Special Section on Laser Damage II

Sensitivity test method for the characterization of laser damage behavior

[+] Author Affiliations
Jonathan W. Arenberg

Northrop Grumman Aerospace Systems, One Space Park Drive, Redondo Beach, California 90278, United States

Michael D. Thomas

Spica Technologies, 18 Clinton Drive #3, Hollis, New Hampshire 03049, United States

Opt. Eng. 53(12), 122517 (Dec 02, 2014). doi:10.1117/1.OE.53.12.122517
History: Received May 9, 2014; Accepted October 16, 2014
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Abstract.  This paper introduces a sensitivity test for characterizing the laser damage behavior of a sample. A sensitivity test analyzes unbinned laser damage test data to estimate the damage probability curve. The means of estimation is by employing a parametric model of the probability of damage and identifying the parameters most likely to produce the observed results using the maximum-likelihood (ML) method. The ML method applied to laser damage measurements is reviewed. The sensitivity test is analyzed for its performance using Monte Carlo methods. A series of laser damage tests are simulated on a test of a hypothetical test optic. A Weibull distribution is selected for the damage probability model, while the virtual test optic was chosen to have a non-Weibull shaped damage probability curve. Damage measurements for varying the number of sites exposed are modeled to show the convergence of the Weibull parameters. This paper concludes by showing how the underlying defect distribution is calculated from results of the sensitivity test.

Figures in this Article

This paper introduces a sensitivity test for the characterization of laser damage behavior of a test sample. During a laser damage measurement, each test site is exposed to a proscribed fluence and then a new site is exposed. There are two outcomes for the laser exposure, damage or no damage. Laser damage tests are typically, “one exposure” or a “go, no-go” test on a single site, similar to other instances where an exposed or stressed test specimen is not available for further testing. Sensitivity tests abound in such areas as: testing of explosives, electronic components, fuses, and toxicology.14

Application of maximum-likelihood (ML) methods has a long history in the study of laser damage, especially laser safety measurements, “laser damage of the eye,” and date back to at least 1970.57 The first application of ML methods to a nonbiological test sample is the determination of a laser damage threshold with a binomial model on multiple shot testing done at China Lake in the late 1970s.8 Recently, there has been a renaissance in the application of ML techniques to various aspects of the laser damage problem by several groups.912 A group from the Lawrence Livermore National Laboratory has used ML damage characterization as part of a procedure to extract the distributions of defects on fused silica, an idea we will revisit later in this work.9 The group from Vilnius University in Lithuania has used ML methods applied to a degenerate defect model on data binned by fluence to study many aspects of the laser damage threshold determination problem.10,11 The results of the Vilnius group show a major increase in the quality of the determination of the threshold over damage frequency methods, frequently used in ISO standard measurements.12 The authors have been studying sensitivity test methods, whose results are also encouraging.1315 All of the recent results emphasize the exciting possibilities that the application of ML methods can bring to the characterization of laser damage; increased threshold determination quality in terms of accuracy and repeatability, comparisons expanded beyond the threshold values, and increased fundamental knowledge of the damage behavior of a sample.

Following this introduction, the ML method will be briefly reviewed. Section 3 introduces the model of the damage performance for our virtual test article namely, selecting the defect distribution and laser illumination profile to formulate the test optic’s damage probability curve PT(ϕ). The candidate probability model that will be optimized using the ML process, P(ϕi,θ), is also introduced. In Sec. 4, the Monte Carlo simulation of the laser damage experiments is shown and the results discussed. Section 5 summarizes the results and provides a look forward.

The results of a hypothetical damage measurement are shown in Fig. 1. At each test site i, the optic is exposed to a single fluence, ϕi and has one of two outcomes; ui, which has the value 1 if the site is damaged and 0 if not. When n spots have been examined, the measurement is completed and the data analyzed. The result of the i-th event or exposure is an ordered pair, (ϕ,ui), containing the information of the fluence level, ϕi and the outcome, ui. The entire set of ordered pairs (ϕ,ui) is called the experiment or measurement and each interrogation of the optic, i, is called an event.

