1. Introduction

KDP crystal is the only kind of nonlinear optical material with the ability of growing in large dimension (>400 mm). As a result, it’s the unique candidate to be large aperture Pockels cells and frequency converters constructing the laser path of inertial confinement fusion (ICF) system [1,2]. Under the irradiation of nanosecond pulse, laser-induced damage occurs at fluences much lower than the intrinsic damage threshold of the material [3,4]. Such phenomenon is mainly ascribed to the precursor defects which might efficiently absorb laser energy at low fluences [5]. Because of the location difference between damage precursors, different types of laser damage might be induced. The most common type of laser damage for KDP crystal is known as bulk damage, which is initiated from vacancies, dislocations or inclusions generated in the crystal growing process [6,7]. Besides, laser-induced surface damage, which mainly arises from cracks, scratches and pits generated in the surface finishing process, is also a major threat to the lifespan of crystal components [8,9]. In the early age, KDP crystals always suffered from serious bulk damage as the laser fluences is only about 5 J/cm² [10,11]. Recently, thanks to the improvement of crystal growth technology and application of laser conditioning technology, bulk damage threshold of KDP crystal can be elevated greatly [12,13]. On the contrary, more and more research results indicate that laser-induced damage on KDP surfaces might occur ahead of bulk damage [14,15]. In addition, surface damage of KDP crystal tends to grow continuously under subsequent laser irradiation, and is more likely to cause catastrophic damage to the crystal components [16]. As a result, laser-induced surface damage of KDP has become a main factor which restricts the output energy of the ICF system.

In order to improve the laser damage performances of KDP crystal, it is crucial to form a clear physical picture of the whole laser-induced damage process. Under nanosecond laser irradiation, it is generally admitted that laser damage of KDP is a consequence of multiple physical processes including energy deposition by precursors, temperature and pressure rise in the absorption center followed by a micro explosion event which results in shockwave formation and fracture of surrounded material [17,18]. Because of its complexity, instead of constructing a unified model describing all these physical details, related research work mainly focuses on some specific stages of the overall damage process [19–21]. For example, rate equation models for describing the excitation of electrons were proposed for studying the laser energy deposition stage [22–24]. Besides, based on thermal diffusion equation, a series of energy absorption models were established for calculating the temperature rise and predicting the damage behavior of KDP [25–27]. By assuming a critical electron density or temperature at which damage occurs, these models were able to predict the laser-induced damage threshold (LIDT) and damage density under different pulse duration as well as pulse shape. However, because of lack of description about the material response after the formation of high temperature and pressure condition, it is impossible to reproduce phenomenon such as shockwave propagation and material fracture observed in the laser damage experiments with the existing models.

Except for studying the laser damage process by theoretical modeling, observing the damage phenomenon and measuring critical parameters involved in this process can also facilitate the understanding of the laser damage mechanism [28,29]. By applying time resolved technique in laser damage experiments, plasma formation, shockwave propagation and material ejection phenomena during the laser damage process could be captured [30,31]. Parameters obtained from these experiments such as absorption size, wave speed and material ejection speed can provide useful information for estimating absorbed energy of the laser damage process. In addition, with the aid of spectral analysis, researchers found that the temporally resolved emission in the laser damage process of several wide band-gap materials, including DKDP crystal, fits the Planckian distribution [32]. According to this phenomenon, it was estimated that the temperature and pressure in the damage process reach as high as ~10000 K and ~25 GPa respectively.

It can be seen that current researches on the laser-induced damage of KDP crystal mainly concentrate on the energy deposition and temperature rising stage of the damage process, while little attention has been paid on the subsequent micro explosion stage. Besides, in comparison with bulk damage, knowledge about the surface damage of KDP crystal is relatively limited. To address these concerns, explosion stage of the laser-induced surface damage process of KDP crystal has been investigated in this paper. By observing and analyzing the microscopic morphologies of surface damage craters, typical precursors inducing KDP surface damage have been identified. Further, equivalent explosion simulation models of the laser-induced damage process have been established via Finite Element (FE) method. Material responses under different explosion energies have been simulated to determine the damage threshold, crater size and shockwave speed during the laser damage process of KDP surface. Finally, laser damage experiments have been carried out to verify the validity of the simulation models.

