1. Introduction

Deuterated potassium dihydrogen phosphate (DKDP) is the isomorph of KDP (KH₂PO₄) crystal. In comparison with KDP, DKDP crystal not only inherits the outstanding nonlinear optical properties from its isomorph, but also possesses lower transverse stimulated Raman scattering (TSRS) gain coefficient, and thus can effectively suppress TSRS damage [1]. As a result, DKDP crystals are widely employed in inertial confinement fusion (ICF) systems, such as the National Ignition Facility (NIF) and Laser Megajoule (LMJ) [2,3]. In these laser systems, DKDP crystals serve as third-harmonic generators (THG), whose function is to convert the 1053 nm (1ω) and 526 nm (2ω) lasers into a 351 nm (3ω) laser. However, experiment results from NIF and other laser systems show that DKDP crystals are the most damage-prone optics under current fluences [4–6]. In other words, laser-induced damage of DKDP is a bottleneck problem that limits the output energy of ICF systems.

During the operation of ICF systems, laser-induced damage always occurs in the bulk or on the rear surfaces of DKDP crystals as the fluence is far below its intrinsic damage threshold. This phenomenon is mainly ascribed to precursor defects which can strongly absorb laser energy at low fluences [7]. Previous studies demonstrated that bulk damage manifesting as a pinpoint was the dominant damage mode for DKDP when the fluence was lower than 5 J/cm², and it had been confirmed that these damage sites were initiated from bulk defects introduced in the crystal growth process [8]. Therefore, a series of techniques such as continuous solution filtration and laser conditioning were employed to improve the bulk damage threshold [9,10]. As the bulk threshold increased, recent damage experiments on DKDP crystals show that surface damage might occur ahead of bulk damage when the fluence exceeds 6 J/cm² [4,5]. In addition, surface damage is more likely to cause serious destruction to crystal components because the damage craters tend to grow continuously during the subsequent laser shot [6,11]. Therefore, determining the precursor defects responsible for surface damage is crucial to improve the damage performance of DKDP components.

Considering the location, it can be inferred that laser-induced surface damage is most probably initiated from microstructural defects introduced by surface finishing processes, such as the single point diamond fly-cutting (SPDF) process for KDP/DKDP crystals [12,13]. During the cutting process, different types of surface microstructures can be generated [14,15]. Hou et al. [16] discovered that ultra-precision fly-cutting can create a lattice misalignment structure (LMS) layer in the subsurface of KDP crystals. GIXRD tests showed that dislocation plays an important role in the evolution of the LMS. Hu et al. [17] conducted an ab initio molecular dynamics study on the properties of LMS. They discovered that the formation of LMS can increase the possibility of material decomposition, which may affect the damage threshold of KDP components. Apart from dislocations and LMS, another common type of microstructural defect on the KDP/DKDP surface is crack/pit caused by material fracture [18,19]. Wang et al. [20] studied the influence of the crystalline orientation on the formation of cracks and proposed a theoretical model for achieving crack-free surfaces. Zhang et al. [21] found that cutting tool geometry is an important factor that affects crack formation in the SPDF process of KDP crystals. In summary, the formation of microstructural defects on KDP surfaces has been studied intensively; however, knowledge of DKDP surface microstructural defects generated by SPDF is rather limited. In addition, the correlation between the formation of surface microstructural defects and degradation of laser-induced damage performance is not yet clear.

To understand the deformation mechanism of KDP/DKDP crystal, a series of indentation and scratch tests have been performed. Guin et al. [22] found that the plastic deformation of KDP/DKDP crystal is mainly due to slip motion by dislocation. Etching experiments showed that there are two slip systems for KDP/DKDP: one is in the (110), (101), (112), and (123) planes with a common Burgers vector of 1/2[111], and the other is in the (010) plane with a Burgers vector of [100]. Kucheyev et al. [23] performed nano-indentation tests on the (001) and (100) planes of DKDP crystal. The results demonstrated that slip is the major mode of plastic deformation in DKDP, and pop-in events in the force-displacement curves are attributed to the initiation of slip. Liu et al. [24] studied the material removal mechanism of KDP crystals at different temperatures using the nano-scratch technique. They found that a large number of nano-grits resulting from crack propagation were generated in the subsurface at room temperature, while at elevated temperatures, some crystallographic LMS and nanocrystals were generated. These studies have made great advances in understanding the deformation behavior of KDP/DKDP in indentation and scratch processes. However, for the actual cutting process, the material deformation mechanism closely related to the generation of DKDP surface, such as plastic chip formation and micro crack propagation, is still not well understood.

In this study, the influence of SPDF parameters on the generation of DKDP surface and the underlying material deformation mechanism were investigated. The formation of surface microstructural defects and their impact on the laser-induced damage performance of DKDP crystals were analyzed. The objective of this study is to reveal the formation mechanism of the DKDP surface processed by SPDF and determine the correlation between surface defects formation and laser-induced damage performance degradation.

