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Abstract
Laser-induced damage growth has often been studied with Gaussian beams in the sub-picosecond regime. However, beams generated by high-power laser facilities do not feature Gaussian profiles, a property that raises questions concerning the reliability of off-line laser-induced damage measurements. Here, we compare laser-induced damage growth dynamics as a function of beam profiles. Experiments on multilayer dielectric mirrors at 1053 nm have been carried out with squared top-hat and Gaussian beams. The results demonstrate that the laser-induced damage growth threshold does not depend on the incident beam profile. A higher damage growth rate, however, has been measured with the top-hat beam. In addition, three different regimes in the growth dynamics were identified above a given fluence. A numerical model has been developed to simulate a complete damage growth sequence for different beam profiles. The numerical results are in good agreement with the observations, three growth regimes were also revealed. These results demonstrate that a linear description of growth cannot be used for the whole growth domain.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the development of the Chirped Pulse Amplification (CPA) technique in 1985 [1], the peak power of lasers has steadily increased worldwide. Since then, numerous projects have led to the creation of high-power laser facilities such as Omega-Ep, ELI-NP, HiLASE or PETAL [2–5]. Recently, the ELI beam lines have reached a power of 10 PW [6]. At these facilities, the energy remains limited by laser-induced damage on optical components located at the final stage of the laser chain.
In the sub-picosecond regime, laser-induced damage is driven by non-linear absorption in the material, which is directly linked to the Electric Field Intensity (EFI). EFI relates the intrinsic damage threshold of a material to the experimental Laser-Induced Damage Threshold (LIDT) [7–9]. Several studies have improved our understanding of damage mechanisms on Multilayer Dielectric (MLD) gratings and mirrors [10–12]. The key role of manufacturing processes and external contamination among other parameters on the LIDT have been highlighted [13–15]. In addition, the intensity and phase modulations present in large beams provide local hot-spots. In the case of the PETAL facility, local hot-spots account for up to 63% of the average target fluence, resulting in laser damage to the transport mirrors [16]. It is difficult to control all these issues and avoid the occurrence of damage sites on large optics. The initial size of these damage sites is generally micrometric which is too small to have detrimental effects on beam propagation. The important question is to understand the behavior of these typical damage sites following a set of successive shots.
The study of damage growth to determine the lifetime of optical components is of prime importance for high-power laser facilities. Damage growth experiments were performed in the nanosecond regime. Genin et al. performed experiments with a Gaussian beam (1.1 mm diameter at 1/e2) centered at 1064 nm on MLD mirrors [17]. They demonstrated different growth behaviors for different initial damage morphologies. Negres et al. performed experiments on fused silica with a large beam of 30 mm diameter in U-V [18]. They demonstrated a linear and an exponential evolution of the damage area. In the sub-picosecond regime, Sozet et al. described a deterministic behavior of growth on MLD mirrors with a threshold lower than the LIDT [19]. It was shown that damage size increases linearly with the number of shots and that the growth rate increases with the incident fluence. Additionally, the influence of EFI distribution on the damage growth phenomenon has been evidenced numerically [20]. On MLD gratings, Hao et al. applied the mono-shot procedure [21] to measure the Laser-Induced Damage Growth Threshold (LIDGT) [22]. They also demonstrated an asymmetrical behavior of damage growth following the beam direction that was connected to asymmetrical EFI distribution [23]. All these sub-picosecond results were obtained through experiments carried out with Gaussian beams. However, as a consequence of local hot-spots in the homogeneous profile of large beam, damage sites are illuminated by both top-hat and Gaussian beam profiles. In a previous work, we reported damage growth experiments with a top-hat beam and investigated the temporal dependency of damage growth [24]. We established that the damage growth rate decreases with increasing pulse duration.
In this work, based on the same methodology used in the previous examples, we compare the damage growth dynamics using top-hat and Gaussian beams. A complete description of damage growth dynamics is given based on experimental data and numerical analysis with a focus on different behaviors of the damage growth phenomenon, which questions the linear characterization of the evolution of the damage area.