2. Laser-induced surface damage phenomenon of KDP

Because of its unique material properties, KDP crystal is recognized as one of the most-difficult-to-process laser optics. After years of exploration, researchers found that single point diamond fly-cutting (SPDF) is the most ideal technology for processing large-aperture KDP crystals [33]. In spite of the fact that great improvements have been made to decrease the roughness and flatness error of KDP surface [34,35], it is still impossible to prevent the formation of surface defects under current fly-cutting condition. Observation of KDP surface defects through atomic force microscope (AFM) shows that these defects can be classified as cracks/fracture pits, surface protuberances and plastic scratches according to their morphology characters, as shown in Fig. 1. As a kind of extremely brittle material, the brittle ductile transition (BDT) depth of KDP is only about hundreds of nanometers [36]. During the fly-cutting process, if the cutting parameters are improperly selected or cutting tool is encountered with undesirable vibration, the crystal will be removed in brittle mode and cracks as shown in Fig. 1(a) will be generated on the finished surface. Another kind of common surface defects is called as surface protuberances, which appear as local convexities with height about 1 μm and width between 20 and 80 μm, as illustrated in Fig. 1(b). Because all the protuberances are distributed on the mounting surfaces, it can be inferred that this kind of surface defects is introduced in the vacuum adsorption stage of the cutting process. In processing KDP crystal by SPDF, large amount of chip is generated and suspended in air. During the subsequent adsorption stage, the chip falling on the chuck or crystal surface is pressed by the adsorption pressure and adheres firmly on KDP surface to form the surface protuberances. Besides, plastic scratches introduced in the wiping and cleaning stages are also a kind of common surface defects, as shown in Fig. 1(c). More details about the characters and formations of KDP surface defects can be found in [15].

 

Fig. 1 Typical KDP surface defects generated in the fly-cutting process: (a) crack, (b) surface protuberance and (c) plastic scratch.

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To figure out the main factors reducing the surface damage threshold of KDP crystal, laser damage performances of typical surface defects have been studied experimentally. By observing the morphology changes of KDP surface defects under the irradiation of nanosecond laser (355nm) with fluences increasing from 3 J/cm² to 30 J/cm², damage initiation position and threshold of each surface defect could be obtained. Besides, damage performances of defects-free KDP surfaces were also studied for comparison, and these results are shown in Fig. 2(a)-2(d). In each of the subfigures, the three images in the left are captured by optical microscope while the magnified morphologies of the damage craters on the right are captured by scanning electron microscope (SEM). To ensure the accuracy of the experimental results, at least ten surface defects belonging to the same type were tested for calculating the LIDT. It can be seen from Fig. 2(a) and 2(b) that damage craters are just originating from the cracks and protuberances when these two kinds of surface defects are shot by laser. Moreover, Fig. 2(e) shows that LIDT of cracks and surface protuberances is 8.2 ± 1.9 J/cm² and 9.4 ± 2.1 J/cm² respectively, which are far below the damage threshold of defects-free KDP surfaces. Therefore, it can be inferred that surface damage in these two cases should be ascribed to energy absorption by cracks and surface protuberances themselves. In contrast, when plastic scratches are under laser irradiation, surface damage occurs at nearby defects-free area rather than the scratches as fluences exceed 18 J/cm², and that value is almost equal to the damage threshold of defects-free KDP surfaces, as shown in Fig. 2(c) and 2(d). SEM images of the damage craters indicate that both damage sites formed on defects-free KDP surfaces and near the plastic scratches contain a central core with dimension about several microns, and that is quite similar to the morphology of the bulk damage sites of KDP reported by Carr et al [37]. As a result, it can be inferred that surface damage in these two cases are actually caused by near-surface bulk damage eruption.

 

Fig. 2 KDP surface damage craters initiating from (a) crack, (b) surface protuberance, (c) plastic scratch, and (d) defects-free surface. (e) Comparison of the LIDT between typical surface defects and defects-free KDP surface.

