1.

INTRODUCTION

In order to make meaningful realistic laser damage tests that do not require huge budgets one usually chooses the S-on-1 protocol.¹ The S-on-1 protocol uses a high number of pulses of constant fluence for the irradiation mimicking thus operation of a typical high power photonics setup or instrument, which is expected to work with constant performance over a long time (and thus a large number of laser pulses). Like the 1-on-1 protocol,² the S-on-1 protocol is based on the measurement of damage probabilities P at different levels of peak fluence F₀ in the beam. The measurement, or estimation, of a probability is always based on a number of statistically independent repetitions of the same experiment. In the case of damage probability measurements we irradiate n sufficiently spaced sites on a sample and decide after inspection under a microscope how many of them, say k, were damaged. Using the maximum-likelihood principle and the numbers n and k of this stochastic experiment with binary outcome, we can then deduce the best hypothesis for the damage probability P = k/n and its uncertainties ΔP⁺ and ΔP^- corresponding to a chosen level of confidence.³

For multi pulse damage tests like the S-on-1 tests we also need to choose how many pulses we want to use in conditions where no damage is detected. This maximum pulse number to use on a single test site, S, will depend on the aim of the experiment: (i) If one only wants to know if the material shows a fatigue effect, a relatively small number of pulses can be sufficient (some hundreds of pulses). (ii) If the test is used to make a validation or to acquire data for an extrapolation of the damage threshold at high pulse numbers, the necessary pulse numbers are much higher.

According to the ISO standard, the S-on-1 damage probability (for a maximum of S pulses per site) is given by the number of sites that damaged at pulse number S or before (k = k( N_d≤S)). The S-on-1 damage probability is thus the complement of the survival probability of the sample if S laser pulses are used.

Using an online damage detection system, one can check for the appearance of damage after each laser pulse and thus record the damaging shot numbers N_d for all damaged sites. The most frequently used online damage detection systems are based on imaging of the irradiated site [ref] or on diffusion measurements [ref] but any pump-probe style measurement giving a response that allows to identify the occurrence of damage may be used. The damaging shot numbers N_d are very information-rich data, as they allow to retrieve any N-on-1 damage probability curve with N ranging from 1 to S. One obtains thus a two dimensional data set P(F₀, N) that can be represented in different ways.

1.1.

Representations of S-on-1 laser damage data and definition of the fatigue defect

For example one can plot all N-on-1 damage curves P(F₀) with the parameter N changing between the curves (Figure 1a). This yields typically in a plot with many curves (N) each containing rather few points (Fluences).⁴

Figure 1:

2000-on-1 data in the bulk of KTP. (a) Full representation of the data set as 2000 P(F) curves. The lowest curve (black squares) is the 1-on-1 curve. (b) Reduced representation of the data set. Only the experimental damage thresholds T are shown as function of the maximum used pulse numbers N. The initial decrease of the T(N) curves indicates a strong fatigue effect. (16 J/cm² for 1-on-1; 9J/cm² for 50-on-1 (= 56% of the 1-on-1 value).

As this plot is usually difficult to read. One often finds a reduced representation that plots the N-on-1 damage thresholds T (the highest fluence for which still P = 0) as a function of the pulse number N (Figure 1b). It is this plot that defines the “fatigue effect”: if T(N) decreases, one says that there is a fatigue effect.⁵ The name of the effect has been chosen in analogy with mechanics, where a metallic work piece that is charged and uncharged repeatedly can break after many cycles, even when a load that is much smaller than the load it can withstand for a single loading/unloading cycle (Figure 2).

Figure 2:

Information on mechanical fatigue damage. (a) Mechanical fatigue damage in Steel.⁶ A chosen stress amplitude is applied and removed for a high number of cycles. Doing the first cycles a small crack is initiated. The crack then grows slowly until the work piece beaks suddenly. (b) Photograph of an aluminum part that was subject to fatigue failure.⁷

Two more, very helpful representations of S-on-1 data exist: (i) Plotting a P(N) curve for each fluence yields some curves with many points that can be used to evaluate if statistical fatigue is present and, if this is the case, one can extract the single-pulse damage probability p₁(F₀) by fitting this plot (Figure 3a). (ii) Marking each damaged site in the N_d - F₀ -plane yields a scatter plot that is a reduced representation of the data (it does not contain the undamaged sites), but that nevertheless allows to estimate which kind of fatigue is present in the sample and yields also information on simultaneous failure modes (Figure 3b).

Figure 3:

Two more representation of the same dataset as in Figure 1. (a) Full representation of the data set as 7 P(N)-curves, one for each Fluence. (b) Reduced representation of the data set. Only damaged sites are shown. Each damaged site is represented by a point in the N_d-F-plane. (Additionally data of a 10 000-on-1 test has been added.