Firstly, the experimental study conducted on MLD mirrors is presented. The experimental procedure is detailed in section 2. Initial damage sites were created by laser ablation, then illuminated with the two beam profiles. Their evolution was recorded and analyzed. Secondly, the experimental results are presented in section 3. Linear damage growth coefficients were determined to compare the damage growth rate according to the top-hat and Gaussian beams. Additionally, a numerical study using the Finite Element Method (FEM) simulation is presented in section 4. The model simulates a complete damage growth sequence in order to improve the interpretation of the observations. Finally, the different results are discussed in section 5 before a conclusion.
2. Experimental procedure
2.1 Laser damage setup
Experiments were carried out on the laser damage test bench illustrated in Fig. 1. The laser source (Amplitude S pulse HP) provides a Gaussian beam with a diameter of 3.25 mm at $1/e^{2}$ operating at 1053 nm and 10 Hz. The pulse length is estimated by an autocorrelator (A.P.E) at 0.8 ps from sech2 fit. The output laser energy is approximately 2 mJ. A fused silica phase plate (FBS-2, Eskma), coupled with a lens of 30 cm focal length provides a 150 $\mu$m at Full Width at Half Maximum (FWHM) squared top-hat beam with a rising edge approximately 70 $\mu$m (distance between the 0 and maximum intensity), as illustrated in Fig. 2(a). Different beam profiles are formed at different planes along the propagation axis, notably a Gaussian beam profile type with a diameter of 170 $\mu$m at $1/e^{2}$ (see Fig. 2(b)). An in-situ visualization system for real-time sample monitoring was implemented, based on a Navitar macroscope working in reflection mode with a motorized zoom coupled to a CCD camera. The camera is synchronized with the laser frequency to capture, shot by shot, the evolution of the damage area.
Fig. 2. Spatial beam profiles of (a) the top-hat and (b) the Gaussian beams captured by the CCD Camera. (c) and (d) are a cross-section of the top-hat and Gaussian beams, respectively.
2.2 Samples
Experiments were carried out on a single sample of MLD mirror deposited by e-beam and the Ion Assisted Deposition (IAD) technique. The design is [BK7 / $(LH)^{15}$ / Air], where L (respectively H) represents a quarter-wave of SiO2 (respectively HfO2) at 45$^{\circ }$ with a refractive index of 1.45 (respectively 1.93) at 1053 nm. The mirror design provides maximum reflection (above 99%) at 1053 nm for both p and s-polarization at 45$^{\circ }$ of incidence.
The intrinsic LIDT of SiO2 and HfO2 layers was measured at 3.09 $\pm$ 0.04 J/cm2 and 2.11 $\pm$ 0.01 J/cm2 respectively [25]. The LIDT was determined with a 1-on-1 test [26] at 45$^{\circ }$ of incidence and 0.8 ps in s-polarization and the LIDT of the mirror was measured at 3.73 $\pm$ 0.44 J/cm2.
2.3 Test procedure
First, a series of shots was performed with the Gaussian beam at a fluence 10% above the LIDT (one shot per site) to initiate identical damage sites (see Fig. 3(a)). The initial damage site represents a delamination of the entire first upper HfO2 layer as evidenced by Atomic Force Microscope (AFM) measurements. The next step consisted in illuminating initial damage sites (beams were centered on the initial damage sites) with successive laser shots and for different fluences. For a given fluence, 3 different sites were illuminated. Figure 3(b) shows a damage site after more than 300 shots with the top-hat beam. All experiments were conducted with a pulse duration of 0.8 ps, 45$^{\circ }$ of incidence, in s-polarization and a laser frequency of 10 Hz. The two beam profiles illustrated in Fig. 2 were used to illuminate initial damage sites. The top-hat and Gaussian beams have effective areas equal to (4.03 $\pm$ 0.12) $\times$ 10−4 cm2 and (1.82 $\pm$ 0.04) $\times$ 10−4 cm2 respectively [27].
Fig. 3. SEM (Scanning Electron microscopy) images of (a) typical initial laser-induced damage site performed with the Gaussian beam shot. The first upper layer of the MLD mirror has been ablated. (b) Final morphology of the damage site after a complete laser damage growth sequence with the top-hat beam. The entire MLD stack is removed.
