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Empirical numerical descriptions of the growth of laser-induced damage have been previously developed. In this work, Monte-Carlo techniques use these descriptions to model the evolution of a population of damage sites. The accuracy of the model is compared against laser damage growth observations. In addition, a machine learning (classification) technique independently predicts site evolution from patterns extracted directly from the data. The results show that both the Monte-Carlo simulation and machine learning classification algorithm can accurately reproduce the growth of a population of damage sites for at least 10 shots, which is extremely valuable for modeling optics lifetime in operating high-energy laser systems. Furthermore, we have also found that machine learning can be used as an important tool to explore and increase our understanding of the growth process.
© 2012 Optical Society of America
1. Introduction
Historically, when an optic on a large aperture laser system were to have damage initiate on its surface the damage would grow un-checked until the optic was deemed consumed. Modern large aperture lasers such as the National Ignition Facility in the USA and the Laser MegaJoule in France utilize, and continue to develop, sophisticated methods of repairing and recycling optics in response to damage [1–6]. A common feature to all of these strategies in dealing with damage is that there is an upper size limit at which they are effective. For this reason, managing the repair and recycle of optics is greatly benefited by the ability to accurately predict how damage sites evolve with laser exposure. This allows one to schedule laser maintenance as well as plan experiments based on the current optics conditions. To date, considerable effort has been directed to understanding how sites evolve under controlled circumstances [7–11]. These previous studies have resulted in the development of empirical descriptions of damage growth (referred herein as rules) which describe the dependencies of damage site growth on laser fluence [7], wavelength [8], pulse duration [9, 10], and current size of the damage site [11]. As these rules become increasingly complex it becomes necessary to develop a framework to use them. In this paper, we will use the most up-to-date growth rule [11] and various computation tools such as Monte-Carlo simulation and machine learning to create predictive models to project the growth of damage sites observed under narrowly constrained conditions.
2. Growth model
The evolution of the Effective Circular Diameter (D) of a damage site on the exit surface of a fused silica optic has been generally described as exponential [7, 10]:
with α being the growth coefficient, ϕ is the measured local fluence, and n is the shot index. Although this work is within the range where Eq. (1) is valid, there is evidence suggesting that this model applies more generally to damage growth on the exit surface growth and for pulses longer than a few ns in duration. In contrast, an additional linear growth term is needed to describe growth on the input surface and/or shorter pulse durations [10]. The growth coefficient is found to follow a Weibull distribution [11] with the probability density function f(α) given by: with k and λ being the shape and scale parameters of the distribution. The mean of the Weibull distribution is given by μ(λ, k) = λΓ(1 + 1/k), with Γ representing the Gamma function, and it describes the average growth rate observed from a population of damage sites under narrowly constrained conditions (site size, pulse duration, fluence). However, these Weibull parameters are also found to be dependent on the laser fluence (ϕ) and pulse duration (τ) as well as the current size (diameter, D) of the damage site. The Weibull scale and shape parameters can be generally parameterized as follows: where b, g are the rates of increase with respect to fluence (in cm2/J) while ϕth, kth are the fluence thresholds (in J/cm2) for the shape and scale of the Weibull distribution respectively. These parameters are determined by clustering the growth measurements (α) in terms of fluence, size and pulse duration, and for each cluster, the Weibull parameters (λ, k) that best represent the statistics of the growth measurements are extracted [9, 11]. For 3ω, 5-ns flat in time (FIT) pulses, the coefficients are listed in Table 1. The errors associated with the growth rule coefficients for sites up to 300 μm and 300–1000 μm are estimated at 10% and 20%, respectively. In particular, the accuracy of the shape parameter coefficients (g and kth) in Table 1 can be further improved with future experimentation due to insufficient data sampling in some regions of the growth parameter space [11]. Furthermore, for pulse durations ranging from ∼2 ns up to ∼20 ns, the Weibull description of growth based on Eqs. (2) and (3) seems to work very well. For pulses shorter than a few ns, where a linear growth behavior has been observed to be dominant on the exit surface of fused silica [10], the validity of a Weibull description is yet to be fully explored.
Table 1. Size dependent Weibull parameters in Eq. (3) for 3ω, 5-ns FIT pulses.
