditions (εθ1 = εθ2 and σr1 = σr2, at r = a). The damage of a sample is assumed to occur when the maximum temperature in host material exceeds the melting point, or the maximum stress exceeds the tensile/compressive strength of the material [29]. $σr(r=a)=−1a36α1E1E2(1+ν2)E1+2(1−2ν1)E2∫0aθ1(r)r2dr$(18) $σθ(r=a)=1a33α1E1E2(1+ν2)E1+2(1−2ν1)E2∫0aθ1(r)r2dr−α2E2(1−ν2)θ2(a)$(19)

With knowledge of the temporal profile of the pulse, the formation and evolution of damage morphologies can be analyzed.

## 3 Analysis of damage features

The experimental setup (Figure 2) used for this work involves a single-mode Nd:YAG laser giving access to 355 nm pulses with Gaussian temporal profile. The fast photodiode is used to observe the temporal profile as shown in Figure 3. The calorimeter is used to measure the pulse energy and the fluence fluctuations have a standard deviation of about 10%. The visible He–Ne laser beam is used to monitor the region of damage under test. The density of precursors on the subsurface of sample is very low, and is less than two orders of magnitude in a square with a 100 μm side [14]. Hence, in order to relate the damage morphologies and specific precursors, 12 μm of beam diameter has been used in our experiment to ensure only one precursor under the irradiation of a laser spot.

Figure 2

Experimental test for damage test

Figure 3

Gaussian temporal profile

The different damage features under laser irradiation at each fluence are observed by optical microscopy (OM) and scanning electron microscopy (SEM). The damage tests of samples are repeated in different regions irradiated by laser pulses at each fluence. As shown in Figure 4, the typical damage features initiated at near LIDT (5 J/cm2) have been observed.

Figure 4

OM and SEM (inset of the figures) images of damage sites observed on the sample after irradiation at near LIDT: (a) The little crack region and (b) the circular damage region with a melted pit

We can see in Figure 4(a) only cracks observed on the surface of the fused silica. Thus, in Figure 4(b), the melted pit can be found in the center of the damage region and the cracks locate around it. As exposed in Ref. [30], the cracks can be formed in a “cold” region, since the local temperature does not exceed the melting point of the matrix. In order to explain the formation of damage morphology, we calculate the thermal stress and temperature under the laser irradiation. The parameters of inclusions and the surrounding matrix (fused silica) used in the calculation are listed in Table 1 [2628].

Table 1

Main parameters used in calculation

If the potential inclusion is assumed to be a spherical Hf particle, the evolution of the damage region can be inferred by calculating the stress and temperature at different times during the pulse duration in Figure 5.

Figure 5

Evolution of stress and temperature during pulse duration. The diameter of Hf inclusion is 60 nm. The laser fluence is 5 J/cm2 (1. Cracks along the radial direction; 2. Melting; 3. spallation)

The negative values of stress represent compressive stress and positive values represent tensile stress [29]. For fused silica, the strength of compressive stress is 1500 Mpa and the strength of tensile stress is 50 Mpa. The hoop stress can cause cracking along the radial direction. The radial stress can cause spallation in the damage region. Thence, the damage features on the surface of fused silica should be accompanied by cracks along the radial direction, melting and spallation, as shown in Figure 4. We can see from Figure 5 that the hoop stress reaches the strength of tensile stress before the temperature reaches the melting point and the radial stress reaches the strength of compressive stress. Therefore, it is assumed that cracks will be initiated before the formation of melted zones and spallation.

There are maximum stresses and temperatures during the pulse as seen in Figure 5, so the maximum stress and temperature with different particle sizes can be obtained in Figure 6.

Figure 6

The maximum stress and temperature as a function of particle diameter. The laser fluence is 5 J/cm2 (1. Cracks along the radial direction; 2. Melting; 3. spallation)

From Figure 6, compared to the condition of melting and spallation, cracks along the radial direction can be initiated by the smaller particle. Thus, by caculating the hoop stress as a function of particle size, the smallest particles to iniate cracks can be obtained. From these results, we can deduce that the damage in Figure 4(a) should be induced by the small particles, because it is characterized by the cracks along the radial direction without other damage features. With an incease in particle size, in addition to cracks along the radial direction, a melted pit will be formed when the temperature reaches the melting point of fused silica. Subsequently, spallation will occur once the radial stress exceed its compressive strength as shown in Figure 4(b).

The maximum hoop stress and temperature of the surrounding matrix along the r direction have been plotted in Figure 7.

Figure 7

The maximum hoop stress and temperature of surrounding matrix along r direction. The diameter of Hf inclusion is 40 nm. The laser fluence is 5 J/cm2

As seen in Figure 7, cracks occur further away from the center of inclusions than the melted regions, in good agreement with obtained experimental result (Figure 4(b)).

The types of potential inclusions have been determined by inductively coupled plasma optical emission spectrometry (ICP-OES) [10]. The sample manufactured by using cerium oxide slurry was applied for our experiment. After accurate weighing, the sample was digested by ultra pure grade hydrofluoric acid (HF) solution in 10 minutes and the average thickness of the sample digested was calculated. By analysis of suitable spectra, the contents of main impurities can be obtained. This process was repeated up to two times and the contents of main impurities from different depth of layers have been detected as per Table 2. Al2O3 particles are weak absorption materials at 355 nm, so we just consider CeO2, Cu, Fe and Hf inclusions in the calculation.

Table 2

The contents of main impurities from different depth of layers (μg/g)

In order to identify the size ranges of potential inlusions that creat different damage morphologies, we calculate the thermal stress and temperature with different sizes of inclusions under the laser irradiation as seen in Figure 8.

Figure 8

Hoop stress and temperature as a funcation of inclusions size

From Figure 8(a), when the hoop stress reaches the strength of the material, there is lower level of temperature which does not exceed the melting point of fused silica. With the increase of inclusion size, the temperature of the surrounding matrix will reach melting point. Additionally, as seen in Figure 8(b), when sufficient heat melts the inclusions, the temperature will increase strongly induced by a significant variation of absorptivity [31]. Consequently, the size ranges of various potential particles are summarized in Table 3.

Table 3

The size ranges of various potential inclusions (nm)

The maximum sizes of all potential inclusions are less than 200 nm. This is reasonable because the conventional optical microscopy whose maximum resolution is 200 nm cannot detect these impurities. As seen in Table 2, for various potential inclusions, the particle sizes to create type 2 are larger than the particle sizes to create type 1. Cu particles can initate cracks with the smallest size compared to other particles. CeO2 inclusions cannot create melted regions as type 2. Hf particles require larger sizes to create type 2 than Cu and Fe particles.

## 4 Conclusion

A model for the heating of absorbing inclusions whose absorptivity is calculated by the Mie theory has been developed in this work. For a measured temporal profile, on the surface of fused silica the temperature and stress distribution induced by various inclusions has been calculated. For various types of damage morphologies, the size range of potential inclusions has been deduced by the limit conditions of temperature and stress of fused silica. The results of our investigations can provide the knowledge of potential inclusion sizes which initiated damage on the subsurface of an optical substrate.

## Acknowledgement

We thank Yajun Zhang for assistance with laser damage testing and Lingling Zhai for measurement of the sample by ICP-OES. This work was supported by National Natural Science Foundation of China (NSFC) (No: U1530109, 61505171, 61505170).

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Accepted: 2017-01-25

Published Online: 2017-04-30

Citation Information: Open Physics, Volume 15, Issue 1, Pages 233–239, ISSN (Online) 2391-5471,

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