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Abstract
The decrease of laser-induced damage threshold (LIDT) when exposed with high number of laser pulses is a well-known phenomenon in dielectrics. In the femtosecond regime this fatigue is usually attributed to the incubation of laser-induced lattice defects. In this work, a computational model is used to combine the data from time-resolved digital holographic microscopy measurements together with results of S-on-1 laser-induced damage threshold test in order to investigate fatigue of ZrO$_2$ single layer coating. Two distinct damage modes were identified and shown to follow different fatigue behaviors: formation of catastrophic damage is highly nonlinear in time, while incubation of color-change mode appears to be linear in time.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The decrease of laser-induced damage threshold (LIDT) of the material when exposed with multiple laser pulses, the fatigue effect, is a well-known phenomenon for all types of solids [1]. The effect can be readily observed [2] by standardized test procedures designed for this specific purpose [3], however the exact nature of the fatigue is not yet fully understood. The gap in understanding of this effect limits our ability to predict LIDT for large numbers of pulses due to the lack of accurate extrapolation models.
In femtosecond regime fatigue is usually attributed to the incubation of laser-induced lattice states or native trap states in the material [1,4,5] which leads to subsequent destabilization of the lattice, nanocracking and subsequent catastrophic damage [6,7]. Generation of such defects in a single pulse case was thoroughly investigated experimentally in bulk materials and coatings during the past few decades using pump-probe techniques [4,8]. The current picture is as follows: an intense laser pulse non-linearly ionizes the material, conduction band electrons relax to self-trapped excitons or other transient states, which are then converted to permanent defects or color centers [4].
However, little is known about the accumulation mechanisms in oxides with lower band gap which are commonly used for thin film production. Previous experiments [5,8] and simulations [9] suggest the existence of laser-induced states, however the information about their origin, characteristic features and resulting damage morphology is extremely scarce.
In this work, we attempt a novel approach of combining experimental data from S-on-1 laser-induced damage threshold test (characteristic damage curves for different damage morphologies) and time-resolved digital holographic microscopy (energy deposition during single laser pulse) into a unified model in order to investigate the fatigue of laser-induced damage in single layer dielectric ZrO$_2$ coating in sub-picosecond regime.
2. Materials and methods
Zirconia (ZrO$_2$) was chosen for the fatigue studies as it is a low absorption, high refractive index material usable for coatings in the near-UV to IR spectral regions that can be used for applications which require high LIDT in the femtosecond regime [10]. Specifically, a 25.4 mm diameter, 1 mm thickness fused silica substrate was coated with a 246 nm (2QWOT @ 1030 nm) zirconia layer by ion-beam sputtering (IBS) for use in the experiments. The refractive index (at 1030 nm) and bandgap of the coating are respectively 2.09 and 5.3 eV. [11].
2.1 LIDT fatigue experiment (S-on-1)
In order to extract the characteristic damage curve of the zirconia coated sample, a modified S-on-1 [3] damage threshold test was performed using a commercial sub-picosecond Yb:KGW laser at a central wavelength of 1030 nm and repetition rate of 50 kHz. The laser produced gaussian pulses with a duration of 304 fs (FWHM) that were focused to 73.6 µm beam diameter (1/e$^2$) at the front surface of the sample. An attenuator consisting of $\lambda /2$ waveplate and two thin film polarizers was used to control irradiation fluence. The laser’s integrated pulse picker was used to pick the required amount of laser pulses for irradiation. Investigated sample was exposed at 45 deg. angle of incidence using linear s polarization. Back-scattered light was measured with a photodiode for in situ damage detection, however, it was not used to stop the irradiation if damage was detected. Additional ex situ damage inspection was performed with differential interference contrast (DIC) Nomarski microscopy with a magnification of 20x.
The standard S-on-1 test procedure was modified by taking into account the deterministic nature of laser-induced damage in the femtosecond regime. Instead of irradiating multiple sample sites with the same fluence value and number of pulses, the sample was irradiated with 1, 10, 10$^2$, 10$^3$, 10$^4$, 10$^5$ and 10$^6$ number of pulses with the same fluence, thus producing a grid in the fluence and number of pulses space. The irradiation was repeated for different fluences with a fluence step of 0.015 J/cm$^2$ starting with the fluence below laser-induced damage and finishing with the fluence above laser-induced damage. A total of 417 sample sites were exposed. This allowed to image the irradiated sites with an ex situ Nomarski microscope in order to retrieve the damage morphology at different numbers of pulses.