3. Experimental results and analysis
During a damage growth sequence, the macroscope captures the damage area after each shot (see Visualization 1). As shown in Fig. 4, the damage evolves over the surface (vertically and horizontally) through its depth. The different layers are successively delaminated. At the end of the damage growth sequence (see Fig. 3(b)), we assume that the entire stack has been removed. The damaged area initially evolves in a preferred direction identical to the laser propagation. With the top-hat beam, from a given fluence, the damage can evolve in the opposite direction to the previous one after several shots (see Fig. 4(c) #15). This phenomenon of asymmetric damage growth is limited with the Gaussian beam probably because of the beam size [23].
Fig. 4. Acquisition of damage areas with the macroscope during growth sequences. The red arrows represent the beam direction. The white arrows point to new damage sites. The white ellipses represent the contour of the initial damage site. Figures (a), (b) and (c) represent damage growth sequences with the top-hat beam respectively at a fluence of 1.11, 1.91 and 3.17 J/cm2. Figures (d), (e) and (f) represent damage growth sequences with the Gaussian beam respectively at a fluence of 0.94, 1.87 and 3.28 J/cm2. See Visualization 1 for complete sequences.
Based on thresholding with image processing, the damage area is determined as function of the number of shots. Figure 5 shows the result of the evolution of the damage area for 3 different sites illuminated with the top-hat beam under the same conditions for a fluence of 1.91 J/cm2 (Fig. 4(b)). A good repeatability is observed in Fig. 5 and for all experiments made under identical experimental conditions (fluence and beam profile). The evolution of the damage area highlights two growth regimes with the top-hat and the Gaussian beams, as shown in Fig. 5(b). However, exclusively with the top-hat beam, above a fluence of 1.4 J/cm2, three regimes were identified as illustrated in Fig. 5(a): initial regime, main regime and saturation regime [24].
Fig. 5. Evolution of the damage area as a function of the number of shots for three different sites illuminated under the same conditions (repeatability) (a) with the top-hat beam at a fluence of 1.91 J/cm2; (b) with the Gaussian beam at a fluence of 1.87 J/cm2.
The initial regime represents a slow evolution of the damage area following the beam direction. The duration (number of shots) of this regime decreases with increasing incident fluence. The initial regime corresponds to 3 shots for high fluence and 50 shots for low fluence. The transition between the initial and the main regime appears to be due to the occurrence of new damage sites located a few microns from the initial damage site. This fact is illustrated in Fig. 4(b) #15, (c) #15) and highlighted by white arrows. These new damage sites grow with the same initial dynamic as the first one. The main regime dynamic corresponds to a multiplication of the initial regime dynamic by the number of occurrences of new damage sites. The saturation regime represents the slowdown of damage growth until an interruption of growth, it occurs as soon as the growth rate of the previous regime begins to decline. We assume that saturation occurs when the damage reaches the limits of the beam size. This regime has not been considered to characterize the damage growth phenomenon, but this aspect will be discussed in the following sections.
Regarding results with the top-hat beam, the evolution of the damage area is non-linear when considering the initial and main regimes together or linear per part when these regimes are considered separately. Previous studies of damage growth have described the evolution of the damage area as a function of the number of shots with an affine relationship that is given in Eq. (1) [19,20,23,24]. The experiments with the Gaussian beam can be described identically. In order to compare the damage growth dynamics as a function of the beam profile, we describe the evolution of damage area with the affine relationship:
$S_{N}$ ($\mu m^{2}$) represents the damage area after N shots, $\beta$ ($\mu m^{2}$/shot) is the linear damage growth coefficient and $S_{0}$ ($\mu m^{2}$) represents the initial damage area. For a given fluence, 3 different initial damage sites were illuminated and we report the average of the three measured values as a $\beta$ coefficient in Fig. 6. In both cases, only the main regime was considered to measure damage growth coefficients. The linear characterization of the damage growth phenomenon will be discussed in the following sections.
Fig. 6. Evolution of damage growth coefficients as a function of the fluence following the top-hat and Gaussian beams. The black dashed line corresponds to the 1-on-1 LIDT of the mirror.
