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Surface laser damage limits the lifetime of optics for systems guiding high fluence pulses, particularly damage in silica optics used for inertial confinement fusion-class lasers (nanosecond-scale high energy pulses at 355 nm/3.5 eV). The density of damage precursors at low fluence has been measured using large beams (1-3 cm); higher fluences cannot be measured easily since the high density of resulting damage initiation sites results in clustering. We developed automated experiments and analysis that allow us to damage test thousands of sites with small beams (10-30 µm), and automatically image the test sites to determine if laser damage occurred. We developed an analysis method that provides a rigorous connection between these small beam damage test results of damage probability versus laser pulse energy and the large beam damage results of damage precursor densities versus fluence. We find that for uncoated and coated fused silica samples, the distribution of precursors nearly flattens at very high fluences, up to 150 J/cm2, providing important constraints on the physical distribution and nature of these precursors.
©2012 Optical Society of America
1. Introduction
Surface laser damage limits the lifetime of optics for systems guiding high fluence pulses, but the underlying nature of the absorbers which lead to damage (damage precursors) is just beginning to be clarified. Since laser-induced surface damage of silica optics in inertial confinement fusion-class lasers leads to costly repairs and/or refinishing, considerable efforts have been exerted in preventing nanosecond-scale laser damage and understanding its origin in systems like this. Most of the laser damage precursors at low fluence (3.5 eV, <20 J/cm2) were found to be associated with fractures and scratches that occur during polishing and finishing [1–4]. A prevention strategy for eliminating these “low-fluence” precursors was developed, but damage initiation at higher fluences appears to be due to another class of precursor which exhibits a threshold behavior near 15J/cm2 for 3ns, 355nm laser pulses. In order to understand the laser damage precursors in this regime of higher fluence and to test the effects of processing changes, we developed a methodology to determine the density of nanosecond-scale (3.5 ns Gaussian beam) extrinsic laser damage precursors at a wide range of UV (3.5eV) laser fluences extending up to 150 J/cm2 for high quality fused silica optical surfaces. A deeper understanding of laser damage processes will be of increasing importance for systems under consideration for energy production [5, 6] or for extending the lifetime or performance of compact high fluence sources. Here, we determine the cumulative probability distribution of precursors as a function of laser fluence on high quality silica surfaces, and show that this distribution does not increase rapidly above the threshold observed between10-20 J/cm2 [3]. All results here apply to 355nm, 3.5ns Gaussian laser pulses. The scaling of damage initiation with pulse shape has been described in [7, 8].
For high energy ns-scale laser pulses, it has proven beneficial to characterize the damage performance of optical surfaces using a density of laser damage precursors rather than a single “damage threshold” [9–13]. This is because the laser damage performance varies significantly over the surface of the sample, and is often due to surface conditions, fractures, scratches, defects or contaminants rather than intrinsic properties of the material. The damage performance is characterized by a cumulative density of damage precursors ρ(Φ), which is the areal density of sites that will cause laser damage up to a fluence Φ. The ρ(Φ) developed this way reveals the property of the material surface and can be compared quantitatively with experiments using large beams and with smaller beams; otherwise, quoting damage thresholds for an experiment on surface damage with nanosecond-scale pulses is highly dependent on the experimental configuration, especially beam size.
Previous studies of the surface laser damage performance for nanosecond-scale lasers have focused on fluences near the average fluence expected for shots performed on the National Ignition Facility and Laser MegaJoule [9–18]. This is clearly the most important fluence range for this application. However, damage performance at higher fluences is also important under these conditions, due to regions where large surface flaws or surface features cause modulations in the beam that may lead to much higher local fluences downstream on the optical path. Additionally, improvements in damage performance at various fluences often are correlated, so that finding methods to improve damage performance at high fluence may also improve damage performance at lower fluences.