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According to these damage phenomena, we propose that typical damage precursors inducing KDP surface damage mainly include cracks, surface protuberances and near-surface bulk defects, as illustrated in Fig. 3(a). Under nanosecond laser irradiation, laser-induced damage process arising from these precursors is pictorially presented in Fig. 3(b). Firstly, interaction between laser and damage precursors produces a large amount of free electrons and results in a fast temperature rise in the energy deposition center. Then, solid material around the deposition center is heated and changed into gas and plasma, making the pressure in this local area reach up to several tens of GPa. On the other hand, the mechanical strength of KDP crystal is only about several hundreds MPa [38], which means the solid material surrounding the phase transition region is unable to withstand such high pressure. Consequently, the absorption energy releases in the form of physical explosion, resulting in the formation of shockwave and fracture of the surrounded solid material to form damage craters. It can be seen that laser-induced surface damage of KDP crystal is a multi-physical coupling problem and it is extremely difficult to establish a comprehensive model for describing the whole process. However, by ignoring the physical details in the energy deposition stage and setting pressure or absorption energy of the gas and plasma as initial condition of the explosion stage, the complicated laser damage process could be simplified as an equivalent explosion process. In this way, laser-induced damage process can be studied through explosion simulation.

 

Fig. 3 Schematic illustrations of (a) typical precursor defects inducing KDP surface damage, (b) main stages and phenomena involved in the laser-induced surface damage process.

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3. Simulation on the laser-induced surface damage of KDP crystal

In this section, equivalent explosion simulation models will be established for studying the laser-induced surface damage of KDP crystal. Formation of damage craters, thresholds of explosion energy as well as propagation of shockwave can be directly extracted from the simulation results.

3.1 Explosion simulation models

As we have mentioned above, laser damage on KDP surfaces are mainly initiated from three kinds of damage precursors: near-surface bulk defects, cracks and surface protuberances. Under the effect of nanosecond laser, these precursors absorb laser energy to form a high-pressure phase transition region. Moreover, it has been reported that diameter of the phase transition region is on the order microns [17,20]. Based on these understandings, equivalent explosion simulation models of the laser damage events originating from near-surface bulk defects, cracks and surface protuberances can be established, as illustrated in Fig. 4(a), 4(b) and 4(c) respectively. In these models, Arbitrary Lagrange Euler (ALE) method is utilized to simulate the fluid behavior of the material in phase transition region, while Lagrangian method combined with stress failure criterion is used for simulating the fracture of KDP, which is realized by element erosion. Besides, air model has also been constructed for studying the shockwave propagation in air. Other details of the simulation models including the material model of KDP crystal, equation of state (EOS) of the phase transition region and estimation of the explosion energy will be elaborated below.

 

Fig. 4 Equivalent explosion simulation models of the laser damage events originating from (a) near-surface bulk defect, (b) crack and (c) surface protuberance.

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3.1.1 Material models of KDP crystal

According to the microscopic morphologies of the damage craters, it can be shown that brittle fracture is the dominant mode in the damage process, while plastic deformation of the crystal is not evident. Therefore, KDP crystal can be regarded as elastic-brittle material in the laser-induced damage process. Moreover, it is well known that KDP crystal is a kind of anisotropic material, and its stress-strain relationship in the elastic range can be written as [39]:

On the other hand, KDP is a kind of brittle material, and its mechanical properties under quasi-static state have been studied extensively [40,41]. However, during the laser-induced damage process, fracture of KDP occurs at much higher strain rate, and the documented parameters are unsuitable for describing the material response under that condition. Aiming at this problem, fracture properties of KDP crystal under higher strain rate are investigated. Through Split Hopkinson Pressure Bar (SHPB) tests and FE simulation, a rate dependent stress failure model for KDP crystal has been obtained, as shown in Eq. (2):

3.1.2 EOS of air and phase transition region

In the simulation model, air near the surface of KDP crystal is regarded as ideal gas and its EOS can be written as:

To describe the extreme material in the phase transition region, the PUFF EOS which interpolates continuously between Mie-Grüneisen solid and perfect gas behavior was proposed by Duchateau et al [42]. In our simulation model, the PUFF EOS model is adopted for studying the laser-induced damage of KDP. The Mie-Grüneisen EOS is expressed as:

3.1.3 Estimation of the explosion energy

As the initial condition of the explosion stage, explosion energy is a crucial parameter of the simulation model. Though it is impossible to know the accurate value of the explosion energy, its variation range can be roughly estimated by [42]:

After these material parameters and initial conditions are properly set, the simulation models would be imported into LS-DYNA solver for simulating the damage process.