2.2 Time-resolved DHM experiments
Time-resolved digital holographic microscopy experiments were performed on the same ZrO$_2$ single layer sample in order to investigate the excitation dynamics of the material during irradiation with laser pulses. Time-resolved DHM experiment, contrary to the time-resolved interferometry technique normally used to characterize electron-hole dynamics in wide bandgap materials[12], allows to perform time-resolved imaging of intensity and phase changes of the microscopic processes due to additional spatial resolution. The experimental setup is described in great detail in previous paper [13] therefore only the experimental parameters will be mentioned in this section. Laser pulses similar to the ones used for the S-on-1 experiment (wavelength 1030 nm, pulse duration 300 fs (FWHM), beam diameter 65 µm at 1/e$^2$ intensity level) were used as the pump pulses in the experiment. Probe pulses were generated using non-collinear optical parametric amplifier (NOPA) and had a central wavelength of 514 nm and pulse duration of 24 fs (FWHM). Beam diameter of the probe pulses was at least 10 times larger than the pump’s. Pumping was performed at 45 deg. angle of incidence while probing was performed at 0 deg. angle of incidence. The geometry of the irradiation is provided in Fig. 1. Both pump and probe pulses had the same linear s polarization. During the experiment, digital holograms were recorded in single shot regime at different time delays (up to 1 ns) using CCD cameras (pixel size 3.45 µm). Fresh test site was used for every delay value. Digital holograms (images of interference patterns) were reconstructed numerically using convolutional algorithm to retrieve bright field and phase shift images. Phase shift caused by Kerr effect in the substrate was removed by fitting it with a two-dimensional Gaussian function. A spatial image region of 50x50 pixels at the center of pump beam was averaged in order to retrieve the temporal phase and transmittance signals.
Relatively high fluence values were chosen for time-resolved DHM to have a better signal to noise ratio. The experiment was performed for three fluence values: directly at 1-on-1 ablation threshold (1.54 J/cm$^2$) and 5 % above and below it (Table 1). 1-on-1 ablation threshold was determined by inspecting test sites with in situ CCD camera after exposing them with different fluence values.
Table 1. Peak fluence and peak intensity values of the single pulse time-resolved DHM experiment.
2.3 Theoretical model
2.3.1 Material response
In order to computationally reproduce both pump-probe and S-on-1 experiments, a theoretical model was used that takes into account multiple transitions in the electronic system. The schematic representation of the simulated energy band system is shown in Fig. 2.
Multiple rate equations [14] were used in order to account for intraband electronic transitions in the conduction band (CB):
The number $k$ of conduction band rate equations is determined by the critical energy $\epsilon _{crit}$ which a conduction band electron must gain in order to collisionally ionize another valence band electron. Critical energy is a combination of bandgap energy $E_{gap}$ and oscillation energy of valence band electrons $\left \langle \epsilon _{osc} \right \rangle$ which they gain in the presence of electric field [14]:
The number of equations is then expressed as a ceiling of critical energy divided by photon energy:
One photon absorption probability $w_{pht}$ is calculated using asymptotic avalanche parameter value [17] and depends on one electron conductivity $\sigma _e$, which is available from Drude model [18]: The material model is described through the polarization vector $\mathbf {P}$. In our case, the polarization is affected by Kerr effect, conduction band electrons and self-trapped exciton state: The polarization of Kerr effect if expressed through the third order electric susceptibility $\chi ^{\left (3\right )}$: The response of conduction band electrons is taken into account by using the Drude model for free electrons [19] with $\omega _{p, CB}$ as plasma frequency:Electric displacement vector $\mathbf {D}$ and complex relative permittivity $\epsilon _{r}$ can then be expressed as:
2.3.2 Electric field in dielectric coating
Transmission of the probe beam is a convolution of the material response and interferometric Fabry-Perot effects. To decouple the response, two dimensional numerical model based on finite-difference time-domain (FDTD) method was developed in order to numerically simulate the pump-probe geometry of DHM experiments. Simulation of pump and probe pulse propagation in dynamic layered system was based on Maxwell’s curl equations [22]:
along with material equations: $\mathbf {B}$ denotes magnetic field, $\mathbf {H}$ – magnetizing field, $\mathbf {E}$ – electric field, $\mathbf {D}$ – displacement field, $\epsilon _{0}$ – vacuum permittivity, $\epsilon _{r}$ – relative permittivity, $\mu _{0}$ – vacuum permeability, $\mu _{r}$ – relative permeability, $\sigma$ – conductivity. These equations were discretized in 2D on a Jee grid [23] for the s polarization using finite-difference time domain (FDTD) method [22] and solved computationally. To reduce computational burden, the beam diameter of the Gaussian pump beam was reduced 10 times (the reduced beam diameter was still 25 times larger than the coating’s thickness) which resulted in the simulation space of 2 µm in the direction of probe beam and 20 µm in the direction perpendicular to probe beam. Probe beam was simulated as an infinite plane wave. The space discretization step was 25 nm (corresponding to time discretization step of 42 as with a Courant factor of 0.5) in order to have at least 10 grid points inside the coating. Absorbing boundary conditions were used in the direction perpendicular to sample surface while periodic boundary conditions were used in the direction parallel to sample surface.3. Results and discussion
3.1 S-on-1 test results
Damage morphology of the S-on-1 experiment’s test sites was qualitatively analyzed in order to identify different damage mechanisms (modes). Damage morphology of the ZrO$_2$ sample varies greatly and depends on both fluence and number of laser pulses used for irradiation, however, two distinct modes were identified: catastrophic failure (Fig. 3) and color-change (Fig. 4).
