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Optical Materials Express

Optical Materials Express

  • Editor: David J. Hagan
  • Vol. 2, Iss. 11 — Nov. 1, 2012
  • pp: 1612–1623

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Defect population variability in deuterated potassium di-hydrogen phosphate crystals

Zhi M. Liao, R. Roussell, J. J. Adams, M. Runkel, W. T. Frenk, J. Luken, and C. W. Carr  »View Author Affiliations


Optical Materials Express, Vol. 2, Issue 11, pp. 1612-1623 (2012)
http://dx.doi.org/10.1364/OME.2.001612


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Abstract

Bulk laser damage variability in deuterated potassium dihydrogen phosphate (DKDP) crystals is well known and makes online conditioning of multiple-beam laser systems difficult to optimize. By using an empirical model, called Absorption Distribution Model (ADM), we were able to map the damage variability of the crystals (boule to boule as well as within same boule) in terms of defect population variations. The defect population variation was found to coalesce into two distinct groupings that can be identified by the defect population in the late growth region of the boule. This result allows us to optimize the conditioning protocol for an arbitrary number of beams with crystals of differing damage quality.

© 2012 OSA

1. Introduction

Potassium dihydrogen phosphate (KDP) and its deuterated analog (DKDP) are nonlinear optical crystals that are uniquely suitable for use in frequency conversion for high-energy large-aperture laser systems such as the National Ignition Facility (NIF) and Laser Megajoule (LMJ). Of all the available non-linear materials only (D)KDP can be grown in boules to the sizes (450 + kg) and clear apertures needed for fusion class lasers. In addition to large size, DKDP has been successfully deployed as a final optic for Third Harmonic Generation (THG) of 351-nm light [1

P. Wegner, J. Auerbach, T. Biesiada Jr, S. Dixit, J. Lawson, J. Menapace, T. Parham, D. Swift, P. Whitman, and W. Williams, “NIF final optics systems: frequency conversion and beam conditioning,” Proc. SPIE 5341, 180–189 (2004). [CrossRef]

] due in no small part to the ability to increase its damage resistance to laser-induced damage through laser conditioning [2

J. Swain, S. Stokowski, D. Milam, and F. Rainer, “Improving the bulk laser damage resistance of potassium dihydrogen phosphate crystals by pulsed laser irradiation,” Appl. Phys. Lett. 40(4), 350–352 (1982). [CrossRef]

–4

M. Runkel, K. Neeb, M. Staggs, J. Auerbach, and A. Burnham, “Results of raster scan laser conditioning studies on DKDP triplers using Nd:YAG and excimer lasers,” Proc. SPIE 4679, 348 (2001).

]. However, although laser conditioning does improve the damage threshold of the crystal, it is not without cost. The safest way to optimally condition a DKDP crystal is to raster scan [4

M. Runkel, K. Neeb, M. Staggs, J. Auerbach, and A. Burnham, “Results of raster scan laser conditioning studies on DKDP triplers using Nd:YAG and excimer lasers,” Proc. SPIE 4679, 348 (2001).

] the crystal in an offline facility using a high quality, flat-top laser beam with optimum spatial homogeneity (minimal contrast), which also means smaller beam area. This can be a time consuming process as the laser spot has to be overlapped sufficiently to ensure uniform fluence, and multiple scans at progressively larger fluence are required. This can be challenging when preparing crystals for use on large-aperture laser systems such as NIF or LMJ where there are hundreds of beams, each with ~1000 cm2 beam area. For example, raster scanning all the crystals used in the NIF using a 10-Hz, ~1-mm2 beam with 0.5-mm2 size steps would take roughly 4 years running at 24 hour/day operation. However, this would have the benefit of being able to condition to higher fluences, and it could be done in parallel or before the online laser facility is completed and without impacting the shot schedule of the online laser.

The fastest way to condition the crystals is to install the optic to the laser and slowly ramp the fluence with full beam area to condition the crystals in situ. However, because of the sensitivity of the crystal growth process, the damage performance can vary significantly from boule to boule and sometimes even within the same boule [5

M. Runkel and A. K. Burnham, “Difference in bulk damage probability distributions between tripler and z-cuts of KDP and DKDP at 355nm,” Proc. SPIE 4347, 408 (2000).

–11

M. Runkel, M. Yan, J. De Yoreo, and N. Zaitseva, “The effect of impurities and stress on the damage distributions of rapidly grown KDP crystals,” Proc. SPIE 3244, 211 (1997).

]. Previous results have indicated that changes in initial growth condition, such as growth temperature or presence of impurities, can potentially affect the damage behavior of the crystal [6

A. K. Burnham, M. Runkel, R. A. Hawley-Fedder, M. L. Carman, R. A. Torres, and P. K. Whitman, “Low-temperature growth of DKDP for improving laser-induced damage resistance at 350nm,” Proc. SPIE 4347, 373 (2000).

]. Furthermore, changes in growth conditions during the growth of KDP rapid-growth boules have also been associated with increasing purity [8

M. Yan, R. Torres, M. Runkel, B. Woods, I. Hutcheon, N. Zaitseva, and J. DeYoreo, “Investigation of impurity and laser-induced damage in the growth sectors of rapidly grown KDP crystals,” Proc. SPIE 2966, 11–16 (1997). [CrossRef]

] or decreasing concentration of emission clusters [7

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys. 85(8), 3988–3992 (1999). [CrossRef]

,9

M. Pommiès, D. Damiani, B. Bertussi, J. Capoulade, H. Piombini, J. Y. Natoli, and H. Mathis, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267(1), 154–161 (2006). [CrossRef]

] from first growth (FG) to late growth (LG) sections of the boule (see Fig. 1 ). Work by Runkel et al. has also suggested strong implications of damage relating to nanoclusters of absorbing impurities that were uniformly present in the growth environment, with damage differences caused by change in the size distribution or absorption mechanism of the precursors [11

M. Runkel, M. Yan, J. De Yoreo, and N. Zaitseva, “The effect of impurities and stress on the damage distributions of rapidly grown KDP crystals,” Proc. SPIE 3244, 211 (1997).