3.2 Simulation results and discussions

3.2.1 Formation of surface damage craters

Figure 5 shows the simulated morphologies of the explosion zone under the effect of different explosion energies. According to these results, explosion damage thresholds (EDT) of KDP surfaces can be determined. For micro explosion events initiating from near-surface bulk defects, it can be seen from Fig. 5(a) that no damage occurs as the explosion energy is lower than 6e⁻¹⁰ J. Then, as the explosion energy reaches up to 8e⁻¹⁰ J, crystal material surrounding the explosion center collapses and bulk damage crater is formed. Subsequently, as the explosion energy further increases up to 1.2e⁻⁹ J, the damage zone would expand to crystal surface and this value is thus determined as the EDT of near-surface bulk defects. In the case of explosion arising from cracks, it can be seen from Fig. 5(b) that though there is no damage as the explosion energy is 2e⁻¹⁰ J, stress concentration appears at the crack tip. Just because of the elevated stress level induced by crack, the neighboring crystal material would fracture at low explosion energy and EDT of crack is only 8e⁻¹⁰ J. When explosion event is initiated from surface protuberance, no damage crater is formed on KDP surface until explosion energy increases up to 1.4e⁻⁹ J, just as shown in Fig. 5(c). In summary, the simulated EDT of KDP crystal is between 8e⁻¹⁰~1.4e⁻⁹ J, which corresponds to an energy density about 200~350 J/cm³. This result is coincident with the absorbed energy density estimated by Demos et al [22,23], and it can prove the rationality of the simulation results.

 

Fig. 5 Morphologies of damage craters caused by explosion events initiated from (a) near-surface bulk defect, (b) crack and (c) surface protuberance.

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Besides, comparison between the simulated EDT and measured LIDT can provide additional evidence for understanding the laser-induced damage mechanism of KDP surface. According to Fig. 5(a) and 5(b), it can be known that EDT of crack is about 0.6 times of the threshold of near-surface bulk defect. On the other hand, Fig. 2(e) shows that LIDT of crack is about 0.4 times of the threshold of defects-free KDP surface. Similarity between these two results indicates that mechanical strength degradation by crack is the main factor to reduce its laser damage resistance. For surface protuberance, it can be seen that its EDT is the highest but its LIDT is almost equal to that of crack. An explanation of such discrepancy is that surface protuberance should possess stronger absorption ability, and it could absorb adequate energy to cause surface damage at much lower laser fluences. To prove this idea, fluorescence characters of KDP surface defects, which can reveal their laser absorption ability, have been studied and the results are shown in Fig. 6. It can be seen that fluorescence intensity of surface protuberance is much stronger than that of the surrounding defects-free surface as well as the crack, and this is coincident with our analysis result. In conclusion, laser damage performance of KDP surface is decided by many factors such the mechanical strength and absorption ability, and any degradation of these characters can lead to the reduction of LIDT.

 

Fig. 6 Fluorescence intensity of (a) KDP surface protuberance is much stronger than (b) crack and the surrounding defects-free area.

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Figure 7 shows the influences of explosion energy on the dimension of damage craters initiating from near-surface bulk defects. Simulation results demonstrate that diameter of surface damage crater D_cra increases with the increment of explosion energy E_exp. Besides, further analysis on the simulation results shows that D_cra is approximately proportional to $\sqrt[3]{E_{exp}}$.This relationship is consistent with the empirical formula proposed by Wang et al [43], and its correctness will be testified in the next section.

 

Fig. 7 Influences of explosion energy on the diameter of damage craters initiating from near surface bulk defects.

