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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 3824–3844
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Growth model for laser-induced damage on the exit surface of fused silica under UV, ns laser irradiation

Raluca A. Negres, David A. Cross, Zhi M. Liao, Manyalibo J. Matthews, and Christopher W. Carr  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 3824-3844 (2014)
http://dx.doi.org/10.1364/OE.22.003824


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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

1. H. Bercegol, A. Boscheron, J. M. Di-Nicola, E. Journot, L. Lamaignerè, J. Néauport, and G. Razé, “Laser damage phenomena relevant to the design and operation of an ICF laser driver,” J. Phys. Conf. Ser. 112, 032013 (2008). [CrossRef]

6

6. L. M. Kegelmeyer, R. Clark, R. R. Leach Jr., D. McGuigan, V. M. Kamm, D. Potter, J. T. Salmon, J. Senecal, A. Conder, M. Nostrand, and P. K. Whitman, “Automated optics inspection analysis for NIF,” Fusion Eng. Des. 87, 2120–2124 (2012). [CrossRef]

]. 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

7. J. O. Porteus and S. C. Seitel, “Absolute onset of optical surface damage using distributed defect ensembles,” Appl. Opt. 23, 3796–3805 (1984). [CrossRef] [PubMed]

9

9. M. J. Runkel and R. Sharp III, “Modeling KDP bulk damage curves for prediction of large-area damage performance,” Proc. SPIE 3902, 436–448 (2000). [CrossRef]

]. 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

10. L. Lamaignère, M. Balas, R. Courchinoux, T. Donval, J. C. Poncetta, S. Reyné, B. Bertussi, and H. Bercegol, “Parametric study of laser-induced surface damage density measurements: toward reproducibility,” J. Appl. Phys. 107, 023105 (2010). [CrossRef]

12

12. T. A. Laurence, J. D. Bude, S. Ly, N. Shen, and M. D. Feit, “Extracting the distribution of laser damage precursors on fused silica surfaces for 351 nm, 3 ns laser pulses at high fluences (20–150 J/cm2),” Opt. Express 20, 11561–11573 (2012). [CrossRef] [PubMed]

]. For example, depending on spot size, fluence and damage site size, incorrect statistics can be derived regarding damage threshold [13

13. M. J. Matthews and M. D. Feit, “Effect of random clustering on surface damage density estimates,” Proc. SPIE 6720, 67201J (2007). [CrossRef]

]. 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

12. T. A. Laurence, J. D. Bude, S. Ly, N. Shen, and M. D. Feit, “Extracting the distribution of laser damage precursors on fused silica surfaces for 351 nm, 3 ns laser pulses at high fluences (20–150 J/cm2),” Opt. Express 20, 11561–11573 (2012). [CrossRef] [PubMed]

, 14

14. C. W. Carr, M. D. Feit, M. C. Nostrand, and J. J. Adams, “Techniques for qualitative and quantitative measurement of aspects of laser-induced damage important for laser beam propagation,” Meas. Sci. Technol. 17, 1958–1962 (2006). [CrossRef]

17

17. C. W. Carr, D. Cross, M. D. Feit, and J. D. Bude, “Using shaped pulses to probe energy deposition during laser-induced damage of SiO2 surfaces,” Proc. SPIE 7132, 71321C (2008). [CrossRef]

]. 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

18. M. A. Norton, L. W. Hrubesh, Z. Wu, E. E. Donohue, M. D. Feit, M. R. Kozlowski, D. Milam, K. P. Neeb, W. A. Molander, A. M. Rubenchik, W. D. Sell, and P. Wegner, “Growth of laser initiated damage in fused silica at 351 nm,” Proc. SPIE 4347, 468 (2001). [CrossRef]

25

25. R. A. Negres, M. A. Norton, D. A. Cross, and C. W. Carr, “Growth behavior of laser-induced damage on fused silica optics under UV, ns laser irradiation,” Opt. Express 18, 19966–19976 (2010). [CrossRef] [PubMed]

]. 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

26. R. A. Negres, G. M. Abdulla, D. A. Cross, Z. M. Liao, and C. W. Carr, “Probability of growth of small damage sites on the exit surface of fused silica optics,” Opt. Express 20, 13030–13039 (2012). [CrossRef] [PubMed]

]. 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/cm2 up to about 23 J/cm2, 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

25. R. A. Negres, M. A. Norton, D. A. Cross, and C. W. Carr, “Growth behavior of laser-induced damage on fused silica optics under UV, ns laser irradiation,” Opt. Express 18, 19966–19976 (2010). [CrossRef] [PubMed]