Fig. 3. Damage morphology images (Nomarski microscopy, 20x objective, high contrast) of the catastrophic mode. In a single pulse case (a) damage is visible as a heat affected zone. After ~10 laser pulses (b) damage is visible as melted and ablated spot about the size of the beam diameter. After 10$^2$-10$^6$ laser pulses (c) damage damage is visible as a melted and ablated spot about two times the size of beam diameter.
Fig. 4. Damage morphology images (Nomarski microscopy, 20x objective, high contrast) of the color-change mode. Damage is visible as red color-change about the size of the beam diameter for wide range of number of pulses (a)-(c).
3.1.1 Morphology of catastrophic failure
The identified catastrophic failure of coating combines a wide range of damage morphologies. After a single laser pulse (Fig. 3(a)) damage is apparent as a change in refractive index most likely caused by melting and resolidification in the pre-ablation regime. At a fluence value where damage occurs after <100 pulses (Fig. 3(b)) it is visible as an ablated spot with a diameter comparable to laser beam diameter. In case of further irradiation (10$^2$–10$^6$ laser pulses) damage grows to a diameter larger than laser beam diameter (Fig. 3(c)).
3.1.2 Morphology of color-change
The abstract term ”color-change” was purposefully chosen for the non catastrophic mode to indicate only the empirical fact of visible change in color (red tint) in the Nomarski microscopy images. The color-change can be caused by a variety of different mechanisms. The most likely mechanism is the formation of color-centers in the coating (which leads to change in material’s refractive index) due to the fact that previous luminescence studies have shown formation of oxygen vacancies in ZrO$_2$ [24]. Any change in optical path not only causes change in color when observed with differential interference contrast Nomarski microscope, but also produces color-change due to different interference conditions. The likelihood of this mechanism is also supported by the fact that damage appears as a deterministic change in color in the shape of the intensity profile of the laser beam. The saturation of the apparent color-change depends on the received dose and correlates well with both the fluence and number of pulses (Fig. 4(a)-(c)) used for irradiation. Quantitative longevity experiments of the color-change mode were not performed, however we did not see any change in the modification of the coating a month after the initial irradiation. Investigation of the exact lattice defects or color-centers that cause color-change mode is outside the scope of the current study, therefore only the fatigue of this damage mode will be investigated.
Previous studies have shown that exposure to sub-picosecond pulses can also lead to formation of high density nanoscale pits [25] on the surface that could cause color-change due to increased scattering, however, these types of damages are visible after irradiation with a single laser pulse, while in this work color-change becomes apparent only after around 10$^3$ laser pulses (Fig. 5). Another type of color-change damage found in literature are plasma scalds formed due to surface imperfections [26], however, this mechanism is the least likely due to the fact that color-change observed in this work does not exhibit the morphology of a ring-like color pattern around a visible precursor which is characteristic to plasma scalds [26].
Fig. 5. Characteristic damage curves for catastrophic and color-change modes in ZrO$_2$. Empirical fatigue model (27) was used to fit the curves. Color-change mode exhibits a much greater fatigue effect than the catastrophic mode (fatigue parameters S are respectively 0.81 and 0.97). Color-change mode becomes LIDT limiting mode after 100 laser pulses.
Even though it is often debatable whether these types of color-change modifications should be considered laser-induced damage, they might indicate the initiation of long term fatigue development. Therefore, the ISO standard defines laser-induced damage as any permanent change to the tested sample [27]. Previous studies have indeed shown that these types of modifications can eventually lead to catastrophic damage [5].