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3.2.2 Propagation of stress wave

Figure 8(a)-8(c) illustrate the propagation of stress wave in KDP crystal under different explosion energy. It should be noted that observation direction is perpendicular to the (001) plane of the crystal. Simulation results show that the stress wave front forms a crossed double-ellipse pattern, and this phenomenon suggests that wave speed is varied with propagating direction. By measuring the propagating distance of the wave front at different time, wave velocity under different explosion energy can be determined, just as shown in Fig. 8(d). In the first 5 nanoseconds, wave velocity increases with the increment of explosion energy, which means the stress propagates in the form of shockwave at the initial period after explosion. Then, wave speed along a specific direction almost keeps constant in the subsequent propagation process, and it doesn’t change with explosion energy. Moreover, it can be found that the simulated wave speed along [100] direction is 5.45 km/s, which is exactly equal to the elastic wave speed along this direction determined by $c_{[100]}=\sqrt{C_{11}/\rho }=5.47\text{km/s}$. As a result, it can be concluded that stress wave in KDP crystal propagate as elastic stress wave at time > 5 ns. These simulation results including the shape of the wave front, wave speed and its variation trend show good agreement with the experiment phenomena reported by H. Jiang et al [30].

 

Fig. 8 Simulated pressure distribution in KDP crystal at different time points as the explosion energy is (a) 2e⁻¹⁰ J, (b) 1e⁻⁹ J and (c) 5e⁻⁹ J. (d) Influences of explosion energy on the propagating speed of stress wave in KDP crystal.

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Figure 9(a), 9(b) and 9(c) show the propagation of air stress wave caused by explosion initiating from cracks. Simulation results indicate that stress wave in air propagates at the same speed along different direction, and that is quite different from the anisotropic propagating mode of the stress wave in KDP crystal. Besides, at different times (5 ns, 20 ns and 50 ns) after the explosion, it can be seen that diameter of the wave front increases with the increment of explosion energy. Such phenomenon indicates that stress in air propagates as shockwave during the entire simulation period (0~50 ns). By measuring the diameter of the wave front at different times, shockwave velocity under different explosion energies can be calculated and the results are shown in Fig. 9(d). It can be seen from these curves that wave velocity reaches the maximum at several nanoseconds after the explosion. For explosion energy ranging from 1e⁻⁹ J to 5e⁻⁹ J, the maximal value of shockwave velocity approximately varies between 5 km/s and 8 km/s. In the subsequent propagation process, wave velocity drops continuously and it is between 1 and 2 km/s at 50 ns. Besides, one can conclude from these simulation results that propagation speed of shockwave in air is sensitive to explosion energy.

 

Fig. 9 Simulated pressure distribution in air at different time points as the explosion energy is (a) 1e⁻⁹ J, (b) 2e⁻⁹ J and (c) 5e⁻⁹ J. (d) Influences of explosion energy on the propagating speed of shockwave in air.

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In summary, EDT of KDP surfaces, formation of damage craters and propagation of stress wave in KDP/air can be obtained conveniently by these simulation models. Hereinto, rationalities of the simulated EDT and correctness of the wave speed in KDP crystal have been proved by comparing with the results reported in literature, while other simulation results will be verified in the following section.

4. Laser damage experiments on KDP surface

In order to validate the correctness of the simulation results as shown in Fig. 7, laser damage experiments on defects-free KDP surfaces have been carried out to investigate the impact of laser fluences on the dimension of damage craters. Before experiment, KDP samples with dimension of 50 × 50 × 10 mm³ were carefully processed by SPDF technique, where the feed rate and cutting depth were respectively set as 10 μm/r and 2 μm for acquiring crack-free surface. During the test, Nd:YAG laser with a wavelength of 355 nm was focused on the front surface of KDP. Because it is known that LIDT of defects-free KDP surface is about 20 J/cm² (355 nm, 3 ns), laser fluences in this experiment was set to 20 J/cm², 30 J/cm², 40 J/cm², 60 J/cm², 80 J/cm² and 100 J/cm². To guarantee the reliability of the experimental results, more than 20 damage tests were performed under each of the laser fluences. After the damage test, microscopic morphologies of the damage craters were captured and the damage area could be determined.