In order to measure the laser damage performance over many orders of magnitude, it is necessary to perform measurements for different experimental conditions or even different experimental setups. Large beam studies have proven more effective at fluences with lower densities of precursors, since the lower density of precursors requires larger areas to obtain significant statistics [11, 19]. Studies with small beams in the millimeter to sub-millimeter range have been useful in probing similar fluences to these large beam studies [10, 12–18].
In order to probe damage performance at higher fluences and damage densities, it is necessary to damage test with beams sizes much smaller than typically performed for this combination of pulse width, wavelength, and fluence (Fig. 1 ). For example, in experiments with 1064 nm laser pulses 5 μm beams were used, showing that the surface damage threshold can reach the intrinsic, bulk threshold in certain areas and under certain surface polishing conditions [20]. Although damage performance at lower fluences is best determined using large aperture beams with fluence registration [9, 11, 19], at high fluences the combination of damage site size and density leads to overlap of multiple damage sites (clumping as described in [21]), obscuring measurements of the density of damage precursors [14]. Smaller damage beams reveal sites that will not damage until much higher fluences. In addition, non-focusing beams at high fluence have significant self-focusing that invalidate the damage testing experiments [22]. In order to probe these higher fluence regions, we have developed an automated damage testing station using 3.5 ns laser pulses focused with fast lenses (100 mm, 200 mm and 300 mm) to significantly smaller areas than typically used for fused silica surfaces. We also develop a rigorous fitting procedure that accurately extracts ρ(Φ) using small beam damage testing.
Fig. 1 (a) Probing the damage performance of optical surfaces with small laser beams isolates the precursors that damage at various fluences. For the densities encountered, laser damage sites will overlap for large beam studies. (b) Large beam studies measure the cumulative density of damage precursors ρ(Φ) up to 25-30 J/cm2. In these experiments we want to determine if there is either a previously unknown surface limitation to damage performance or what populations of precursors are present in this fluence range. If there is a fixed damage threshold for fused silica surfaces, we expect a rapid increase in ρ(Φ) as shown by the extrapolated black curve. If we find a shallow increase, the precursor distribution must be dominated by extrinsic, non-uniform features such as defects or contaminants.
By measuring the behavior of ρ(Φ) at higher fluences (Fig. 1(b)), we determine whether the laser damage precursors are of rapidly increasing density or even intrinsic to surfaces (black line) or they are more isolated (red line). The automation employed allows these experiments to be performed quickly, which will also enable the monitoring of changes in damage performance at high fluence due to improvements in surface preparation processes.
2. Materials and methods
2.1 Fused silica sample preparation
Fused silica plates (50 mm in diameter by 10 mm thick; Corning 7980 (Corning Inc., Canton, NY) or Heraeus Suprasil 314 (Heraeus Quartz America, Buford, GA) were prepared as described for the advance mitigation process (AMP) in Ref [3]. Briefly, each sample undergoes a 28 μm buffered oxide etch that etches open fractures caused during polishing; the etch removes a defect layer which leads to laser damage at lower fluences (< 15J/cm2). In the experimental results section below, Protocol 1 is as described in Ref [3]. Protocol 2 uses a 60 °C Nitric acid rinse intended to clean the surface of contaminants. Protocol 3 is the same as Protocol 1, except that the samples are dip-coated with a silica sol-gel quarter-wave anti-reflection coating for 351 nm: they are coated using the process in Ref [23]. using colloidal silica sol prepared according to [24].
Selected sol-gel coated samples were intentionally contaminated with organic chemicals in an effort to understand the effects of such contamination on the damage performance of coated optics. Silica sol-gel anti-reflective coatings, which have very large surface areas, are known to be extremely sensitive to organic contaminants. When contaminants are adsorbed onto the coatings the effective refractive index of the coatings changes, thereby increasing the reflectivity and reducing the transmission through the optic. For our experiments, we placed a silica sol-gel coated disk of silica optical glass in a vacuum chamber at a pressure of 10−4 torr or less with 1 gram of squalane hydrocarbon lubricant for 100 hours. After this exposure, the transmission through the sample at 351 nm decreased from 99.7% before to 93.3% after, a change of 6.4%, indicating that the coating was saturated with the contaminant. Analysis of residues rinsed from the optic using methylene chloride solvent identified the contaminant as squalane.
