Predictive modeling techniques for nanosecond-laser damage growth in fused silica optics

Predictive modeling techniques fornanosecond-laser damage growth infused silica opticsZhi M.Liao,∗Ghaleb M.Abdulla,Raluca A.Negres,David A.Cross,and Christopher W.CarrLawrence Livermore National Laboratory,7000East Avenue,Livermore,California94550,USA∗liao2@Abstract:Empirical numerical descriptions of the growth of laser-induceddamage have been previously developed.In this work,Monte-Carlo tech-niques use these descriptions to model the evolution of a population ofdamage sites.The accuracy of the model is compared against laser damagegrowth observations.In addition,a machine learning(classification)tech-nique independently predicts site evolution from patterns extracted directlyfrom the data.The results show that both the Monte-Carlo simulationand machine learning classification algorithm can accurately reproducethe growth of a population of damage sites for at least10shots,which isextremely valuable for modeling optics lifetime in operating high-energylaser systems.Furthermore,we have also found that machine learning canbe used as an important tool to explore and increase our understanding ofthe growth process.©2012Optical Society of AmericaOCIS codes:(160.4670)Optical materials;(140.3330)Laser damage.References and links1.S.T.Yang,M.J.Matthews,S.Elhadj,D.Cooke,G.M.Guss,V.G.Draggoo,and P.J.Wegner,“Comparingthe use of mid-infrared versus far-infrared lasers for mitigating damage growth on fused silica,”Appl.Opt.49, 2606–2616(2010).2.S.T.Yang,M.J.Matthews,S.Elhadj,V.G.Draggoo,and S.E.Bisson,“Thermal transport in CO2laser irradiatedfused silica:In situ measurements and analysis,”J.Appl.Phys.106,103106(2009).3.S.Elhadj,M.J.Matthews,S.T.Yang,and D.J.Cooke,“Evaporation kinetics of laser heated silica in reactiveand inert gases based on near-equilibrium dynamics,”Opt.Express20,1575–1587(2012).4. 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C.W.Carr,M.D.Feit,M.C.Nostrand,and J.J.Adams,“Techniques for qualitative and quantitative measure-ment of aspects of laser-induced damage important for laser beam propagation,”Meas.Sci.Technol.17,1958–1962(2006).14.R.A.Negres,G.M.Abdulla,D.A.Cross,Z.M.Liao,and C.W.Carr,“Probability of growth of small damagesites on the exit surface of fused silica optics,”Opt.Express20,13030–13039(2012).15.I.H.Witten and E.Frank,Data Mining:Practical Machine Learning Tools and Techniques,2nd ed.(MorganKaufmann,2005).16.J.R.Quinlan,“Learning with continuous classes,”in Proceedings AI’92,Adams and Sterling,eds.(World Sci-entific,1992).1.IntroductionHistorically,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,man-aging 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 laserflu-ence[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 computa-tion 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 modelThe 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]:D n(φ)=D n−1·exp[α(φ)],(1) withαbeing the growth coefficient,φis the measured localfluence,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:f(α;λ,k)=kλ αλk−1exp − αλ k ,(2)#166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15570with k andλbeing the shape and scale parameters of the distribution.The mean of the Weibull distribution is given byµ(λ,k)=λΓ(1+1/k),withΓrepresenting 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 laserfluence(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:λ(φ,τ,D)=b(φ,τ,D)·[φ−φth(τ,D)],(3)k(φ,τ,D)=1+g(φ,τ,D)·[φ−k th(τ,D)]where b,g are the rates of increase with respect tofluence(in cm2/J)whileφth,k th are theflu-ence 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 offluence,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].For3ω,5-nsflat in time(FIT) pulses,the coefficients are listed in Table1.The errors associated with the growth rule coeffi-cients for sites up to300µm and300-1000µm are estimated at10%and20%,respectively.In particular,the accuracy of the shape parameter coefficients(g and k th)in Table1can 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∼2ns 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.Table1.Size dependent Weibull parameters in Eq.(3)for3ω,5-ns FIT pulses.Size range(µm)bφth g k th50-1000.040 4.920.28 5.37100-3000.037 4.720.28 5.14300-10000.029 4.610.28 5.003.DataThe experimental approach has been described in detail elsewhere[9,10].In brief,on the order of100damage sites with diameters between25-80µm were initiated in a regular array with spacing of∼3mm using a single pulse from a355-nm,Nd:Y AG table top laser with an8-ns near Gaussian temporal profile focused to a spatial Gaussian spot of∼450µm(diameter at1/e2of intensity)on the exit surface of a1-cm thick silica substrate.By maintaining the grid spacing, we can expose all sites simultaneously with the3-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 localfluences around the beam averagefluence.Alignment beamfiducials are also placed on the same surface using a CO2laser technique and aid in the accurate registration of the localfluence to an individual site on every laser shot to within100µm.More details on thefluence calibration methods can be found in[9,13].We found no measurable cross-talk between adjacent damage sites with diameters up to about1mm.Individual site diameters are measured after each laser exposure using a robotic microscope under various illuminations with optical resolution as high as0.86µm.This highly parallel technique greatly enhances data collection rate while maintaining precisions not typically available in-situ[4].Although#166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15571Fig.1.Plot of average site size and laserfluence at351-nm as a function of shot numberin the