## Abstract

We present a comprehensive statistical model which includes both the probability of growth and growth rate to describe the evolution of exit surface damage sites on fused silica optics over multiple laser shots spanning a wide range of fluences. We focus primarily on the parameterization of growth rate distributions versus site size and laser fluence using Weibull statistics and show how this model is consistent with established fracture mechanics concepts describing brittle materials. Key growth behaviors and prediction errors associated with the present model are also discussed.

© 2014 Optical Society of America

## 1. Introduction

The energy density at which large, inertial-confinement fusion (ICF) class lasers may be operated is determined by the robustness of the optics and how well they resist laser-induced damage. By operating these lasers in the regime where damage is not just allowed, but orchestrated to match a desired repair and maintenance tempo, the achievable energy densities increase by over an order of magnitude [1–6]. Managing the level of damage with the precision needed to successfully operate in such a dynamic equilibrium has required a much more sophisticated ability to anticipate the appearance and evolution of laser-induced damage than previously available.

Aspects of laser-induced damage have at times been described as either deterministic or stochastic. For many years, the standard measure of optical materials’ susceptibility to damage initiation was the S/1 measurement in which the propensity of a sample to damage was described as the probability of detecting damage as a function of laser exposure fluence [7–9]. Moreover, the lowest fluence for which damage was observed was often cited as the damage threshold. This methodology, while extremely valuable for its relatively simple implementation and its utility in comparing the relative damage resistance among samples, has been recognized as inherently limited to qualitative measurements under narrow circumstances. A number of works have shown that this definition of threshold is highly dependent on sample (laser spot) size [10–12]. For example, depending on spot size, fluence and damage site size, incorrect statistics can be derived regarding damage threshold [13]. It has been shown that a much more deterministic measurement is the damage density as a function of fluence, or *ρ*(*ϕ*), which is a function of pulse duration, wavelength, surface preparation, but not, once sufficient area of the optic has been sampled, spot size [12, 14–17]. The spot size dependence of the two measurements can be observed upon increasing the sampled area. Namely, for the S/1 measurement, the average value of the damage threshold decreases with increased area while, for the *ρ*(*ϕ*) measurement, both the minimum density accessible and the uncertainty at a given fluence are reduced.

Conversely, damage growth (i.e., evolution of damage upon post-initiation laser shots) has largely been treated as a deterministic process in which sites started growing at a fixed fluence threshold and then continued to evolve at a rate (with some uncertainty) defined by the laser exposure fluence [18–25]. Recently, we have shown that the fluence at which a damage site can be prompted to grow depends both on the size of the damage site as well as the laser conditions, and is in fact a stochastic but well characterized function of fluence and site size [26]. In this work, we complete the inversion of how damage is described; namely, we show that the rate at which a damage site grows is not deterministic with fluence but a stochastic function of both laser fluence and site size with distinct trends (average behaviors).

We present a comprehensive statistical model to describe the expected rate of increase in the lateral dimensions of exit surface damage sites on fused silica optics. The growth model presented here is based on ∼4300 instances of damage site evolution under exposure to 351-nm, 5-ns flat-in-time (FIT) pulses. The data set includes growth observations from multiple experiments performed at fixed laser fluences, spanning from ∼5 J/cm^{2} up to about 23 J/cm^{2}, with exit surface sites 30 microns and larger. In this work, we introduce a new mixed joint probability model which includes both the probability of growth and growth rate with focus primarily on the parameterization of growth rate distributions vs. site size and laser fluence. Namely, we model the growth rate as a continuous random variable drawn from a Weibull distribution and show how this model is consistent with established fracture mechanics in fused silica. In addition, we apply the model to forecast the evolution of damage sites over multiple laser shots spanning a wide range of fluences. Salient growth behaviors and prediction errors associated with the present model are also discussed.

## 2. Experimental procedures

Details of the experimental approach have been described elsewhere [25, 27]. In brief, on the order of 100 damage sites per sample were pre-initiated in a regular array using a table top Nd:YAG laser system with ∼400 *μ*m 1/e^{2} diameter spot focused on the exit surface of the test sample by ramping up in fluence at each location until damage occurs. In this work, preinitiated sites (or as-initiated sites) will refer to sites produced by this procedure without further modification. Prior to damage initiation, all substrates (2-inch diameter, 1-cm thick 7980 Corning glass) were prepared with high damage resistant surfaces, representative of large area optics [28]. The grid spacing varied across samples/experiments according to the intended shot sequence; we found no measurable cross talk between adjacent damage sites for a grid spacing of at least 3X the projected final damage site diameters.

The growth experiments used the 3-cm diameter Optical Science Laboratory (OSL) laser beam with ∼17% spatial beam contrast [29] to simultaneously expose all the sites on a single sample. We thus tested sites with a range of local fluences that vary within ∼2–3 J/cm^{2} around the beam average fluence and computed the local mean fluence in a ∼1 mm patch with better than 5% uncertainty using fluence registration methods outlined in [27]. Individual site diameters are then measured using a robotic microscope under various illuminations with optical resolution as high as ∼1 *μ*m. Although we have conducted experiments under a wide variety of laser conditions, this work will focus on growth observations from exit surface damage sites on fused silica with 351 nm, 5 ns flat-in-time (FIT) pulses in high vacuum (10^{−5} torr) and at room temperature. Specifically, we include results from 18 separate experiments (samples) in which we monitored the evolution of pre-initiated damage sites from ∼30 *μ*m up to hundreds of microns and even several millimeters under multiple laser exposures (5 or more) at fixed laser fluences, spanning from ∼5 J/cm^{2} up to about 23 J/cm^{2}.