2.2 Automated damage testing
We developed a damage testing station that automates many of the procedures for damage testing. It is similar to the experimental setup described in Ref [14]. The primary differences are that we do not monitor the laser beam profile with every pulse, we use faster lenses (10, 20, and 30 cm), and we use a removable microscope system to perform the post mortem damage analysis. The pulsed laser (Ekspla NL132, Vilnius, Lithuania) is allowed to run at 10Hz (351 nm, 2.9 ns pulse width, FWHM, 3.5 ns equivalent Gaussian as computed by EquivInit [7, 8]). Pulses are selected by synchronizing a shutter (SH05 shutter controlled by SC10 shutter controller, Thorlabs, Newton, New Jersey) to the laser pulses, allowing only a specified number of laser pulses to pass through at each site. A counter-timer on a computer DAQ card (PCI-6602, National Instruments, Austin, Texas) is triggered by the external trigger output of the laser, producing a synchronized pulse controlling the shutter that is of sufficient duration to allow the specified number of pulses to pass through.
The pulse energy is selected using a λ/2 waveplate in a motorized rotation stage (495 ACC, Newport Corporation, Irvine, California) controlled by the computer through a motion controller (ESP301 Newport Corporation, Irvine, California). A Brewster angle polarizer dumps the majority of the beam energy to a beam dump, while the fraction that is allowed to pass through is reflected, and passes through an iris (set to 2.8 mm diameter) to clean up the laser mode. A beam splitter is used to direct a fraction of the remaining beam to an energy meter (LabMaxTop, Coherent, Santa Clara, California) that is monitored by the computer. The readings of the energy meter recorded by the computer are synchronized by using the same counter-timer card as the shutter. In this case, re-triggerable pulses from the PCI-6602 are used that produce laser pulse triggers only when the shutter is open.
The energy readings for the energy meter were calibrated by comparing readings for the same laser pulses from a second LabMaxTop energy meter placed in front of the lens. The energies passing through the waveplate/polarizer pair were calibrated by stepping through angles from 0 to 90 degrees from the minimum in 1 degree increments, measuring the energy meter readings for 20 pulses at each step.
Three removable plano-convex lenses of focal lengths 100 mm, 200 mm and 300 mm (UV grade fused silica, LA4380, LA4102, and LA4579, Thorlabs, Newton, New Jersey) are placed on stages that allow independent focusing of the three lenses. The sample is mounted on a xyz stage where the x and y axes (transverse to the optical axis) are motorized (LTA-HS, Newport Corporation, Irvine, California), and controlled by the same ESP301 as the rotational mount. The sites tested for damage susceptibility are placed in a grid pattern (Fig. 2(b) ).
Fig. 2 (a) Experimental configuration for automated small beam damage testing and detection. (b) Damage testing pattern. Fiducials are placed along the x and y axes. The damage test proceeds by rows with increasing pulse energy. (c) After the damage test is performed, the laser shutter is closed, the microscope is put into place, and each site is automatically imaged to determine if damage occurred.
We ensured that the rear surface of the silica sample was at the focus of the beam by first focusing the removable microscope system on a sample coated with aluminum on half of the surface. The illumination was provided by the 351 nm laser pulses (to eliminate issues with chromatic aberration). The lenses were then positioned to provide the smallest beam at this focus. The beam profile was then measured using a beam profiling camera (BeamView, Coherent, Santa Clara, California); the beam was magnified by the microscope used for automated imaging. The pixel size was calibrated using the motorized stage.
