Experimental set-up

In order to provide meaningful predictions of optics lifetime, it is essential to collect accurate experimental data first. A conventional LIDT test bench was adapted for a long-term measurement at company LIDARIS for this purpose. There were two major concerns to address: long-term stability of laser source and appropriate sensitivity of online damage detection (registration of time to failure events In situ). As it has been already demonstrated, LIDT statistics could be affected by nonrepeatable pulses of different temporal shapes (distinct longitudinal modes)². To avoid accidental multi-mode lasing within a long-lasting light burst the injection seeded Nd:YAG diode pumped solid state (DPSS) laser source (NL940-100-SH/TH, EKSPLA) was exclusively designed. It generates single longitudinal mode pulses at 1064 nm, with 100 Hz repetition rate and effective pulse duration ~10 ns (at full width half maximum - FWHM). The light is then frequency doubled and tripled (355 nm) by using nonlinear crystals. Optical beamline of LIDT test bench consists of the three key elements:

The beam diameter for particular testing was set to 209 ± 3 μm at 1/e² level of peak fluence. The sample is placed onto 2 axes (X-Y) motorized translation stage at 45 degrees with respect to laser beam line. Damage events were monitored by two techniques, namely:

Fig. 1.

The schematics of an experimental LIDT test bench. λ/2 is a half-wave plate, M1, M2 are steering mirrors, W - a wedge, L - lens, PD - photodiode, S - mechanical shutter, Photo det. is online DIC microscopy detection unit.

Experimental set of samples

For this experimental study large set of high reflectivity (HR) mirror coatings was prepared. All multilayer dielectric coatings were deposited by using Ion Beam Sputtering (IBS) technique on conventionally polished 25.4 mm in diameter fused silica substrates. The central wavelength of 355 nm was designed to operate at a 45-degree angle of incidence (AOI). Further results for single mirror experiments are described thus representing typical behavior of observed sample set.

Collection of data

In order to predict lifetime, we started from the classical ISO S-on-1 test approach, that is frequently used to benchmark optical materials for laser industry³. On the other hand, this kind of test collects data that has a meaning of “time to failure - TTF” events thus we can also interpret them in terms of accelerated lifetime testing. The idea behind S-on-1 is simple: entrance surface of the sample is divided into a virtual matrix of sites to be irradiated with a quite small beam (compared to the whole surface). Each test site is then irradiated with maximum onset laser pulses (for example S = 10 000 in our case) at various fluence level (sometimes also called as stress level). At least 10 sites are irradiated for each fixed fluence. During such irradiation three key variables are recorded:

Calculation of damage probabilities

The data collected as described above can be then interpreted in different ways (either Bayesian approach of damage probability analysis as a function of fluence⁴ or Weibull/Survival approach of damage/survival probability as a function of lifetime⁵, frequently used with accelerated lifetime testing). Herewith we limit ourselves to damage probability approach. After collection of TTF data at various fluences damage probability is calculated for each class of laser pulses. Damage probability is calculated as a ratio of damaged and total irradiated sites for each onset fluence level (stress) of particular pulse class of interest. It is then plotted as a function of laser fluence (see Fig. 3 for example). By fitting the empirical damage probability curve in the Bayesian way, we get the most probable estimate of threshold fluence (maximum fluence with zero damage probability) that corresponds to laser-induced damage threshold (LIDT). An increased dose of light (number of incident laser pulses) typically ends up in lower LIDTs (fatigue effect). On this point, we have to recall that precise fitting model of damage probability could differ for different samples. In the range of nanosecond laser pulse durations, damage event is usually related to external defects inherent to the manufacturing process (cutting, polishing, cleaning, coating and etc.). As a rule of thumb, such defects are randomly distributed on the sample surface (within first few microns below the surface). By the means of threshold fluence, these defects are often described in form of so called defect ensemble^6–10. As the distribution of defect thresholds in UV is rather narrow we consider them as identical (all have a constant damage threshold), described by so-called degenerate defect ensemble. Even if this assumption is not absolute it was convenient for the exemplification of the proposed approach. Fit function based on degenerate defect ensemble has two fit parameters: defect density and threshold fluence (LIDT). As it was already mentioned fitting procedure of the damage probability curve is based on the Bayesian approach (Bayesian interference) and is described in¹¹. As a result of such data processing, we get a new set of data that will be used for further fatigue analysis: probability density functions PDF(F, N) for each pulse class. Here N – corresponds to Nth laser pulse class, and F – to onset fluence respectively. PDFs for higher pulse classes can be significantly constrained or modified by using prior knowledge: Bayes’s theorem says that the posterior probability density is the product of the prior probability density and the likelihood function (times a constant). Or in other words, it defines how to update the probabilities of hypotheses when given evidence. For pulse classes N > 1 a prior knowledge can be also used:

The prior distribution is some internal knowledge about the data. For instance, prior can define inequalities such as LIDT for infinite exposure of laser pulses is less than LIDT for single shot irradiation (F_inf ≤ F₁). In the case of LIDT testing, the maximum-likelihood function is constructed based on two key things:

By using Bayesian fitting approach so-called characteristic damage curve (CDC) is obtained, where threshold data point for each pulse class has estimated likelihood distribution. We consider this distribution as a new PDF function when interpreting aging behavior of characteristic damage curve-.

