1.

Introduction

For large-aperture, high-power laser systems, such as the National Ignition Facility in the United States,^1,2 Laser Megajoule in France,³ and the SGIII laser facility in China,⁴ the ultraviolet optical lifetime of fused silica must be increased. The polishing contaminants in the near-surface region of optical components can absorb sub-band gap light and produce a local heating that can initiate a material damage.⁵ Many experimental facts have shown that absorbing nanometer-sized inclusions are responsible for the initiation of the damage process: an increase of the damage thresholds with purification of subsurface of fused silica;^6,7 a spatial variation of the damage threshold on the surface or in bulk of optical substrates;^8,9 and a dependence of the damage threshold on the irradiation spot-size and wavelength.^10,11 However, in most cases, the impurities are not identified by modern optical techniques since they are nanoscale size and are distributed at low concentration.¹²

It is obvious that the inclusion-initiated damage has a statistical character because of the spatial distribution of inclusions in a sample.¹⁰ The theoretical studies of inclusion-initiated damage were based on the resolution of Fourier equation. ^13–16 However, these models have not been substantiated enough to explain the statistical character in experiments. The information on damage density and damage threshold of precursors can be extracted from the experimental curves of damage probability.^11,17 Feit and Rubenchik have presented a model¹⁸ that the size distribution of nanoabsorbers is related to the damage density and damage probability, which predicts the dependence of damage density on pulse duration.

In this paper, we go further to relate the contents of various impurities measured from the subsurface layers of different samples to damage probability. In Sec. 2, based on calculation of absorption of spherical particles and then solving the heat equation, for various particles, the critical fluence required to initiate damage can be calculated. Considering the fit distribution parameters, the laser damage probability on the surface of fused silica has been calculated. In Sec. 3, the subsurface components of impurities for different samples are determined by inductively coupled plasma optical emission spectrometry (ICP-OES) and the data points of laser damage probability have been measured. Subsequently, the theoretical model presented has been used for analyzing the effect of various impurities on damage probability.

2.

Theoretical Model

2.1.

Critical Fluence

Contaminants detected include the major polishing compound components (Ce or Zr from ${\mathrm{CeO}}_2$ or ${\mathrm{ZrO}}_2$), and other metals (Fe, Cu, Cr) induced by the polishing step or earlier grinding steps. Al is present largely because of the use of ${\mathrm{Al}}_2{\mathrm{O}}_3$ in the final cleaning process. ${\mathrm{Al}}_2{\mathrm{O}}_3$ and ${\mathrm{ZrO}}_2$ are nonabsorbing materials at 355 nm, so we just consider ${\mathrm{CeO}}_2$, Cu, Fe, and Cr particles in the simulation. With the improvement of surface-micromachining process, few 100-nm particles can be identified by classical optical techniques and can be removed from the subsurface of fused silica, so the particle radius of $<100\mathrm{nm}$ was simulated in the model. For simplification, we only consider the shape of a sphere, although it is not necessarily needed in all cases.¹⁹ The temperature distribution is necessary for evaluating the critical fluence required to initiate damage, and the spherical particle heating under the laser radiation is described by the equation of heat conduction.

Eq. (1)

$$C_i(T){\rho }_i\frac{\partial T_i}{\partial t}=\nabla [{\chi }_i(T)\nabla T_i]+\frac{\sigma I}{V}f(t)\theta (R-r),$$

where $\rho$, $\chi (T)$, and $C(T)$ present density, thermal conductivity, and thermal capacity, respectively [values for $T$ up to 2200 K (Ref. 19)]. $I$ and $f(t)$ are maximum intensity and temporal shape of laser pulse. A subscript $i$ has two values: $i=p$ for an inclusion and $i=h$ for a host material. We consider a Gaussian temporal profile, $f(t)=\exp [-4(t^2/{\tau }^2)]$, for consistency with the experimental condition. $\theta (x)$ is a function defined as $\theta (x)=0$ at $x<0$ and $\theta (x)=1$ at $x\ge 0$. $\sigma$ is the absorption cross-section of the inclusion, and $V=(4/3)\pi R^3$ is the inclusion volume. The material thermal and optical parameters for calculation are exposed in Refs. 20 and 21. The absorption cross-section $\sigma$ is calculated with the Mie theory.²²