, 27

27. R. A. Negres, M. A. Norton, Z. M. Liao, D. A. Cross, J. D. Bude, and C. W. Carr, “The effect of pulse duration on the growth rate of laser-induced damage sites at 351 nm on fused silica surfaces,” Proc. SPIE 7504, 750412 (2009). [CrossRef]

]. 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 μm 1/e2 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

28. T. I. Suratwala, P. E. Miller, J. D. Bude, W. A. Steele, N. Shen, M. V. Monticelli, M. D. Feit, T. A. Laurence, M. A. Norton, C. W. Carr, and L. L. Wong, “HF-based etching processes for improving laser damage resistance of fused silica optical surfaces,” J. Am. Cer. Soc. 94, 416–428 (2011). [CrossRef]

]. 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

29. M. C. Nostrand, T. L. Weiland, R. L. Luthi, J. L. Vickers, W. D. Sell, J. A. Stanley, J. Honig, J. Auerbach, R. P. Hackel, and P. Wegner, “A large aperture, high energy laser system for optics and optical components testing,” Proc. SPIE 5273, 325–333 (2004). [CrossRef]

] 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/cm2 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

27. R. A. Negres, M. A. Norton, Z. M. Liao, D. A. Cross, J. D. Bude, and C. W. Carr, “The effect of pulse duration on the growth rate of laser-induced damage sites at 351 nm on fused silica surfaces,” Proc. SPIE 7504, 750412 (2009). [CrossRef]

]. Individual site diameters are then measured using a robotic microscope under various illuminations with optical resolution as high as ∼1 μm. 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 μm up to hundreds of microns and even several millimeters under multiple laser exposures (5 or more) at fixed laser fluences, spanning from ∼5 J/cm2 up to about 23 J/cm2.

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 μm and grown damage sites (30 μm 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

15. C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE 6403, 64030K (2007). [CrossRef]

,30

30. J. Wong, J. L. Ferriera, E. F. Lindsey, D. L. Haupt, I. D. Hutcheon, and J. H. Kinney, “Morphology and microstructure in fused silica induced by high fluence ultraviolet 3ω(355 nm) laser pulses,” J. Non-cryst. Solids 352, 255–272 (2006). [CrossRef]

,31

31. G. Hu, Y. Zhao, D. Li, Q. Xiao, J. Shao, and Z. Fan, “Studies of laser damage morphology reveal subsurface feature in fused silica,” Surf. Interface Anal. 42, 1465–1468 (2010). [CrossRef]

]) 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

15. C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE 6403, 64030K (2007). [CrossRef]

, 32

32. G. M. Guss, I. L. Bass, R. P. Hackel, C. Mailhiot, and S. G. Demos, “In situ monitoring of surface postprocessing in large-aperture fused silica optics with optical coherence tomography,” Appl. Opt. 47, 4569–4573 (2008). [CrossRef] [PubMed]

, 33

33. B. Bertussi, P. Cormont, S. Palmier, P. Legros, and J. L. Rullier, “Initiation of laser-induced damage sites in fused silica optical components,” Opt. Express 17, 11469–11479 (2009). [CrossRef] [PubMed]

]), with diameters ranging from about 5 μm up to about 30 μm. 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

30. J. Wong, J. L. Ferriera, E. F. Lindsey, D. L. Haupt, I. D. Hutcheon, and J. H. Kinney, “Morphology and microstructure in fused silica induced by high fluence ultraviolet 3ω(355 nm) laser pulses,” J. Non-cryst. Solids 352, 255–272 (2006). [CrossRef]

, 34

34. S. G. Demos, M. Staggs, and M. R. Kozlowski, “Investigation of processes leading to damage growth in optical materials for large-aperture lasers,” Appl. Opt. 41, 3628–3633 (2002). [CrossRef] [PubMed]

, 35

35. M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett. 99, 151109 (2011). [CrossRef]

] in addition to multiple fracture surfaces (historically referred to as pansies [15

15. C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE 6403, 64030K (2007). [CrossRef]

, 32

32. G. M. Guss, I. L. Bass, R. P. Hackel, C. Mailhiot, and S. G. Demos, “In situ monitoring of surface postprocessing in large-aperture fused silica optics with optical coherence tomography,” Appl. Opt. 47, 4569–4573 (2008). [CrossRef] [PubMed]

, 33

33. B. Bertussi, P. Cormont, S. Palmier, P. Legros, and J. L. Rullier, “Initiation of laser-induced damage sites in fused silica optical components,” Opt. Express 17, 11469–11479 (2009). [CrossRef] [PubMed]

]). 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 μm which relates primarily to the laser pulse temporal duration and shape used for damage initiation (10-ns, near Gaussian) [15

15. C. W. Carr, M. J. Matthews, J. D. Bude, and M. L. Spaeth, “The effect of laser pulse duration on laser-induced damage in KDP and SiO2,” Proc. SPIE 6403, 64030K (2007). [CrossRef]

,16

16. C. W. Carr, J. B. Trenholme, and M. L. Spaeth, “Effect of temporal pulse shape on optical damage,” Appl. Phys. Lett 90, 041110 (2007). [CrossRef]

]. 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.