Figure 10(a), 10(b) and 10(c) illustrate typical morphologies of KDP surface damage craters as the fluences are respectively 20 J/cm², 60 J/cm² and 80 J/cm². It can be seen that there is an increasing trend of damage area with the raising of laser fluences, and such increment is more apparent as the fluences exceeds 60 J/cm². Moreover, both of the damage craters formed in the cases of 20 J/cm² and 60 J/cm² contain only one central core, while multiple cores are generated as the fluences is 80 J/cm². Figure 10(d) shows the relationship between the square root of damage area A_cra and laser fluences F. Fitting result indicates that $\sqrt{A_{cra}}$ is approximately proportional to $\sqrt[3]{F}$ as the laser fluences is lower than 60 J/cm². This relation shows good agreement with the fitting curve as shown in Fig. 7 and thus has proved the correctness of the simulation result. However, the measured damage dimensions deviate from the fitting curve as the fluences is higher than 60 J/cm². This is mainly because several damage precursors are activated and multiple explosion events happen simultaneously at high laser fluences, as can be seen from the SEM image of the damage crater. Under that condition, single core explosion model as shown in Fig. 4 is not appropriate for describing the damage process. Instead, multicore explosion model should be established for studying the damage behavior of KDP surface irradiated by laser with high fluences, and it will be considered in the future research.

 

Fig. 10 SEM images of KDP surface damage craters formed under laser fluences of (a) 20 J/cm², (b) 60 J/cm² and (c) 80 J/cm². (d) Influences of laser fluences on the dimension of damage craters.

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Besides, time resolved pump and probe (TRPP) technology is utilized to study the shockwave propagation phenomenon during the laser-induced damage of KDP surface. Figure 11 illustrates the schematic layout of the TRPP experiment setup. The Nd:YAG laser with 355 nm wavelength and 4.5 ns pulse duration is used as the pump light, and the probe light is a Nd:YLF laser with wavelength of 527 nm and duration of 50 ps. Delay between the pump and probe laser is achieved by using a digital delay generator, and the time interval can be varied arbitrarily between 1ns and 100 μs. A CCD camera equipped with a zooming lens module is utilized to capture the shadow images of the damage process. In this experiment, shockwave generated in the damage event arising from surface cracks is investigated for comparing with the simulation results in Fig. 9. To this end, large amount of cracks are made on KDP surface by setting the feed rate as 30 μm/r [36]. For damage event under specific laser fluence, shockwave at 5 ns, 15 ns, 30 ns and 50 ns delay are recorded, and more than 5 images at each moment are captured to ensure the reliability of the test results.

 

Fig. 11 Schematic layout of the TRPP setup for observing the shockwave propagation phenomenon.

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Figure 12(a), 12(b) and 12(c) shows the image series of shockwave propagating in air as the laser fluences is 15 J/cm², 20 J/cm² and 30 J/cm² respectively. Experimental results prove that air shockwave forms a circular front, and dimension of the wave front increases with the increment of laser fluences. By measuring the propagating distance of the shockwave at different times, average velocity of the shockwave can be determined and the results are shown in Fig. 12(d). It can be seen that the measured shockwave velocity ranges from 6 to 10 km/s at the first few nanoseconds and decreases rapidly to 1-2 km/s at 50 ns. Besides, the variation trend and the decay rule of the measured wave velocity curves show good agreement with the simulation results as shown in Fig. 9, and thus has proved the validity of the simulation models.

 

Fig. 12 Transient image of air shockwave at 5 ns, 30 ns and 50 ns as the laser fluences is (a) 15 J/cm², (b) 20 J/cm² and (c) 30 J/cm². (d) Impact of laser fluences on the air shockwave velocity.

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5. Conclusion

According to the laser-induced damage phenomenon of KDP crystal, it is known that typical damage precursors leading to surface damage mainly include cracks, surface protuberances and near-surface bulk defects. By simplifying laser-induced damage arising from these precursors as equivalent explosion events, simulation models of the damage process are established and phenomena such as damage crater formation and shock wave propagation are successfully reproduced. Simulation results indicate that mechanical strength degradation by crack is the main factor to reduce its damage resistance, while strong absorption by surface protuberance should be responsible for the reduction of LIDT. Besides, stress wave in KDP crystal forms a crossed double-ellipse pattern and it propagates as elastic wave several nanoseconds after the damage occurs. In contrast, stress propagates in the form of shockwave in air. By performing laser damage tests and TRPP experiments on KDP surface, impact of laser fluences on the dimension of damage craters as well as propagation of shockwave in air are investigated. Comparison between the simulation and experimental results indicate that single core explosion model can predict the damage dimension variation trend and shockwave velocity decay rule as fluences is below 60 J/cm², while a multicore explosion model is needed to reliably simulate the damage area at higher fluences. In conclusion, studying the laser-induced damage of KDP crystal by explosion simulation is feasible, and such method can be expected to provide more insight into understanding the laser damage mechanism of the material.