3.1.3 Characteristic damage curves
Laser-induced damage thresholds were evaluated separately for each damage mode. Characteristic damage curves of the S-on-1 test are provided in Fig. 5. No saturation of LIDT is seen for up to 10$^6$ laser pulses for the color-change mode, while LIDT of the catastrophic mode appears to saturate after around 10$^5$ laser pulses. In order to compare the fatigue effect of both modes experimental data was fitted using a widely used empirical model [23]:
where N – number of laser pulses, $F_{N}$ – LIDT after N laser pulses, $F_{1}$ – LIDT after single laser pulse, $S$ – incubation parameter. The identified modes exhibit different fatigue behaviour, namely, LIDT of color-change mode drops faster with increased number of laser pulses than LIDT of catastrophic mode (fatigue parameters are $S=0.81$ and $S=0.97$ respectively, with $S=1$ representing no fatigue at all).3.2 Time-resolved DHM results
The empirical model described in previous section does not provide any insight into why damage modes on the same sample exhibit completely different fatigue behaviour. Therefore, time-resolved digital holographic microscopy experiment was performed in order to investigate processes that take place when laser pulse interacts with coating’s material and to approximate the amount of energy deposited into the material during a single pulse.
The experimental relative transmittance and phase shift curves for all tested fluences (Table 1) are provided in Fig. 6. Increase of conduction band electrons due to non-linear ionization at the peak of the pulse results in a sharp drop of transmittance and positive phase shift. Subsequent immediate trapping process is also evident: transmittance increases (due to decrease of conduction band electron density) and phase shift becomes negative. Temporal characteristics of the signals do not appear to be exactly the same: phase shift (Fig. 6(b)) saturates much faster than the transmittance change (Fig. 6(a)). In order to identify different types of transitions, all curves were fitted with simple exponential decay curves (see Table 2 and solid lines in Fig. 6). Both phase shift and relative transmittance curves have a common temporal component of 0.65 ps, however, relative transmittance has an additional term of 10.3 ps. Similar temporal transitions were identified for transmittance change in TiO$_2$, Ta$_2$O$_5$ and HfO$_2$ [8]. The shorter component is believed to be caused by the formation of self-trapped excitons (STEs), while the longer component accounts for carrier interband recombination and the evolution of STEs [8]. However, it is not yet entirely clear why only one of these components is apparent in the phase shift signal.
Fig. 6. Experimental data of relative transmittance and phase shift in ZrO$_2$. Solid lines represent exponential relaxation fits (see Table 2). (All curves are shifted by 1 ps in order to plot them on the logarithmic scale.)
Table 2. Exponential decay functions which where used to fit for change in transmittance and phase shift (solid lines in Fig. 6)
The experimental data of time resolved DHM experiment for fluences up to ablation threshold was simulated using previously described material response and FDTD model. Simulation was performed in a 2.5 ps window to investigate energy deposition during a single laser pulse. Experimental data and simulations of pump-probe experiments are compared in Fig. 7(a) and Fig. 7(b) for relative transmittance and phase shift respectively. All simulated curves are calculated using a single set of material parameters (Table 3) which was retrieved by simultaneously fitting experimental data of relative transmittance and phase shift for both fluences. Simulations and experiments are in good agreement, therefore the same material parameters were used to evaluate the amount of deposited energy for for fluences up to the ablation threshold.
Fig. 7. Experimental data and simulation of relative transmittance and phase shift in ZrO$_2$. (All curves are shifted by 1 ps in order to plot them on the logarithmic scale.
Table 3. Fit parameters of the pump-probe model.
The amount of energy deposited during a single pulse $d_1$ (normalized to amount of energy deposited during 1-on-1 ablation) was evaluated by making an assumption that it is proportional to the ionized electron density (for the given laser parameters) and simulated for fluence values up to the ablation threshold (Fig. 8). The total density of ionized electrons at ablation threshold was $2 \cdot 10^{21}$ m$^{-3}$. Energy deposition profile is determined largely by the order of multiphoton absorption with only slight corrections at higher fluences due to full Keldysh model, impact ionization and interference effects.
Fig. 8. Deposited energy for a single laser pulse of a given fluence. Energy deposition profile is determined largely by the order of multiphoton absorption with only slight corrections at higher fluences due to full Keldysh ionization model and impact ionization.
3.3 Fatigue models
The evaluated relative amount of energy deposited during a first laser pulse was used together with an additional incubation assumptions in order to reproduce the measured characteristic damage curves for different damage modes. In this section two different incubation models are discussed that supplement each other in an attempt to describe fatigue of zirconia coating.