3. Data
The experimental approach has been described in detail elsewhere [9, 10]. In brief, on the order of 100 damage sites with diameters between 25–80 μm were initiated in a regular array with spacing of ∼3 mm using a single pulse from a 355-nm, Nd:YAG table top laser with an 8-ns near Gaussian temporal profile focused to a spatial Gaussian spot of ∼450 μm (diameter at 1/e2 of intensity) on the exit surface of a 1-cm thick silica substrate. By maintaining the grid spacing, we can expose all sites simultaneously with the 3-cm diameter Optical Science Laboratory (OSL) laser beam [12]. We take advantage of the spatial beam contrast in OSL to expose sites with a range of local fluences around the beam average fluence. Alignment beam fiducials are also placed on the same surface using a CO2 laser technique and aid in the accurate registration of the local fluence to an individual site on every laser shot to within 100 μm. More details on the fluence calibration methods can be found in [9, 13]. We found no measurable cross-talk between adjacent damage sites with diameters up to about 1 mm. Individual site diameters are measured after each laser exposure using a robotic microscope under various illuminations with optical resolution as high as 0.86 μm. This highly parallel technique greatly enhances data collection rate while maintaining precisions not typically available in-situ [4]. Although we have conducted experiments under a wide variety of laser conditions, this work will focus on sites exposed on the exit surface of SiO2 samples in high-vacuum, at room temperature with 3ω, 5-ns FIT pulses. Specifically, 58 pre-initiated damage sites on a 2-inch silica substrate were subjected to a series of nearly identical 29 laser shots at the nominal fluence of ∼7 J/cm2 and standard deviation of 0.9 J/cm2 from all the sites. A tabulated data set was compiled for this sample where each entry contains at a minimum the site ID, shot number, current site size, pre-shot site size, single-shot growth rate (according to Eq. (1)), local mean fluence, and a number of other attributes (derived or measured parameters) which will be discussed shortly corresponding to one observation of a site on a specific laser shot. Figure 1 summarizes the evolution of the mean site size (left axis) and fluence (right axis) exposures from 58 sites as a function of shot number (1 to 29), respectively. The dashed lines represent the standard deviation of the mean size and fluence for this population of sites, respectively. As the sites grow the mean size increases but also the size distribution gets wider shot-to-shot (as seen from Fig. 1).
Fig. 1 Plot of average site size and laser fluence at 351-nm as a function of shot number in the data set corresponding to 58 sites and 29 laser shots. The dashed lines represent the standard deviation of the mean size and fluence, respectively.
4. Analytical predictive model
Since the growth process is analyzed as a random process, the most common method of modeling such a process is through the use of Monte-Carlo (MC) simulations. Specifically, we start with the initial site sizes for S number of sites at shot 0 (i.e., 1D0 ... SD0) as the initial condition and the measured local fluences ϕ1 on the first shot to calculate the Weibull distribution of the growth coefficient α according to Eqs. (2)–(3) for each site. Recall that each site may have a different starting size and be exposed to a different fluence and therefore have a different α distribution. Next, each site has M=2000 random growth coefficients (α) generated using the Weibull distribution and, for each randomly generated α, a calculated growth size D (i.e., 1D1,1 ... 1D1,M) is obtained using Eq. (1). In other words, 2000 randomly generated outcomes to the first site and its first shot exposure are generated. Each of these 2000 outcomes (new size) is then propagated with a new alpha distribution for each shot based on its exposure fluence and its size projected from the previous shot. This process is repeated for all N=29 laser shots of the data set (see Figure 2) and results in 2000 trajectories for the first site. The process is then repeated for each of the 58 sites. At the end of the simulation (shot N), each site i will have M=2000 possible sizes; from this size distribution we can calculate the expected size of the prediction <iDN>.