## 3. Damage site morphology

Typical damage site (as-initiated and grown) morphologies using high-resolution scanning-electron-microscope (SEM) imaging are illustrated in Fig. 1. As-initiated damage sites with diameters up to about 50 *μ*m and grown damage sites (30 *μ*m and larger) fall into one of three general categories. Type I damage consists of smooth, typically sub-micron features (historically referred to as gray haze [15,30,31]) and are indicated by white arrows in the image at the top, left hand side column of Fig. 1. Type I sites are much more prevalent on low damage resistance surfaces and often accompany the other types of damage. These features only rarely have associated fracture and do not evolve under nanosecond laser exposure in the range of fluences examined in this work. Type II damage (on the left hand side column of Fig. 1) comprise a wide range of morphologies with either single or multiple smooth fracture surfaces lacking a central damage core region (historically referred to as mussels [15, 32, 33]), with diameters ranging from about 5 *μ*m up to about 30 *μ*m. Type III damage consists of larger sites, as-initiated or grown (right hand side column of Fig. 1), which include a central damage core containing highly modified material, rich in defects [30, 34, 35] in addition to multiple fracture surfaces (historically referred to as pansies [15, 32, 33]). Moreover, type III sites are largely self-similar for a given pulse duration, with comparable morphologies at different spatial scales. Growing sites typically evolve from type II to type III after one or more laser exposures (depending on the laser parameters and individual site morphology). In our experiments, we note a fairly sharp transition from type II to type III damage for site diameters of ∼30±5 *μ*m which relates primarily to the laser pulse temporal duration and shape used for damage initiation (10-ns, near Gaussian) [15,16]. This type II-type III damage boundary region shifts to smaller site sizes with reduced pulses duration and begins to fail for pulses shorter than ∼2 ns.

Growth of type II sites is greatly influenced by their morphology and is intermittent in nature – their evolution over a number of shots is dominated by a probability of growth rather than growth rate, as discussed in detail in [26]. This behavior can be related to the fact that the effective fracture toughness of a site will scale with the square root of the ratio of crack tip radius and crack length [36]. In damage sites with smooth broad features as with type I/II sites, the toughness is high. In fact, it is difficult to establish a cleavage feature in which a crack tip would reside, at least on the length scale of the SEM images. In contrast, type III sites grow more consistently shot-to-shot, with a probability of growth in excess of 50% (90%) for laser fluences above 5 (8) J/cm^{2}. Type III damage morphology is consistent with this behavior: high concentration of long crack lengths, and the presence of many cleavage features presumably containing small crack tip radii. Experiments are ongoing to expand our database on small site growth behaviors to include the effects of site morphology. In this work, we focus on the parameterization of growth rate vs. site size and laser fluence for type III damage sites with diameters larger than 30 *μ*m.

## 4. Statistical growth model

We propose to model the growth behavior of exit surface damage sites in fused silica under repeated UV, ns laser exposures using a mixed joint probability density function (PDF, describing the probability for the variable to assume a value from the population of all possible outcomes). The evolution of a damage site based on the expectation of exponential growth between laser shots (with 5 ns FIT pulses) has historically been given by:

where*d*

_{n−1},

*d*are the effective circular diameters (ECD) before and after the

_{n}*n*th shot, though some research groups have used the area of the site instead [1, 19, 23, 37]. The growth coefficients

*η*and

*α*are random variables describing the probability of growth and growth rate, respectively.

We model the probability of growth coefficient, *η*, as a discrete random variable which takes two distinct values, either 0 or 1, corresponding to *α* = 0 (no-growth events) and *α* > 0 (growth events), respectively. We use the binomial distribution to model the probability of growth process with a probability mass function (PMF) of getting exactly *X* number of successes (i.e., *η* =1) in *m* trials (experiments) given by:

*C*(

*m*,

*X*) is the binomial coefficient and

*p*is the probability of success in each trial – the measured probability of growth for type III damage sites as a function of site size and exposure laser fluence [26]. In practice, we use Monte-Carlo methods to simulate

*m*growth experiments (a large number) for each damage site [38] and generate a random, binomial distributed probability of growth vector

*η*of length

*m*having a number of successes close to the mean of the binomial distribution (i.e., the expected value of

*X*,

*E*[

*X*] =

*m*·

*p*) [39].

We model the growth rate, *α*, as a continuous random variable. The goal is to find a unique, continuous PDF that approximates the experimental growth rate distributions. The distribution will be parameterized in terms of site size and local laser fluence. The general methodology for fitting distributions to the data includes four steps: 1) model/function choice: hypothesize families of distributions; 2) estimate parameters; 3) evaluate quality of fit; 4) goodness-of-fit statistical tests. The statistical methods are not in themselves novel but are discussed in more details in **??**.

Here we summarize the main properties of the 2-parameter Weibull distribution chosen for our model based on exploratory data analysis (see 6 and 9). The PDF of a Weibull random variable *X* (i.e., growth rate, *α*) is:

*k*,

*λ*> 0 are the shape and scale parameters of the distribution, respectively [40]. The form of the PDF in Eq. (3) changes drastically with the value of

*k*; it starts heavily right-skewed for

*k*< 1 and becomes nearly symmetric for 2 <

*k*< 7 (illustrated later in Section 8). The mean (or expected value) and variance of a Weibull random variable

*X*can be expressed as: and

The Weibull distribution is widely used in reliability engineering and failure analysis where the variable *X* is a “time-to-failure”; here the growth rate, *α*, relates to the amount of failure and therefore the inverse of the damage site lifetime. As a matter of fact, our difficulties in dealing with no-growth events (*α* = 0) and the implications associated with the development of a lifetime metric (i.e., infinite lifetime) are the reasons for including the probability of growth in our model (as discussed above).