].

Fig. 1 Schematic of a DKDP boule with growth regions and an illustration of precursor defects with different sizes (a) and densities (α). Filled-in circles denote Type 1 precursors and open circles denote Type 2 precursors (see theory section).

This means that the parameters for optimum in-line conditioning (number of shots and fluences to use for these shots) will vary between parts cut from different boules and possibly between parts cut from FG and LG regions of the same boule. The optimum conditioning protocol is the one using the smallest number of shots possible and the lowest fluences possible to achieve conditioning to the specifications given, for example, maximum allowed number of damage initiation sites at operating fluence. To determine this optimum conditioning protocol, it is necessary to know the number of damage sites, X, that will be initiated by an operating fluence ϕOP on a crystal that has previously been irradiated (and thus conditioned) up to fluence ϕC. This set of measurements X(ϕOP, ϕC ), i.e., damage sites as a function of operating fluence and conditioning fluence, can be conveniently represented as a “conditioning map” that would come from precise measurement of the damage density, ρ(ϕ) [12

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(7), 1958–1962 (2006). [CrossRef]

], for pristine and conditioned material from the same boule. Furthermore, damage density measurements can be complicated to obtain and require large beam size (~cm2) so that damage can be appropriately sampled [12

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(7), 1958–1962 (2006). [CrossRef]

]. Instead, most of the sample parts are measured using a faster damage probability measurement such as a single-fluence multiple-shot test (S/1) or ramped-fluence multiple-shot test (R/1) [13

J. Adams, J. A. Jarboe, M. Feit, and R. P. Hackel, “Comparison between S/1 and R/1 tests and damage density vs. fluence (rho(phi)) results for unconditioned and sub-nanosecond laser-conditioned KD2PO4 crystals,” Proc. SPIE 6720, 672014 (2008).

]. Since damage probability measurement only provides probability of damage reaching the minimum detectable damage density, not the number of initiated sites, it cannot be used to optimize conditioning ramps without a model that can transform it into damage density measurements. The concept of damage threshold itself is ambiguous and must be taken in the context of the testing protocol, in particular, the testing beam volume and the number of sites tested. The testing beam volume determines the minimum detectable damage density (MDDD), which is defined as a single damage site in the total irradiated volume of the test beam at a given fluence (irradiated volume of the test beam times the number of tested sites per fluence). The number of test sites determines the resolution of the probability data. For example, a test protocol that has 20 sites would typically have a probability resolution (and to a certain extent, a quick estimate of 1/n error bar) of 5%. As a result, damage threshold fluence is typically the minimal fluence at the probability resolution limit needed to produce the MDDD. In this work, the damage probability data is collected using 10 sites per fluence with a small beam volume (~0.75 mm 1/e2 diameter) at a thickness of 10 mm, which corresponds to a MDDD of at most ~2 mm−3 with an error bar on probability value of ± 10%.

Recent development of an Absorption Distribution Model (ADM) [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

] has significantly improved the ability to predict the conditioned and unconditioned damage behavior of DKDP crystals using the standard S/1 and R/1 damage probability tests. Since we have far more samples with damage probability test results than damage density data, we can leverage these data into constructing detailed conditioning maps that can be used to develop a systematic conditioning protocol optimized to minimize the number of shots required. In this work, we have analyzed a large number of damage probability tests and damage density tests from over a dozen different DKDP boules using ADM to assess the defect population variability from boule to boule as well as from within the same boule. We will use the relationship of growth differences and ADM to generate conditioning maps that can be used to optimize online-conditioning protocols for multiple beamlines simultaneously.

2. Theory

ADM was developed by Spaeth et al. [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

,15

M. Spaeth, “Absorption distribution model,” LLNL Internal Presentation (2007).

] at NIF as a self-consistent empirical model that can explain a diverse set of damage-related data that includes damage probability [13

J. Adams, J. A. Jarboe, M. Feit, and R. P. Hackel, “Comparison between S/1 and R/1 tests and damage density vs. fluence (rho(phi)) results for unconditioned and sub-nanosecond laser-conditioned KD2PO4 crystals,” Proc. SPIE 6720, 672014 (2008).

] and damage density measurements [12

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(7), 1958–1962 (2006). [CrossRef]

]. It assumes that the precursor defects are not homogenous but made up of (at least) two distinct populations of defect clusters, one of which absorbs linearly (Type 1) and the other nonlinearly (Type 2) (see Fig. 1). These precursors can be nano-scale absorbing defect clusters that transform into macroscopic damage through thermal runaway in the bulk material and propagating absorbing front; similar effects have been recently predicted and verified experimentally [16

C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B 82(18), 184304 (2010). [CrossRef]

]. The precursors have a range of sizes and each size can have a range of absorption values (α) through various densities of the individual defect clusters (Fig. 1). However, the absorption values of the two types of defects are assumed to be completely correlated [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

] and are depicted in Fig. 1 as corresponding filled-in and open circles. The figure shows where there is a propensity for a high density of Type 1 defects, there is also a corresponding high density of Type 2 defects (it is not intended to show the two types as being mixed clusters). The details of the model derivation and application can be found elsewhere [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

]. In essence, ADM assumes R/1 data (ramped-fluence single-shot tests) to correspond to fully conditioned material containing only Type 1 defect. Thus R/1 data allows using thermal diffusion calculations to obtain the absorption characteristics of the Type 1 defects using a critical damage temperature of 220°C. S/1 data (single-fluence multiple-shot test) contains influences from Type 1 and Type 2 defects and allows extracting the absorption characteristics of the Type 2 defects using a critical conditioning temperature of 135°C. The damage probability P(ϕ,a) at fluence ϕ for damage precursors of radius, a, is expressed as:
Ρ(ϕ,a)= ∫ αx(ϕ,a)∞ f(α)⋅dα
(1)
where P(ϕ,a) is the probability of damage, αx is the threshold absorption necessary for a precursor of radius, a, to achieve damage temperature Θx = 225 °C at fluence ϕ via thermal diffusion. Laser conditioning occurs when a precursor is heated over the conditioning temperature (ΘC) but below the damage temperature (Θx), at which point the affected Type 2 absorption reduces, therefore increasing the damage threshold [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

].