Fig. 1 Typical exit surface damage site morphologies revealed by SEM imaging. (Left) Type I – approximately 1 μm smooth pits, indicated by white arrows. Type II – single or multiple smooth fracture surfaces lacking damage cores, with overall diameters up to about 30 μm. (Right) Type III – multiple fracture surfaces surrounding damage cores, with overall diameters larger than about 30 μm. Here we examine the growth behavior of type III sites (as-initiated and grown). For all individual images, the scale bars (black horizontal lines) represent 20 μm.

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:
dn=dn1eηα,
(1)
where dn−1, dn are the effective circular diameters (ECD) before and after the nth shot, though some research groups have used the area of the site instead [1

1. H. Bercegol, A. Boscheron, J. M. Di-Nicola, E. Journot, L. Lamaignerè, J. Néauport, and G. Razé, “Laser damage phenomena relevant to the design and operation of an ICF laser driver,” J. Phys. Conf. Ser. 112, 032013 (2008). [CrossRef]

, 19

19. G. Razé, J. M. Morchain, M. Loiseau, L. Lamaignère, M. Josse, and H. Bercegol, “Parametric study of the growth of damage sites on the rear surface of fused silica windows,” Proc. SPIE 4932, 127–135 (2003). [CrossRef]

, 23

23. L. Lamaignère, S. Reyné, M. Loiseau, J. C. Poncetta, and H. Bercegol, “Effects of wavelengths combination on initiation and growth of laser-induced surface damage in SiO2,” Proc. SPIE 6720, 67200F (2007). [CrossRef]

, 37

37. L. Lamaignère, G. Dupuy, A. Bourgeade, A. Benoist, A. Roques, and R. Courchinoux, “Damage growth in fused silica optics at 351 nm: refined modeling of large-beam experiments,” Appl. Phys. B, 1–10 (2013). [CrossRef]

]. 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:
B(X;m,p)=C(m,X)pX(1p)mX,
(2)
where 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

26. R. A. Negres, G. M. Abdulla, D. A. Cross, Z. M. Liao, and C. W. Carr, “Probability of growth of small damage sites on the exit surface of fused silica optics,” Opt. Express 20, 13030–13039 (2012). [CrossRef] [PubMed]

]. In practice, we use Monte-Carlo methods to simulate m growth experiments (a large number) for each damage site [38

38. Z. M. Liao, G. M. Abdulla, R. A. Negres, D. A. Cross, and C. W. Carr, “Predictive modeling techniques for nanosecond-laser damage growth in fused silica optics,” Opt. Express 20, 15569–15579 (2012). [CrossRef] [PubMed]

] 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

39. V. Kachitvichyanukul and B. W. Schmeiser, “Binomial random variate generation,” Commun. ACM 31, 216–222 (1988). [CrossRef]

].

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:
f(X;λ,k)={kλ(Xλ)k1e(X/λ)kX0,0X<0,
(3)
where k, λ > 0 are the shape and scale parameters of the distribution, respectively [40

40. N. L. Johnson, S. Kotz, and N. Balakrishnan, “Weibull distributions,” in Continuous Univariate Distributions, 2nd ed. (Wiley, 1994), Vol. 1, pp. 628–722.

]. 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:
E[X]=λΓ(1+1/k),
(4)
and
var[X]=λ2[Γ(1+2/k)(Γ(1+1/k))2],
(5)
where Γ denotes the Gamma function, respectively.

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 μm [26

26. R. A. Negres, G. M. Abdulla, D. A. Cross, Z. M. Liao, and C. W. Carr, “Probability of growth of small damage sites on the exit surface of fused silica optics,” Opt. Express 20, 13030–13039 (2012). [CrossRef] [PubMed]

, 41

41. R. A. Negres, Z. M. Liao, G. M. Abdulla, D. A. Cross, M. A. Norton, and C. W. Carr, “Exploration of the multi-parameter space of nanosecond-laser damage growth in fused silica optics,” Appl. Opt. 50, D12–D20 (2011). [CrossRef] [PubMed]

] and translates into a size dependent error in the growth rate, i.e., about 6% to 4% for 30–50 μm sites and less for larger sites. For perceived changes in ECD less than 2 μm 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.

Table 1. Number of observations from growth rate experiments distributed over size and fluence bins (1 J/cm2 fluence bin width up to 11 J/cm2 and wider at higher fluences).