2.3 Automated imaging
Before removing the sample that is damage tested, the imaging microscope is put into place after the shutter is closed. An image of each site is acquired (Fig. 2(c)) with off axis illumination (in a pseudo-dark field mode). The objective used was a 20X, 0.42NA M Plan Apo (Mitutoyo, Kawasaki, Japan) with a long working distance, and the images were acquired by a CCD camera (Watec WAT-902C, Middletown, New York). During the data analysis, the position of the laser within the images is determined by averaging over all of the images. A region of interest is selected; if more than 100 pixels have values of 100 or more (8 bit frame grabber), damage is tabulated as having occurred. This information is recorded with the recorded pulse energy.
3. Theory and data analysis
3.1 Estimations of self-focusing
In laser beams that reach high intensities, self-focusing may occur that changes the beam focus parameters [25, 26]. In order to determine the extent of self-focusing in our sample testing configuration, we solved the paraxial scalar wave equation including self-focusing with non-linear coefficient γ,
The value of γ used was taken from Ref [27].The beam mode is cleaned up by placing an iris (radius of 1.4 mm) before the focusing lens (Fig. 2(a)). The beam focus is in the far-field, so that we can treat the focusing beam as a Gaussian beam. In this approximation, the far-field of a circular aperture of radius r is approximated as a Gaussian with intensity 1/e radius of r/1.915 (Fig. 3(a) ). Using these calculations, Fig. 3(b) shows how much the beam is intensified as a function of pulse energy. The 30 cm focus has stronger self-focusing due to the longer Rayleigh range of this beam with a slower focus. The maximum pulse energies for which data are obtained are 1.4 mJ for the 30 cm focus, and 0.4 mJ for the 10 cm focus. Under these conditions, the self-focusing decreases the area by 3% or less. These effects are insignificant in the energy ranges we measure, and will not be considered further.
Fig. 3 (a) Comparison of radial distribution of circular distribution of radius 1.4 mm and Gaussian with intensity 1/e radius of a/1.915. (b) Intensification expected as a function of pulse energy for beams focused with the 10 cm (red) and 30 cm (blue) focusing lenses. Since the pulse energies used were below 2 mJ for the 30 cm lens, and below 1 mJ for the 10 cm lens, we do not expect significant intensification or self focusing in our measurements.
3.2 Fitting model
The data resulting from these measurements consist of a series of damage tested areas which experienced varying pulse energies. Each site is imaged to determine if damage occurred. The data are then the probability that a site damages as a function of pulse energy. We analyze the data by extracting ρ(Φ), the cumulative density of defects that will damage up to fluence Φ. We perform this procedure using a formula generalized from Refs [10, 15],
P(Epulse) is the probability of causing damage with a laser pulse of energy Epulse and fluence distribution at the surface of Φ(x,y). By changing variables, dxdy = dA = (dA/dΦ)dΦ, we can rewrite Eq. (2) asThe function dA/dΦ is determined from the beam profile images (shown in Fig. 4 ) scaled by the measured pulse energy Epulse to obtain the fluence distribution.Fig. 4 Beam Profiles are used to convert pulse energy to dA/dΦ, the amount of area covered by a certain fluence. Green squares are 50 μm for each image. (a) Beam profile for f = 300 mm with 50 μm box shown for scale. (b) Beam profile for f = 100 mm. (c) Calculation of dA/dΦ from images in (a) and (b).
We describe ρ(Φ) as a piecewise continuous series of exponentially growing fluence ranges (Fig. 5 ),
By recursively calculating ρ(Φi) as a function of ρ(Φi-1), ρ(Φi) = ρ(Φi-1) exp [ri(Φi -Φi-1)], we obtainIn using this model during a fit, we fix the transition positions Φi, and allow the rates ri to vary in the fitting procedure (described in the Appendix). The choice of using a piecewise exponential model was suggested by measurements [3] that show exponentially growing ρ(Φ) for lower fluence. We needed to add flexibility to the model to ensure that any changes in shape of ρ(Φ) (e.g., sub-exponential behavior over longer fluence ranges) could be captured during the fitting procedure; hence, given a sufficient number of nodes (Φi), the model fit can be made very general. Also, by restricting the values of the exponential growth rates ri to be strictly positive, the model Eq. (5) is strictly increasing, as is mathematically required for a cumulative distribution.Fig. 5 The model for the cumulative distribution of the laser damage precursors is a piecewise continuous, exponentially growing function. We restrict the values of different exponential growth rates to be strictly positive. The fluence positions of transitions between ri are fixed during the fit.