The accuracy of the simulation can be compared to the measured data by plotting the cumulative density function (CDF) of the measured and expected values for all the sites in the set from the Monte-Carlo runs (see Fig. 3). Results in Fig. 3 suggest that the model does an excellent job in reproducing the data for n=10 shots. Part of the reason for the high accuracy is that we are predicting the final state of the ensemble of sites. In other words, if <iD10> for site i is lower than measurement and <jD10> for site j is higher than measurement, the errors cancel one another when both are incorporated into the CDF. The uncertainty in predicting for an individual site is discussed below in section 5.2. At n=20 shots, the simulation results start to deviate from the measured data on the larger size ranges (∼250 μm to 450 μm). At n=29 shots, the simulation results continue to further deviate, at this point it is difficult to evaluate whether the deviation is a result of compounding residual errors that started at shot n∼20 or reflects the accuracy of the growth model for that size range. This is because the coefficients used for our growth model in Table 1 are mostly based on experimental data for sites with diameters in the 50–250 μm range, as noted in Section 2. As a result, our MC simulation can potentially have a larger error bar on the larger sizes. Despite these limitations, the predicted largest size is very close to the largest measured size up to 18 shots (see inset graph in Fig. 3). This observation has critical practical implications for operations as the largest few sites are the main driver for optics repair and replacement strategies. Furthermore, the measured data shows that the smallest size (i.e., CDF∼0.02) changes very little from shot 0 to shot n=29, this is not well captured from the Monte-Carlo simulation.
Fig. 3 Cumulative density function (CDF) of measured sizes data (symbol) and Monte-Carlo projected sizes (solid line) as a function of number of laser shots. The inset graph shows the measured (symbol) and predicted (line) maximum size as a function of shot number.
It is possible that the current model excludes other potentially important aspects such as the history of fluence exposure and other site parameters that make up the growth behaviors of individual sites and therefore may affect the growth rate model. Here we assume that each data entry (site/shot) is an independent event with current site size and local fluence are all that matter in determining growth. However, we recently discussed one example of laser exposure history and how it affects the probability of growth for small damage sites [14]. Although present work is focused on utilizing current models to make predictions, insights helpful to developing future models could be gained by examining other derived attributes for individual sites in our data set. For example, we have computed the total growth factor G, defined as the ratio of the final to the initial size of a site (i.e., G=D29/D0), in an attempt to capture the total growth behavior. Similarly, each site has been exposed to a cumulative (total) fluence over the 29 shots. We then compared how well different attributes are able to capture, to a first order, the growth trends of individual sites. Scatter plots in Figs. 4(a)–(b) illustrate two of these relationships, namely final vs. starting sizes and total growth factor vs. cumulative fluence for all 58 sites, respectively. It is evident from Fig. 4(b) that a fairly good correlation exists between G and total fluence while the correlation is very weak between starting and final sizes as plotted in Fig. 4(a). It is possible that the co-dependency of these attributes is not linear and as such it is beyond the simple 2D scatter plots. In section 5 we will discuss how additional measured attributes could be employed to further improve the model accuracy by using machine learning.
Fig. 4 Scatter plot of (a) starting size vs. final size and (b) cumulative fluence vs. total growth factor for all 58 sites, respectively.
5. Machine learning model
In this section we use supervised machine learning, more specifically classification technique [15] to build a model that can predict future damage site sizes. Unlike the previous section which used MC calculations as a framework to implement a number of empirically derived rules, this method uses a subset of the data to derive rules (or patterns) to predict growth. Classification is the method of determining which categories/classes a new observation belongs to based on a set of training data containing observations with known categories/classes [15]. The goal of this particular study is to determine the damage size; our observations are direct measurements, i.e., attributes, that we have outlined in Section 3 such as previous damage size, local fluence on the site, etc. We can also include derived quantities like the cumulative fluence on a site. The heart of the classification method is to use statistical tools such as logistic regression to predict outcomes (i.e., dependent variable) based on attributes (i.e., independent variables). Mathematically, this is no different than some of the rules that we have previously proposed [7–11] where we have isolated and drawn dependencies of growth on local fluence and previous size, etc. The difference is that whereas human analysis would be able to isolate and correlate a few key, dominant attributes (such as local fluence) to the outcome, the classifier algorithm is able to examine concurrently a large array of attributes and output a linear combination of all the attributes which best describes the growth behaviors. This is especially powerful when there isn’t a clear dominant attribute for human analysis; as a matter of fact, this type of scheme has the advantage in that investigators need not know initially which attributes are important. Indeed, including large numbers of irrelevant attributes will not degrade the final prediction. However, both human analysis and classifier algorithm can only draw correlations between dependent and independent variables based on the observations, not causality. This is especially important when the analysis results are applied to a different data set (i.e. from different samples, using different laser parameters, etc.) where the measurements and observations are not apparently different. For example, our results could show that damage growth is strongly correlated with laser fluence within our specified experimental conditions, however extending the same rule (or using classification) to predict damage growth under different laser parameters, e.g., multiple wavelengths or different pulse duration/shape, would not work. Again, the results show correlation with fluence but not causality, which might include different fundamental mechanisms responsible for growth under multi-wavelength excitation or fluence vs. intensity dependent energy deposition processes.