Eq. (2)

$$Q_{\text{ext}}=\frac{2\pi }{k^2}\sum _{n=1}^{\infty }(2n+1)\mathrm{Re}(a_n+b_n),$$

Eq. (3)

$$Q_{\text{sca}}=\frac{2\pi }{k^2}\sum _{n=1}^{\infty }(2n+1)({|a_n|}^2+{|b_n|}^2),$$

Eq. (4)

$$\sigma =Q_{\text{ext}}-Q_{\text{sca}},$$

where $Q_{\text{ext}}$ and $Q_{\text{sca}}$, respectively, are the extinction cross-section and scattering cross-section. $k=2\pi N/\lambda$, where $N$ is the optical index of host material and $\lambda$ is the wavelength of irradiation. $a_n$ and $b_n$ are the scattering coefficients determined with continuity relations.

We plotted in Fig. 1 the absorption cross-section of various particles (${\mathrm{CeO}}_2$, Cu, Fe, and Cr) embedded in fused silica. We can see from Fig. 2 that the absorptivity of ${\mathrm{CeO}}_2$ particles is much lower than others (Cu, Fe, and Cr) with the same size.

Fig. 1

The absorption cross-section calculated at 355 nm as a function of particle radius for various particles.

Fig. 2

Critical fluence calculated as a function of particle radius for various particles.

Considering the Fourier transform of the temperature, Eq. (1) can be written as

Eq. (5)

$$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial {\widehat{T}}_i}{\partial t}\right)+{\alpha }_i^2{\widehat{T}}_i=-\frac{\sigma I\theta (R-r)}{2V{\chi }_p(T)}\sqrt{\pi }\tau \exp (-\frac{{\tau }^2{\omega }^2}{16}),$$

where ${\alpha }_i={[\omega {\rho }_i{C}_i(T)/2{\chi }_i(T)]}^{1/2}(1+i)$. Applying the limit condition, the solutions can be expressed as

Eq. (6)

$${\widehat{T}}_p(r,\omega )=\frac{A_{p}I}{r}[\exp (i{\alpha }_{p}r)-\exp (-i{\alpha }_{p}r)]-\frac{\sigma I}{2V{\alpha }_p^2{\chi }_p(T)}\sqrt{\pi }\tau \exp (-\frac{{\tau }^2{\omega }^2}{16}),$$

Eq. (7)

$${\widehat{T}}_h(r,\omega )=\frac{A_{h}I}{r}\exp (i{\alpha }_{h}r),$$

where $A_p$ and $A_h$ can be obtained by use of the boundary condition [${\widehat{T}}_p(R)={\widehat{T}}_h(R)$, ${\chi }_p(T)\partial {\widehat{T}}_p/\partial r{|}_{r=R}={\chi }_h(T)\partial {\widehat{T}}_h/\partial r{|}_{r=R}$). Then, the temperature $T_i$ in the inclusion and host material can be obtained by numerical inverse Fourier transform. Damage is assumed to take place when maximum temperature at the particle–host material interfaces reaches a critical value ($\sim 2200\mathrm{K}$).²³ Thus, the critical fluence $F_c$ required to reach the critical temperature can be expressed as

Eq. (8)

$$F_c={\int }_{-\infty }^{+\infty }If(t)\mathrm{d}t={(\pi )}^{\frac{3}{2}}R\tau T_c{[\underset{t}{\max }(\sum _{\omega =-N}^N{A}_h\exp (i{\alpha }_{h}R-i\omega t)\mathrm{\Delta }\omega )]}^{-1}\mathrm{.}$$

We consider that various particles embedded in fused silica are irradiated at 355 nm during pulse duration of 10 ns, and the critical fluence as a function of particle radius has been plotted in Fig. 2.

We can see from Fig. 2 that ${\mathrm{CeO}}_2$ particles require higher fluence to initiate damage when the particle radius is $<100\mathrm{nm}$.

2.2.