Figure 6 shows the evolution of the damage growth coefficients as a function of the fluence according to the two beam profiles. Experiments have shown that for fluences lower than 0.8 J/cm2, the initial damage does not grow. The damage growth coefficient is set at 0. This fluence corresponds to the LIDGT and represents approximately 20% of the LIDT ($3.84 \pm 0.40$ J/cm2). The LIDGT does not depend on the beam profile. For fluences between 0.8 and 1.4 J/cm2, a systematic evolution of the damage site was noticed, which represents an increase of a few micrometers of the initial damage size (less than 80 $\mu$m). In this case, the first layer was removed along the beam direction as shown in Fig. 4(a) and (d). Regarding the top-hat beam, the final damage size is smaller than the beam size. With a perfect top-hat beam, the final damage site dimensions should reach the limit of the beam dimensions, but the dynamic should be identical. This point will be discussed in the next section. For this range of fluences (0.8 to 1.4 J/cm2), the damage growth coefficients are between 0 and 30 $\mu m^{2}$/shot. For large optical components, these values could be acceptable. Approximately 30,000 shots will be necessary to reach a damage area of 1 $mm^{2}$. A functional LIDGT could be set at a fluence of 1.4 J/cm2 which represents approximately 35% of LIDT (see Fig. 6). Above this fluence of 1.4 J/cm2, asymmetrical damage growth was observed, with a preferential direction following the beam direction, as shown in Fig. 4(c) and (f). The damage growth coefficients increase strongly with the fluence and the main result is that the rate is much higher with the top-hat beam. The intensity gradient of the Gaussian beam appears to limit the damage growth rate. However, the size of the beam can also influence the result notably for the Gaussian beam with another intensity gradient. This point will be discussed in the following sections.
To summarize this section, we measured in both cases a LIDGT equal to 20% of LIDT and a limited evolution of the damage area up to 35% of LIDT. Above this value, the damage growth dynamics appear to be determined by the incident beam profile. With the top-hat beam, three growth regimes were revealed, raising questions about the linear description of damage growth. We measured a higher growth rate with the top-hat beam than the Gaussian beam, though other factors may influence these results such as beam size. In order to improve our understanding, we developed a numerical model to simulate an entire damage growth sequence.
4. Numerical modeling of damage growth with the finite element method
4.1 Structure of the model
A numerical model was developed with the Wave Optics module and the electromagnetic waves, frequency domain physics interface of the Comsol Multiphysics software (Version 5.6) in 2D to simulate damage growth experiments. The LIDT depends on the electric field intensity (EFI) distribution for MLD mirrors or MLD gratings in the sub-picosecond regime [8,12]. Therefore, we assume that damage growth is driven by the same physical process, i.e a localization of energy deposition related to EFI distribution.
Sozet et al. used a FEM model to simulate damage growth experiments through EFI distribution and reported a good correlation between the simulations and laboratory observations for a dozen shots [20]. Here, the main objective was to simulate a complete damage growth sequence.
The geometry implemented in the model is illustrated in Fig. 7(a). This represents the entire design of the mirror used for the experiment, described in section 2.2. The layers surrounding the mirror design and underlined in purple represent the Perfect Matched Layers (PML). This function is implemented in Comsol to avoid reflection of incident waves on the edges of the domain. The PML thickness was set to 1 $\mu$m. The dimensions of the system under study were set to 400 $\times$ 9.16 $\mu m^{2}$. The geometry is meshed with the free triangular function implemented in Comsol and a maximum cell size equal to $\lambda$/6 ($\lambda$ represents the incident wavelength of 1053 nm). A rectangular hole (filled with air) in the first HfO2 layer was selected as initial damage with a length of 45 $\mu$m corresponding to AFM observations.
Fig. 7. (a) Representation of the geometry implemented in the FEM model. The y-axis is represented with a magnification of x20. The domains underlined in purple are defined as Perfect Matched Layers (PML). The entire mirror design presented in section 2.2 is represented. (b) Beam functions implemented at the input port from the top side of the air domain. Representation of one iteration for the simulation of laser-induced damage in the FEM model. The color scale represents the normalized EFI and the gray scale represents the refractive index. (c) The EFI distribution is calculated for a perfect mirror. (d) The previous result is used as a source term to calculate the EFI distribution for the damaged mirror. (e) Local EFI values superior to the electric field value which corresponds to the intrinsic LIDT of each material. (f) Evolution of the damaged structure according to the upgrade of optical indexes.