data set corresponding to58sites and29laser shots.The dashed lines represent thestandard deviation of the mean size andfluence,respectively.we have conducted experiments under a wide variety of laser conditions,this work will focus on sites exposed on the exit surface of SiO2samples in high-vacuum,at room temperature with 3ω,5-ns FIT pulses.Specifically,58pre-initiated damage sites on a2-inch silica substrate were subjected to a series of nearly identical29laser shots at the nominalfluence of∼7J/cm2and standard deviation of0.9J/cm2from 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 meanfluence,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.Figure1summarizes the evolution of the mean site size(left axis)andfluence(right axis)exposures from58sites as a function of shot number(1to29),respectively.The dashed lines represent the standard deviation of the mean size andfluence 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).4.Analytical predictive modelSince the growth process is analyzed as a random process,the most common method of mod-eling 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 shot0(i.e.,1D0...S D0)as the initial condi-tion and the measured localfluencesφ1on thefirst shot to calculate the Weibull distribution of the growth coefficientαaccording to Eqs.(2)-(3)for each site.Recall that each site may have a different starting size and be exposed to a differentfluence and therefore have a differentαdistribution.Next,each site has M=2000random growth coefficients(α)generated using the Weibull distribution and,for each randomly generatedα,a calculated growth size D(i.e.,1D1,1 ...1D1,M)is obtained using Eq.(1).In other words,2000randomly generated outcomes to the first site and itsfirst shot exposure are generated.Each of these2000outcomes(new size)is then propagated with a new alpha distribution for each shot based on its exposurefluence and its size projected from the previous shot.This process is repeated for all N=29laser shots of#166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012(C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15572Fig.2.Schematic of the Monte-Carlo simulation with S Sites,N shots,and M simulations.the data set(see Figure2)and results in2000trajectories for thefirst site.The process is then repeated for each of the58sites.At the end of the simulation(shot N),each site i will have M=2000possible sizes;from this size distribution we can calculate the expected size of the prediction<i D N>.The accuracy of the simulation can be compared to the measured data by plotting the cu-mulative 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.3suggest that the model does an excellent job in reproducing the data for n=10shots.Part of the reason for the high accuracy is that we are predicting thefinal state of the ensemble of sites.In other words,if<i D10>for site i is lower than measurement and<j D10>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 section5.2.At n=20shots,the simulation results start to deviate from the measured data on the larger size ranges(∼250µm to450µm).At n=29shots, 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∼20or reflects the accuracy of the growth model for that size range.This is because the coefficients used for our growth model in Table1are mostly based on experimental data for sites with diameters in the 50-250µm range,as noted in Section2.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 to18shots(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 small-est size(i.e.,CDF∼0.02)changes very little from shot0to shot n=29,this is not well captured from the Monte-Carlo simulation.It is possible that the current model excludes other potentially important aspects such as #166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012(C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15573Fig.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 graphshows the measured(symbol)and predicted(line)maximum size as a function of shotnumber.the history offluence 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 localfluence 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 thefinal 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 the29shots.We then compared how well different attributes are able to capture,to afirst order,the growth trends of individual sites.Scatter plots in Figs.4(a)-(b)illustrate two of these relationships,namelyfinal vs.starting sizes and total growth factor vs.cumulativefluence for all58sites,respectively.It is evident from Fig.4(b)that a fairly good correlation exists between G and totalfluence while the correlation is very weak between starting andfinal 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 simple2D scatter plots.In section5we will discuss how additional measured attributes could be employed to further improve the model accuracy by using machine learning.5.Machine learning modelIn 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.Clas-sification 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]. #166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15574Fig.4.Scatter plot of(a)starting size vs.final size and(b)cumulativefluence vs.totalgrowth factor for all58sites,respectively.