Each defect type absorption distribution can be characterized by its mean (μ) and standard deviation (σ) as follows:
f(α)= 1 σ 2π e − (α−μ)2 2 σ2
(2)
With the total absorption α = α1 + α2 (c0 + c1I + … + cnIn) where α1 is the Type 1 linear absorption, α2 is the Type 2 nonlinear absorption, I is the laser intensity, and cn’s are the coefficients of different orders of intensity-dependent nonlinear absorption. As a result, ADM can extract from each set of R/1 measurements the Type 1 defect absorption parameters (μ1, σ1) and from each set of S/1 measurements the Type 2 defect absorption parameters (μ2, σ2), a total of 4 parameters for each set of damage probability data (see Fig. 2 ). Note in Fig. 2(a), the damage probability is shown for precursor of maximum size (amax ~500nm); this is because due to size dependence on heat diffusion, maximum size precursors are the first to damage.

Fig. 2 Damage probability measurement (S/1-circle, R/1-filled dot) (a) and corresponding parameters of Type 1 and Type 2 absorption distribution as extracted by ADM model (b).

ADM also can calculate the damage density measurement (ρ(ϕ)) by using the following equation:
ρ(ϕ)= ∫ a min a max n(a)Ρ(ϕ,a)da
(3)
with amin and amax being the minimum and maximum precursor sizes respectively, and n(a) being the precursor size distribution given by [17

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulselength scaling and laser conditioning,” Proc. SPIE 5273, 74–82 (2003).

,18

J. B. Trenholme, M. D. Feit, and A. M. Rubenchik, “Size-selection initiation model extended to include shape and random factors,” Proc. SPIE 5991, 59910X (2005). [CrossRef]

]
n(a)= N(b−1) ab( a min 1−b− a max 1−b)
(4)
where N is the total density of the precursor and b is the size-dependent scaling power coefficient. The coefficients amin and amax are usually set as 50 and 500 nm respectively, which are consistent with the observed sizes of damage sites and the smallest sizes suitable for absorbing energy in sufficient density [17

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulselength scaling and laser conditioning,” Proc. SPIE 5273, 74–82 (2003).

]. The size-dependent scaling power coefficient, b, is set to be ~3 because this is a typical value for characterizing size variation in optics contamination [18

J. B. Trenholme, M. D. Feit, and A. M. Rubenchik, “Size-selection initiation model extended to include shape and random factors,” Proc. SPIE 5991, 59910X (2005). [CrossRef]

]. This implies that ADM extracts only 1 additional parameter (N) for each set of ρ(ϕ) measurements (see Fig. 3 ). This also means that N could vary with each different boule and therefore potentially limit the usefulness of ADM (since it is necessary to measure damage density in order to extract N), unless we can find some generalization of N using damage probability data (S/1 and R/1) or growth data.

Fig. 3 Damage density ρ(ϕ) measurement with ADM prediction (a) and corresponding precursor density distribution used by ADM (b).

3. Data analysis

Gooch & Housego (Ohio), LLC, has grown DKDP boules using the “conventional” or slow-growth method since 2000 for third harmonic frequency conversion crystals for NIF producing UV light at 351 nm. NIF has slightly more than 200 THG optics that were cut from various boules produced by Gooch & Housego. Usually, at least one “witness sample” (~5 x 5 cm square) from each boule has been damage tested using the damage probability test at LLNL (i.e. S/1 and R/1) for quality assurance and these results have been carefully recorded in an extensive database [13

J. Adams, J. A. Jarboe, M. Feit, and R. P. Hackel, “Comparison between S/1 and R/1 tests and damage density vs. fluence (rho(phi)) results for unconditioned and sub-nanosecond laser-conditioned KD2PO4 crystals,” Proc. SPIE 6720, 672014 (2008).

,19

M. Runkel, M. Yan, J. De Yoreo, and N. Zaitseva, “The effect of impurities and stress on the damage distributions of rapidly grown KDP crystals,” Proc. SPIE 3244, 211 (1997).

]. When possible, more than 1 sample from a given boule was damage tested and these samples are labeled with an identifier which denotes if they were first growth (FG) or late growth (LG) material (see Fig. 1). We have analyzed over 50 different S/1 and R/1 damage test results as well as over a dozen damage density measurements (see Table 1 ).

Table 1  Boule samples with various damage testing data
Samples with identified regionSamples with both FG and LG regionSamples with ρ(ϕ) data
Test Data Set>50~39~20
Number of boules>16~14~13

3.1 Precursor defect population variation over all boules

For each sample a with damage probability test data (i.e., S/1, R/1), we used ADM to extract the precursor defect parameters (μ1, σ1, μ2, σ2). Figure 4 shows the histogram of these defect parameters for all samples in the inventory. It is evident from Fig. 4 that there is a range of absorption mean and standard deviation values for all samples in inventory. Type 2 precursors have in general a broader distribution (i.e., larger variance) both in the absorption mean and the absorption standard deviation of any given sample.