The model is set to simulate the propagation of a uniform incident wave coming from the top in s-polarization. The propagation constant is calculated for an angle of incidence of 45$^{\circ }$. The maximum intensity is calculated from a given fluence. The EFI distribution of the damaged structure is realized in two steps. First, the EFI distribution is calculated for a perfect mirror (without damage) by solving Maxwell’s equations with the FEM (see Fig. 7(c)). Next, the EFI distribution for the damaged structure is calculated with the scattered field method for which the previous calculation is considered as a source term (see Fig. 7(d)). At this point, the model is able to localize nodes where the electric field is above the value corresponding to the intrinsic LIDT of each material, (see Fig. 7(e)). The calculation of the EFI distribution is characterized by the presence of high peaks of EFI in the depth of the MLD stack (see Fig. 7(d)). Nodes with EFI higher than the LIDT, but located deep within the stack (embedded more than 2 layers under air domains) were not considered as damaged. We set the intrinsic LIDT to 3.09 J/cm2 and 2.11 J/cm2 respectively for SiO2 and HfO2 layers. The intrinsic LIDT of each material was determined with experimental damage tests on monolayers coming from the same fabrication batch presented in [28]. To simulate a laser-induced damage process, the refractive index of each layer is set to 1 (like air domain, see Fig. 7(f)). All these steps represent one iteration and the next iteration starts with the new damaged structure.
To investigate the influence of the beam profile numerically, we introduce functions to modulate the uniform intensity at the input port. To simulate the top-hat beam, we set a top-hat profile of 200 $\mu$m at Full Width at Half Maximum (FWHM) and rising edges of 60 $\mu$m (distance between 0 and maximum intensity), as illustrated in Fig. 7. We chose to define a Gaussian beam profile with a waist equal to 114.5 $\mu$m (see Fig. 7(b)). The Gaussian and top-hat functions have an equal integral value to maintain the proportion of equivalent incident energy in both cases. Different computation sequences were performed for different maximum intensities (fluence) according to the two beam profiles (see Visualization 2).
4.2 Numerical results and analysis
Figure 8 presents the result of a complete numerical damage growth sequence according to the top-hat and Gaussian beams. The maximum intensity corresponds to a fluence of 1.2 J/cm2. The beam is coming from the top of the air domain and the constant propagation is calculated for an angle of incidence of 45$^{\circ }$. In both cases, during the first few iterations, the initial damage grows according to beam direction and the first or the first three layers only are damaged. Then, we notice the occurrence of new damage sites in the first upper layers, visible in Fig. 8(a) #25, (b) #60), along with an asymmetric growth. These features have already been observed experimentally. At the end, the damage site reaches the edges of the beam size and almost the entire MLD stack is removed, as illustrated in Fig. 8(a) #200, (b) #200). For all fluences, the damage reaches at least one boundary of the beam at the end of the computation. The simulation confirms that, with a perfect top-hat beam profile, damage growth is limited only by beam size. The saturation observed experimentally for low fluence after a growth of few microns could be a consequence of the inhomogeneity of the top-hat beam.
Fig. 8. Simulations of a complete damage growth sequence of 200 iterations with (a) the top hat and (b) Gaussian beams. The maximum intensity corresponds to a fluence of 1.2 J/cm2. The beam is coming from the top with an angle of incidence of 45$^{\circ }$. The gray scale represents the refractive index and the color scale represents the EFI distribution. The white arrows point to new damage sites. See Visualization 2 for the complete sequences.
For each iteration, the length of the damaged sites on the top layer was reported as a function of the number of shots in order to compare it with the experimental results shown in Fig. 9. The damage length represents the total integrated length of the main damage site and the new and dissociated damage sites. The distance between the main site and the new damage sites is not considered, analogous to experimental measurements of a single damage area value that comprises the main and any new separate damage sites. However, due to the 3D distribution of the EFI, we expect that the 2D calculation of the EFI distribution will not be identical. In addition, the 2D model only considers the in-plane propagation and all out-of-plane propagation, related to diffraction on the damaged structure, is neglected. We assume that the evolution of the damage length in the 2D model is representative of the evolution of the experimental damage zone. The main result is that we retrieve numerically, above a fluence of 1.2 J/cm2, the three growth regimes that have been observed experimentally for both top-hat and Gaussian beams. The simulation confirms the transition between the initial and main regimes with the occurrence of new damage sites. The duration (number of shots) of the initial regime decreases with increasing fluence. The damage growth rates during the initial regime are identical for the two beams and for each fluence. The initial regime, however, is longer for the Gaussian beam and the rates during the main regime are higher for the top-hat beam. Moreover, for low fluence, the simulations describe a nonlinear evolution of the damage length as illustrated in Fig. 9. This point will be discussed in the next section.