5.1. Data preparation
To apply the classification technique to the problem of damage growth, we have added additional, derived attributes of cumulative fluence (∑ϕ) and total growth factor (G) to the measured data set described in Section 3 (which includes shot number n, previous size Dn−1, current size Dn, local fluence ϕ). Cumulative fluence is the total fluence that the site has seen and captures the amount of energy deposited at each site up to that instance in time (i.e., shot number). Figure 4b showed that total growth factor G appears to trend reasonably well with ∑ϕ after 29 shots, which is not unexpected. We have now supplemented our data set with these derived attributes after each shot number. We then divided the data into two sets, training and test data. The data was treated as a time series data and the first 20 shots (2/3) of the aggregate data were used for training purposes while the last 10 shots (1/3) were used for testing. The last third of the data includes the more aggressive growth behavior and we will start by predicting that specific region of the data. The training data set contained a total of 1161 instances and covers shots 1 through 20. The test data contained 523 instances and covers shots 21 through 29; these were the shots we needed to predict the sizes for. We developed an algorithm to simulate an online prediction algorithm where the actual size can be measured for an arbitrary number of shots (n). We used the first (n=20) number of shots for training since they represent two thirds of the data which is the percentage recommended by the data characteristics for building the predictive model. To deploy such a model in a practical situation, the model should be built with as many historical instances as possible to increase the accuracy of the prediction.
5.2. Model results
The result of the 29th shot supervised machine learning prediction is plotted in Fig. 5(a) along with the measured final size and the Monte-Carlo simulation results. The latter results shown in Fig. 5(a) are different from those presented in Fig. 3 in that the MC simulation starts with initial sizes after shot 20 and runs for 9 shots. The results show that both supervised machine learning and Monte-Carlo simulation were able to accurately reproduce the measured sizes after the last 9 laser shots. It is worthwhile to note that this Monte-Carlo result is not as accurate as the 10th shot prediction in Fig. 3, where the ensemble damage sizes are substantially smaller. Furthermore, machine learning is doing a slightly better job on the extreme tails of the size distribution. Although both models predict the final size population as a whole, it does not mean that both models have similar accuracy in predicting any specific site. In Fig. 5(b), we plot the difference of the measured and predicted size for individual sites after 9 shots. It is evident that ML produces the better individual site prediction as it has a narrower error distribution. This is not surprising as the attributes used by the classifier algorithm draw on the past growth behavior (the first 20 shots) of the site it is predicting for. In contrast, the Monte-Carlo simulation uses a model that was derived from aggregate data collected from several samples and predicts the average behavior of any site but not necessarily a specific site.
Fig. 5 Initial and final cumulative size distribution (CDF) (a) as well as the probability size density (PDF) (b) for site specific error (measured-predicted) for measured data and prediction results using Monte-Carlo (MC) simulation as well as supervised machine learning (ML), respectively.
5.3. Model discovery
The classifier algorithm uses the training data to derive a statistical model for predicting the next size based on the attributes. We used a supervised modeling algorithm that is based on model trees [16]. The model deals with continuous class problems and it is a good fit for time series data. It provides a structured representation (conventional decision tree structure) of the data and piecewise linear fit (function) of the class at the leaves instead of discrete class labels. For details on how the tree generation works please refer to [16]. Below is an example rule that was generated by the model tree:
The current size (Dn) is predicted using a linear combination of the attributes given (i.e., shot number n, previous size Dn−1, fluence ϕ, cumulative fluence ∑ϕ, etc.) with the coefficient of each attribute generated by the model to accurately predict the size. Furthermore, we can normalize each coefficient to the maximum value of the attribute; hence the value of each attribute will have the same range of 0 to 1. It is then possible to rank the attributes based on their weighting factors. Below we will show how this can be used to measure the importance of each attribute and track if the attribute contribution changes as the sites grow. The steps of deriving a generalized weighted rule for predicting damage size are exemplified below:
where xi is the i-th attribute value (i.e., fluence, shot number, etc.), x̂i is the maximum value of i-th attribute value, and x̄i is the normalized value of i-th attribute (ranges from 0 to 1). The terms ci and wi are the un-normalized and normalized weighting coefficients of each attribute, respectively. Lastly, we can rank the attributes by the magnitude of the normalized weighting coefficients such that w1 has the highest contribution and wk has the lowest k-th contribution to predicting the size. In addition to providing predictions without initially knowing which attributes are important, this type of classification approach can provide insight into which parameters are relevant to a prediction. For example, Table 2 shows the rank order (i) and the weighting coefficient (wi) for the shot number attribute for each of the size-dependent rules. The table shows a relatively strong dependence on shot number that was not captured in the previously derived rules (Eqs. (2)–(3)). Specifically, the shot number (n) becomes more important as size increases, as suggested by the increasing rank order in Table 2. Furthermore, for large sizes, the weighting coefficient is actually negative, which seems to imply retardation of growth with shot number.Table 2. The shot number dependence to predicted size according to the machine learning algorithm and Eq. (5).