The likelihood or probability, L, of observing a given experiment, all events being independent, is the product of the probabilities of the occurrence each event, Ei, Display Formula

L=i=1nEi.(1)

The probability of an event i occurring is given by the parameterized model for the probability of damage at the fluence for the i-th event, ϕi, P(ϕi,θ), where θ are the model parameters. For the case of a laser damage test, Ei is given by Display Formula

Ei(ϕ,θ)=(1ui)[1P(ϕi,θ)]+uiP(ϕi,θ).(2)

The model parameters, θ, are varied to determine the parameter values that give the maximum value of L.

Finding the maximal value of Eq. (1) is typically arithmetically cumbersome; therefore, the natural logarithm of L, L, is usually the expression that is analyzed,16 namely Display Formula

L(ϕ,θ)=i=1nn[Ei(ϕ,θ)].(3)

Substitution of Eq. (2) into Eq. (3) gives the final expression for LDisplay Formula

L(ϕ,θ)=i=1nn{(1ui)[1P(ϕi,θ)]+uiP(ϕi,θ)}.(4)

The models for damage probability, P(ϕ,θ), are substituted into Eq. (4) and the model parameters θ that maximize L are called the ML solution.

The two-parameter Weibull distribution is the damage probability model chosen for this study. This distribution is known for its flexibility17 and is written as Display Formula

P(ϕ;η,β)=1e(ϕη)β.(5)

Equation (5) is substituted into Eq. (4) giving the expression that is optimized to find the ordered pair, (η,β) that maximizes L for a given equation Display Formula

L(ϕ;η,β)=i=1nn[(1ui)(e(ϕiη)β)+ui(1e(ϕiη)β)].(6)

This section introduces the probability of damage of the test optic under illumination, PT(ϕ) and the Monte Carlo simulation of the damage measurement.

PT(ϕ) is derived from the conditions of illuminating a Gaussian distribution of defects in fluence18 with a flat top beam. Our particular assumptions result in a probability of damage curve with the following equation Display Formula

PT(ϕ)=1exp{K[erf(μ2σ)erf((μϕ)2σ)]}.(7)

The parameters selected for this analysis are K=5, μ=30 and σ=7J/cm2. A plot of Eq. (7) using the listed parameter values is shown in Fig. 2. It should be noted that PT(ϕ) was deliberately chosen to have a form different from PT(ϕ), since in a real experiment the defect distribution is not known a priori.

Graphic Jump LocationF2 :

Damage probability for test optic, PT(ϕ).

To simulate a laser damage measurement with n events, the test fluences, ϕi, are randomly chosen with uniform density on the interval [0,Φmax]. Φmax is selected to be sufficiently high so that PT(Φmax)=1. For this study, Φmax=75J/cm2. For each event, a second random number, wi, uniformly distributed on the interval [0,1] is selected. If wi is greater than PT(ϕi), the i-th site is recorded as having damage and ui=1, ui=0 otherwise. Figure 3 illustrates a Monte Carlo realization of a measurement and shows the (ϕi,ui).

Graphic Jump LocationF3 :

Test optic probability of damage curve and typical damage test results.

The Monte Carlo model described in Sec 3 was evaluated for 100 trials for three values of n, 100, 300, and 600 and (η,β) was determined. Figures 456 show bubble plots of the (η,β) pairs determined from the maximization of L. In these plots, the size of the bubble is proportional to the frequency of occurrence and the ellipse enclosing two standard deviations, 2σ, of the joint distribution of (η,β) is shown by the dashed curve. Figure 7 shows that as n increases, the area enclosed by the ellipse dramatically decreases. The decrease in the area of the error ellipse with increasing n is a clear trend. The error ellipses also decrease in eccentricity as n increases, indicating greater statistical independence between η and β. Figure 7 also shows that the ellipses fit inside one another, meaning that. in this case, the sensitivity method converges to an ultimate (η,β) pair with increasing n.

Graphic Jump LocationF4 :

(η,β) and 2σ ellipse for n=100.

Graphic Jump LocationF5 :

(η,β) and 2σ ellipse for n=300.

Graphic Jump LocationF6 :

(η,β) and 2σ ellipse for n=600.