Fig. 9. Simulation of the evolution of the length of the damage site on the upper layer according to the top-hat and Gaussian beams as a function of the number of shots. The damage length represents the length of the main damage site and the new and dissociated damage sites. However, the distance between the main site and the new damage sites was not considered. These data correspond to the growth sequences shown in Fig. 8.
In the same way as presented in section 4, damage growth coefficients were measured to compare the damage growth dynamics with top-hat and Gaussian beams numerically. Only the main regime was considered, and the following relationship was used:
$L_{N}$ ($\mu$m) represents the length of the damage site after N shots, $\alpha$ ($\mu$m/shot) is the linear damage growth coefficient and $L_{0}$ ($\mu$m) represents the initial damage length.
Figure 10 reports the evolution of numerical damage growth coefficients as a function of fluence. The results evidenced similar trends to those observed experimentally. A LIDGT of approximately 0.6 J/cm2 was determined numerically. A slow evolution of the length of the damage site as a function of the number of shots was reported under a fluence of 1.2 J/cm2. This fluence represents the numerical functional LIDGT (see Fig. 10). Above this fluence, the damage growth rate with the top-hat beam is higher than with the Gaussian beam.
Fig. 10. Comparison of the damage growth coefficients obtained with the FEM model according to the top-hat and Gaussian beams.
The main goal of the FEM model is not to reproduce exactly the experiments, but rather to evaluate trends or to illustrate some experimental observations. The input parameters of the model consider a perfect top-hat beam and a perfect mirror design. These contrast with the experimental conditions, with a non-perfect top-hat beam and deposited mirror manufacturing errors. In addition, the incubation and deposition of debris resulting from the damage growth were not considered [29,30]. These features could explain the difference between the simulations and the experiments. The model provides some answers for various hypotheses that will be discussed in the next section.
5. Discussion
The study of damage growth aims to define guidelines for the lifetime of optical components operating in high-power laser facilities. The size of the beam is generally in the order of a few centimeters. In the laboratory, it is very difficult to reproduce such a large beam. The various experiments were carried out with micrometric beams and the representativeness is therefore questionable.
5.1 Characterization of the damage growth phenomenon with a Gaussian beam
We observed, both experimentally and numerically, a higher damage growth rate with the top-hat beam than with the Gaussian beam (see Fig. 6). We assume that the size of the Gaussian beam and especially the intensity gradient could influence these results.
Numerical damage growth sequences with Gaussian beams of waist equal to 70 and 114.5 $\mu$m at 1/e were performed. The comparison was made in the case where the intensity maxima were equal and calculated from a fluence of 1.2 J/cm2. The pulse total energy and the intensity gradients were therefore different. Figure 11 shows the evolution of the damage length as a function of the number of shots according to the two beam sizes. In both cases, the damage length evolves with the same dynamic, but the damage site with a 70 $\mu$m beam waist saturates after 90 shots, while the other one saturates after 180 shots. This fact could explain the significant difference in damage growth dynamics with the top-hat and Gaussian beams observed experimentally and the small difference obtained numerically. It will be interesting to perform experiments with a top-hat and a Gaussian beam with an identical equivalent surface. In conclusion, we show that the size and the intensity gradient of Gaussian beam impact the measurement of damage growth dynamics.
Fig. 11. Evolution of the length of the damage site on the top layer according to the two different Gaussian beam sizes as a function of the number of shots. The two beams have the same maximum intensity calculated for a fluence of 1.2 J/cm2.
5.2 Linear characterization of damage growth phenomenon
If we consider the initial and the main regimes together, the affine relationship is not relevant. Moreover, we notice experimentally and numerically for some cases that the linear characterization of the main regime is questionable as illustrated in Fig. 9. The damage length was measured with the FEM model as a function of the number of shots according to the right and left sides of the initial damage site (see Fig. 12). In this case, the graph shows that the damage growth dynamic is a consequence of asymmetric growth. This fact has been already observed on MLD gratings [23]. The damage growth saturates on the right side while, at the same time, the damage growth starts on the left side with a different dynamic. With the increase of the incident fluence, the delay (number of shots) between the beginning of the damage growth on both sides is reduced. For high fluences, the delay is almost equal to zero and the main regime can be described with an affine relationship. This result shows that the damage growth phenomenon is not linear in its totality but linear per part. It should be necessary to describe damage growth dynamics with another method, for example, by determining the number of shots to reach a certain dimension related to incident beam size.