Fig. 4 Defect parameter histogram for absorption mean (μ) on left (a) and for absorption standard deviation (σ) on (b) for both Type 1 (filled) and Type 2 (not-filled) precursor types.

It is worthwhile to plot the extracted defect population parameters for each sample as a scattering plot with the x axis associated with Type 1 defect absorption parameters and y axis associated with Type 2 defect absorption parameters (see Fig. 5(a) ). In this type of plot, the Type 1 and Type 2 defect absorption parameters are associated loosely with the MDDD for fully conditioned materials and MDDD for unconditioned material respectively. For example, a decrease of the Type 1 defect absorption is associated with an increase in the fluence needed to produce the MDDD in fully conditioned material. Although the MDDD fluence is a function of both Type 1 and Type 2 absorption defects, in reality, Type 2 absorption defects dominate this process because of nonlinear absorption. As a result, a decrease of the Type 2 defect absorption is associated with an increasing damaging (or conditioning) fluence since the difference between damaging and conditioning is only the maximal local temperature reached [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

].

Fig. 5 (a) Plot of Type 1 vs. Type 2 absorption parameters (mean, μ - open symbol, standard deviation, σ - filled symbol) for all data on all boules. (b) grouped plot of Type 1 vs. Type 2 absorption parameter for FG (circle) and LG for LL16 boule only.

Plotting the entire test results from different boules on the same graph (see Fig. 5(a)) shows the variability of defect parameters for all samples as well as the relationships between Type 1 and Type 2 precursors. Although the mean absorption value for both Type 1 and Type 2 are similar (19.7 and 19.1 respectively), Type 2 defect parameters have a larger variance than Type 1 defect parameters (both mean μ and standard deviation σ). There are two potential interpretations for this; one is that there is in general a larger variation of Type 2 defect parameters, which is plausible since Type 2 defects can be conditioned, which might imply they are “extrinsic defects” such as contaminations, whereas Type 1 defects can be thought of more as “intrinsic defects” that might be associated with the stoichiometry of the growth condition and therefore, showing less variations within the same location. The second interpretation for the larger Type 2 defect parameter variance is that since the Type 2 defect parameter is extracted from S/1 measurements, which usually only have about ~7 data points per data set, and Type 1 is extracted from R/1 measurements, which have ~30-70 data points, the variation is related to the resolution (i.e., error bar) of the test data. In either case, what we infer is that in general, damage behavior for boule to boule is more consistent than conditioning behavior. As a result, greater care is warranted when attempting to condition multiple optics and a systematic approach is required.

3.2 Precursor defect population variations over single boule

We will focus mainly on test results from FG and LG from this point forward as we had little data from the other growth regions. Most of the test samples have 1 or 2 damage probability measurements from each boule’s growth region; with the most extensive being the LL16 boule, which has 10 damage probability measurements from the FG growth region and 4 damage probability measurements from the LG growth region. Figure 5(b) is a plot of the Type 1 vs. Type 2 defect parameters for both FG and LG samples for the LL16 boule only. Both the FG and LG defect parameters for LL16 are close to the mean absorption of all the boules (see Fig. 5(a)), indicating that this boule is not an outlier. It is also apparent that the Type 2 defect parameters (both mean and standard deviation) have more variance than the Type 1 parameters (also keeping in trend with the sample population), with LG data having less variation than FG data. It is also interesting to note that a Type 2 defect mean absorption (μ2) decreases as the boule transitions from FG to LG, which agrees with previous results that show an increasing “purity” [8

M. Yan, R. Torres, M. Runkel, B. Woods, I. Hutcheon, N. Zaitseva, and J. DeYoreo, “Investigation of impurity and laser-induced damage in the growth sectors of rapidly grown KDP crystals,” Proc. SPIE 2966, 11–16 (1997). [CrossRef]

] or a decrease in “emission clusters” (i.e., contaminant) [7

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys. 85(8), 3988–3992 (1999). [CrossRef]

,9

M. Pommiès, D. Damiani, B. Bertussi, J. Capoulade, H. Piombini, J. Y. Natoli, and H. Mathis, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267(1), 154–161 (2006). [CrossRef]

] as the boule grows. The fact that previous studies [7

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys. 85(8), 3988–3992 (1999). [CrossRef]

–9

M. Pommiès, D. Damiani, B. Bertussi, J. Capoulade, H. Piombini, J. Y. Natoli, and H. Mathis, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267(1), 154–161 (2006). [CrossRef]

] didn’t find significant difference in damage behavior in FG vs. LG also wasn’t surprising from this result. Although the mean absorption of the FG sample is larger than LG, the variance is large enough that a significant amount of testing (in our case, ~8 tests) would be required to demonstrate the trend, whereas previous studies typically tested only a single sample.

3.3 Precursor defect grouping by boule type

To examine the possible relationship of precursor defect parameters for samples of FG vs. LG from the same boule, a graph is generated (see Fig. 6 ) in which Type 1 defect mean absorption (μ1) is plotted vs. Type 2 defect mean absorption (μ2). In the figure, each boule is represented by an arrow vector whose arrow end represents the value for the LG portion of the boule, and the starting point of the vector represents the FG portion of the boule. The Type 1 mean absorption (μ1) is extracted using Eqs. (1) and (2) from R/1 damage probability test data and likewise, the Type 2 mean absorption (μ2) is extracted using S/1 damage probability test data. As a result, the length of the vector denotes the degree of heterogeneity of the boule (from FG to LG), and the direction of the arrow vector indicates the change in the boule “purity” as it grows. What emerges from this graph is a grouping of the boules that shows at least 2 distinct behaviors that we have labeled as Group A and B (see Fig. 6).