3.3 Obtaining error estimates in the fits using a bootstrapping procedure
We use a bootstrapping procedure to estimate errors in the fitted model [28], which allows us to rigorously estimate the errors in the fitted ρ(Φ) in a straightforward manner without performing undue approximations. This procedure was used because of the non-trivial relationship between the data acquired (damage probability as a function of pulse energy) and the extracted ρ(Φ) curves. Multiple instances of similar data sets are simulated, and the same fitting procedure was applied to each instance. The distribution in fitted ρ(Φ) over all of the instances is the error estimate for the fit of the actual data subject to the constraints inherent in the model.
The procedure begins by fitting the original data sets as described in the previous section.
Second, a series of new, simulated data sets are generated where the number of bins and their spacing was kept the same as the original data set. The value for each bin is simulated from a pseudo-random number generator for the binomial distribution, where the number of damage tests ni is taken directly from the original data set, and the probability of damage pi is taken from the fit of the original data. An alternative method whereby pi is estimated from the original data as the number of damage events divided by the number of damage tests was used. This other method did not produce substantially different results.
Third, each simulated instance of the data was fitted to the same model, producing an instance of data with the same statistical characteristics as the original data set. In order to accurately probe the possible solution space, initial fitting parameters were chosen to have wide variation. We randomly chose new fitting parameters based on the range in fitted parameters from the original data set. If the same initial parameters were used for every bootstrap instance, we found that parameters that were unimportant to the fit were often unchanged, even though true uncertainty at that point was much higher. After fitting each instance, the uncertainty of the original extracted ρ(Φ) was computed as a standard deviation of bootstrap instances (Fig. 6 ).
Fig. 6 (a) The uncertainty in the fitted ρ(Φ) is estimated as a bootstrapping procedure. The original extracted ρ(Φ) is shown in black, and the extracted ρ(Φ) of 11 bootstrap instances are shown as well. At each point, the standard deviation was calculated to estimated the uncertainty in the values. (b) Error bars are calculated as the standard deviation of all bootstrap instances.
One source of error that is not accounted for using this procedure is slow spatial variations in the density of damage precursors ρ(Φ). This is sometimes observed as a change in the damage probability within the rows of damage test with the same pulse energy. If rows of increasing pulse energy are coincident with regions of lower precursor density, then the measurement can lead to anomalous results with decreasing damage probability as fluence increases. We have not observed this hypothetical result, but we have observed increased variability. Repeated measurements on the same or a similar sample is used to remove uncertainties resulting from observed spatial variations.
3. Simulation results
In order to show that our methodology can reliably extract ρ(Φ) for general precursor distributions, we simulated data sets with various functional forms, and fit them using the extraction method described above. The simulations followed the same damage test statistics as used for the experimental data in the next section. The results are in Fig. 7 . The extraction methodology was able to accurately reproduce the various functional forms used for ρ(Φ). The cubic and exponential forms are reproduced well (Fig. 7(a)), as are the forms with exponential increases interrupted by plateaus (Fig. 7(b)). Note that the sharp corners are somewhat rounded off in the fitted form for Fig. 7(b), suggesting the limits on resolution of sharp features in ρ(Φ).
Fig. 7 Extracted ρ(Φ) for various functional forms. (a) Cubic ρ(Φ) and fit are shown in red. Exponential ρ(Φ) and fit are shown in black. (b) ρ(Φ) with exponential increases interrupted by plateaus are shown in red and black. Corresponding fits are shown by symbols.
