1. Introduction

Laser-induced damage on the exit surface of fused silica optics is a topic of considerable interest for large aperture, high-power laser systems such as the National Ignition Facility (NIF) and the Laser MegaJoule (LMJ) [1–3]. Laser damage can be discussed in terms of two key problems: damage initiation by a single pulse and damage growth due to subsequent laser pulses. In recent years, advances in the manufacturing and post-processing steps lead to significant improvements in the surface quality of fused silica components and thus a notable reduction in the number of initiations per optic [4]. The size of damage initiation sites created by nanosecond (ns) pulses is strongly influenced by the pulse duration, shape, and fluence but ranges from less than 1 μm to 30 μm [5–9]. The overall site morphology, including modified material, stresses and fractures, plays an important role in the future evolution of such a site under subsequent laser exposure. In general terms, the damage process is likely to re-ignite at the damage site location causing an expansion of the damaged volume, referred to as damage growth [10–17]. Fundamental understanding of the damage growth process and ability to predict laser optics performance in regards to optical damage are of critical importance.

Earlier studies brought forward the complexity of the damage growth process due to the many laser, material and experimental parameters involved and described how aggressive a site will evolve as a function of key parameters [18–32]. Furthermore, a more recent study focused on the uncertainty in growth rate even under identical laser conditions and quantified the effects of current site size on growth rate [33]. In addition, this latter study briefly introduced a salient growth trend associated with small damage sites. Specifically, small sites (with diameters less than ∼100 μm) in general require a higher laser fluence to grow, and their behavior is better described by a probability of growth.

In this work we investigate the growth probability of laser induced damage sites on the exit surface of SiO₂ optics under exposure to carefully controlled and characterized laser pulses. We employ descriptive statistics to summarize the evolution of a large ensemble of sites. This method allows quantification of the effects of site size and laser fluence on probability of growth from data sets collected at fixed pulse durations.

2. Experimental and data analysis techniques

Our experimental approach to growth studies has been described in detail elsewhere [18, 19]. In brief, we take advantage of the 3-cm diameter Optical Science Laboratory (OSL) laser beam [34] to simultaneously test a large number of sites with local fluences that vary within ±2–3 J/cm² (due to the ∼17% spatial beam contrast) around the beam average fluence. For each site, we compute the local mean fluence in a ∼0.5 mm patch with 10% uncertainty using the fluence registration method outlined in [19,35]. Individual site diameters are measured after each laser shot using a robotic microscope under various illuminations (back- and oblique incidence-light resulting in bright and dark field images, respectively) with optical resolution as high as 0.86 μm. This highly parallel technique greatly enhances data collection rate while maintaining precision not typically available in-situ [2, 32]. This work will focus on sites exposed on the exit surface of SiO₂ samples in high-vacuum with 351-nm, 5-ns Flat-In-Time (FIT) pulses.

As we will be estimating an unknown parameter (probability of growth) from a limited number of observations, it is important to consider the limitations on accuracy imposed by our sample sizes. The percent error on a probability measurement based on the number of observations, n, can be estimated from:

For the above reasons, we have used a raster scan initiation method to prepare a large number of damage sites with sizes in the ∼10–80 μm range on a single substrate to be tested simultaneously. These sites were initiated in a regular array with nominal spacing of 1 mm with a single pulse from a 355-nm, Nd:YAG table top laser with an 8-ns near Gaussian temporal profile focused to a spatial Gaussian spot of ∼450 μm (diameter at 1/e² of intensity) on the exit surface of a 1-cm thick silica substrate (Corning 7980 glass). The laser fluence was tailored to a level such that the single-shot (S/1) probability to initiate damage at any given location did not exceed 30%. For our bare silica substrates prepared with high damage resistance surfaces [4], this lower fluence corresponds to 40–45 J/cm² while a fluence of 60–70 J/cm² was necessary for 100% initiation probability under similar focusing conditions (peak fluence values are quoted here). There are several benefits of using a raster approach in combination with lower initiation fluence. Not only are many more small sites generated on a given sample compared to past sample preparations but the size distribution (and the overall damage morphology) produced by this method is very similar to that observed from large beam initiation, i.e., mostly single pit damage sites with diameters up to ∼40 μm [9]. However, due to inherent shot-to-shot laser fluence fluctuations and/or non-uniformities in the substrate surface, a small number of larger (up to 80 μm) and/or multi-pit sites were also initiated. We thus ensure that the sites are representative of those encountered in high power, large aperture laser systems and the results of this work are applicable to the development of predictive models on laser damage evolution within such systems. The sample was translated in a raster scan pattern and a single laser pulse was fired at each of ∼900 locations from a preset grid with 1 mm spacing, resulting in ∼300 initiated sites within the 3-cm OSL beam aperture. If a higher throughput of initiation sites is desired while preserving the site morphologies, a sample could be scanned twice using interleaved grids, i.e., minimum site separation of 0.5 mm. The combination of a preset grid and a small focused beam used for initiation minimizes the cross talk between adjacent damage sites with diameters up to about ∼500 μm, which is not a limitation considering that only a small fraction of these small sites will grow at fluences up to about 12 J/cm² [33].