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 Section3such as previous damage size, localfluence on the site,etc.We can also include derived quantities like the cumulativeflu-ence 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 localfluence 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 localfluence)to the outcome,the clas-sifier 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 thefinal 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 obser-vations are not apparently different.For example,our results could show that damage growth is strongly correlated with laserfluence within our specified experimental conditions,how-ever 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 withfluence but not causality,which might include different fundamental mechanisms responsible for growth under multi-wavelength excitation orfluence vs.intensity dependent energy deposition processes.5.1.Data preparationTo apply the classification technique to the problem of damage growth,we have added addi-tional,derived attributes of cumulativefluence(∑φ)and total growth factor(G)to the measured data set described in Section3(which includes shot number n,previous size D n−1,current size #166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15575Fig.5.Initial andfinal cumulative size distribution(CDF)(a)as well as the probability sizedensity(PDF)(b)for site specific error(measured-predicted)for measured data and pre-diction results using Monte-Carlo(MC)simulation as well as supervised machine learning(ML),respectively.D n,localfluenceφ).Cumulativefluence is the totalfluence 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∑φafter29shots, 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 thefirst20shots(2/3)of the aggregate data were used for training purposes while the last10shots(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 of1161instances and covers shots1 through20.The test data contained523instances and covers shots21through29;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 thefirst(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 resultsThe result of the29th shot supervised machine learning prediction is plotted in Fig.5(a)along with the measuredfinal size and the Monte-Carlo simulation results.The latter results shown in Fig.5(a)are different from those presented in Fig.3in that the MC simulation starts with initial sizes after shot20and runs for9shots.The results show that both supervised machine learning and Monte-Carlo simulation were able to accurately reproduce the measured sizes after the last9laser shots.It is worthwhile to note that this Monte-Carlo result is not as accurate as the10th shot prediction in Fig.3,where the ensemble damage sizes are substantially smaller. #166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15576Furthermore,machine learning is doing a slightly better job on the extreme tails of the size distribution.Although both models predict thefinal 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 after9shots.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 (thefirst20shots)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.5.3.Model discoveryThe 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 goodfit for time series data.It provides a structured representation(conventional decision tree structure)of the data and piecewise linearfit(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:IFD n−1>331andD n−1<430(4)THEND n=13.11·(1.0·φ−0.5·n+0.074·D n−1+0.061·∑φ)−51.21.The current size(D n)is predicted using a linear combination of the attributes given(i.e., shot number n,previous size D n−1,fluenceφ,cumulativefluence∑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 of0to1.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:D n=A(c1x1+c2x2+..+c k x k)+BD n=A(c1ˆx1¯x1+c2ˆx2¯x2+..+c kˆx k¯x k)+B,(5)D n=A(w1¯x1+w2¯x2+..+w k¯x k)+Bwhere x i is the i-th attribute value(i.e.,fluence,shot number,etc.),ˆx i is the maximum value of i-th attribute value,and¯x i is the normalized value of i-th attribute(ranges from0to1).The terms c i and w i are the un-normalized and normalized weighting coefficients of each attribute, stly,we can rank the attributes by the magnitude of the normalized weighting coefficients such that w1has the highest contribution and w k 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 pa-rameters are relevant to a prediction.For example,Table2shows the rank order(i)and the weighting coefficient(w i)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 impor-tant as size increases,as suggested by the increasing rank order in Table2.Furthermore,for#166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15577large sizes,the weighting coefficient is actually negative,which seems to imply retardation of growth with shot number.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 on17th shot(n=17)while the other is on the10th shot(n=10),this could simply mean that the site that is on the17th shot is growing slower(if the starting size is similar)than the one that is on the10th 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 differentfluence histories(i.e.,higherfluencefirst vs.lowerfluencefirst).Table2.The shot number dependence to predicted size according to the machine learningalgorithm and Eq.(5).Size range Shot number Shot number(µm)rank order(i)weighting coefficient(w i)<14050.01140-17640.21176-24940.37249-31630331-4303-0.46>3842-0.396.DiscussionIt is evident from comparing the Monte-Carlo and machine learning algorithms used for dam-age 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 limi-tations 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,experi-mental 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.,2J/cm2)shots among the last9shots discussed above(using the same sample and laser parameters).These additional laser shots most probably do not lead to damage growth[11,14]and thefinal 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.Fur-thermore,if the shot number dependence as discovered by the machine learning classifier can #166998 - $15.00 USD Received 18 Apr 2012; revised 7 Jun 2012; accepted 8 Jun 2012; published 26 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15578。