## Growth model for laser-induced damage on the exit surface of fused silica under UV, ns laser irradiation |

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

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. 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]

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. 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]

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. 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/cm^{2}),” Opt. Express **20**, 11561–11573 (2012). [CrossRef] [PubMed]

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

*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/cm^{2}),” Opt. Express **20**, 11561–11573 (2012). [CrossRef] [PubMed]

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. C. W. Carr, D. Cross, M. D. Feit, and J. D. Bude, “Using shaped pulses to probe energy deposition during laser-induced damage of SiO_{2} surfaces,” Proc. SPIE **7132**, 71321C (2008). [CrossRef]

*ρ*(

*ϕ*) measurement, both the minimum density accessible and the uncertainty at a given fluence are reduced.

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. 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]

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]

^{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

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. 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]

*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

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]

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]

^{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

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]

*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

*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

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 SiO_{2},” Proc. SPIE **6403**, 64030K (2007). [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]

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 SiO_{2},” Proc. SPIE **6403**, 64030K (2007). [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]

*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

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. 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]

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 SiO_{2},” Proc. SPIE **6403**, 64030K (2007). [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]

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

_{2},” Proc. SPIE **6403**, 64030K (2007). [CrossRef]

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]

## 4. Statistical growth model

*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

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. 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. 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 SiO_{2},” Proc. SPIE **6720**, 67200F (2007). [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]

*η*and

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

*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: 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]

*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]

*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]

*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

**??**.

*X*(i.e., growth rate,

*α*) is: where

*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 where Γ denotes the Gamma function, respectively.

*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

*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

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. 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]

*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.

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]

*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.

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

*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.

^{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

*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.

*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: Here fitting parameters

*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

**20**, 13030–13039 (2012). [CrossRef] [PubMed]

^{2}at the 95% confidence level.

^{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

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. 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]

21. M. A. Norton, E. E. Donohue, M. D. Feit, R. P. Hackel, W. G. Hollingsworth, A. M. Rubenchik, and M. L. Spaeth, “Growth of laser damage in sio2 under multiple wavelength irradiation,” Proc. SPIE **5991**, 599108 (2005). [CrossRef]

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]

*α*(

*ϕ*) [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]

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]

*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.

*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.

^{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.

^{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%.

^{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: where Θ(

*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).

^{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).

*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.

## 7. Model validation

**20**, 13030–13039 (2012). [CrossRef] [PubMed]

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]

*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

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 CO_{2} laser,” Proc. SPIE **7842**, 784220 (2010). [CrossRef]

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]

^{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.

*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

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]

^{2}.

## 8. Relationship to energy absorption and fracture mechanics

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]

*σ*(under mode I loading) exceeds a critical value given by: where

*ℓ*is the characteristic crack length.

**50**, D12–D20 (2011). [CrossRef] [PubMed]

*α*〉 increases linearly with fluence as: then 〈

*χ*〉 ≈ 2

*γB*and 〈

*ϕ*〉 ≈

_{c}*ϕ*. Ref. [41

_{th}**50**, D12–D20 (2011). [CrossRef] [PubMed]

*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

**50**, D12–D20 (2011). [CrossRef] [PubMed]

*γ*= 4.3 J/m

^{2}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]

*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.

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]

*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*, 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}*ϕ*

_{0}and standard deviation

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

_{ϕ}*α*is then given by:

*f*described by Eq. (12) for different parameters of the fluence PDF and fixed coupling efficiency,

_{α}*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

*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

**20**, 13030–13039 (2012). [CrossRef] [PubMed]

*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.

## Appendix

*x̄*(estimates the population mean) and the standard deviation,

*SD*(

*SD*

^{2}estimates the population variance) computed as: 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:

_{x̄}*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: 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 [50–52]. 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.

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

*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).

*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

## References and links

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9. | M. J. Runkel and R. Sharp III, “Modeling KDP bulk damage curves for prediction of large-area damage performance,” Proc. SPIE |

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. |

11. | L. Lamaignère, G. Dupuy, T. Donval, P. Grua, and H. Bercegol, “Comparison of laser-induced surface damage density measurements with small and large beams: toward representativeness,” Appl. Opt. |

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/cm |

13. | M. J. Matthews and M. D. Feit, “Effect of random clustering on surface damage density estimates,” Proc. SPIE |

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. |

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 SiO |

16. | C. W. Carr, J. B. Trenholme, and M. L. Spaeth, “Effect of temporal pulse shape on optical damage,” Appl. Phys. Lett |

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 SiO |

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 |

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 |

20. | M. A. Norton, E. E. Donohue, W. G. Hollingsworth, J. N. McElroy, and R. P. Hackel, “Growth of laser initiated damage in fused silica at 527 nm,” Proc. SPIE |

21. | M. A. Norton, E. E. Donohue, M. D. Feit, R. P. Hackel, W. G. Hollingsworth, A. M. Rubenchik, and M. L. Spaeth, “Growth of laser damage in sio2 under multiple wavelength irradiation,” Proc. SPIE |

22. | M. A. Norton, E. E. Donohue, M. D. Feit, R. P. Hackel, W. G. Hollingsworth, A. M. Rubenchik, and M. L. Spaeth, “Growth of laser damage on the input surface of SiO |

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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 |

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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 |

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. |

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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 |

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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. |

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. |

36. | B. R. Lawn, |

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 |

39. | V. Kachitvichyanukul and B. W. Schmeiser, “Binomial random variate generation,” Commun. ACM |

40. | N. L. Johnson, S. Kotz, and N. Balakrishnan, “Weibull distributions,” in |

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. |

42. | C. P. Robert and G. Casella, |

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. |

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. |

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 |

47. | A. A. Griffith, “The phenomena of rupture and flow in solids,” Philos. Trans. R. Soc. London A |

48. | S. M. Wiederhorn, “Fracture surface energy of glass,” J. Am. Cer. Soc. |

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 |

50. | R Core Team, |

51. | A. C. Davison and D. V. Hinkley, |

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, |

**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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### References

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- A. Conder, J. Chang, L. Kegelmeyer, M. Spaeth, P. Whitman, “Final optics damage inspection (FODI) for the National Ignition Facility,” Proc. SPIE 7797, 77970P (2010). [CrossRef]
- I. L. Bass, G. M. Guss, M. J. Nostrand, 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]
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