## 5. Data reduction

The single-shot growth rate (coefficient), *α*, is based on the measured change in the lateral dimensions (ECD) of a site after each laser exposure according to Eq. (1). The experimental error associated with the ECD measurements is about 2 *μ*m [26, 41] and translates into a size dependent error in the growth rate, i.e., about 6% to 4% for 30–50 *μ*m sites and less for larger sites. For perceived changes in ECD less than 2 *μ*m we set *α* (and *η*) values to zero, i.e., no-growth events within the experimental errors. Here we examine the growth rate distributions and therefore consider only growth events with *α* > 0. Individual site growth responses are tabulated with several attributes, including site and sample IDs, shot number, single-shot growth rate and local mean fluence on the corresponding laser shot.

In order to quantify the effects of site size and laser fluence on growth rate, we divide our data set into two-dimensional bins with narrowly defined experimental conditions, e.g., similar in size sites exposed to nearly identical fluences, and examine the sampled (by experimentation) growth rates from each bin. Table 1 summarizes the distribution of ∼4300 observations by size and fluence. The number of data points in each category (bin) is the sample size which governs the errors in estimating the properties of the underlying distribution (mean, variance, skewness, etc.). The specific bounds of the bins were chosen based on the growth trends documented in previous studies and the sampling frequency achieved in our data set. Namely, the growth rate exhibits a nonlinear dependence on site size (see B coefficient in [41]) – declining rapidly with site size up to about 100 *μ*m followed by saturation as the sites get larger. Hence, we vary the size bin width from narrow (small sites) to wide (large sites) to better capture the size effects (see Table 1). In contrast, past results documented a linear dependence of growth rate on laser fluence, thus growth distributions with even sampling of the fluence can be used. Table 1 indicates that not all bins are sampled equally. Most of our observations were collected for sites with *d* < 500 *μ*m exposed to fluences in the ∼5–12 J/cm^{2} range and support the analysis of growth rate distributions based on narrow, 1 J/cm^{2} fluence bins (having more than 50 samples per bin, with a few exceptions). In contrast, less data is available at higher fluences and/or with larger sites due to inherently low yields in any single experiment – fewer sites are placed within the beam aperture to prevent cross talk between adjacent sites. At higher fluences, we group the observations in wider fluence bins (see Table 1) to improve statistics and establish growth trends.

A discussion of the effects of N (number of samples) on the growth rate model accuracy is warranted. For this purpose, we used Monte-Carlo methods to simulate experiments with repeated random N-sampling of the Weibull distribution [42]. We found that the shape of the distribution (*k* parameter) determines, to a first order, the uncertainty (standard errors) in the parameter estimates as a function of N. The influence of the shape parameter can be understood if we consider the flexibility of the Weibull distribution: with a small sample size, if the shape parameter value estimate is slightly off, the estimated shape of the distribution (illustrated in Section 8) may be altered considerably and these slight differences can affect our conclusions. In contrast, the errors in the sample means (estimate the population means) and other descriptive statistics are less sensitive to N (similar errors are obtained for roughly N/2). For a target error of ∼10% or better associated with the distribution parameter estimates from each bin, we estimate the optimal sampling at a minimum N∼100 and N∼50 for 0.6 < *k* < 1 and *k* > 1, respectively.

Similarly, fluence binning – the width of the bin – can also affect model accuracy in ways that are more difficult to quantify. The question is whether or not coarser fluence bins (2 J/cm^{2} versus 1 J/cm^{2} at higher fluences, see Table 1) preserve (interpolate) the underlying trends in the data. For example, the union distribution of adjacent 1 J/cm^{2} fluence bins has higher variance than that of individual bins and may distort the shape of the sampling distribution. These effects are however reduced at higher fluences where the sampling distribution parameters were found to vary slowly as a function of fluence. In what follows we use 2 J/cm^{2} fluence bins at higher fluences to derive sample means and only sparsely to estimate distribution parameters.

## 6. Results

In this section we i) present key findings from exploratory data analysis which support the growth rate model based on Weibull distribution, and ii) summarize the size and fluence dependent Weibull shape and scale parameters which allow growth parameterization. Additional results and example distribution fits to the data can be found in 9.

Figure 2 illustrates the single-shot growth rates measured from type III exit surface damage sites as a function of laser fluence. Sites are binned by their pre-exposure ECD in Figs. 2(a)–2(d) as 30–50, 50–100, 100–500, and 500–5000 *μ*m, respectively. All graphs have the same *x–y* scales to aid the comparison of growth trends vs. size. The density of data points in Fig. 2 conveys the coverage of a large parameter space, however with different sample sizes (presented in Table 1). The data illustrates the large variability in growth rates (much beyond measurement errors, estimated at less than ∼6% and ∼5% in growth rate and local mean fluence, respectively) recorded from similar in size sites under exposure to nearly identical laser conditions; this spread is also laser fluence and size dependent.