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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

42. C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed. (Springer, 2004). [CrossRef]

]. 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.

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.

Fig. 2 Measured single-shot growth rate as a function of laser fluence for type III exit surface damage sites following exposure to 355-nm, 5-ns FIT laser pulses. Sites are grouped by their pre-exposure diameter in four size bins with d=30–50, 50–100, 100–500, and 500–5000 μm corresponding to panels (a) through (d), respectively.

Fig. 3 Mean growth rate based on the estimated sample means (solid data points) and standard errors (vertical error bars, less than 15% of mean values except for data points indicated by arrows) as a function of laser fluence and site size. Logistic fits to the data using Eq. (6) and parameters from Table 2 are shown by the corresponding solid lines for each size bin.

Table 2. Logistic best fit parameters (estimated values and standard errors) to mean growth rate vs. size range vs. laser fluence in Fig. 3 using Eq. (6) and A1 = 0.

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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 μm 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/cm2) 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/cm2 and gradually evolves into a two-sided distribution, nearly symmetric at fluences beyond ∼11–13 J/cm2. 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/cm2 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.

Fig. 4 Experimental growth rate distributions vs. laser fluence for damage sites with d= 50–100 μm. Data from each fluence bin (a categorical x-variable as bin center±1/2 width, e.g., 11±0.5 J/cm2) are represented as one-dimensional dot plots along the y-axis (green, closed circles with systematic jittering). For comparison, the Weibull PDF curves with parameter estimates from the data are also shown (blue, solid lines re-scaled to 120% height of the experimental distributions).

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/cm2 fluence bins are shown by the solid data points and span fluences up to ∼11–12 J/cm2 and ∼15 J/cm2 for sites with d=30–100 μm and d=100–500 μm, respectively. In addition, open data points in Fig. 5 represent results from 2 J/cm2 fluence bins. All uncertainties in the parameter estimates are within ∼10%.

Fig. 5 Weibull shape parameter estimates vs. laser fluence for exit surface damage sites with diameters up to 500 μm (three size bins shown in the legend) using MLE fitting of experimental growth distributions with 1 and 2 J/cm2 fluence bins (solid and open data points, respectively). The estimated standard errors (vertical error bars) are within 10% for all fluences. Dotted lines (green and blue, for d=30–50 and 50–500 μm, respectively) represent the analytical approximation for k(ϕ) given by Eq. (7).

For all site sizes, results in Fig. 5 suggest that the Weibull shape parameter is fairly constant up to about 6–6.5 J/cm2 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:
k(ϕ)={(0.70±0.03)+0.28(ϕ6.5)Θ(ϕ6.5),d=3050μm,(1.00±0.06)+0.30(ϕ6.0)Θ(ϕ6.0),d=50500μm,
(7)
where Θ(ϕ) is the Heaviside step function and ϕ is the fluence in J/cm2. 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 μm, respectively); the model for k(ϕ) is well supported by the k-estimates at low fluences from 1 J/cm2 fluence bins (solid data points) and extrapolates up to ∼23 J/cm2 based on estimates from 2 J/cm2 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/cm2 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/cm2 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] = λΓ(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/cm2. 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).

Fig. 6 Weibull scale parameter estimates vs. laser fluence for exit surface damage sites with diameters up to 500 μm (three size bins shown in the legend) using experimental growth distributions with 1 J/cm2 fluence bins (solid data points, via MLE fitting) and 2 J/cm2 fluence bins (open data points, via Eq. (4)), respectively. Vertical error bars represent the estimated standard errors. Logistic fits to the data using Eq. (6) and parameters from Table 3 are shown by the corresponding dotted lines for each size bin and represent the analytical approximation for λ(ϕ).

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/cm2. The best fit parameters (using Eq. (6) with A1 = 0) and their size dependence are shown in Table 3.

Table 3. Logistic best fit parameters (estimated values and standard errors) to Weibull scale parameter vs. fluence as a function of size range in Fig. 6 using Eq. (6) and A1 = 0.

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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

Table 4. Sample sets used for the Monte-Carlo simulation. The size and fluence ranges are summarized as (mean values ± standard deviation).

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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 μm, and the targeted mean final sizes for the simulation were typically 250 μm. A target size of ∼250 μm was chosen because it approaches but is still below the critical size of ∼300–500 μm dictated by current laser damage mitigation techniques [3

3. I. L. Bass, G. M. Guss, M. J. Nostrand, and P. J. Wegner, “An improved method of mitigating laser-induced surface damage growth in fused silica using a rastered pulsed CO2 laser,” Proc. SPIE 7842, 784220 (2010). [CrossRef]

,5

5. R. Beeler Jr., A. Casey, A. Conder, R. Fallejo, M. Flegel, M. Hutton, K. Jancaitis, V. Lakamsani, D. Potter, S. Reisdorf, J. Tappero, P. Whitman, W. Carr, and Z. Liao, “Shot planning and analysis tools on the NIF project,” Fusion Eng. Des. 87, 2020–2023 (2012). [CrossRef]

]. However, because the average fluence in each experiment differed in order to span the 5–23 J/cm2 range, the number of shots to reach a final mean size of ∼250 μm 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.