Prediction of optics lifetime: fitting and extrapolation of characteristic damage curve

Each estimated LIDT value in the CDC has a PDF. CDC is then fitted using different mathematical models available in the scientific literature, which are associated with optics lifetime degradation (fatigue). It is based on the Bayesian approach. First of all, we compose likelihood function out of experimentally measured PDF’s and candidate models simulating experimental characteristic damage curves. The general form of likelihood could be described as follows:

Here, S – amount of pulse classes of interest, PDF_i – estimated likelihood distribution of threshold parameter for i_th pulse class as described in⁴, F_th(N) – chosen aging model of characteristic damage curve, N_i – number of incident laser pulses for particular class, θ(N_i)- set of experimental data: empirical probability and chosen model of defects to fit damage probability. To find best fit (maximum L) for each F_th(N) function parameters should be varied. The LIDT values estimated from damage probability curves are plotted versus the incident number of laser pulses - CDC. In order to predict optics lifetime CDC should be extrapolated by using best estimates of fitted model parameters. However, best fit says nothing about the measurement uncertainty. In order to estimate the uncertainty of extrapolation Markov Chain Monte Carlo Method (MCMC) is used¹³. MCMC procedure provides a sampled distribution of extrapolated model parameters. Two statistical techniques were used to interpret empirical extrapolated distribution, namely assumed distribution fitting and Kernel Density Estimation (KDE)^14,15. Distributions of the best estimates are determined by MCMC procedure. However, this time, LIDT is not (is not necessary to be) the parameter of the fit model. In order to extract LIDT distribution a large number of random extrapolation curves are generated using MCMC samples. Then obtained sample distribution is used to calculate Cumulative Distribution Function (CDF and Percent Point Function (PPD)). Percent Point function is an inverse CDF. In this way, limited experimental data could be extracted to the high number of laser pulses. This method was validated experimentally.

Failure mode separation

Two types of permanent changes (damage mechanisms/modes) have been identified in recorded images for the tested sample (fig. 2), namely:

Also, as it is shown in Fig. 2 (bottom row), different damage modes can co-exist together, thus also screening each other. In this particular case, the test site is considered as damaged after 1 pulse when estimating “defect driven” damages and is only damaged after 3000 pulses when estimating “color change” mode. Interestingly, for the investigated test sample, defect driven damages always appeared within the first 1-10 laser pulses while most of the color change type of damages started only after 3000 laser pulses or more at the same laser fluence level. The “color change” was the only type of damage, which occurred for the higher number of laser pulses (100k, 1M, and 10M). The S(10.000)-on-1 test results were interpreted separately for each failure mode. To investigate trends in laser-induced damage threshold both failure modes should be identified and interpreted in isolation from each other.

Fig. 2.

Evolution of the typical laser-induced damage morphologies obtained by Nomarski (DIC, 10x) microscopy for the fluence slightly below single shot damage threshold.

S(10.000)-on-1 LIDT results

After visual morphology analysis damage statistics were separated into two groups and probability curves were calculated in isolation (for each failure mode) for every particular laser pulse class. Fluence dependencies of separated failure mode probabilities are shown in (Fig. 3). The statistics of catastrophic defect driven damage is rather random and is almost “frozen” after 10 laser pulses. Damage associated with color change features quite deterministic damage behavior that is “moving” toward left side when the amount of irradiated pulses is increasing. This behavior suggests a non-saturated incubation effect.

Fig. 3.

Damage probabilities at various pulse classes calculated by using different LID criteria: defect driven damage – on the left, color change – on the right.

By fitting damage probability curves with degenerate model characteristic damage curves calculated for pulse classes that contain meaningful data. For such fitting maximum likelihood approach is used, that has both S(10.000)-on-1 LIDT test for each damage mode (or detection method) are shown in the Fig. 4. They are visualized as lines with scatter either square or dot. Calculated LIDT values are summarized in Table 2.

Table 2.

Calculated LIDT values from S(10.000)-on-1 LIDT test using 2 different LID criteria.

Fig. 4.

Characteristic damage curves calculated LIDT values from S(10.000)-on-1 LIDT test using 2 different LID criteria.

Results show that defect driven damage mode (green) features an abrupt decrease in LIDT within a first few laser pulses and after approaches almost asymptotic constant level. This was a typical observation defect limited characteristic damage curve. On the contrary “Color change” type of mode starts quite late at around 3000 laser pulses. The LIDTs at 10k laser pulses corresponding to both failure modes are quite similar, but the trends and shapes of the CDC are very different. Color changing mode goes down without saturation thus suggesting a further monotonic decay.