In Fig. 3 we summarize the data trends in Fig. 2 using common descriptive statistics (see 9 for the growth rate populations from discrete size and fluence bins. Specifically, we plot the sample means with standard errors (solid data points and vertical error bars, respectively) as a function of laser fluence (bin center shown on the *x*-axis) and site size (shown in the legend). Here we used wider fluence bins to compute the sample means at higher fluences with better statistics (as indicated in Table 1); we did however compare the results in Fig. 3 to those obtained from the corresponding 1 J/cm^{2} fluence bins (not shown here) and found no statistically significant differences between the data sets. The uncertainty in the mean is less than ∼10–15% for all data points in Fig. 3 with three exceptions denoted by arrows, which are included to better define the growth trends. Results from each size bin suggest that the mean growth rate is monotonically increasing with laser fluence, ascending faster in the beginning followed by a slower rate of increase (a stretched out “S” shape with distinct phases of growth). Hence, we use logistic curve fits to summarize the trends, as shown by the solid lines in Fig. 3 (with adjusted R-square greater than 0.98) of the general form:

*A*

_{1},

*A*

_{2},

*x*

_{0}, and

*S*represent the initial and final (saturation) growth rate values, center (the value of

*x*at midpoint, in fluence units) and shape (relates to the inflection point), respectively. For completeness, Fig. 3 includes the sample means from the highest size bin (magenta, solid diamonds data points) but data are insufficient to support a similar logistic fit at this time. Pending future experimentation with large sites, growth rates from the two highest size bins are deemed comparable within the statistical errors. For all other size bins, we set

*A*

_{1}= 0 as the growth rate drops off naturally towards zero at fluences below ∼5 J/cm

^{2}[26]. Table 2 shows the best fit parameters and their size dependence; these parameters allow interpolation of the mean growth rate at arbitrary fluences. It should be noted that fits to the data from different size bins in Fig. 3 are distinct for all fluences in the range ∼5–23 J/cm

^{2}at the 95% confidence level.

It should be noted that most previous studies reported multi-shot growth rates of individual sites with modest statistics at laser fluences up to about 12 J/cm^{2}; as such, linear fits to *α*(*ϕ*) (trending upwards with fluence) approximated the average (over all sizes) growth behaviors, with the onset of observable growth (growth threshold) near ∼5 J/cm^{2} and slopes between 0.03–0.04 (depending on the laser parameters) [1,19,21,25]. More recently, we have refined the laser pulse duration and site size dependence to the growth threshold and slope coefficients in *α*(*ϕ*) [27, 41]. In comparison, results in Fig. 3 summarize trends in the single shot growth rate with considerable improved statistics over an extended fluence range and reveal three different stages of growth in *α*(*ϕ*) – nearly exponential, linear and sub-linear increase at low (∼5–6.5 J/cm^{2}), intermediate (∼7–13 J/cm^{2}) and high (∼15–23 J/cm^{2}) fluences, respectively. We will discuss these behaviors and their significance in more detail later.

Next, we use graphical techniques to gain insight into the shape of the underlying growth distributions. For example, frequency density plots (or histograms) estimate the underlying probability density function and can be compared to the fundamental shapes associated with standard analytic distributions. This procedure is illustrated in Fig. 4 for sites with diameters in the 50–100 *μ*m range (similar results were obtained for the other size bins, not shown here). Specifically, the growth rate observations from each fluence bin (here a categorical *x*-variable, bin center±1/2 width in J/cm^{2}) are displayed as one-dimensional dot plots with systematic jittering (green, closed circles) along the *y*-axis and convey the shape of the experimental growth rate distributions. In addition, we overlay the Weibull PDF curves with parameters estimated from the data (blue, solid lines in Fig. 4). Results suggest that the shape of the experimental distribution changes significantly with increasing fluence – starts as heavily right-skewed (with most growth rate values near zero) at fluences around ∼5–6 J/cm^{2} and gradually evolves into a two-sided distribution, nearly symmetric at fluences beyond ∼11–13 J/cm^{2}. Since the shape of the theoretical Weibull distribution is similarly flexible, we hypothesize a growth model based on the Weibull distribution with fluence dependent shape parameter. Indeed, the qualitative agreement between the experimental distributions and the estimated Weibull PDF curves in Fig. 4 supports this hypothesis. Moreover, data suggests that the shape parameter, *k*, increases monotonically with fluence from ∼1 to ∼5 over the ∼5–23 J/cm^{2} fluence range. Additional results from exploratory data analysis to support the Weibull model (e.g., the fluence dependence of a more complex shape indicator) are presented in 9.

Having chosen a model based on exploratory data analysis (key findings presented above), we now proceed to parameterize the growth behaviors by fitting Weibull distributions to the data from individual size and fluence bins with adequate sample size (i.e., uncertainties in the estimated distribution parameters on the order of 10%). As discussed in the previous section, coarser fluence binning, in addition to sample size, may also distort the shape of the distributions. Therefore, we base our model primarily on results derived from 1 J/cm^{2}-fluence bins (fluences up to ∼15 J/cm^{2}) and supplement at higher fluences with results from 2 J/cm^{2}-fluence bins to better define the trends. The statistical methods (e.g., maximum-likelihood-estimation (MLE) of distribution parameters, goodness-of-fit tests) and representative fits to experimental distributions are discussed in more detail in 9.

In Fig. 5 we summarize the estimated shape parameter values with standard errors (vertical error bars) from selected experimental growth distributions. The results based on 1 J/cm^{2} fluence bins are shown by the solid data points and span fluences up to ∼11–12 J/cm^{2} and ∼15 J/cm^{2} for sites with d=30–100 *μ*m and d=100–500 *μ*m, respectively. In addition, open data points in Fig. 5 represent results from 2 J/cm^{2} fluence bins. All uncertainties in the parameter estimates are within ∼10%.