Laser Damage Probability on the Surface of Optical Materials

We assume that the breakdown is reached if a particle is irradiated with fluence higher than $F_c$, and the damage probability can be theoretically calculated based on the distribution law of particles. When the damage precursors are assumed to be subsurface inclusions, the laser damage probability can be expressed as a function of fluence $F$.¹⁰

Eq. (9)

$$P(F)=1-\exp [-{\int }_0^{F}g(F_c)S_{F_c}(F)\mathrm{d}{F}_c],$$

where $S_{Fc}(F)$ is the region within the spot size with fluence $F$ greater than critical fluence $F_c$, $S_{Fc}(F)=(\pi {\omega }_0^2/2)\ln (F/F_c)$, with ${\omega }_0$ the beam radius. $g(F_c)$ presents the number of defects per unit area that damage at fluence between $F_c$ and $F_c+d{F}_c$. However, $F_c$ depends on the particle size, and the size distribution of particles is unknown. Hence, we consider a power law distribution since this type of variation is typically found for clusters.¹⁹

Eq. (10)

$$n(R)=\frac{(\gamma -1)d_0}{R_{\min }^{1-\gamma }-R_{\max }^{1-\gamma }}{R}^{-\gamma },$$

where $\gamma$ is a constant (its value often is 2 to 4 for natural processes, such as optics contamination²⁴), and $d_0$ is the density of particles per unit of surface. Based on the relationship between critical fluence and particle size, the upper limit $R_{\max }$ can be obtained from measured damage threshold and the lower limit $R_{\min }$ can be obtained where the experimental damage probability is 1. The relationship between $g(F_c)$ and density of particles $d_0$ is

Eq. (11)

$${\int }_0^{\infty }g(F_c)\mathrm{d}{F}_c=d_0.$$

With this model we have the ability to describe laser damage on the surface of fused silica as function of fluence $F$ by choosing two physical characteristics: the size distribution of particles $\gamma$ and their density $d_0$ on the subsurface of optical materials. By choosing the fit distribution parameters $d_0$ and $\gamma$, we can insert the $R_{\min }$ and $R_{\max }$ from sample S1 (see Table 1) to calculate the laser-induced damage probability based on the relationship between critical fluence and particle radius.

Table 1

The values for Rmin and Rmax of different particles from samples S1 to S4.

$R_{\min }$, $R_{\max }$ (nm)	${\mathrm{CeO}}_2$	Cu	Fe	Cr
S1	37, 50	9, 13	11, 15	13, 16
S2	33, 45	8, 11	10, 13	12, 14
S3	32, 38	7, 10	9, 12	11, 13
S4	30, 36	6, 9	8, 11	10, 12

Figure 3 shows that damage probability initiated by ${\mathrm{CeO}}_2$ particles increases as the density of particles $d_0$ increases, and decreases as the parameter of size distribution $\gamma$ increases. In order to identify the influence of various particles on damage probability, we plotted in Fig. 4 the curves of laser damage probability initiated by various particles calculated with same parameters $d_0=1\times {10}^6{\mathrm{mm}}^2$ and $\gamma =3$. From Fig. 4, we can see that considering the size distribution from sample S1 as seen in Table 1, ${\mathrm{CeO}}_2$ particles have a greater damage probability than others (Cu, Fe, and Cr) with the same distribution parameters $d_0$ and $\gamma$.

Fig. 3

Laser damage probability initiated by ${\mathrm{CeO}}_2$ particles with different distribution parameters $d_0$ and $\gamma$.

Fig. 4

Laser damage probability initiated by various particles with same distribution parameters ($d_0=1\times {10}^6{\mathrm{mm}}^2$ and $\gamma =3$).

3.

Experiment

The experimental setup for laser-damage test has been described in detail elsewhere,^11,17 and only a brief description is given here. The data points of laser damage probability are measured at 355 nm using injected Nd:YAG laser with the Gaussian temporal profile. The effective pulse duration (at $1/\mathrm{e}$) is 10 ns. In order to obtain typical damage probability in larger range of fluence, the small spot diameter of 8 μm (at $1/e^2$) is chosen in the test. The error of measured spot diameter is $\sim 140\mathrm{nm}$. The damage test 1-on-1 is applied with a large number of points to obtain a reliable measurement. We observe the 50 different regions under the laser irradiation at each fluence $F$, and each data point $P(F)$ is plotted by counting the number of damage regions at each fluence $F$. Energy of the incident beam is measured with a calorimeter, and the fluence fluctuations have a standard deviation of $\sim 10\%$. To have a good accuracy of measurement, the test procedure of damage probability is repeated 10 times and the deviation $▵P$ of average value is $<0.08$. In order to identify the effect of the contents of various impurities on laser damage probability, the components of impurities from subsurface layer are determined by ICP-OES and the data points of damage probability have been measured.