3.3.1 Nonlinear incubation model
The LIDT fatigue of the catastrophic mode was addressed by numerous experimental and theoretical studies [1,2,5,9,28]. If laser fluence is above a certain threshold, cyclic stress induced by laser pulses will lead to formation of microscopic cracks at the stress concentrators (defects, inhomogeneities). Eventually a crack will form and slowly reach a critical size before propagating suddenly and causing catastrophic damage. This process is highly nonlinear in time. Catastrophic damage was shown to follow a generic phenomenological model for fatigue applicable to a large class of materials [28]. The model is based on increase of energy deposition per single laser pulse and additional decrease in critical energy required to damage the material. To check the applicability of this model to our experimental results, the increase in energy deposition can be expressed as:
where $d_n$ – energy deposited after $n$ pulses, $d_{\infty }$ – maximum energy deposition ($d_{\infty }$ > $d_{1}$), $d_{1}$ – energy deposited after single pulse, $r_{d}$ – energy deposition growth parameter ($r_d$ > 1). Maximum energy deposition is related to the single pulse energy deposition through energy deposition increase ratio $\Delta d$: The decrease in threshold energy required to damage the material can be expressed as: where $t_n$ – threshold energy after $n$ pulses, $t_{\infty }$ – saturation threshold energy ($t_{\infty }$ < $t_{1}$), $t_{1}$ – threshold energy for first pulse, $r_{t}$ – threshold decrease parameter ($r_t$ < 1). Number of pulses at which the damage will occur can thus be found by solving the inequality: The fit ($\Delta d = 0.9$, $r_{d} = 1.04$, $t_{1} = 0.41$, $t_{\infty } = 0.17$, $r_{t} = 0.999985$) of catastrophic damage mode with nonlinear fatigue model is provided in Fig. 9. The model correctly reproduces two distinct drops in LIDT between 10$^2$-10$^3$ and 10$^4$-10$^5$ laser pulses as was predicted [28].Fig. 9. Fits of fatigue models for characteristic damage curves of catastrophic failure and color-change modes.
3.3.2 Linear incubation model
Fatigue of color-change mode appears to follow completely different principles than fatigue of catastrophic mode: it doesn’t approach any saturation fluence asymptotically and there are no sudden drops in LIDT (characteristic damage curve is linear on a log-log scale, see Fig. 5). We propose that the incubation of the color-change mode is linear in nature, i.e. energy deposition per laser pulse is approximately constant for every consecutive laser pulse until color-change becomes apparent. If we assume that energy deposition is constant with each laser pulse, we get a simple arithmetic progression for energy deposition during multiple pulse exposure:
where $d_n$ – energy deposited after $n$ pulses, $d_{1}$ – energy deposited after single pulse. Number of pulses at which the damage will occur can thus be found by solving the inequality where $t$ is a constant threshold energy at which color-change becomes apparent. The fit ($t=140$) of color-change damage mode with linear fatigue model is provided in Fig. 9. The fit is in good agreement with the experimental results. It is evident that color-change mode is screened by the catastrophic failure mode up to 10$^3$ laser pulses, however, it becomes the LIDT limiting mode afterwards.4. Conclusions
Two distinct damage modes in ZrO$_2$ optical coating were identified from damage morphology images of modified S-on-1 experiment. These damage modes exhibited contrasting fatigue behavior: catastrophic failure mode saturated faster than the color-change mode (fatigue parameters of the empirical fatigue model were respectively $S=0.97$ and $S=0.81$), however, the color-change mode became apparent only after 10$^3$ laser pulses. The fact that different damage modes on the same sample can be screened and become apparent only after prolonged irradiation should be taken into account when attempting to extrapolate the lifetime of optical components.
Time-resolved digital holographic microscopy experiment was performed to reveal the excitation dynamics of ZrO$_2$ single layer coating. Temporal component of 0.65 ps visible in both phase shift and transmittance signals was identified and attributed to generation of self-trapped excitons. An additional process visible only in transmittance with a temporal component of 10.3 ps was identified, however, further studies are need to identify the underlying physical mechanism of this term. Numerical material response model was used to extract the relative amount of deposited energy per single laser pulse from time-resolved DHM data. Additional incubation models were used to showcase the differences in fatigue behaviour of identified damage modes: incubation of catastrophic damage is highly nonlinear in regards to the irradiation time, while incubation of color-change mode appears to be linear in time up to the visibility threshold. The apparent linear incubation of color-change mode challenges the notion that characteristic damage curves can be easily extrapolated to a threshold fluence value that does not cause damage at infinite irradiation, since the characteristic damage curve of this model approaches zero. Further studies are needed with even longer irradiation times in order to assert whether any threshold fluence exists for such color-change modes.
Funding
European Regional Development Fund (ERDF) (01.2.2-LMT-K-718-01-0014).
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