Fig. 12. Evolution of the total length of the damage site to the right and left side of the initial damage as a function of the number of shots with a top-hat beam of 200 $\mu$m at FMHM. The maximum intensity was calculated for a fluence of 1.2 J/cm2.
5.3 Influence of the beam size
One sensitive aspect of the damage growth phenomenon is the incident beam size. The top-hat beam profile simplifies the characterization of the damage growth phenomenon, but we have seen that beam size could limit the damage growth dynamics. We investigated the impact of beam size with the FEM model. Damage growth sequences were performed with different top-hat beam sizes. The incident intensity is equal in all cases and equivalent to a fluence of 1.8 J/cm2. The width of the geometry was increased to 1.4 mm to preserve the same mesh grid between each simulation. Figure 13 shows the evolution of the damage length following different top-hat widths at FWHM. The initial regime dynamic is the same for all beam sizes and the transition to the main regime occurs after the same number of shots or the same damage length. In contrast, the main regime dynamic differs markedly according to beam size. The simulations show that larger beams cause more occurrences of new damage sites than the smallest beams and could explain the difference in growth dynamics. With regard to high-power lasers, this result once again raises questions concerning the representativeness of damage growth experiments with micrometric beams. Throughout the simulation however, beam size appeared to have no impact on the dynamic and the duration of the initial regime. It could be sufficient to describe the damage growth dynamic and determine the lifetime of optical components by considering the initial regime only. Performing damage growth experiments with millimetric beams will be necessary and important to improve the understanding of damage growth phenomenon.
Fig. 13. Evolution of the length of the damage site according to the width at FWHM of top-hat beams simulated with the FEM model. The maximum intensity is calculated from a fluence of 1.8 J/cm2. The geometry was expanded to a width of 1.4 mm.
6. Conclusion
To conclude, this study investigated the impact of beam profile on damage growth dynamics for improving the understanding of damage growth phenomenon. The experiments were performed on a MLD mirror with a top-hat and a Gaussian beam in s-polarization. The damage area was measured as a function of the number of shots.
Three different regimes were clearly identified both experimentally and numerically in the damage growth dynamics with the top-hat beam, in agreement with that presented first in [24]. Highlighting three regimes in the damage growth dynamics, however, raises questions concerning the linear characterization of the damage growth phenomenon and should motivate a novel method for characterizing damage growth experiments. To compare the impact of beam profile on damage growth dynamics, linear damage growth coefficients were determined by considering the main regime only with an affine relationship [19,20,23,24]. First, we evidenced that beam profile has no impact on LIDGT. Moreover, we determined a functional LIDGT below which damage growth is slow and could be acceptable for large optics. Next, we measured a higher growth dynamic with the top-hat beam than with the Gaussian beam.
A 2D numerical model has also been developed to reproduce a complete damage growth sequence. The evolution of damage length in the first layer of the MLD stack was considered to evaluate trends and to make comparisons with the experimental results. The same values of LIDGT were estimated in both cases. The model reveals the same three regimes observed experimentally in the damage growth dynamic for the top-hat and Gaussian beams. The model also confirms a higher growth dynamic with the top-hat beam than with the Gaussian beam, in good agreement with the experimental results. To optimize the representativity, a 3D model will be necessary. The use of a 3D model should facilitate the interpretations and comparison, but we were limited because of the computational power required for the volume of interest.
One of the important points that emerges from this work is that homogeneous beams seem more representative than Gaussian beams for describing damage growth dynamics. All the results presented in this article were obtained for one given mirror design. It will be interesting to measure damage growth dynamics with different mirror designs and extend the numerical study to MLD gratings which are also critical components in high-power laser facilities. This work will be benefit from the experimental and numerical tools that have been developed to study the issue of laser damage growth for multilayer dielectric stacks.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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