A typical damage site initiation layout as well as the size distribution of individual pits prepared by the raster-scan method (one pass) outlined above are illustrated in Figs. 1(a)–1(b), respectively. In this case, damage initiation occurred at 220 locations within the 3-cm OSL beam aperture centered on a 2-inch silica substrate. As damage sites typically have aspect ratio close to 1 but are not circular, we will describe their diameter throughout this work in terms of an effective circular diameter, ECD, defined as $2\sqrt{A/\pi }$, where A is the area of the site. The site diameters ranged from ∼7 up to 80 μm. Three samples were prepared in a similar fashion and sites were characterized using the robotic microscope.

 

Fig. 1 Typical (a) layout of damage sites and in-beam CO₂ fiducials and (b) size distribution of individual pits produced by a single pass, raster scan initiation method at a preset grid with 1 mm spacing on the exit surface of SiO₂ substrates for probability of growth experiments.

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Based on the OSL beam contrast, we have designed probability of growth experiments at fixed fluences to cover the range of 4–12 J/cm² (in high-vacuum, 351-nm, 5-ns FIT pulses). Specifically, three samples were prepared with up to 300 small damage sites per sample using the raster scan method outlined above and exposed for 4–5 shots at 5, 8.5 and 10.5 J/cm² nominal fluences, respectively. Furthermore, in order to study the effects of laser exposure history, we ramped the fluence on one of the samples (#1) from low to high using small fluence increments (5, 6, 8.5 and 10.5 J/cm² with 4, 5, 2, and 2 shots at each fluence step, respectively).

Upon completion of the shot sequences and the acquisition of micrographs for individual sites on a given sample, the data was reduced as follows. The pre- and post-shot micrographs of individual sites were compared and a binary growth decision was made, either 1 or 0 if the lateral dimensions of the sites, including sub-surface fracture, did or did not change following the laser shot, respectively. We note that this growth classification was made by a human rather than based on an image thresholding routine. Although very labor intensive, human evaluation allowed us to virtually eliminate the experimental errors (including instrument and image processing) in the detection of growth. The growth classification procedure after one laser exposure is illustrated in Fig. 2 for three damage sites (A, B, and C). Namely, careful inspection by a human of the pre- and post-shot micrographs of these sites revealed no changes in the lateral dimensions of site A; in contrast, minor changes (limited to 1–2 pixels) at the periphery of site B and a relatively large change for site C occurred, as indicated by the red arrows in Fig. 2 (top). Therefore, the binary growth decision corresponding to sites A, B, and C was 0, 1, and 1, respectively, also noted in Fig. 2 (bottom). Throughout this work, this exact classification procedure was employed for all sites/shots to detect growth with high confidence. In contrast, detection of growth based on the measured changes in the sites’ diameter from shot to shot may yield different results, in particular for small damage sites undergoing minimal or no changes, as illustrated for sites B and A in Fig. 2 (bottom), respectively. As discussed in [33], the uncertainty associated with diameter measurements (automated based on image thresholding routines) is generally about ±2 μm and still affects subsequent growth rate calculations (not discussed in this work). Furthermore, for each laser exposure, damage sites were grouped in several size bins according to their pre-shot measured ECD (e.g., ∼7–20, 20–30, 30–50, 50–70, 70–100, etc.). This grouping allowed us to track the effects of current site size on the probability of growth. We note that 10 μm-wide bins were initially employed, however similar growth trends observed from adjacent bins, e.g., 30–40 and 40–50 μm, allowed wider bins to be used with improved statistics (effectively more observations, therefore better accuracy on the probability measurement).

 

Fig. 2 (Top) Micrographs of typical small damage sites acquired by the robotic microscope using back illumination before and after one laser exposure (images are on the same scale). The red arrows indicate the locations where changes in the sites’ lateral dimensions occurred after laser exposure. (Bottom) The binary growth decisions based on visual inspection compared to the measured changes in the sites’ diameter.