Fig. 6 Plot of Type 1 vs. Type 2 defect mean absorption for each boule and the relationship between the FG and LG as indicated by the direction of the arrow.
  • 1. Group A consists of 8 boules that have LG Type 2 mean absorption μ2 (LG) < 19 cm−1. All of these boules have a higher Type 2 absorption mean for FG vs. LG. These boules behave exactly like LL16, which we have presented in Fig. 5(b), where we have seen an increasing “purity” as the boule is grown, which is consistent with previous findings [7

    S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys. 85(8), 3988–3992 (1999). [CrossRef]

    –9

    M. Pommiès, D. Damiani, B. Bertussi, J. Capoulade, H. Piombini, J. Y. Natoli, and H. Mathis, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267(1), 154–161 (2006). [CrossRef]

    ]. These boules in general have a better damage performance because of the lower Type 2 absorption mean.
  • 2. Group B consists of 6 boules that have LG Type 2 mean absorption μ2 (LG) > 19 cm−1. The primary difference of Group B boules in contrast to Group A boules, is that all boules have a lower Type 2 mean absorption value for FG vs. LG. As a result, Group B boules in general exhibit a decreasing “purity” as the boule is grown. Since these boules in general have a higher Type 2 mean absorption value, these boules also exhibit a poorer damage performance.

In terms of conditioning or damage performance, the boules from Group B would have the lower damage threshold, but they also would have a lower conditioning threshold. As for a fully conditioned damage threshold (which depends strongly on Type 1 mean absorption), the nominal boule from Group A and B would have similar performance. An interesting observation is that unlike the Type 1 absorption value in which there is a 60-40 split with respect to increasing vs. decreasing purity (i.e., arrow pointing up or down in Fig. 6) going from FG to LG, all of the boules exhibit decreasing purity in intrinsic precursors (i.e., Type 1) except for two boules that we have identified using double lines in Fig. 6. These two boules are outliers in that although they both belong to Group A, one of them has the lowest R/1 damage fluence (highest Type 1 value) and the other one has the highest R/1 damage fluence (lowest Type 1 value) of all the boules. They represent the opposite ends of the spectrum, and if we can determine what caused this difference, it might just provide the key identifier in growing better damage-resistant boules. It is clear that these correlations play an important role in the amount and kind of precursors that the boules inherit from the growth process. It was confirmed by the manufacturer of these boules after reviewing this classification that the distinction of Group A vs. Group B generally corresponds to growth parameters they have long suspected play an important role in damage performance.

3.4 Precursor density grouping by boule type

Damage density ρ(ϕ) tests have been performed on a number of the same samples over the years at LLNL for pulse-width dependence and conditioning studies [3

J. J. Adams, C. W. Carr, M. D. Feit, A. M. Rubenchik, M. L. Spaeth, and R. P. Hackel, “Pulse length dependence of laser conditioning and bulk damage in KD2PO4,” Proc. SPIE 5647, 265 (2004).

,12

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(7), 1958–1962 (2006). [CrossRef]

,20

M. Runkel, J. Bruere, W. Sell, T. Weiland, D. Milam, D. Hahn, and M. Nostrand, “Effects of pulse duration of bulk laser damage in 350-nm raster-scanned DKDP,” Proc. SPIE 4392, 405 (2002).

]. Although boule IDs have always been recorded, the growth regions of the samples were usually not recorded. ADM analysis using Eq. (3) to extract the total precursor density (N) from each ρ(ϕ) measurement was performed for all available data. For samples that did not have growth regions identified, we found that in every case, only one set of absorption parameters from either FG or LG probability data produced ρ(ϕ) that fit the data; as a matter of fact, the model could not converge on a solution using the other set of absorption values. This is an important revelation that shows the self-consistency of ADM and its ability to discriminate erroneous data.

Figure 7 shows a plot of extracted total precursor density N vs. Type 1 absorption mean μ1. At first glance, there doesn’t seem to be any relationship between the precursor density N and the defect absorption parameters, but when the data was grouped according to the results from Fig. 6, a trend emerges. An exponential dependence emerged in Fig. 7 between the Type 1 mean absorption μ1 and the precursor density N for Boule Type A. The first dependence (blue line) centers around data from Group A boules and other boules (marked with green triangles) that we were not able to classify because of only having damage probability data from the FG growth region (remember it is the LG Type 2 mean absorption μ2 that differentiates group A boules from group B). The four data points from Type B boules in Fig. 7 are closely clustered so that it is impossible to draw any conclusion as to whether or not the dependence of N is constant, linear, or exponential from that data alone. However, in light of the strong exponential dependence of the data from the Type A boules, we argue that the Type B data should follow a similar trend (red dashed line). It is worthwhile to note that the value of N can change from FG to LG in a given boule just as its absorption parameters change within the same boule. As a matter of fact, the four data points from the Type B boules in Fig. 7 come from just two boules. The middle two values (in terms of defect density N) come from samples from the LG portion of same boule and are virtually identical. The other two values come from samples from the FG portion of another boule with N being slightly different. The existence of these dependencies along the same boule groupings using an independent set of measurements (damage density vs. damage probability) underlines the possibility that these groupings do not just provide empirical convenience, but have a fundamental relationship with each other in terms growth conditions. This implies that we can now estimate the precursor density N from just the Type 1 absorption mean, μ1, using the following fit
N( μ1)= { 7⋅ 104⋅ e −0.194⋅ μ1 Type A 8⋅ 103⋅ e −0.194⋅ μ1 Type B
(5)
Note that the Type A boules have a stronger dependence on the Type 1 absorption mean μ1, than Type B boules; this, however, does not necessary mean that Type A boules will perform better in damage than Type B boules since damage performance is dependent on both absorption distribution and defect density (N). The absorption distribution can have a higher impact with respect to damage threshold, while defect density (N) is in general associated with density of damage sites. Group A boules have better damage performance in that it took higher fluences for damage to start because of the lower absorption values, but there could potentially be more damage at higher fluences.