The range of validity of the fit, as determined by the error bars and distance from actual ρ(Φ), is more restrictive along the y-axis, the density of precursors. The range over which the fits are valid is from about 3x103 and 1x106 cm−2. This is primarily because the area tested determines the density of precursors that is able to be probed. If we consider the top 10% of the beam energy (where most damage occurs), the area tested for each fluence bin ranged from ~10−4 cm2 for larger fluences (above) 50 J/cm2 to ~10−3 cm2 for fluences lower than 50 J/cm2. With a precursor density of 3x103 cm−2, the lower area would lead to ~3 damage sites, which is a reasonable lower limit for providing statistically meaningful information.
4. Experimental results
We demonstrated this procedure for extracting damage precursor density on samples prepared using the procedures described in Ref [3]. (Fig. 8 ). Three data sets were acquired using the automated laser damage testing station with different beam sizes (Fig. 8(a)). The focal lengths of the lenses used are listed, rather than the beam size. The raw data of the experiment consists of a series of three curves showing the probability of laser damage versus measured pulse energy in mJ. The simplest procedure is to convert the pulse energy into the peak fluence of each pulse, which is calculated as the fluence of the top 10% of the beam (Fig. 8(b)) [10]. The observation that the beams focused with smaller focal lengths have higher damage thresholds shows that these are not intrinsic silica damage. The damage results from non-uniform surface flaws or contamination.
Fig. 8 Example of extraction of precursor damage density by combining damage test results using lenses with three different focal lengths. (a) The raw data of the experiment consists of a series of three curves showing the probability of laser damage versus measured pulse energy in mJ. (b) Data from part A, but with the pulse energy is converted to the fluence of the top 10% of the beam. (c) The distribution of the laser damage precursors ρ(Φ) as a function of fluence (black line). These results can be directly compared to the large beam results at lower fluence (cyan line). (d) ρ(Φ) extracted for each focal length separately plotted with the ρ(Φ) extracted when all information is taken into account.
The simple conversion to fluence does not take into account the variations in fluence over the whole beam, and is not ideal for extracting ρ(Φ). By using the rigorous procedure described in previous sections, we extract ρ(Φ), the distribution of the laser damage precursors, as a function of fluence (black line, Fig. 8(c)). These results can be directly compared to the large beam results at lower fluence (cyan line, Fig. 8(c)) [19]. The agreement between the two measurements from different laboratories and data analysis techniques gives confidence in the resulting curves, and eliminates any effects of the beam sizes on the damage test results.
Extracting ρ(Φ) for each focal length separately shows where each data set contributes to the final result, and shows that the results agree with each other before combining the results (Fig. 8(d)). The larger beam sizes contribute data toward lower fluences, since the larger area tested with each laser pulse is more likely to capture the regions with low damage probability. The smaller beam sizes contribute to the higher fluence data. By combining all three measurements, it is possible to extract ρ(Φ) to well over 100 J/cm2.
Sample processing dramatically affects the damage performance of the fused silica surfaces (Fig. 9 ). Figure 9(a) shows ρ(Φ) extracted for uncoated fused silica parts. The first set (Samples 1 and 2) are prepared using the advanced mitigation process (AMP) as described in Ref [3]. (Protocol 1). Sample 3 is prepared only with a 60 °C Nitric acid rinse intended to clean the surface of contaminants. The precursor density changes dramatically under these conditions.
Fig. 9 (a) Extracted ρ(Φ) for uncoated silica samples. Samples 1 and 2 are samples undergoing the full treatment listed in Ref [3]. (b) The damage performance samples with applied antireflective silica sol-gel coatings are able to be similar to that of uncoated parts. Sample 4 was processed similarly to sample 1, except the sol-gel coating was applied. Sample 5 was the same as Sample 4, except a squalane hydrocarbon contamination which impairs the antireflective properties of the coating was deliberately added.
Samples processed using AMP and then coated with silica sol-gel antireflective coatings can have similar damage performance to uncoated parts up to the levels observed here (Samples 4 and 5 in Fig. 9(b); uncoated Sample 1 is shown for comparison). Prior to damage testing, Sample 5 underwent a contamination process whereby a volatile compound (squalane, see Materials and Methods) is placed in a vacuum chamber with the part, allowing the voids in the coating to be filled. The coatings antireflective properties are destroyed, and we damage tested the sample to determine if damage performance degraded as well. Interestingly, no significant change in damage performance was observed.