For all site sizes, results in Fig. 5 suggest that the Weibull shape parameter is fairly constant up to about 6–6.5 J/cm^{2} and increases monotonically with fluence beyond that point – for simplicity, we will assume a linear fluence dependence. The latter behavior is consistent with the results in Fig. 4 and Fig. 9 (see 9). In addition, the estimated *k*-values from the two highest size bins largely overlap within the error bars (black squares vs. blue triangle data points). Based on these trends, we propose a simple analytical approximation for *k*(*ϕ*) in Fig. 5 as:

*ϕ*) is the Heaviside step function and

*ϕ*is the fluence in J/cm

^{2}. Equations (7) represent best fits to the data in Fig. 5 which are depicted by the dotted lines (green and blue, for d=30–50 and 50–500

*μ*m, respectively); the model for

*k*(

*ϕ*) is well supported by the

*k*-estimates at low fluences from 1 J/cm

^{2}fluence bins (solid data points) and extrapolates up to ∼23 J/cm

^{2}based on estimates from 2 J/cm

^{2}fluence bins (open data points).

In Fig. 6 we summarize the estimated scale parameter values with standard errors (vertical error bars) from selected experimental growth distributions. The results based on 1 J/cm^{2} fluence bins using MLE fitting are shown by the solid data points (the same bins used for *k*(*ϕ*) in Fig. 5). At higher fluences, the results from 2 J/cm^{2} fluence bins were unreliable (estimated errors greater than 20%) due to insufficient sample sizes. We used the properties of the Weibull distribution and Gamma function to work around this problem. Specifically, Eq. (4) indicates that the mean of the Weibull PDF is directly proportional to its scale parameter, i.e., *E*[*X*] = *λ*Γ(1 + 1/*k*). The sample means, in contrast to sample distribution parameters, are less sensitive to bin sample size and can be estimated with lower uncertainties (see Table 1 and results in Fig. 3). In addition, results in Figs. 4 and 5 suggested that 2 < *k*(*ϕ*) < 6 for laser fluences beyond ∼11 J/cm^{2}. Under these conditions, the quantity Γ(1 + 1/*k*) is slowly varying and constrained to the interval (0.89, 0.93), independent of the exact functional form of *k*(*ϕ*) (here assumed to be linear). Thus Eq. (4) in conjunction with Eqs. (7) can be used to estimate the scale parameter values at higher fluences for all size bins (as shown by the open data points in Fig. 6) with only slightly larger uncertainties than those associated with sample means (increased variance due to Γ(1 + 1/*k*) term).

For all size bins, results in Fig. 6 suggest that *λ*(*ϕ*) exhibits a similar “S” shape behavior to that of the mean growth rate vs. fluence in Fig. 3. Therefore, we use logistic curve fits to summarize the data trends, as depicted by the dotted lines in Fig. 6 (with adjusted R-square greater than 0.98); these fits allow interpolation of the Weibull scale parameter at arbitrary fluences in the range ∼5–23 J/cm^{2}. The best fit parameters (using Eq. (6) with *A*_{1} = 0) and their size dependence are shown in Table 3.

In summary, we have parameterized the measured growth rate distributions from type III exit surface damage sites in fused silica following exposure to 355-nm, 5 ns FIT pulses using a Weibull distribution model. We have estimated the means, shape and scale parameters of the distributions as a function of laser fluence and site size and provided simple, analytical approximations for use in predictive models. In addition, these approximations were verified to be internally self-consistent, i.e., the logistic fits to the sample means (Eq. (6) and Table 2) are consistent with derived distribution mean values via Eq. (4) using the logistic fits to the scale parameter (Eq. (6) and Table 3) and linear fits to the shape parameter (Eqs. (7)).

## 7. Model validation

The ultimate goal of this study is to construct a growth model that can forecast the evolution (trajectory) of damage sites over multiple laser shots spanning a wide range of fluences. The data for probability of growth [26] and growth rate presented here were reduced using single-shot observations and growth trends analyzed as a function of laser exposure fluence and site size; this approach implicitly assumes independent and identically distributed single shot events. In order to validate the growth model in the context of multiple laser shot/varied fluence conditions, we use Monte-Carlo methods to simulate the evolution of a collection of damage sites with a distribution of sizes over the laser shot sequence (from our experiments) and compare the predicted vs. measured final size distributions [38]. The initial site size distributions, average multi-shot laser fluences, as well as the number of sites/laser shots from separate experiments are summarized in Table 4.

The simulation uses the initial size distribution (at shot 0) and randomly picks the growth behavior of 2000 simulations based on the size and fluence dependent probability of growth, *η*, and growth rate, *α*, in Eq. (1) for each shot thereafter. The starting site sizes for each of these samples are similar, ranging from ∼30 to 100 *μ*m, and the targeted mean final sizes for the simulation were typically 250 *μ*m. A target size of ∼250 *μ*m was chosen because it approaches but is still below the critical size of ∼300–500 *μ*m dictated by current laser damage mitigation techniques [3,5]. However, because the average fluence in each experiment differed in order to span the 5–23 J/cm^{2} range, the number of shots to reach a final mean size of ∼250 *μ*m varies from sample to sample as seen in Table 4, i.e., 20, 6 and 4 shots for low, intermediate and high fluences, respectively. The measured (from experiments) size distributions (initial and final) for the ensemble of sites are quantified in terms of the cumulative size distribution (CDF). In addition, since the Monte-Carlo simulation for individual sites yields a distribution of sizes from 2000 random trials, the predicted final size distribution for the ensemble of sites is expressed as the CDF of (mean values ± standard deviation) from individual site predictions.