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We then proceeded with a preliminary, single-shot probability of growth analysis by computing the fraction of growing sites in each size bin after each shot for a series of laser shots with same average fluence (not using the local mean fluence for each site yet, i.e., all sites are grouped in one fluence bin). It should be noted that the starting size bin membership of individual sites may or may not be preserved upon subsequent shots depending on their shot-to-shot diameter change. Figure 3 summarizes the single-shot probability of growth vs. size vs. shot number obtained from sample #1 using the first four shots in the test sequence at ∼5 J/cm². The errors bars in Fig. 3 are derived based on Eq. (1) and are estimated at 15%, 10% and 6% for the ∼7–20, 20–30 and 30–50 μm size bins, respectively. Larger size bins are not shown here due to poor statistics though the existent data does follow the same general trends. These results confirm that site size is an important parameter in determining the likelihood of a damage site to grow upon laser exposure, i.e., growth probability increases with size. Furthermore, results in Fig. 3 suggest that the probability of growth is reasonably constant shot-to-shot within the frame of this preliminary analysis, namely fluence averaging over the entire beam profile. This behavior was also observed from sites on other samples exposed to series of similar shots at higher laser fluences and has important implications for the fundamental growth mechanisms. Namely, a shot independent probability of growth implies that damage sites are more or less constantly evolving with each laser exposure. This evolution may not necessarily lead to a perceived change in diameter (lateral dimensions) on every shot, defined as growth in this work and detected by a human examining an optical micrograph with ∼1 μm resolution, but may involve other small internal changes in the site morphology or stress fields surrounding the site. The measurement of such subtle changes can be accomplished using additional diagnostics, such as polarimetry, optical coherence tomography, and scanning electron microscopy; this will be investigated in future studies.

 

Fig. 3 Preliminary results on single-shot probability of growth vs. size vs. shot number (1 to 4) for damage sites exposed to nearly identical laser exposures with average fluence of ∼5 J/cm² (sample #1).

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For the remainder of this work, we will adopt the hypothesis of a shot independent probability of growth for series of nearly identical laser exposures and its direct implications to data analysis. The latter is considerably simplified as we combine all shots as if it were one shot with effectively many more sites exposed at once to their local mean fluence. Indeed, instead of tracking the evolution of a limited number of sites in each size bin from shot to shot, i.e., a single-shot approach which keeps track of shot history and individual sites, we will now group together all observations of sites/shots (corresponding to a size bin) from the tabulated data set associated with an experiment/sample. Each entry (row) in our data set contains at a minimum the site ID, shot number, current site size, previous site size (determines the membership to a size bin for probability measurements), local mean fluence, growth decision (1 or 0), and possibly other attributes (derived or measured parameters) corresponding to one observation of a site on a specific laser shot. The outcome of this analysis is augmented statistics within each size bin which now includes the original sites (as-initiated) and new members (in all but the lowest size bin) as their diameter is changing upon subsequent laser shots. In other words, this cumulative single-shot approach mixes observations of sites with similar size but different laser exposure history (one or more shots at fixed fluences) and provides a first order account of the effects of site size on the probability of growth.

3. Results and discussion

Data analysis was performed for the two experimental scenarios presented in Section 2, namely small damage sites exposed to series of i) nearly constant (at 5, 8.5 and 10.5 Jcm² from three separate but similarly prepared samples, respectively) and ii) ramp-up fluence (at 5, 6, 8.5 and 10.5 J/cm² from sample #1 only) sequence of shots. We note that only the first shots at 5 J/cm² on sample #1 are included in the data set for constant fluence. In addition, these experiments have effectively covered a wide range of fluences from ∼2 J/cm² up to ∼13 J/cm² due to the OSL beam contrast and multiple samples (experiments). Because each of the ∼300 sites on a part may see a different fluence on each shot, every site/exposure is treated as an independent experiment. This highly parallel technique has allowed us to amass data sets in excess of 3000 and 4000 entries for constant fluence and ramped fluence experiments, respectively. The results of the cumulative single-shot analysis approach are presented in Figs. 4(a)–4(b) as the probability of growth as a function of laser fluence (in 1 J/cm² bins, using the local mean fluence for each site) and site size for the two experimental scenarios described above.

 

Fig. 4 Cumulative single-shot probability of growth vs. site size vs. laser fluence for damage sites exposed to (a) nearly constant and (b) ramp-up fluence sequence of shots, respectively. The solid lines represent the best fits to the data using logistic curves (Eq. (2)).