Fig. 7 Type 1 defect mean absorption (μ1) vs. precursor density N for Group A boules (data – circular blue, fit blue line) and Group B boules (data – red square, fit red line). Data from ungrouped boules (lack of LG data) were marked as green triangles.

4. Modeling results

As discussed in the earlier section, damage probability data is adequate for quality assurance or relative comparison, but it is not as useful in damage prediction in managing an optic’s lifetime. A well designed laser system would have a specification that sets limits on the number of damage sites or the maximum size of damage sites. As a result, it is not practical to set the operation of a laser using the damage probability data; a 10% damage probability does not correlate to the actual number of damage initiations. Furthermore, although R/1 damage probability data can help provide the threshold for an optimal (i.e., fully) conditioned damage sample, it is difficult to extract the consequence of conditioning ramps or the number of damage initiations if the sample is only partially conditioned.

An optimal conditioning protocol can be calculated based on the results of this study. For example, if a DKDP crystal was required to be conditioned to operate at 8 J/cm2 at 3ω using a 5-ns flat-in-time (FIT) pulse without exceeding a given amount of bulk sites as dictated by the optic’s scattering budget, this can be accomplished only by knowing the crystal’s boule type (A or B) from the probability of damage test data (S/1, R/1) from LG samples or possibly from growth conditions. ADM is able to calculate an optic’s damage density (ρ(ϕ)) as a function of operating fluence and conditioning fluence (see Eq. (3) with precursor parameters (μ1, μ2) and total precursor density, which is now a function of μ1 and boule type (see Eq. (5). The expected number of initiations X that a laser shot can cause on a crystal can then be calculated as
〈 X( ϕ OP, ϕC)〉=V⋅ ∫ ρ( ϕ OP; ϕC)⋅f(ϕ; ϕ OP, ϕσ)⋅dϕ
(6)
with ϕOP being the mean operating fluence, ϕσ being the standard deviation of the damage fluence, ϕC being the conditioning fluence, and V being the volume of the crystal. The fluence is assumed to be from a 3-ns Gaussian—for other pulse shapes, an equivalent conversion factor would need to be calculated [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

,21

C. W. Carr, J. B. Trenholme, and M. L. Spaeth, “Effect of temporal pulse shape on optical damage,” Appl. Phys. Lett. 90(4), 041110 (2007). [CrossRef]

]. The first term inside the integral is the calculated conditioned ρ(ϕOP; ϕC) (see Eqs. (3) and (4)) using the Type 1 and Type 2 defect parameters from damage probability test data (S/1, R/1) and the conditioned fluence ϕC (which modifies the Type 2 defect parameter) to which the optic has been exposed [14

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

]. The precursor density N that is used to calculate the damage density is derived using the trend lines from Eq. (5). The second term in the integral is the laser fluence distribution, which for most laser systems can be modeled as having a Gaussian (or Rician) fluence distribution with a mean fluence ϕOP and a standard deviation ϕσ that is directly related to beam contrast [22

Z. M. Liao, J. Huebel, J. Trenholme, K. Manes, and C. W. Carr, “Modeling max-of-N fluence distribution using measured shot-to-shot beam contrast,” Appl. Opt. 50(20), 3547–3552 (2011). [CrossRef] [PubMed]

]. For simplicity, we will assume that the conditioning laser fluences are uniform and that the current shot has a fixed contrast of 10%. As a result, we can now calculate the conditioning matrix (i.e., conditioning map), which is the expected number of initiation sites as a function of operating mean fluence (ϕOP) and conditioned fluence (ϕC). Figure 8 shows an example of the conditioning map of two boules, one labeled “Type A nominal,” which has defect parameters (μ1, μ2) consistent with Group A (like LL16) and one labeled “Type B outlier,” which has defect parameters similar to the boule on the very top of the graph in Fig. 8, also one of the lowest damage resistant boules in the study. Figure 8 shows the number of expected initiations using contours in log value as a function of conditioned fluence (ϕC) in the horizontal axis and operating fluence (ϕOP) in the vertical axis. If the specification is the total initiation sites, Xmax = 105 sites, then an optimal conditioning sequence can be individually calculated for each boule (see solid red line). The conditioning sequence is optimized by maximizing the damage fluence of each shot (vertical axis) given the current conditioning fluence (horizontal axis), so that the accumulated initiation sites (X1 + X2+..Xn where Xn is the number of initiations at the nth shot including the desired operating fluence shot) are kept below Xmax until the desired operating fluence is achieved (8 J/cm2 for this example).

Fig. 8 Contour plot of the conditioning map showing the number of damage initiations (in log units) for a nominal boule from Group A (left graph) vs. the worst outlier boule from Group B (right graph), the optimal conditioning sequence to reach 8 J/cm2 for each boule is shown in red solid line.

Figure 8 clearly shows that a nominal boule from group A will only need two shots to operate safely at 8 J/cm2, while the worst boule would need at least three shots to operate safely. This is a simple illustration to demonstrate the potential of calculating optimal conditioning protocols for an arbitrary number of boules. From here, it is a straightforward optimization problem to calculate the optimal conditioning for an arbitrary number of crystals given that the probability damage data on FG and LG parts are available. However, since we are using damage probability to sample the precursors, our sampling is only as good as the sampling resolution of the test (i.e., the number of sites or the area/volume of the testing). For example, most damage probability data is collected using 10 sites per fluence with a small beam volume (~0.75 mm 1/e2 diameter), which corresponds to a dynamic range of 0.5 sites/mm3 to 4 sites/mm3 for a 10-mm-thick part. This implies our calculation is intended for a range of 103 to 105 for a volume of ~104 mm3, which is consistent with NIF specifications.