The initial ensemble size CDF of all sites from all samples is shown in Fig. 7 for reference (blue, solid triangle-line). The equivalent ensemble CDF of all sites after they have been grown may also be seen in Fig. 7 (red, solid circle-line). The multi-shot simulation results (Fig. 7, black dashed line) match well with the experimental results at both ends of the final size distribution (up to 100 *μ*m and beyond 350 *μ*m) and lag everywhere else by ∼50, 35 and 10 *μ*m at CDF∼0.10, 0.50, and 0.8, respectively. In our previous study, where we simulated one experiment at low fluence (sample A) without including the probability of growth, the prediction results matched the final sizes for CDF∼0.2–0.5 and overestimated everywhere else (see Fig. 3 in Ref. [38]). The addition of probability of growth to the model (presented here) has the expected effect on model predictions, i.e., decreases both the mean and the variance (by reducing the tails) of the final size distribution. With respect to the accuracy of predictive operational models, results in Fig. 7 validate the mixed joint probability model for multiple laser shots with an expanded fluence range up to ∼23 J/cm^{2}.

## 8. Relationship to energy absorption and fracture mechanics

We now discuss the physical interpretation of the parameters gathered from the Weibull model. Additional details about the model discussed below are available elsewhere [43]. Because the key measurable of our experiments is the extension of cracks emanating from a damaged area, the ultimate governing physics is that of quasi-brittle fracture mechanics. It has previously been shown that laser induced damage events can result in high temperatures and pressures [44] and large stress loads in and around the damage site leading to generation of a shockwave, onset and growth of radial, circumferential and axial cracks, and prolonged material ejection [45,46]. However, because of the mechanical strength of the material, a critical stress loading is required for crack propagation. It can be shown using the Griffith energy balance criterion between elastic strain energy and free surface energy [47] that crack propagation will occur when the applied stress *σ* (under mode I loading) exceeds a critical value given by:

*ℓ*is the characteristic crack length.

Energy absorbed from a laser pulse that generates stress fields *σ* < *σ _{c}* will cause heating, acoustic waves and thermal emission but are expected to leave the elastic body intact. For higher fluence levels such that

*σ*>

*σ*, fracture will occur leading to a lateral (and axial) expansion of cracks which can be identified as damage growth. The amount of damage or new surface area created by the laser pulse is related to the excess absorbed energy above the critical energy which yields the fracture criterion above through the material surface energy,

_{c}*γ*. If we define

*χ*as the fraction of energy absorbed (or ’coupling efficiency’) that contributes to the elastic strain field surrounding a crack in a damage site, the fractional change in surface area can be expressed in terms of fluence as:

*ϕ*corresponds to the condition

_{c}*σ*=

*σ*. Recasting

_{c}*α*in terms of the change in damage surface area Δ

*A*and initial surface area

*A*=

*πd*

^{2}/4, and inserting the expression above for Δ

*A/A*, Eq. (1) becomes:

*α*(

*ϕ*) in Eq. (10) naturally leads to a linear increase at low fluences (previously documented, see [1,19,21,25,27,41]) since ln(

*z*+ 1) ≈

*z*for

*z*≪ 1 and a sub-linear dependence at higher fluence (presented here). Both the coupling efficiency

*χ*and the critical threshold

*ϕ*in Eq. (10) are expected to depend on damage site morphology, the former through the density of highly absorbing fracture surfaces and the latter indirectly through the crack length dependence implicit in the Griffith criterion of Eq. (8).

_{c}We can then estimate the parameters introduced in Eq. (9) from experiments performed at single pulse duration, i.e., 5-ns FIT, and similar in size damage sites [41]. If the average growth rate 〈*α*〉 increases linearly with fluence as:

*χ*〉 ≈ 2

*γB*and 〈

*ϕ*〉 ≈

_{c}*ϕ*. Ref. [41] discussed the size dependence of the rate of increase in growth rate with fluence (

_{th}*B*coefficient or slope, in cm

^{2}/J) and fluence threshold for growth (

*ϕ*, in J/cm

_{th}^{2}), i.e., both decreasing with site size. Since smaller damage sites have, on average, smaller crack lengths than large ones, higher stresses will be required to open them up. This means that more energy is absorbed, and, assuming that the majority of this energy is used to create a new fractured site, larger relative areas will be created for small sites as compared to large ones, and this can lead to larger growth rates for smaller sites. Using the experimental values of the coefficients in Eq. (11) [41] and

*γ*= 4.3 J/m

^{2}as the surface energy for fused silica [48], we can estimate

*χ*and

*ϕ*for individual size bins in Table 1, e.g., ∼3.4·10

_{c}^{−5}and ∼7.5 J/cm

^{2}, and ∼2.7·10

^{−5}and ∼5 J/cm

^{2}for sites with

*d*=30–50 and 100–500

*μ*m, respectively.