1.2

Precursor encounter models for single pulse nanosecond laser damage

Nanosecond laser damage is generally admitted to be caused by fabrication defects in the optical material. The idea came up after studying the morphology of laser damages close to the threshold [ref laurent et Feit] and comparing the electric field strength at the laser damage threshold to the electric field strength at dielectric breakdown. The idea is now widely accepted since models that hypothesize the laser damage probability to be the probability to find a damage precursor in the high fluence region of the laser beam succeed to describe the dependency of 1-on-1 laser damage tests on the laser beam size.^8,9 Several variations of these “precursor encounter models” exist that differ mainly in the shape of the statistical distribution function g(T) describing the distribution of the precursor thresholds in the ensemble.^9–11 All of them consider a homogeneous distribution of the damage precursors in the host material and suppose a deterministic reaction (the occurrence of damage) if a precursor is irradiated at a fluence exceeding its threshold.

Equation 1 gives the general expression of P(F₀) for these models. A(T/F₀) is the surface area that is irradiated by a fluence exceeding the threshold T if the peak fluence of the beam is F₀. In particular, for a spatially Gaussian test beam one obtains:

With d the 1/e² diameter of the spatially Gaussian laser beam.

2.

MATERIAL MODIFICATION FATIGUE – THE ANALOGY TO MECHANICS

In mechanics, the mechanism causing the fatigue effect was readily discovered: During the first cycles a small crack forms and grows slowly until the work piece is weak enough to suddenly break during the last and damaging cycle. In laser damage, the most widely spread hypothesis follows a similar argumentation line.⁵ One suspects that the first pulses (the incubation pulses) cause cumulative material modifications that “weaken” the material up to a state where it can be broken by the last and damaging laser pulse. However, most of the time this hypothesis is proposed solely based on decreasing T(N), i.e. also in cases where no direct evidence of cumulative material modifications during the incubation pulses is presented. Even if experimental evidence for cumulative material modifications is provided,¹² the precise physical mechanisms that finally lead to damage are not easy to determine in the case of laser induced damage and determining them usually needs long and detailed investigations. Two groups of references should be mentioned in this context (refs^13,14 and references therein), but others have also been reviewed.⁵

3.

MODELS FOR FATIGUE LASER INDUCED DAMAGE

The successful single-pulse damage models for the nanosecond regime encourage us to model the optical material under test as a host material with homogeneously distributed damage precursors. We will ground the following analysis of possible models for fatigue laser induced damage on this image of the optical material.

3.3

Statistical fatigue

However not all experiments for which a fatigue effect was observed can be understood by cumulative laser-induced material modifications (Figure 4). The presence of a strong fatigue effect at 1064 nm and the absence of fatigue at 532 nm or 355 nm would mean that the material modification are caused by IR photons but not by 532nm-photons or 355nm-photons, which is difficult to conceive.

Figure 4:

T(N) data obtained at different test wavelengths in the bulk of nonlinear optical crystals. (a) KTP and RTP at 1064 nm and 532 nm. (b) LBO at 1064 nm, 532 nm and 355 nm. In all three materials the fatigue effects disappears at shorter wavelength which is contradictory to light-induced, cumulative material modifications causing the fatigue effect.

The situation can be understood if one assumes that there exists a single pulse damage probability p₁ at 1064 nm, and that the interaction becomes deterministic for shorter wavelengths. In case of statistical interaction, the single pulse damage probability p₁ is independent of the pulse number N, meaning that there are no material modifications. An S-on-1 damage experiment can then be understood as statistically independent resampling and the P(N) curves can be modeled by the simple formula P(N) = 1 – (1 – p₁)^N (Figure 5a).²⁶

Figure 5:

P(N) data obtained in the bulk of KTP and the corresponding fits using the statistical fatigue model. (a) Measurement with a multiple longitudinal mode (MLM) laser. (b) Measurement with a single longitudinal mode (SLM) laser.

The stable single pulse damage probability may possibly be caused by the laser statistics of multi longitudinal mode (MLM) lasers or it may be caused by the light matter interaction itself.²⁷ Experimental data with validated single longitudinal mode (SLM) operation (each pulse has to be SLM) is still a bit rare (Figure 5b), but statistical fatigue seems to be very frequent for IR experiments.^26–28 By the way, this statistical model has first been proposed in the early days of laser damage research by Bass et al.²⁹ and has been abandoned later on (for a discussion see refs^4,5).

In terms of defects, the constant single pulse damage probability can be understood as a consequence of the light-induced formation of defects that may cause damage with a certain probability, but that are of short lifetime, so that the material is indistinguishable from the pristine material before the next laser pulse.

4.

DISTINGUISHING THE DIFFERENT TYPES OF FATIGUE

The P(N)-curves for statistical fatigue and material modification fatigue have completely different shapes, so that it is simple to distinguish the two processes by observing the P(N) data. For statistical fatigue, the P(N)-curves start steep and flatten gradually (Figure 6b). For material modification fatigue caused by an ideal laser (and neglecting any single pulse damage), the damage probability is zero as long as the material modifications did not reach the critical point. For a perfectly stable laser and material modifications that are operated in the host material, this critical modification will be reached after the same number of pulses for all tested sites (critical pulse number). The damage probability P(N) thus switches suddenly from 0 to 1 when exceeding the critical pulse number (Figure 6c).³⁰ If no fatigue effect is present, all sites either damage at the first pulse or do not damage at all and the corresponding P(N) curves will be horizontal lines (Figure 6a).