Graphic Jump LocationF7 :

Error ellipses for all values of n.

Figures 8910 show the probability of damage curves, P(ϕ), determined from the (η,β) solutions. The P(ϕ) are calculated for each of the 100 (η,β) pairs for a given n and the minimum, mean and maximum values of P at a given ϕ determined and plotted. Figures 8910 also include the PT(ϕ), shown as the solid line trace. As n increases, the distance between the maximum and minimum values of P(ϕ) decreases. For n=100 and 300, the range envelops PT(ϕ). At n=600, PT(ϕ) and P(ϕ) agree well at small fluences but differ at higher fluences. The fact that there are some regions where PT(ϕ) and P(ϕ) disagree should be expected, as they do not have the same mathematical form.

Graphic Jump LocationF8 :

Probability of damage from (η,β) for n=100.

Graphic Jump LocationF9 :

Probability of damage from (η,β) for n=100.

Graphic Jump LocationF10 :

Probability of damage from (η,β) for n=100.

Damage measurements made in the manner described in this paper also can yield important insights into the underlying defect distribution, allowing for comparisons among samples beyond the comparison of damage thresholds.9 Consider the formulation of the probability of damage curve, Pr(D), from Poisson statistics, and expressed in terms of the number of defects above threshold within the laser damage spot, s19Display Formula

Pr(D)=1es.(8)

The value of s is Display Formula

s=0ϕmaxa(ϕ)f(ϕ)dϕ,(9)
where a(ϕ) is the area in the test beam whose local fluence is equal or greater than ϕ, and f(ϕ) is the areal density of defects that damage at ϕ.20 Equating (5) with (8) gives Display Formula
(ϕη)β=s.(10)

Substitution of the result for s given by Eq. (9) into Eq. (10) gives Display Formula

0ϕmaxa(ϕ)f(ϕ)dϕ=(ϕη)β.(11)

Since our hypothetical beam is a flat top, a(ϕ) is given Display Formula

a(ϕ)=πw2.(12)
where w is the beam radius and is a constant. For a flat top beam, ϕ=ϕmax, Eq. (11) now becomes Display Formula
0ϕf(ϕ)dϕ=1πw2(ϕη)β.(13)

The defect density distribution, f(ϕ), is ultimately found by differentiation of Eq. (13) by ϕ. Equation (13) shows that a laser damage sensitivity test does provide an efficient, simple method to determine the defect density distribution, which is seminal to understanding the damage behavior of an optic under any fluence profile.20

Figure 11 shows a plot of s using the mean probability of damage for n=100, 300, and 600 and solving Eq. (8). The estimates of s are very similar for the three values of n for fluences less than 40J/cm2. To calculate the range of s, Eq. (8) is again solved for s, but using the maximum and minimum probabilities at each fluence. Figure 12 shows the range of s and shows a strong trend of decreased range with increasing n.

Graphic Jump LocationF11 :

Estimated number of defects, s above the damage threshold within the test spot.

Graphic Jump LocationF12 :

Range of estimated number of defects, s above the damage threshold within the test spot.

This paper has introduced a sensitivity test for laser damage measurements. This technique processes unbinned damage results and uses ML methods to determine a parametric model for the probability of damage. We have shown how the results derived from this model can be used to directly extract the defect density distribution providing fundamental insight into damage behavior. This study has also demonstrated that the results of the sensitivity test converge with increasing the number of sites exposed. The reader can glean a hint of robustness of the sensitivity method by virtue of the fact that PT(ϕ) and P(ϕi) had different forms, yet a reasonable result was obtained. All of these factors make it clear that the further development of a sensitivity test for laser damage is likely to produce useful outcomes for the entire laser damage community.

In our study, ϕi+1 has no relation to the fluence of any of the previous i events. This simple implementation is not the most efficient in terms of convergence of the answer with n. The selection of fluence or stress levels in similar types of tests has been an active area of research for many years in the literature of sensitivity testing.2123 We plan to investigate the manner in which fluence is selected based on the test history as a means to increase the rate of convergence of the model parameters with n.