In terms of incident fluence distribution, we expect that the irregular damage site morphology associated with molten and fractured surfaces will lead to large variations in local fluence. Indeed, Génin et al. showed that diffraction along conical cracks under uniform illumination can lead to ∼100X intensity fluctuations [49]. We therefore consider that the *local* fluence within a given damage site leading to laser absorption and breakdown and appearing in Eq. (10) can be described by a probability density function (PDF). The fluence PDF, *f _{ϕ}*, would naturally take into account the variation in morphology across all damage sites tested and could alternatively include the probability of overlap between light intensity and absorptivity (which is also expected to vary within a damage site). Moreover, the fluence distribution gives rise to the size and fluence dependent probability of growth and growth rate distributions via Eq. (10). Here we focus on the latter and assume a normal fluence distribution with mean

*ϕ*

_{0}and standard deviation

*σ*. It can be shown [43] that the PDF for

_{ϕ}*α*is then given by:

In Fig. 8(a) we plot *f _{α}* described by Eq. (12) for different parameters of the fluence PDF and fixed coupling efficiency,

*χ*=3.4·10

^{−5}. At low fluences near the growth threshold (Δ

*ϕ*=

*ϕ*

_{0}−

*ϕ*∼ 0), the growth distribution is right skewed, with most growth rate values near zero. This behavior is to be compared with Weibull distributions with

_{c}*k*≤ 1 as shown in Fig. 4 and Fig. 8(b). In contrast, the distribution is two-sided, nearly symmetric for fluences above the threshold, scaling with the ’excess’ fluence Δ

*ϕ*, which can now be identified with the Weibull parameter

*λ*. The uncertainty in the growth responses (width of the distribution) increases with larger

*σ*, as expected. The characteristic behaviors in Fig. 8(a) agree qualitatively with the measured growth behaviors. The similarities between the fracture-derived and Weibull PDFs in Figs. 8(a) and 8(b), in particular at fluences above the growth threshold, suggest that the Weibull statistics is consistent with our physical model based on established fracture mechanics in fused silica and support the use of the latter to streamline and expand data analysis of experimental growth distributions over a wider range of damage site sizes and laser fluences (see Table 1). Note however that at fluences near the growth threshold, unlike the Weibull distribution, peaked solutions for

_{ϕ}*f*are finite for arbitrarily small

_{α}*α*values (in agreement with data) and decay much faster than

*k*≤ 1 Weibull functions. That is, the Weibull PDF possesses a higher kurtosis than that of the fracture PDF, and in turn describes the measured distribution better for large

*α*values. Moreover, Weibull statistics will enable future parameterization of growth rate for a variety of experimental conditions (laser pulse duration/exposure history, damage initiation conditions, environment, etc.) and help fine-tuning of the multi-parameter fracture model.

## 9. Summary

The statistical model presented here focuses on growth parameterization at fixed wavelength and laser pulse duration and shape. We use the Weibull family of distributions to quantify the dependencies of the growth rate on the laser fluence and site size based on a large number of observations. The model captures the salient behaviors associated with exponential damage growth on the exit surface of fused silica optics under ns laser exposure as follows:

- The mean growth rate from similar size sites is monotonically increasing with laser fluence, ascending faster in the beginning followed by a slower rate of increase – this behavior is reflected in the fluence dependence of the Weibull scale parameter,
*λ*. - For a given size range, the variance of the growth rates increases with fluence which can be related to competing effects associated with
*λ*and*k*. - At the same fluence, smaller sites tend to grow with larger growth rates, on average, than larger sites (Fig. 3) and possess a higher fluence threshold – the intercept and slope of the characteristic logistic “S” curves is modulated by both Weibull parameters,
*λ*and*k*. - Probability of growth is intimately tied to the shape parameter
*k: k*values near or less than 1 imply a decaying exponential PDF near the growth threshold that is associated with a highly stochastic growth behavior.

Expanding on the latter behavior, the results presented in Fig. 5 indicate two distinct behaviors in the functional dependence of *k*(*ϕ*), i.e., a plateau (*k* ∼constant) near the growth threshold followed by a linear increase in which *k* tracks the mean fluence, *ϕ*. In particular, the behavior at low fluences, up to about 6–6.5 J/cm^{2}, coincides with the region where the probability of growth is less than 100% [26]. In contrast, our statistical model only includes growing sites which by definition assumes 100% probability of growth. The constant, low k-values (*k* ≤ 1) associated with the growth distributions in this range of fluences are indicative of intermittent growth behavior between laser shots and/or different failure mechanisms at low vs. high fluence.

We also introduce a mixed joint probability model which includes both the probability of growth and growth rate and show how it can be used to forecast the evolution of damage sites over multiple laser shots spanning a wide range of fluences.

Finally, we discuss the physical interpretation of the damage growth data in the framework of established fracture mechanics in fused silica and derive an alternative, fracture-based PDF to approximate the measured growth rate distributions. The salient features of the Weibull PDF analysis can then be viewed from a physical perspective. In particular, the slope of *α*(*ϕ*) near threshold was found to scale as *χ*/2*γ*, implying that smaller sites more efficiently couple laser energy into elastic strain energy. Smaller damage sites also tended to possess larger damage thresholds, consistent with the idea that the critical stress leading to brittle failure varies inversely to square root of the fracture (crack) length scale. The transition near threshold from a *k* < 1 to *k* > 1 behavior in Weibull statistics can be related to the fracture model condition *ϕ* > *ϕ _{c}*, with the width of the

*α*distribution being related to the fluence standard deviation. However, the higher kurtosis of the Weibull near threshold better describes the experimental data. Nonetheless, similarities between the fracture and Weibull PDFs, in particular at fluences above the threshold, support the use of Weibull statistics to streamline and expand growth parameterization for a variety of experimental conditions which will enable future development of the multi-parameter fracture model.

## Appendix

The general methodology for fitting distributions to the data includes four steps: 1) model/function choice: hypothesize families of distributions; 2) estimate parameters; 3) evaluate quality of fit; 4) goodness-of-fit statistical tests. We used Origin commercial software (from OriginLab, Northampton, MA) and R language [50] for data visualization and statistical computing, respectively.