Figure 6:

Theoretical P(N)-curves for: (a) a sample without fatigue effect; (b) a sample with statistical fatigue and (c) a sample in which cumulative material modifications are at the origin of the fatigue effect.

Simply plotting all damaged sites in the Nd - F0 - plane represents a simpler and faster (but less quantitative) means to distinguish the different situations: For the deterministic material modification fatigue all points form a well-defined curve (Figure 7c). In the absence of fatigue all points are concentrated on the first pulse numbers (Figure 7a) and the gradual increase of the damage probability with the number of pulses during statistical fatigue translates to a scattering of the N_d values over typically more than one decade (Figure 7b).

Figure 7:

Theoretical N_d(F)-curves for: (a) a sample without fatigue effect; (b) a sample with statistical fatigue and (c) a sample in which cumulative material modifications are at the origin of the fatigue effect.

5.

THE BEAM SIZE EFFECT

Data acquired in the UV on thin films with different beam sizes clearly shows that no fatigue effect is observed with sufficiently large laser beams.³⁰ This is understandable in view of the above discussion. If the laser beam diameter is of the same size as the mean distance between fabrication defects, some sites will always damage during the first laser pulses because the probability of encounter between the beam and the fabrication defects is high. Depending on the irradiation conditions the few sites without fabrication defects may damage later but this does not change the threshold of the measurement that is determined by the fabrication defects.

6.

THE WAVELENGTH EFFECT

Measurements at different wavelengths show that for longer wavelengths, where the laser-matter interaction is weak, statistical fatigue is obtained whereas material modification fatigue is obtained for shorter wavelengths and thus stronger and more deterministic laser-matter interaction. An example for this behaviour is shown in Figure 8 where Suprasil I^® has been tested at 1064 nm and 355 nm. Plotting the P(N) curves clearly shows the differences in the fatigue mechanisms dominating the observations in the IR or the UV. We should mention here that both measurements were done with (different) MLM lasers. The laser fluctuations were especially high for the UV laser²⁵ inducing an enlarged transition from P = 0 to P = 1 (Figure 8b). The signature of the cumulative material modification fatigue is anyway easily recognizable. Thus this measurement makes also clear that even strong laser fluctuations do not cause statistical fatigue if cumulative material modifications appear with the given wavelength in the sample under test. Similar conclusions can be drawn from measurements in thin films.³⁰

Figure 8:

S-on-1 tests in the bulk of high-OH synthetic fused silica (Suprasil^® 1). (a) For a test at 1064 nm and (b) for a test at 355 nm. In the IR statistical fatigue is observed and in the UV material modification fatigue is observed.

7.

SIMULTANEOUS FAILURE MODES

An advantage of the N_d(F) scatter plots over the P(N)-plots is that the first easily allow to recognize the simultaneous presence of different failure modes. An example is given in Figure 9 where an ion beam sputtered silica layer has been tested at 266 nm using a beam diameter of 30 µm.³⁰ At the given beam diameter one always encounters fabrication defects that damage during the first pulses. These fabrication defects determine the S-on-1 damage thresholds so that in terms of the T(N)-curve one would say that there is no fatigue effect. The sites that do not contain fabrication defects however show that material modification fatigue is simultaneously present in this sample.

Figure 9:

500-on-1 test of an ion beam sputtered silica layer at 266 nm.

Considering that laser conditioning affects the fabrication defects, one could say that ‘fatigue’ and ‘conditioning’ may be present in one and the same sample as these two effects are based on two different types of defects and two different types of physical mechanisms leading to damage.

8.

SUMMARY AND CONCLUSIONS

In summary, we provided an overview of nanosecond multiple pulse measurements and discussed different empirical models for these experiments. We concluded that precursor encounter models should not be used to describe S-on-1 measurements showing material modification fatigue. The light-induced material modifications that finally lead to damage are most probably located in the host material and not in the pre-existing fabrication defects. The defects leading to material modification fatigue are light-induced and of long life-time so that they accumulate during the incubation pulses. We also showed that for weak laser-matter interaction, like it is found at IR wavelengths, statistical fatigue is frequently observed. The defects associated with statistical fatigue are light-induced and of short lifetime so that they do not accumulate.

In order to distinguish between material modification fatigue and statistical fatigue one should plot the damage probability P as function of maximum pulse number N (or S). For statistical fatigue these data are well described by the statistical model and for material modification fatigue the incubation pulses (with P = 0) are followed by a quick transition from P = 0 to P = 1. It is also possible to distinguish material modification fatigue and statistical fatigue from N_d(F) scatter plots of the damaged sites, as the N_d values obtained upon statistical fatigue are distributed over at least a decade whereas they are more concentrated (ideally deterministic) for material modification fatigue.

N_d(F) scatter plots of the damaged sites also allow us to quickly understand if there are two parallel failure modes. For example for some sites fabrication defects may cause damage during the first pulses and for the sites without fabrication defects intrinsic light-host material interaction may cause material modification fatigue causing damage at higher pulse numbers.