It is important to note that although the classifier algorithm indicates that shot number (n) correlates with damage size for larger size, it does not necessary mean that there is a strong causal relationship between the number of shots and damage site size. For example, if two damage sites are in the same size bin and if one damage site is on 17th shot (n=17) while the other is on the 10th shot (n=10), this could simply mean that the site that is on the 17th shot is growing slower (if the starting size is similar) than the one that is on the 10th shot. As a result, the classifier algorithm is simply adjusting the predicting size for sites with large shot number to account for a slower growth history. Furthermore, the fact that this dependency gets stronger with larger sizes could just be that it took sites to get large enough to accumulate a growth history. This example shows that although shot number correlates with damage size, the number of shots did not directly cause the growth of the site to slow down. As a matter of fact, the cause of this difference could be that these two sites have a different damage morphology that forms when different precursors are initiated or that they have different growth trajectories caused by different fluence histories (i.e., higher fluence first vs. lower fluence first).
6. Discussion
It is evident from comparing the Monte-Carlo and machine learning algorithms used for damage prediction that the classifier algorithm benefits from its ability to use all observations (i.e., attributes) and learn extensively about a particular data set. This however also imposes limitations on the use of the classifier algorithm in that its predictive model is exclusively derived from the training data; if prediction of a new test data (e.g., with different attributes, experimental parameters or laser exposure history) is attempted, the model’s accuracy will be greatly compromised. For example, let us assume that a new test data is simply generated by adding sporadic, low-fluence (e.g., 2 J/cm2) shots among the last 9 shots discussed above (using the same sample and laser parameters). These additional laser shots most probably do not lead to damage growth [11, 14] and the final damage site sizes will be very similar to the outcome of the original experiment; however, such drop-out laser shots were not present in the training set used by the classifier algorithm. As a result, the predictive model derived above would fail to achieve similar accuracy with the new testing data since the shot number attribute would no longer have the same significance. The Monte-Carlo simulation, on the other hand, would have similar accuracy in predicting growth in this new scenario because it can account for no-growth in the case of low-fluence laser shots (based on the empirical growth rules, Eqs. (2)–(3)). This flexibility makes the Monte-Carlo method a compelling case to use for growth predictions. Furthermore, if the shot number dependence as discovered by the machine learning classifier can be analyzed and added to the existing rules then we would expect the resultant simulation with Monte Carlo to be much closer to the classifier algorithm. As a matter of fact, the ideal use of machine learning classifier is to discover and refine attributes which can then be isolated and analyzed in single-parameter studies to more accurately account for their contribution into a Monte-Carlo simulation model.
7. Conclusion
We have shown that both Monte-Carlo simulation and supervised machine learning can accurately reproduce the evolution of a population of damage sites over 10 or more laser shots, depending on the size range. Although the Monte-Carlo technique is more flexible in terms of applying to different data sets (since there is an implicit understanding on the depending variables), its outcome may not be as accurate as that of the machine learning classifier technique. However, the classifier technique would require stronger oversight to ensure the training data and prediction data are consistent with each other. In addition, we have also shown that machine learning can be a powerful predictive technique as well as an important tool to increase our understanding of the growth process.
Acknowledgments
The authors would like to thank the OSL crew for their dedication and high standards. The authors would also like to thank Brian Gallagher for his valuable comments. 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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