STEP 1: Damage Probability Curve
In this subsection, we will explain how damage probability curve is constructed from the measurement data.
Visualize the surface of an optical sample, which is divided into a virtual matrix of sites to be tested (see fig. below). Based on the testing procedure each test site is exposed to one (1on1 test) [ref. 1] or multiple laser pulses (Son1 test) [ref. 2]. We will begin with the 1on1 case and then explain how the same principles are applied in the Son1 case.
For now let us say, that a constant number of sites n is irradiated by a single laser pulse at a constant laser fluence (see fig. above). Spatial and temporal characteristics of the applied laser pulses are measured prior to the test. After exposure, each site is observed by Nomarski microscope in order to inspect for the visual change (damage) of the surface or volume. Consequently, each tested site is given a status: damaged (dark blue) or nondamaged (light blue). Then the probability of damage at specified fluence level P_{i} is calculated as the ratio of damaged k and total irradiated sites n:
P_{i} = k_{i}/n_{i}
The same procedure is repeated at increased laser fluence. Following this algorithm, the whole optical element surface is exposed. The obtained probabilities of damage are then plotted as a function of averaged fluence. For instance in the example above, at fluence F_{1} there were no damage points, thus damage probability is 0. Respectively, damage probabilities at fluencies F_{2}, F_{3}, F_{4} and F_{5} are 0.4 (2/5), 0.6 (3/5), 0.8 (4/5) and 1 (5/5).
This is the socalled damage probability curve. Typically it is shown in reversed axis (see fig. below on the right).
A measurement result is only complete if it is accompanied by a statement of the uncertainty in the measurement. Thus, for each measured point, fluence (horizontal axis) and damage probability (vertical axis) uncertainty interval should be established. Pulsetopulse fluence variation is assumed to be of the normal distribution (see fig. below on the left). Thus the horizontal – fluence uncertainty of the measurement is evaluated as: F_{mean }+ (2 · σ_{Fmean)} 

Since the amplitude of fluctuations is directly proportional to the average laser fluence F_{mean}, the standard deviation varies in absolute scale for increasing fluence. In fact, three different factors affect fluence (horizontal) uncertainty: laser pulse energy, the beam diameter and the shape variation on a pulsetopulse basis. Pulsetopulse energy variation is known to be of a normal distribution. Other effects, however, can be treated in a similar manner. 
In order to understand the uncertainty of measured damage probability (vertical uncertainty), you should treat damage probability just as any another variable you want to measure (for instance, weight or length). The question that we want to answer is: what was a probability of measuring the damage probability value, which we have had recorded. Just as what is the accuracy of any other measured variable. To know that, you have to know the probability density function (PDF) of your variable – PDF_{pest.}, where p_{est.} – is the probability estimated from the measurement.
Basically, LIDT measurements might be considered as a Bernoulli experiment with possible “1 – damage” and “0 – no damage” result as an outcome. The probability to receive exactly k successful outcomes after n trials, assuming the probability of success p, is defined by the binomial PDF:
PDF_{pest.(kn, p)} = C(n, k)·p^{k}·(1p)^{nk}
where C(n, k) is a binomial coefficient.
In reality, p is never known prior to the measurement. If it was, there would be no need to do the test. All we know is probability estimated p_{est.} after the measurement which is not necessarily equal to the p – the true value. However, we can reverse the question and ask which binomial PDF when p varies from 0 to 1 is most likely to explain the p_{est.} (see fig. below on the left). By reversing a question we are applying principles of the statistical technique known as maximum likelihood [ref. 3].
In this particular case, PDF is always maximized at p = p_{est.}. This can be proven numerically. Variation of p will draw a distribution that depends on the numbers of damaged and total tested sites, k and n. It shows how probable the tested p_{est.} value is. Integration over 95% area of the defined distribution can be used for uncertainty interval calculation. This could be done for each damage probability data point obtained at different fluence (see fig. above right). The distributions obtained following the binomial formula are asymmetric when the damage probability is close to 0 or 1. Asymmetric distributions suit well the physical damage probability model. No negative or higher than 1 damage probability values are possible. Firstly, such interpretation binomial data was published in the work of A. Hildenbrand and coworkers [ref. 4].
FUN to know:
Most of the early LIDT measurements were reported without any evaluation of uncertainty. Characterization of statistical uncertainty puzzled laserinduced damage scientific society for quite while. The wide use of the statistical uncertainty started only in the last decade since 2008.
Son1 case
Most optical elements are intended to be exposed at laser shots and different laser fluencies. Herewith we will briefly discuss how damage probability curve is constructed in the Son1 test [ref. 2] testing regime.
Just as in the 1on1 case, in a multipulse irradiation regime, the constant amount of sites is exposed with a constant laser fluence. However, this time each site is irradiated with a burst of laser pulses (see fig. below on the left). The number of pulses S can vary for different applications. As a rule of thumb, maximum of 1000 pulses are often chosen to irradiate every site at low repetition rate regime (1100 Hz). If the optical element is damaged in the middle of the laser pulse burst, the exposure should be stopped immediately in order to prevent further destruction and contamination of the sample by ablation products. Most of the LIDT test stations have online damage detection systems. Typically those systems are based on monitoring the backscattered light. Whenever the damage occurs, the signal of backscatter light changes: it increases or decreases. So, after exposure, at least three parameters are registered for each tested site: fluence, the number of pulses before damage and the status (damage or nondamaged). Then the fluence is increased and the sequence is repeated for other sites. Following this algorithm, all sites of the test matrix are exposed. The typical Son1 test results are shown in the fig below on the right.
The damage probability is constructed for the socalled pulse classes. Pulse class is simply a number of pulses at which you want to know probability for damage to occur. Mostly damage probability is reported at 1, 10, 100 and 1000 pulses. Measurement data are classified into laser pulse classes based on two conditions (see fig below on the right):
 all test sites which are damaged at pulse number higher than particularly selected class are considered to be nondamaged;
 all test sites which are damaged at pulse number of interest or before are considered to be damaged for the class of interest.
Then for each fluence level F_{i}, there is a set of damaged and nondamaged points. This set is used to calculate damage probability in the same war as it was described for the 1on1 case (see fig below on the left). Damage probability curves are used to extract laserinduced damage threshold value. Read more on chapter 3.2 LIDT evaluation.
Reference
[1] Lasers and laserrelated equipment – test methods for laserinduced damage threshold Part 1: Definitions and general principles (ISO 212541:2011).
[2] Lasers and laserrelated equipment – test methods for laserinduced damage threshold Part 2: Threshold determination (ISO 212541:2011).
[3] I. J. Myung, Tutorial on maximum likelihood estimation, J. Math. Psychol. 47, 90–100 (2003).
[4] A. Hildenbrand, F. R. Wagner, H. Akhouayri, J. Y. Natoli, and M. Commandré, Accurate metrology for laser damage measurements in nonlinear crystals, Opt. Eng. 47(8), 083603 (2008).
If you are interested in laserinduced damage phenomena we also highly recommend to read:
[4] D. Ristau, LaserInduced Damage in Optical Materials (CRC press, Taylor &Francis Group, Florida, 2014).
[5] R. M. Wood, Laser damage in optical materials (A. Hilger, Bristol, 1986)