5. Discussion
We have developed an automated damage testing station in combination with a rigorous data fitting procedure that extracts the precursor damage density as a function of fluence over a large range of fluences. This allows for a self-consistent way to associate small-beam damage test measurements which measure the probability of damage vs. fluence with large-beam damage test measurements which measure the density of damage precursors. We have shown that the damage test results match in the region of overlap for these measurements. This experimental procedure and analysis can be applied to a wide variety of optical materials to determine their damage behavior over wide fluence ranges.
We have shown that this density of surface damage precursors in high quality silica parts exhibits a large increase in density near 20-30 J/cm2, but does not increase more dramatically beyond this level. This has important implications in the identification of the damage precursors for high fluence damage. These results imply that the damage precursors are localized even up to very high fluences. In using methodologies to identify damage precursors, this will allow us to determine if any observed features have the appropriate areal density to be identified as candidates for laser damage precursors.
Similar performance is able to be achieved with samples coated with antireflective, silica sol-gel coatings and uncoated samples. This indicates that there are no intrinsic properties of the sol-gel coatings which prevent their use for higher damage threshold optics. Contamination of the coating with this organic compound did not have a significant effect on the damage performance at high fluence. Clearly, not all contaminants or manners of contamination will cause a decrease in damage threshold.
6. Appendix: fitting using the maximum likelihood estimator for the binomial distribution
As discussed in Ref [29], it is possible to modify the L-M fitting procedure to minimize the MLE for Poisson deviates rather than the least squares measure. We have done the same for the binomial distribution, following the discussion in the supplementary note of Ref [29]. We replace the least squares calculation in the L-M routine with the MLE estimator for the binomial distribution, make corresponding changes in the gradients calculated, and achieve the desired procedure.
For a data set x = (x1,x2,…,xm) where each xi is the number of “successes” for bin i; xi is then non-negative integer. For each i, there is a binomial distribution with success probability pi, and ni trials. The observed data set x is randomly distributed around a mean value at each point ni pi,
For our data set, the various test fluences are grouped into bins i. Each ni represents the number of sites tested for damage within the fluence range for bin i, and the number of damage events observed is xi. The probability of damage at the fluence for bin i is pi. The goal is now to fit a model function f = (f1,f2,…,fn) that depends on parameters a = (a1,a2,…,am) in such a way that fi is as close as possible to pi.The principle of maximum likelihood asserts that the most likely model f is the one that maximizes the likelihood function,
The likelihood function is essentially the joint probability distribution for the entire data set. It is customary to divide by the maximum possible likelihood , and minimize twice the negative logarithm of this ratio in order to change the repeated products into sums. By minimizing this function, which we call , we obtain the maximum likelihood estimator (MLE) for the binomial distribution,This function is minimized to find the best fit. Note that each fi is required to be greater than 0 and less than 1; any fitting procedure must adjust a in such a way that f is contrained to these limits. Otherwise, the minimization will run into problems.We extended the L-M algorithm for use with the MLE for the binomial distribution in bins [Eq. (8)] following many of the arguments made in Marquardt’s original paper and in the book Numerical Recipes [30], and in our discussion of Poisson-distributed bins [29]. By calculated the gradients of the expression in Eq. (8), the L-M algorithm proceeds as previously described [29–31]. Improvements in the L-M algorithm have been implemented and extended to other maximum likelihood estimation problems, and may be used in this application as well, although we have not implemented them here [32].
Acknowledgments
We thank Wren Carr, Zhi Liao, Stavros Demos and Mary Norton for many useful discussions and for assistance with equipment loans and pulse length measurements. We also thank George Hampton and William Gourdin for their assistance in preparing the squalane contaminated sample. This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
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