Fig. 7 Measured vs. predicted (mean ± SD) cumulative final size distributions (CDF) using Monte-Carlo simulations of growth behavior over a wide range of local fluences and different number of shots for 150 sites with starting size range of ∼30–100 μm (initial CDF) (see Table 4).

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

43. M. J. Matthews, R. A. Negres, C. W. Carr, and A. M. Rubenchik, Lawrence Livermore National Laboratory, are preparing a manuscript to be called “Probability distribution model for multi-shot laser damage on fused silica surfaces.”

]. 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

44. C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004). [CrossRef] [PubMed]

] 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

45. S. G. Demos, R. A. Negres, R. N. Raman, A. M. Rubenchik, and M. D. Feit, “Material response during nanosecond laser induced breakdown inside of the exit surface of fused silica,” Laser Photonics Rev. 7, 444–452 (2013). [CrossRef]

,46

46. S. G. Demos, R. N. Raman, and R. A. Negres, “Time-resolved imaging of processes associated with exit-surface damage growth in fused silica following exposure to nanosecond laser pulses,” Opt. Express 21, 4875–4888 (2013). [CrossRef] [PubMed]

]. 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

47. A. A. Griffith, “The phenomena of rupture and flow in solids,” Philos. Trans. R. Soc. London A 221, 163–198 (1921). [CrossRef]

] that crack propagation will occur when the applied stress σ (under mode I loading) exceeds a critical value given by:
σ>σc=KIc,
(8)
where KIc=0.75MPam is the (mode I) fracture toughness of silica and is the characteristic crack length.

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

41. R. A. Negres, Z. M. Liao, G. M. Abdulla, D. A. Cross, M. A. Norton, and C. W. Carr, “Exploration of the multi-parameter space of nanosecond-laser damage growth in fused silica optics,” Appl. Opt. 50, D12–D20 (2011). [CrossRef] [PubMed]

]. If the average growth rate 〈α〉 increases linearly with fluence as:
α=B(ϕϕth),
(11)
then 〈χ〉 ≈ 2γB and 〈ϕc〉 ≈ ϕth. Ref. [41

41. R. A. Negres, Z. M. Liao, G. M. Abdulla, D. A. Cross, M. A. Norton, and C. W. Carr, “Exploration of the multi-parameter space of nanosecond-laser damage growth in fused silica optics,” Appl. Opt. 50, D12–D20 (2011). [CrossRef] [PubMed]

] discussed the size dependence of the rate of increase in growth rate with fluence (B coefficient or slope, in cm2/J) and fluence threshold for growth (ϕth, in J/cm2), 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

41. R. A. Negres, Z. M. Liao, G. M. Abdulla, D. A. Cross, M. A. Norton, and C. W. Carr, “Exploration of the multi-parameter space of nanosecond-laser damage growth in fused silica optics,” Appl. Opt. 50, D12–D20 (2011). [CrossRef] [PubMed]

] and γ = 4.3 J/m2 as the surface energy for fused silica [48

48. S. M. Wiederhorn, “Fracture surface energy of glass,” J. Am. Cer. Soc. 52, 99–105 (1969). [CrossRef]

], we can estimate χ and ϕc for individual size bins in Table 1, e.g., ∼3.4·10−5 and ∼7.5 J/cm2, and ∼2.7·10−5 and ∼5 J/cm2 for sites with d=30–50 and 100–500 μm, 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

49. F. Y. Gènin, A. Salleo, T. V. Pistor, and L. L. Chase, “Role of light intensification by cracks in optical breakdown on surfaces,” J. Opt. Soc. Am. A 18, 2607–2616 (2001). [CrossRef]

]. 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, fϕ, 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 ϕ0 and standard deviation σϕ. It can be shown [43

43. M. J. Matthews, R. A. Negres, C. W. Carr, and A. M. Rubenchik, Lawrence Livermore National Laboratory, are preparing a manuscript to be called “Probability distribution model for multi-shot laser damage on fused silica surfaces.”