The model for P(ϕi) used in this work is a two-parameter Weibull distribution. It was chosen for its simplicity and flexibility. Recall that for large values of n, the true value for PT(ϕ) lies outside the tolerance limits for P(ϕ), see Fig. 10. Since the form of f(ϕ), which determines PT(ϕ) is not generally known, determining a more flexible form than our current assumption will be another avenue of process development. Increased flexibility will come with the computational burden of additional model parameters, which will drive up computing time and will likely require a longer test to glean the additional information required to make a more complex P(ϕ) worthwhile.

Our research will mainly proceed along these two axes of advancement. We look forward to sharing our results in these pages.

The authors would like to acknowledge the helpful discussions and disagreements with our many colleagues at the Boulder Damage Symposium over the years. These interactions have inspired and guided this work. These discussions, questions, and challenges are of immense value to us, but too numerous to list, lest we forget someone. We would also like to acknowledge the two anonymous reviewers who made helpful and important suggestions for improvements in this paper. Thank you.

Langlie  H. J., “A reliability test method for “One-Shot” items,” Ford Aerospace Report U-1792, Defense Technical Information Center (DTIL) (1962).
Dixon  W. J., Wood  M., “A method for obtaining and analyzing sensitivity data,” J. Am. Stat. Assoc.. 43, , 109 –126 (1948). 0003-1291 CrossRef
Garwood  R., “The application of maximum likelihood to dosage-mortality,” Biometrica. 32, , 46 –58 (1941). 0006-3444 CrossRef
Neyer  B. T., “New developments in sensitivity testing,” Presented at  45th Ann. Mtg on Fuse Section ,  National Defense Industrial Association ,  Long Beach, California  ( April 2001).
Frisch  G. D., “Quantal response analysis as applied to laser damage threshold studies,” Memorandum Report M70-27-1 of the Joint AMRDC-AMC Laser Safety Team, Department of the Army, Frankford Arsenal, Philadelphia, Pennsylvania (1970).
Lund  B. J., “The probitfit program to analyze data from laser damage threshold studies,” Walter Reed Army Institute of Research Report USAMRD/TR-2006–0001, Defense Technical Information Center (DTIL) (2006).
Cain  C. P., Noojin  G. D., Stolarski  D. J., “Near-infrared ultrashort pulse laser bioeffects studies,” AFRL-HE.BR.TR-2003–0029 Final Report, Defense Technical Information Center (DTIL) (2003).
Porteus  J. O., Jernigan  J. L., Faith  W. N., “Multithreshold measurement and analysis of pulsed laser damage on optical materials,” Damage in Laser Materials: 1977, Nat. Bur. Stand. (U.S.). , Special Publication 509, pp. 507 –515,  U.S. Government Printing Office ,  Washington, DC  (1977).
Laurence  T. A. et al., “Extracting the distribution of laser damage precursors on fused silica surfaces for 351 nm, 3 ns laser pulses at high fluences (20−150  J/cm2),” Opt. Express. 20, (10 ), 11561 –11573 (2012). 1094-4087 CrossRef
Batavičiutė  G. et al., “Bayesian approach of laser-induced damage threshold analysis and determination of error bars,” Proc. SPIE. 8530, , 85301L  (2012). 0277-786X CrossRef
Batavičiutė et al., “Revision of laser-induced damage threshold evaluation from damage probability data,” Rev. Sci. Instrum.. 84, , 045108  (2013). 0034-6748 CrossRef
ISO 21254-2:2011, Lasers and laser-related equipment––Test methods for laser-induced damage threshold—Part 2: Threshold determination, http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=43002 (9  May 2014).
Arenberg  J. W., Thomas  M. D. “Improving laser damage measurements: an explosive analogy,” Proc. SPIE. 8530, , 85301L  (2012). 0277-786X CrossRef
Arenberg  J. W., Thomas  M. D., “Laser damage threshold measurements via maximum likelihood estimation,” Proc. SPIE. 8885, , 88850O  (2013). 0277-786X CrossRef
Arenberg  J. W., Thomas  M. D., “A maximum likelihood method for the measurement of laser damage behavior,” Proc. SPIE. 9237, , 923717  (2014). 0277-786X CrossRef
Myung  I. J., “Tutorial on maximum likelihood estimation,” J. Math. Psychol.. 47, , 90 –100 (2003). 0022-2496 CrossRef
Weibull  W., “A statistical distribution function of wide utility,” J. Appl. Mech.. 18, , 293 –297 (1951). 0021-8936 
Krol  H., “Investigation of nanoprecursors threshold distribution in laser-damage testing,” Opt. Commun.. 256, (1–3 ), 184 –189 (2005). 0030-4018 CrossRef
Picard  R. H., Milam  D., Bradbury  R. A., “Statistical analysis of defect caused damage in thin films,” Appl. Opt.. 16, (6 ), 1563  (1977). 0003-6935 CrossRef
Porteus  J. O., Seitel  S. C., “Absolute onset of optical surface damage using distributed defect ensembles,” Appl. Opt.. 23, (21 ), 3796 –3805 (1984). 0003-6935 CrossRef
Neyer  B. T. “A d-optimality based sensitivity test,” Technometrics. 36, , 61 –70 (1994). 0040-1706 CrossRef
Neyer  B. T., “New developments in sensitivity testing,” Presented at  45th Ann. Mtg on Fuse Section of the National Defense Industrial Association ,  Long Beach, California  (2001).
Neyer  B. T., Gageby  J., “ISO 14304 Annex B all-fire/no-fire test and analysis methods,” in  Proc. 17th Symp. on Explosives and Pyrotechnics ,  Essington, Pennsylvania  (1999).