Appendix

To obtain material parameters indispensable for constructing the explosion simulation model, SHPB experiment combined with FE simulation are performed to study the fracture properties of KDP crystal. Fig. 13 illustrates the construction and principle of the SHPB experiment system. Under the effect of impact load, a compressional wave is generated in the incident bar. When this stress wave propagates to the sample, reflected wave and transmitted wave is respectively generated in the incident bar and transmitted bar. By measuring the reflected signal ε_r and transmitted signal ε_t with strain gauge, the stress and strain of KDP sample during the test can be calculated:

(6)$$\begin{array}{l}{\dot{\epsilon }}_{\text{s}}=2c·{\epsilon }_r/L_s\\ {\epsilon }_s=2c·{\displaystyle {\int }_0^t{\epsilon }_{r}dt}/L_{\text{S}}\\ {\sigma }_s=S_B·E·{\epsilon }_t/S_S\end{array}$$

where ${\dot{\epsilon }}_{\text{s}}$, ${\epsilon }_s$ and ${\sigma }_s$is the strain rate, strain and stress of the sample, c represents the velocity of the compressional wave, E is the Young’s modulus of the bar, L_S is the length of the sample and S_S is its cross-section area. By adjusting the impact velocity, the stress-strain relation of KDP crystal under different strain rate can be obtained.

 

Fig. 13 Schematic layout of the SHPB experiment setup.

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Besides, FE simulation model of the SHPB experiment has been constructed, as illustrated in Fig. 14. In view of the symmetric character of the problem, only 1/4 model of the impact process has been established for reducing computation time. To ensure that the simulation and experiment conditions are consistent, all the geometry parameters in the model are identical with the experiment parameters. Other details of the model such as mesh discretization and boundary conditions are demonstrated in the figure. For simulating the damage behavior of KDP crystal, a strain rate dependent failure model is used and it is expressed as ${\sigma }_F=A(1+B\ln \dot{\epsilon })$. In this model, A and B are unknown parameters need to be determined.

 

Fig. 14 FE simulation model of the SHPB experiment.

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By comparing the simulation and the experimental results, fracture parameters of KDP can be determined. Firstly, reflected and transmitted strain can be obtained directly through SHPB experiment, just as shown in Fig. 15(a) and 15(b). According to these results, stress-strain relation of KDP under different strain rate can be determined, as illustrated in Fig. 15(c). Experiment result indicates that the nominal failure stress (the maximal point in the stress-strain curve) of KDP decrease with the increment of strain rate. On the other hand, fracture parameters A and B are adjusted repeatedly in the simulation model, and the simulation results under different parameter combinations are compared with the experimental results. Finally, it is found that the simulated and measured strain curves coincide well as A is 117 MPa and B is −0.017, as shown in Fig. 15(d) and 15(e). As a result, the failure stress of KDP is ultimately determined as ${\sigma }_F=117(1-0.017\ln \dot{\epsilon })\text{MPa}$. In addition, fracture process of KDP crystal during the SHPB test can be obtained, as shown in Fig. 15(f).

 

Fig. 15 SHPB experiment results of KDP: (a) reflected strain, (b) transmitted strain and (c) stress-strain relation. SHPB simulation results: (d) reflected strain, (e) transmitted strain and (f) fracture process of crystal.

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Funding

National Natural Science Foundation of China (51702302); Science Challenge Project (TZ2016006-0503-02).

Acknowledgment

The authors gratefully thank Dr. Feng Jiang and Mr. Xiaosheng Luan for providing help in SHPB test, both of whom are from Huaqiao University, P.R. China.

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