] that the PDF for α is then given by:
fα(α)=2γχσϕ2πexp{2α(12(σϕ)2)[γχ(e2α1)+ϕcϕ0]2}.
(12)

In Fig. 8(a) we plot fα 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 (Δϕ = ϕ0ϕc ∼ 0), the growth distribution is right skewed, with most growth rate values near zero. This behavior is to be compared with Weibull distributions with 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 Δϕ, which can now be identified with the Weibull parameter λ. The uncertainty in the growth responses (width of the distribution) increases with larger σϕ, 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 fα are finite for arbitrarily small 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.

Fig. 8 (a) Growth rate fracture-derived PDF, fα, assuming a normal local fluence distribution with different mean, ϕ0, and standard deviation, σϕ, parameters (see Eq. (12)) with fixed coupling efficiency, χ =3.4·10−5). (b) The effect of the shape parameter, k, on the Weibull PDF with fixed scale parameter, λ =0.2.

9. Summary

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/cm2, coincides with the region where the probability of growth is less than 100% [26

26. R. A. Negres, G. M. Abdulla, D. A. Cross, Z. M. Liao, and C. W. Carr, “Probability of growth of small damage sites on the exit surface of fused silica optics,” Opt. Express 20, 13030–13039 (2012). [CrossRef] [PubMed]

]. 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.

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

50. R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, 2013).

] 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, (estimates the population mean) and the standard deviation, SD (SD2 estimates the population variance) computed as:
x¯=1Ni=1Nxi,SD=1N1i=1N(xix¯)2,
(13)
where 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:
SEx¯=SDN.
(14)

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:
[1+var(X)/E(X)2]=Γ(1+2/k)[Γ(1+1/k)]2.
(15)
The corresponding sample statistic is [1 + (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

50. R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, 2013).

52

52. A. Canty and B. Ripley, boot: Bootstrap R (S-Plus) Functions(2012), R package version 1.3–7.

]. An example is illustrated in Fig. 9 for size bin with d=50–100 μm. 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

50. R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, 2013).

,53

53. W. N. Venables and B. D. Ripley, Modern Applied Statistics With S, 4th ed. (Springer, 2002). [CrossRef]

] to estimate the shape and scale parameters as well as their standard errors (from the variance matrix).

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 μm, ϕ =6±0.5 J/cm2, (b) d=50–100 μm, ϕ =8±0.5 J/cm2, and (c) d=100–500 μm, ϕ =11±0.5 J/cm2, 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).

Fig. 9 Estimated sample statistic [1 + (SD/x̄)2] vs. laser fluence for size bin with d=50–100 μm (solid data points with vertical error bars representing the mean values and standard errors, respectively). (Inset) The corresponding Weibull distribution estimand vs. k, see Eq. (15).
Fig. 10 Weibull Q-Q plots for three experimental distributions with size and fluence bins (d, ϕ) shown at the top of each graph. The MLE estimates for the Weibull shape and scale parameters (k and λ in Eq. (3)), sample size (N) and p-values from goodness-of-fit tests are also noted.

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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M. C. Nostrand, T. L. Weiland, R. L. Luthi, J. L. Vickers, W. D. Sell, J. A. Stanley, J. Honig, J. Auerbach, R. P. Hackel, and P. Wegner, “A large aperture, high energy laser system for optics and optical components testing,” Proc. SPIE 5273, 325–333 (2004). [CrossRef]

30.

J. Wong, J. L. Ferriera, E. F. Lindsey, D. L. Haupt, I. D. Hutcheon, and J. H. Kinney, “Morphology and microstructure in fused silica induced by high fluence ultraviolet 3ω(355 nm) laser pulses,” J. Non-cryst. Solids 352, 255–272 (2006). [CrossRef]

31.

G. Hu, Y. Zhao, D. Li, Q. Xiao, J. Shao, and Z. Fan, “Studies of laser damage morphology reveal subsurface feature in fused silica,” Surf. Interface Anal. 42, 1465–1468 (2010). [CrossRef]

32.

G. M. Guss, I. L. Bass, R. P. Hackel, C. Mailhiot, and S. G. Demos, “In situ monitoring of surface postprocessing in large-aperture fused silica optics with optical coherence tomography,” Appl. Opt. 47, 4569–4573 (2008). [CrossRef] [PubMed]

33.

B. Bertussi, P. Cormont, S. Palmier, P. Legros, and J. L. Rullier, “Initiation of laser-induced damage sites in fused silica optical components,” Opt. Express 17, 11469–11479 (2009). [CrossRef] [PubMed]

34.

S. G. Demos, M. Staggs, and M. R. Kozlowski, “Investigation of processes leading to damage growth in optical materials for large-aperture lasers,” Appl. Opt. 41, 3628–3633 (2002). [CrossRef] [PubMed]

35.

M. J. Matthews, C. W. Carr, H. A. Bechtel, and R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett. 99, 151109 (2011). [CrossRef]

36.