5. Conclusion

ADM was used to analyze damage test results from over a dozen DKDP boules to investigate variations of defect populations from boule to boule as well as different growth regions within a boule. Large variations of defect populations of both types were found from boule to boule. In addition, two distinct groupings of the boules were found that were predicated on the defect absorption parameters of the last growth regions (μ2(LG)). This grouping is also important in determining the relationship between the Type 1 mean absorption and the total defect precursor density. Understanding this grouping can potentially refine growth conditions to produce more damage-resistant boules, as well as help formulate optimal conditioning protocols that can significantly reduce shot time while avoiding damage.

Acknowledgments

The authors thank Cindy Cassady for help in editing this manuscript. This work is performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and funded through LLNL office of LDRD (LLNL-JRNL-576732).

References and links

1.

P. Wegner, J. Auerbach, T. Biesiada Jr, S. Dixit, J. Lawson, J. Menapace, T. Parham, D. Swift, P. Whitman, and W. Williams, “NIF final optics systems: frequency conversion and beam conditioning,” Proc. SPIE 5341, 180–189 (2004). [CrossRef]

2.

J. Swain, S. Stokowski, D. Milam, and F. Rainer, “Improving the bulk laser damage resistance of potassium dihydrogen phosphate crystals by pulsed laser irradiation,” Appl. Phys. Lett. 40(4), 350–352 (1982). [CrossRef]

3.

J. J. Adams, C. W. Carr, M. D. Feit, A. M. Rubenchik, M. L. Spaeth, and R. P. Hackel, “Pulse length dependence of laser conditioning and bulk damage in KD2PO4,” Proc. SPIE 5647, 265 (2004).

4.

M. Runkel, K. Neeb, M. Staggs, J. Auerbach, and A. Burnham, “Results of raster scan laser conditioning studies on DKDP triplers using Nd:YAG and excimer lasers,” Proc. SPIE 4679, 348 (2001).

5.

M. Runkel and A. K. Burnham, “Difference in bulk damage probability distributions between tripler and z-cuts of KDP and DKDP at 355nm,” Proc. SPIE 4347, 408 (2000).

6.

A. K. Burnham, M. Runkel, R. A. Hawley-Fedder, M. L. Carman, R. A. Torres, and P. K. Whitman, “Low-temperature growth of DKDP for improving laser-induced damage resistance at 350nm,” Proc. SPIE 4347, 373 (2000).

7.

S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys. 85(8), 3988–3992 (1999). [CrossRef]

8.

M. Yan, R. Torres, M. Runkel, B. Woods, I. Hutcheon, N. Zaitseva, and J. DeYoreo, “Investigation of impurity and laser-induced damage in the growth sectors of rapidly grown KDP crystals,” Proc. SPIE 2966, 11–16 (1997). [CrossRef]

9.

M. Pommiès, D. Damiani, B. Bertussi, J. Capoulade, H. Piombini, J. Y. Natoli, and H. Mathis, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun. 267(1), 154–161 (2006). [CrossRef]

10.

R. A. Negres, N. P. Zaitseva, P. DeMange, and S. G. Demos, “An expedited approach to evaluate the importance of different crystal growth parameters on laser damage performance in KDP and DKDP,” Proc. SPIE 6403, 64031S (2007).

11.

M. Runkel, M. Yan, J. De Yoreo, and N. Zaitseva, “The effect of impurities and stress on the damage distributions of rapidly grown KDP crystals,” Proc. SPIE 3244, 211 (1997).

12.

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(7), 1958–1962 (2006). [CrossRef]

13.

J. Adams, J. A. Jarboe, M. Feit, and R. P. Hackel, “Comparison between S/1 and R/1 tests and damage density vs. fluence (rho(phi)) results for unconditioned and sub-nanosecond laser-conditioned KD2PO4 crystals,” Proc. SPIE 6720, 672014 (2008).

14.

Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett. 35(15), 2538–2540 (2010). [CrossRef] [PubMed]

15.

M. Spaeth, “Absorption distribution model,” LLNL Internal Presentation (2007).

16.

C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B 82(18), 184304 (2010). [CrossRef]

17.

M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulselength scaling and laser conditioning,” Proc. SPIE 5273, 74–82 (2003).

18.

J. B. Trenholme, M. D. Feit, and A. M. Rubenchik, “Size-selection initiation model extended to include shape and random factors,” Proc. SPIE 5991, 59910X (2005). [CrossRef]

19.

M. Runkel, M. Yan, J. De Yoreo, and N. Zaitseva, “The effect of impurities and stress on the damage distributions of rapidly grown KDP crystals,” Proc. SPIE 3244, 211 (1997).

20.

M. Runkel, J. Bruere, W. Sell, T. Weiland, D. Milam, D. Hahn, and M. Nostrand, “Effects of pulse duration of bulk laser damage in 350-nm raster-scanned DKDP,” Proc. SPIE 4392, 405 (2002).

21.

C. W. Carr, J. B. Trenholme, and M. L. Spaeth, “Effect of temporal pulse shape on optical damage,” Appl. Phys. Lett. 90(4), 041110 (2007). [CrossRef]

22.