The fused silica samples (S1, S2, S3, S4) polished by cerium oxide slurry with different polishing levels were used in the experiment. Because of insufficient polishing process, there are more structural defects (per area), such as submicroscopic cracks, pores, and indentations, observed on the surface of samples S3 and S4. The size of the samples is $35\times 35\times 3\mathrm{mm}$. After accurate weighing and thickness measurement, $\sim 1\mu \text{m}$ of fused silica was digested by ultrapure grade hydrofluoric acid solution during 7 min. The masses of subsurface layer digested, respectively, were 0.00215, 0.00243, 0.00256, and 0.002695 g. The contents of impurities can be obtained by suitable spectral analysis. The contents of ${\mathrm{CeO}}_2$ and ${\mathrm{Al}}_2{\mathrm{O}}_3$ incorporated during polishing and cleaning process can be calculated based on the contents of Ce and Al measured by ICP-OES. Table 2 gives the contents of main impurities from the subsurface layer.

Table 2

The contents of main impurities from subsurface layer of fused silica (μg/g).

We can see from Table 2 that the contents of ${\mathrm{CeO}}_2$ impurities have much more than others and have large distinction in different samples. In order to relate the contents of various impurities to damage probability, the impurities are assumed to be spherical and their mass $m$ (per area) has a homogeneous distribution on the subsurface of fused silica. Thereby, the density (per area) of particles $d_0$ can be calculated from Eq. (10).

Substituting Eq. (13) into Eq. (9), the curves of damage probability from samples S1 to S4 can be calculated. The scheme for calculation has been presented in Fig. 5.

Fig. 5

The scheme for calculating damage probability.

Figure 6 shows the experimental data points of damage probability measured on the surface of fused silica and theoretical curves initiated by impurities. As seen in Fig. 6, the smaller particle is required to absorb more fluence to reach breakdown. Thus, the damage threshold increases from samples S1 to S4 because the upper limit $R_{\max }$ decreases as seen in Table 1. Cu and Cr impurities have a very weak influence on experimental damage probability since their contents on the subsurface of the samples are very low. On the contrary, ${\mathrm{CeO}}_2$ and Fe impurities are closely related to the damage probability when the levels of contents are high as seen in sample S1. We can also notice that for the samples with low ${\mathrm{CeO}}_2$ contents (S2, S3, and S4), this correlation is weaker, and it has a good agreement with experimental data on ${\mathrm{CeO}}_2$ contents dependence of damage density.²⁶ In the case of ${\mathrm{CeO}}_2$ impurities, as the dramatic decrease of the contents from samples S1 to S4, the damage density will decrease according to our calculation. As a consequence, the damage probability induced by the laser pulse with same fluence will decrease. Obviously, a large discrepancy is found between theory and experiment in samples S3 and S4 since there are more structural defects located on the subsurface of samples from insufficient polishing process. These structural defects with a spatial distribution add the absorbing centers near the surface²⁷ and cause more damage sites than expected from the distribution of impurities, so the measured laser damage probability is found to be larger than theoretical calculation.

Fig. 6

The data points of damage probability measured on the surface of fused silica and theoretical curves initiated by impurities for different samples. (a) S1. (b) S2. (c) S3. (d) S4.

4.

Conclusion

A model has been presented in order to relate the distribution properties of various impurities on the subsurface of fused silica to damage probability. The theoretical curves of damage probability initiated by the impurities having a given density and size distribution have been obtained. The data points of damage probability on the surface of fused silica have been measured. Meanwhile, the contents of impurities from the subsurface layer of fused silica have been determined by ICP-OES. The correlation of different contents of impurities to damage probability has been analyzed, and it has a good agreement with obtained results. This model is of interest for identifying the influence of various impurities induced by polishing, grinding, and cleaning processes on laser damage probability, and it can also be applied to investigate laser damage on surface of other optical substrates or films.