B. R. Lawn, Fracture of Brittle Solids, 2nd ed. (Cambridge University, 1993). [CrossRef]

37.

L. Lamaignère, G. Dupuy, A. Bourgeade, A. Benoist, A. Roques, and R. Courchinoux, “Damage growth in fused silica optics at 351 nm: refined modeling of large-beam experiments,” Appl. Phys. B, 1–10 (2013). [CrossRef]

38.

Z. M. Liao, G. M. Abdulla, R. A. Negres, D. A. Cross, and C. W. Carr, “Predictive modeling techniques for nanosecond-laser damage growth in fused silica optics,” Opt. Express 20, 15569–15579 (2012). [CrossRef] [PubMed]

39.

V. Kachitvichyanukul and B. W. Schmeiser, “Binomial random variate generation,” Commun. ACM 31, 216–222 (1988). [CrossRef]

40.

N. L. Johnson, S. Kotz, and N. Balakrishnan, “Weibull distributions,” in Continuous Univariate Distributions, 2nd ed. (Wiley, 1994), Vol. 1, pp. 628–722.

41.

R. A. Negres, Z. M. Liao, G. M. Abdulla, D. A. Cross, M. A. Norton, and C. W. Carr, “Exploration of the multi-parameter space of nanosecond-laser damage growth in fused silica optics,” Appl. Opt. 50, D12–D20 (2011). [CrossRef] [PubMed]

42.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed. (Springer, 2004). [CrossRef]

43.

M. J. Matthews, R. A. Negres, C. W. Carr, and A. M. Rubenchik, Lawrence Livermore National Laboratory, are preparing a manuscript to be called “Probability distribution model for multi-shot laser damage on fused silica surfaces.”

44.

C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, and S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004). [CrossRef] [PubMed]

45.

S. G. Demos, R. A. Negres, R. N. Raman, A. M. Rubenchik, and M. D. Feit, “Material response during nanosecond laser induced breakdown inside of the exit surface of fused silica,” Laser Photonics Rev. 7, 444–452 (2013). [CrossRef]

46.

S. G. Demos, R. N. Raman, and R. A. Negres, “Time-resolved imaging of processes associated with exit-surface damage growth in fused silica following exposure to nanosecond laser pulses,” Opt. Express 21, 4875–4888 (2013). [CrossRef] [PubMed]

47.

A. A. Griffith, “The phenomena of rupture and flow in solids,” Philos. Trans. R. Soc. London A 221, 163–198 (1921). [CrossRef]

48.

S. M. Wiederhorn, “Fracture surface energy of glass,” J. Am. Cer. Soc. 52, 99–105 (1969). [CrossRef]

49.

F. Y. Gènin, A. Salleo, T. V. Pistor, and L. L. Chase, “Role of light intensification by cracks in optical breakdown on surfaces,” J. Opt. Soc. Am. A 18, 2607–2616 (2001). [CrossRef]

50.

R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, 2013).

51.

A. C. Davison and D. V. Hinkley, Bootstrap Methods and Their Applications (Cambridge University, 1997). [CrossRef]

52.

A. Canty and B. Ripley, boot: Bootstrap R (S-Plus) Functions(2012), R package version 1.3–7.

53.

W. N. Venables and B. D. Ripley, Modern Applied Statistics With S, 4th ed. (Springer, 2002). [CrossRef]

OCIS Codes
(140.3330) Lasers and laser optics : Laser damage
(160.4670) Materials : Optical materials
(160.6030) Materials : Silica
(240.6700) Optics at surfaces : Surfaces
(260.2160) Physical optics : Energy transfer

ToC Category:
Materials

History
Original Manuscript: November 28, 2013
Revised Manuscript: January 27, 2014
Manuscript Accepted: January 28, 2014
Published: February 11, 2014

Citation
Raluca A. Negres, David A. Cross, Zhi M. Liao, Manyalibo J. Matthews, and Christopher W. Carr, "Growth model for laser-induced damage on the exit surface of fused silica under UV, ns laser irradiation," Opt. Express 22, 3824-3844 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-3824