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Due to the nature of the raster scan initiation method (with the goal of producing a large number of small sites) and the range of fluences explored in this preliminary study (up to ∼13 J/cm²), the largest size bins were sparsely populated while the smallest sites (up to 30 μm) rarely grew. Future experimentation with larger damage sites and fluences in excess of 15 J/cm² is warranted to supplement the probability of growth results depicted in Fig. 4. All data points in Figs. 4(a)–4(b) represent size/fluence bins with a minimum of 50 observations, therefore the errors on the probability values (not shown) prescribed by Eq. (1) are less than 15%, and often below 10%. Other errors due to fluence and size binning are convoluted in these probability values, but as such errors are only a few percent. the overall limit of the measurement errors is ∼15% or less.

Results shown in Fig. 4(a) illustrate the growth behaviors of small damage sites for the simplest example of laser exposure history where all sites have been exposed for up to 5 shots at nearly identical fluences (all other laser parameters being the same). The next straightforward example of laser exposure history is illustrated in Fig. 4(b) where all sites have been exposed to gradually increasing fluences. The latter is true based on the repeatable sample positioning with respect to the laser beam and the reasonably static beam contrast, i.e., the local mean fluence experienced by any damage site follows closely the variation in the average beam fluence for all shots. There are both similarities and differences between the two scenarios presented in Figs. 4(a)–4(b) and are discussed next.

First, let us discuss the general trends that apply to both experiments. The probability of growth is strongly dependent on site size and fluence, i.e., increasing for larger sites and higher fluences. In particular, size effects dominate the growth behavior of sites with diameters up to 50 μm, i.e., the rate at which the probability increases from 0 to 1 (slope) is very distinct for the first three size bins. To better quantify these size effects, we use a 4-parameter logistic function to fit the probability of growth data vs. fluence for each individual size bin (solid and dashed curves in Figs. 4(a)–4(b), respectively). For reasons noted above, the fits to the data for the first and last two size bins are only qualitative at this point pending future experiments. The logistic function depicts a sigmoid curve and is widely used for growth modeling (for example, dose-response in pharmacology/chemistry) as follows:

Secondly, the effects of laser exposure history can be clearly seen by comparing the shape of the probability vs. fluence curves for the same size bins from Figs. 4(a) and 4(b). Specifically, the ramp-up fluence scenario (Fig. 4(b)) resulted in a systematic decrease in the slope of the sigmoid curves, for all size bins up to 70 μm, compared to those observed from the fixed fluence scenario (Fig. 4(a)). In other words, there is a “conditioning” effect for small sites manifested as a reduction in their likelihood to grow at higher fluences following pre-exposure to lower fluences. To better illustrate this effect, the logistic fitting curves corresponding to 20–30 μm and 30–50 μm sites from Figs. 4(a)–4(b) are overlaid in Fig. 5(a) (solid and dashed lines, respectively). Furthermore, in Fig. 5(b) we plot the fluence at which the probability of growth is equal to 0.5 (x₀ fitting parameter) from all curves in Figs. 4(a)–4(b) as a function of size bin (the center value) and laser exposure history. The error bars represent the uncertainty associated with the logistic curve fits, larger for the extreme size bins (more data needed). It is important to note that the results in Fig. 5(b) both retain the ∼5 J/cm² “growth threshold” which is well accepted for large sites (see for example [20]) and are in excellent agreement with our previous growth studies which reported the effect of site size on probability and rate of growth [9, 33]. The trends illustrated in Fig. 5 also demonstrate that the conditioning effect (for damage initiation) long known to be present in KDP [36, 37] is also present to a lesser extent in SiO₂ (for damage growth), though the fundamental mechanisms associated with these processes are almost certainly very distinct.

 

Fig. 5 Comparison of (a) the probability of growth vs. fluence for 20–30 μm and 30–50 μm size bins and (b) the fluence at which the probability of growth is equal to 0.5, x₀, vs. damage site size (bin center) for two laser exposure scenarios, fixed fluence (solid lines) and ramp-up fluence (dashed lines), respectively. The logistic curves and fitting parameters are based on the results shown in Figs. 4(a)–4(b).

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4. Summary

We have shown here that the onset of damage growth is not governed by a threshold, but is probabilistic in nature and depends both on the current size of a damage site and the laser fluence to which it is exposed. Furthermore, we have shown that the history of laser exposure also influences the behavior of individual sites. This taken with the shot to shot independence to the probability of growth clearly indicates that internal features of the damage sites evolve with each laser exposure even if such changes do not manifest as observable changes to the site diameter. Further investigation of the evolution of the internals of non-growing damage sites under laser irradiation is ongoing. These results have important implications for the development of predictive models on laser damage evolution from initiation to a preset size imposed by various damage repair protocols. It is likely that detailed knowledge of how damage sites respond to laser exposure as a function of size can be used in adding efforts to post-process optics in order to make them more robust.