Jonathan W. Arenberg holds a BS degree in physics and an MS and PhD degrees in engineering, all from the University of California, Los Angeles. He has been with Northrop Grumman Aerospace Systems since 1989. His work experience includes optical, space, and laser systems. He has worked on major high-energy and tactical laser systems, laser component engineering, and metrology issues. He is currently the chief engineer for the James Webb Space Telescope Program.

Michael D. Thomas received his BS degree in optical engineering from the University of Rochester, in 1983. Upon graduation, He joined Sanders Associates of Nashua NH, where he was involved in solid state laser design and research and thin film design. He has been president of Spica Technologies since the company was founded in 1990 and is currently involved in laser damage testing and performance of precision optical measurements.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation

Jonathan W. Arenberg and Michael D. Thomas
"Sensitivity test method for the characterization of laser damage behavior", Opt. Eng. 53(12), 122517 (Dec 02, 2014). ; http://dx.doi.org/10.1117/1.OE.53.12.122517


Figures

Graphic Jump LocationF2 :

Damage probability for test optic, PT(ϕ).

Graphic Jump LocationF3 :

Test optic probability of damage curve and typical damage test results.

Graphic Jump LocationF4 :

(η,β) and 2σ ellipse for n=100.

Graphic Jump LocationF5 :

(η,β) and 2σ ellipse for n=300.

Graphic Jump LocationF6 :

(η,β) and 2σ ellipse for n=600.

Graphic Jump LocationF7 :

Error ellipses for all values of n.

Graphic Jump LocationF8 :

Probability of damage from (η,β) for n=100.

Graphic Jump LocationF9 :

Probability of damage from (η,β) for n=100.

Graphic Jump LocationF10 :

Probability of damage from (η,β) for n=100.

Graphic Jump LocationF11 :

Estimated number of defects, s above the damage threshold within the test spot.

Graphic Jump LocationF12 :

Range of estimated number of defects, s above the damage threshold within the test spot.