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  30. J. Wong, J. L. Ferriera, E. F. Lindsey, D. L. Haupt, I. D. Hutcheon, J. H. Kinney, “Morphology and microstructure in fused silica induced by high fluence ultraviolet 3ω(355 nm) laser pulses,” J. Non-cryst. Solids 352, 255–272 (2006). [CrossRef]
  31. G. Hu, Y. Zhao, D. Li, Q. Xiao, J. Shao, Z. Fan, “Studies of laser damage morphology reveal subsurface feature in fused silica,” Surf. Interface Anal. 42, 1465–1468 (2010). [CrossRef]
  32. G. M. Guss, I. L. Bass, R. P. Hackel, C. Mailhiot, S. G. Demos, “In situ monitoring of surface postprocessing in large-aperture fused silica optics with optical coherence tomography,” Appl. Opt. 47, 4569–4573 (2008). [CrossRef] [PubMed]
  33. B. Bertussi, P. Cormont, S. Palmier, P. Legros, J. L. Rullier, “Initiation of laser-induced damage sites in fused silica optical components,” Opt. Express 17, 11469–11479 (2009). [CrossRef] [PubMed]
  34. S. G. Demos, M. Staggs, M. R. Kozlowski, “Investigation of processes leading to damage growth in optical materials for large-aperture lasers,” Appl. Opt. 41, 3628–3633 (2002). [CrossRef] [PubMed]
  35. M. J. Matthews, C. W. Carr, H. A. Bechtel, R. N. Raman, “Synchrotron radiation infrared microscopic study of non-bridging oxygen modes associated with laser-induced breakdown of fused silica,” Appl. Phys. Lett. 99, 151109 (2011). [CrossRef]
  36. B. R. Lawn, Fracture of Brittle Solids, 2nd ed. (Cambridge University, 1993). [CrossRef]
  37. L. Lamaignère, G. Dupuy, A. Bourgeade, A. Benoist, A. Roques, R. Courchinoux, “Damage growth in fused silica optics at 351 nm: refined modeling of large-beam experiments,” Appl. Phys. B, 1–10 (2013). [CrossRef]
  38. Z. M. Liao, G. M. Abdulla, R. A. Negres, D. A. Cross, C. W. Carr, “Predictive modeling techniques for nanosecond-laser damage growth in fused silica optics,” Opt. Express 20, 15569–15579 (2012). [CrossRef] [PubMed]
  39. V. Kachitvichyanukul, B. W. Schmeiser, “Binomial random variate generation,” Commun. ACM 31, 216–222 (1988). [CrossRef]
  40. N. L. Johnson, S. Kotz, N. Balakrishnan, “Weibull distributions,” in Continuous Univariate Distributions, 2nd ed. (Wiley, 1994), Vol. 1, pp. 628–722.
  41. R. A. Negres, Z. M. Liao, G. M. Abdulla, D. A. Cross, M. A. Norton, C. W. Carr, “Exploration of the multi-parameter space of nanosecond-laser damage growth in fused silica optics,” Appl. Opt. 50, D12–D20 (2011). [CrossRef] [PubMed]
  42. C. P. Robert, G. Casella, Monte Carlo Statistical Methods, 2nd ed. (Springer, 2004). [CrossRef]
  43. M. J. Matthews, R. A. Negres, C. W. Carr, A. M. Rubenchik, Lawrence Livermore National Laboratory, are preparing a manuscript to be called “Probability distribution model for multi-shot laser damage on fused silica surfaces.”
  44. C. W. Carr, H. B. Radousky, A. M. Rubenchik, M. D. Feit, S. G. Demos, “Localized dynamics during laser-induced damage in optical materials,” Phys. Rev. Lett. 92, 087401 (2004). [CrossRef] [PubMed]
  45. S. G. Demos, R. A. Negres, R. N. Raman, A. M. Rubenchik, M. D. Feit, “Material response during nanosecond laser induced breakdown inside of the exit surface of fused silica,” Laser Photonics Rev. 7, 444–452 (2013). [CrossRef]
  46. S. G. Demos, R. N. Raman, R. A. Negres, “Time-resolved imaging of processes associated with exit-surface damage growth in fused silica following exposure to nanosecond laser pulses,” Opt. Express 21, 4875–4888 (2013). [CrossRef] [PubMed]
  47. A. A. Griffith, “The phenomena of rupture and flow in solids,” Philos. Trans. R. Soc. London A 221, 163–198 (1921). [CrossRef]
  48. S. M. Wiederhorn, “Fracture surface energy of glass,” J. Am. Cer. Soc. 52, 99–105 (1969). [CrossRef]
  49. F. Y. Gènin, A. Salleo, T. V. Pistor, L. L. Chase, “Role of light intensification by cracks in optical breakdown on surfaces,” J. Opt. Soc. Am. A 18, 2607–2616 (2001). [CrossRef]
  50. R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, 2013).
  51. A. C. Davison, D. V. Hinkley, Bootstrap Methods and Their Applications (Cambridge University, 1997). [CrossRef]
  52. A. Canty, B. Ripley, boot: Bootstrap R (S-Plus) Functions(2012), R package version 1.3–7.
  53. W. N. Venables, B. D. Ripley, Modern Applied Statistics With S, 4th ed. (Springer, 2002). [CrossRef]

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