Exploratory data analysis can be the first step in suggesting the kind of probability density function (PDF) to use for a statistical model; this step includes getting descriptive statistics and using graphical techniques (histograms, density estimates, box plots, etc.). All descriptive statistics provide numerical summaries of the data, such as measures of central tendency or variability, independent of the underlying sample’s distribution (see Fig. 3).

The most common statistics for sampling of a population are the sample mean, *x̄* (estimates the population mean) and the standard deviation, *SD* (*SD*^{2} estimates the population variance) computed as:

*N*is the number of observations. In addition, the standard error of the mean,

*SE*, estimates the standard deviation of the error in the sample mean relative to the true mean (by virtue of the central limit theorem) and is given by:

_{x̄}To gain insight into the shape of the underlying distributions, we used graphical techniques (such as frequency density plots or histograms, one-dimensional dot plots) to visualize the experimental growth distributions and compare them to standard analytic distributions (see Fig. 4). Results showed that the shape of the experimental distributions change significantly with fluence, resembling the flexibility of the theoretical Weibull distribution vs. shape parameter. For reference, in Fig. 8(b) we illustrated the Weibull PDF curves prescribed by Eq. (3) with fixed scale (*λ* = 0.2) and various shape parameters, *k*=0.7, 1, 2 and 5, respectively. We also found an additional, more complex shape indicator (descriptive statistic) which depends solely on the shape parameter, *k*. Using Eqs. (4)–(5) for the mean and variance of the Weibull distribution, respectively, it can be shown that:

*SD/x̄*)

^{2}] and estimates the Weibull distribution statistic from Eq. (15). We used bootstrap resampling methods to estimate the mean and standard error of this complex shape estimator based on the sample distributions [50–52]. An example is illustrated in Fig. 9 for size bin with d=50–100

*μ*m. Each data point represents the mean and standard error of the sample statistic from individual fluence bins. For comparison, the Weibull distribution statistic is plotted in the inset graph. Results in Fig. 9 indicate a great deal of similarities between the functional forms of the estimator vs. fluence and the estimand vs.

*k*and further support a Weibull distribution model with a fluence-dependent shape parameter, in agreement with results in Fig. 4 (using graphical techniques), i.e.,

*k*is monotonically increasing with fluence.

Based on the results of exploratory data analysis outlined above, we hypothesized a statistical model based on the family of Weibull distributions.

At the second step, we inferred the Weibull distribution parameters from the data collected for various size and fluence bins. We used the maximum-likelihood-estimation (MLE) methods for fitting of univariate distributions [50,53] to estimate the shape and scale parameters as well as their standard errors (from the variance matrix).

Next, we evaluated the quality of the fits from previous step using Weibull Quantile-Quantile (Q-Q) plots. A Q-Q plot is a graphical technique (qualitative rather than a numerical summary) for determining if a data set comes from a known population (Weibull distributed in this case). Specifically, we plot the empirical and theoretical Weibull quantiles on the *y*- and *x*-axis, respectively. A 45-degree reference line is also plotted. If the empirical data come from the population with the chosen distribution, the points should fall approximately along this reference line. It should be noted that linear regression based on Q-Q plots is often an alternative method (to MLE) for estimating the distribution parameters due to its relatively simple implementation (available in Origin).

At the final step, we used Kolmogorov-Smirnov (KS) goodness-of-fit statistical tests (quantitative) to decide if the sample data comes from a population with the Weibull parameters (see R-program, function ks.test()) inferred from MLE (step 2 above). The test statistic quantifies a distance between the empirical and theoretical cumulative distribution functions. The p-value of the KS test is compared to significance levels usually referred in statistical literature (0.05); if the p-value is higher than the latter, we fail to reject the null hypothesis that the sample data follow a Weibull distribution.

Representative fits to experimental distributions are illustrated in Figs. 10(a)–10(c) for three different size and fluence bins as (a) d=30–50 *μ*m, *ϕ* =6±0.5 J/cm^{2}, (b) d=50–100 *μ*m, *ϕ* =8±0.5 J/cm^{2}, and (c) d=100–500 *μ*m, *ϕ* =11±0.5 J/cm^{2}, respectively. For each distribution fit in Fig. 10, we note the estimated Weibull shape and scale parameters (*k*, *λ*) with standard errors, sample sizes (N) and p-values (p). Results in Fig. 10 indicate good overall agreement between the data (green, solid circles) and the Weibull model (red, solid line), as reflected in both the proximity of the data to the theoretical reference line and high p-values (greater than the statistical significance level, 0.05). We note however the effect of sample size on the estimated standard errors for the Weibull parameters (discussed in 5). Namely, for the case of *k* < 1 in Fig. 10(a), the uncertainties in the parameter estimates reveal under-sampling for this bin (*N* < 100); in contrast, the uncertainties in Figs. 10(b) and 10(c) are within ∼10% (*k* > 1, *N* > 50). The same MLE fitting procedure was applied to the data from all size and fluence bins and results were further filtered as to maintain uncertainties in the estimated distribution parameters on the order of 10%; for these selected bins, all p-values were greater than 0.20, with 70% of values greater than 0.50, thus supporting our hypothesis (model choice).

## Acknowledgments

We thank W. A. Steele, J. J. Adams, G. M. Guss and the OSL team for assistance in sample preparation and execution of the experiments. This work was performed under the auspices of the U.S. Department of Energy (DOE) by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.

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