Tables

References

Langlie  H. J., “A reliability test method for “One-Shot” items,” Ford Aerospace Report U-1792, Defense Technical Information Center (DTIL) (1962).
Dixon  W. J., Wood  M., “A method for obtaining and analyzing sensitivity data,” J. Am. Stat. Assoc.. 43, , 109 –126 (1948). 0003-1291 CrossRef
Garwood  R., “The application of maximum likelihood to dosage-mortality,” Biometrica. 32, , 46 –58 (1941). 0006-3444 CrossRef
Neyer  B. T., “New developments in sensitivity testing,” Presented at  45th Ann. Mtg on Fuse Section ,  National Defense Industrial Association ,  Long Beach, California  ( April 2001).
Frisch  G. D., “Quantal response analysis as applied to laser damage threshold studies,” Memorandum Report M70-27-1 of the Joint AMRDC-AMC Laser Safety Team, Department of the Army, Frankford Arsenal, Philadelphia, Pennsylvania (1970).
Lund  B. J., “The probitfit program to analyze data from laser damage threshold studies,” Walter Reed Army Institute of Research Report USAMRD/TR-2006–0001, Defense Technical Information Center (DTIL) (2006).
Cain  C. P., Noojin  G. D., Stolarski  D. J., “Near-infrared ultrashort pulse laser bioeffects studies,” AFRL-HE.BR.TR-2003–0029 Final Report, Defense Technical Information Center (DTIL) (2003).
Porteus  J. O., Jernigan  J. L., Faith  W. N., “Multithreshold measurement and analysis of pulsed laser damage on optical materials,” Damage in Laser Materials: 1977, Nat. Bur. Stand. (U.S.). , Special Publication 509, pp. 507 –515,  U.S. Government Printing Office ,  Washington, DC  (1977).
Laurence  T. A. et al., “Extracting the distribution of laser damage precursors on fused silica surfaces for 351 nm, 3 ns laser pulses at high fluences (20−150  J/cm2),” Opt. Express. 20, (10 ), 11561 –11573 (2012). 1094-4087 CrossRef
Batavičiutė  G. et al., “Bayesian approach of laser-induced damage threshold analysis and determination of error bars,” Proc. SPIE. 8530, , 85301L  (2012). 0277-786X CrossRef
Batavičiutė et al., “Revision of laser-induced damage threshold evaluation from damage probability data,” Rev. Sci. Instrum.. 84, , 045108  (2013). 0034-6748 CrossRef
ISO 21254-2:2011, Lasers and laser-related equipment––Test methods for laser-induced damage threshold—Part 2: Threshold determination, http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=43002 (9  May 2014).
Arenberg  J. W., Thomas  M. D. “Improving laser damage measurements: an explosive analogy,” Proc. SPIE. 8530, , 85301L  (2012). 0277-786X CrossRef
Arenberg  J. W., Thomas  M. D., “Laser damage threshold measurements via maximum likelihood estimation,” Proc. SPIE. 8885, , 88850O  (2013). 0277-786X CrossRef
Arenberg  J. W., Thomas  M. D., “A maximum likelihood method for the measurement of laser damage behavior,” Proc. SPIE. 9237, , 923717  (2014). 0277-786X CrossRef
Myung  I. J., “Tutorial on maximum likelihood estimation,” J. Math. Psychol.. 47, , 90 –100 (2003). 0022-2496 CrossRef
Weibull  W., “A statistical distribution function of wide utility,” J. Appl. Mech.. 18, , 293 –297 (1951). 0021-8936 
Krol  H., “Investigation of nanoprecursors threshold distribution in laser-damage testing,” Opt. Commun.. 256, (1–3 ), 184 –189 (2005). 0030-4018 CrossRef
Picard  R. H., Milam  D., Bradbury  R. A., “Statistical analysis of defect caused damage in thin films,” Appl. Opt.. 16, (6 ), 1563  (1977). 0003-6935 CrossRef
Porteus  J. O., Seitel  S. C., “Absolute onset of optical surface damage using distributed defect ensembles,” Appl. Opt.. 23, (21 ), 3796 –3805 (1984). 0003-6935 CrossRef
Neyer  B. T. “A d-optimality based sensitivity test,” Technometrics. 36, , 61 –70 (1994). 0040-1706 CrossRef
Neyer  B. T., “New developments in sensitivity testing,” Presented at  45th Ann. Mtg on Fuse Section of the National Defense Industrial Association ,  Long Beach, California  (2001).
Neyer  B. T., Gageby  J., “ISO 14304 Annex B all-fire/no-fire test and analysis methods,” in  Proc. 17th Symp. on Explosives and Pyrotechnics ,  Essington, Pennsylvania  (1999).

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