Z. M. Liao, J. Huebel, J. Trenholme, K. Manes, and C. W. Carr, “Modeling max-of-N fluence distribution using measured shot-to-shot beam contrast,” Appl. Opt. 50(20), 3547–3552 (2011). [CrossRef] [PubMed]

OCIS Codes
(140.3330) Lasers and laser optics : Laser damage
(140.3390) Lasers and laser optics : Laser materials processing
(190.4400) Nonlinear optics : Nonlinear optics, materials

ToC Category:
Nonlinear Optical Materials

History
Original Manuscript: July 11, 2012
Revised Manuscript: September 28, 2012
Manuscript Accepted: October 8, 2012
Published: October 15, 2012

Citation
Zhi M. Liao, R. Roussell, J. J. Adams, M. Runkel, W. T. Frenk, J. Luken, and C. W. Carr, "Defect population variability in deuterated potassium di-hydrogen phosphate crystals," Opt. Mater. Express 2, 1612-1623 (2012)
http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-2-11-1612


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References

  1. P. Wegner, J. Auerbach, T. Biesiada, S. Dixit, J. Lawson, J. Menapace, T. Parham, D. Swift, P. Whitman, and W. Williams, “NIF final optics systems: frequency conversion and beam conditioning,” Proc. SPIE5341, 180–189 (2004). [CrossRef]
  2. J. Swain, S. Stokowski, D. Milam, and F. Rainer, “Improving the bulk laser damage resistance of potassium dihydrogen phosphate crystals by pulsed laser irradiation,” Appl. Phys. Lett.40(4), 350–352 (1982). [CrossRef]
  3. J. J. Adams, C. W. Carr, M. D. Feit, A. M. Rubenchik, M. L. Spaeth, and R. P. Hackel, “Pulse length dependence of laser conditioning and bulk damage in KD2PO4,” Proc. SPIE5647, 265 (2004).
  4. M. Runkel, K. Neeb, M. Staggs, J. Auerbach, and A. Burnham, “Results of raster scan laser conditioning studies on DKDP triplers using Nd:YAG and excimer lasers,” Proc. SPIE4679, 348 (2001).
  5. M. Runkel and A. K. Burnham, “Difference in bulk damage probability distributions between tripler and z-cuts of KDP and DKDP at 355nm,” Proc. SPIE4347, 408 (2000).
  6. A. K. Burnham, M. Runkel, R. A. Hawley-Fedder, M. L. Carman, R. A. Torres, and P. K. Whitman, “Low-temperature growth of DKDP for improving laser-induced damage resistance at 350nm,” Proc. SPIE4347, 373 (2000).
  7. S. G. Demos, M. Staggs, M. Yan, H. B. Radousky, and J. J. De Yoreo, “Investigation of optically active defect clusters in KH2PO4 under laser photoexcitation,” J. Appl. Phys.85(8), 3988–3992 (1999). [CrossRef]
  8. M. Yan, R. Torres, M. Runkel, B. Woods, I. Hutcheon, N. Zaitseva, and J. DeYoreo, “Investigation of impurity and laser-induced damage in the growth sectors of rapidly grown KDP crystals,” Proc. SPIE2966, 11–16 (1997). [CrossRef]
  9. M. Pommiès, D. Damiani, B. Bertussi, J. Capoulade, H. Piombini, J. Y. Natoli, and H. Mathis, “Detection and characterization of absorption heterogeneities in KH2PO4 crystals,” Opt. Commun.267(1), 154–161 (2006). [CrossRef]
  10. R. A. Negres, N. P. Zaitseva, P. DeMange, and S. G. Demos, “An expedited approach to evaluate the importance of different crystal growth parameters on laser damage performance in KDP and DKDP,” Proc. SPIE6403, 64031S (2007).
  11. M. Runkel, M. Yan, J. De Yoreo, and N. Zaitseva, “The effect of impurities and stress on the damage distributions of rapidly grown KDP crystals,” Proc. SPIE3244, 211 (1997).
  12. 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(7), 1958–1962 (2006). [CrossRef]
  13. J. Adams, J. A. Jarboe, M. Feit, and R. P. Hackel, “Comparison between S/1 and R/1 tests and damage density vs. fluence (rho(phi)) results for unconditioned and sub-nanosecond laser-conditioned KD2PO4 crystals,” Proc. SPIE6720, 672014 (2008).
  14. Z. M. Liao, M. L. Spaeth, K. Manes, J. J. Adams, and C. W. Carr, “Predicting laser-induced bulk damage and conditioning for deuterated potassium dihydrogen phosphate crystals using an absorption distribution model,” Opt. Lett.35(15), 2538–2540 (2010). [CrossRef] [PubMed]
  15. M. Spaeth, “Absorption distribution model,” LLNL Internal Presentation (2007).
  16. C. W. Carr, J. D. Bude, and P. DeMange, “Laser-supported solid-state absorption fronts in silica,” Phys. Rev. B82(18), 184304 (2010). [CrossRef]
  17. M. D. Feit and A. M. Rubenchik, “Implications of nanoabsorber initiators for damage probability curves, pulselength scaling and laser conditioning,” Proc. SPIE5273, 74–82 (2003).
  18. J. B. Trenholme, M. D. Feit, and A. M. Rubenchik, “Size-selection initiation model extended to include shape and random factors,” Proc. SPIE5991, 59910X (2005). [CrossRef]
  19. M. Runkel, M. Yan, J. De Yoreo, and N. Zaitseva, “The effect of impurities and stress on the damage distributions of rapidly grown KDP crystals,” Proc. SPIE3244, 211 (1997).
  20. M. Runkel, J. Bruere, W. Sell, T. Weiland, D. Milam, D. Hahn, and M. Nostrand, “Effects of pulse duration of bulk laser damage in 350-nm raster-scanned DKDP,” Proc. SPIE4392, 405 (2002).
  21. C. W. Carr, J. B. Trenholme, and M. L. Spaeth, “Effect of temporal pulse shape on optical damage,” Appl. Phys. Lett.90(4), 041110 (2007). [CrossRef]
  22. Z. M. Liao, J. Huebel, J. Trenholme, K. Manes, and C. W. Carr, “Modeling max-of-N fluence distribution using measured shot-to-shot beam contrast,” Appl. Opt.50(20), 3547–3552 (2011). [CrossRef] [PubMed]

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