2. Material and experiments

2.1 Material

Single crystal DKDP is a soft and brittle material belonging to the tetragonal system with the I-42d space group. In the crystal lattice of DKDP, (PO₄) tetrahedrons are linked through O—D…O bonds to form a complicated network, as illustrated in Fig. 1(a). To realize third-harmonic generation required in the ICF system, the DKDP crystal must be sliced along a specific crystalline orientation, as shown in Fig. 1(b). In this study, DKDP crystals (grown by Shangdong University, China) with 95% deuterium content were sliced into THG samples with dimension of 50 mm × 50 mm × 10 mm. Then, all the samples were delicately processed by SPDF to generate smooth surfaces with roughness Ra < 1.5 nm for the subsequent SPDF and laser-induced damage experiments.

 

Fig. 1. (a) Lattice structure of a pure DKDP crystal; (b) crystalline orientation of THG component shown in the material coordinate system.

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2.2 SPDF experiments

Figure 2 illustrates the principle of processing DKDP crystals by SPDF. A circular diamond cutter mounted at the edge of the fly-cutting disc rotates with the spindle to achieve the principal cutting movement. Because the diameter of the fly-cutting disc is much larger than that of the DKDP sample, the cutting direction is almost along a constant crystalline orientation, which is beneficial to obtain consistent surface quality for anisotropic materials. In this cutting process, the material removal mode, and the resulting surface microstructures may be influenced by the feed rate f, cutting depth a_p, cutting direction θ (as defined in Fig. 1(b)), etc. Therefore, different combinations of fly-cutting parameters, as listed in Table 1, were used in the experiments to create DKDP surfaces with various microstructures.

 

Fig. 2. Schematic illustration of the SPDF process.

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Table 1. Process parameters used in the SPDF experiments

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2.3 Laser-induced damage tests

After the fly-cutting experiments, the laser-induced damage performance of the DKDP surfaces was studied using a small-aperture laser damage test workbench. A Gaussian laser beam with a wavelength of 355 nm was focused on the rear surface, where the spot diameter was approximately 400 µm. To determine the laser-induced damage thresholds, R-on-1 laser-induced damage tests were performed with a starting fluence of ∼ 4 J/cm² and an increment of ∼ 2 J/cm². During the tests, the damage was captured using a CCD camera equipped with a long-working distance microscope. For each tested surface, the damage thresholds at ten different sites were measured. Then, the obtained 7 ns results were converted into 3 ns thresholds using the empirical formula of τ^0.45 to generate damage probability curves.

2.4 Surface characterization methods

To obtain detailed information about the morphological features of the DKDP surface microstructures, the samples processed under different SPDF conditions were characterized using an atomic force microscope (AFM, Bruker Dimension Icon) and scanning electronic microscope equipped with a focused ion beam (SEM-FIB, ZEISS Crossbeam 540). Furthermore, GIXRD (Bruker D8) and nano-indentation/nano-scratch (Hysitron, TI950) tests were performed to reveal the formation mechanism of DKDP surfaces.

3. Results and discussions

3.1 DKDP surfaces created under different SPDF conditions

As the feed rate increased from 5 µm/r to 40 µm/r, while the cutting depth and cutting direction were set to 2 µm and 0°, respectively, the variation in the DKDP surface morphologies were captured. It can be seen from Fig. 3(a) that a smooth surface with roughness R_a= 0.75 nm is generated when the feed rate is 5 µm/r. Under this condition, no surface defects were formed, except for the tool marks along the cutting direction. When the feed rate was increased to 10 µm/r, micro ripples with a period of ∼ 9 µm and an amplitude of several nanometers emerged, resulting in an increase in the surface roughness to 1.43 nm, as shown in Fig. 3(b). Moreover, ripples with the same period and a larger amplitude can also be observed in the cases of f = 20 µm/r and 30 µm/r, as illustrated in Figs. 3(c) and (d). For all the cases, the intersection angle between the ripples and cutting direction is equal to the complementary angle of the cutting edge inclination (35°), which means that the ripples are parallel to the projection of the cutting edge. In previous studies, similar surface ripples were observed in the scratch processes of KDP and other materials, which were attributed to the “stick-slip motion” between the indenter and surface material [25,26].

 

Fig. 3. AFM images of DKDP surfaces processed by SPDF with different feed rates. (a)–(d) Images with a scanning size of 80 µm × 80 µm: (a) f = 5 µm/r; (b) f = 10 µm/r; (c) f = 20 µm/r; (d) f = 30 µm/r, where the blue lines at the bottom show the 2D profiles of the surfaces. (e)–(h) Images with a scanning size of 5 µm × 5 µm: (e) f = 5 µm/r; (f) f = 10 µm/r; (g) f = 20 µm/r; (h) f = 30 µm/r.

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In addition to micro ripples, another type of microstructure generated on the machined DKDP surface is micro grain with size ranging from tens to hundreds of nanometers. It can be observed from Fig. 3(e) that such a defect is inconspicuous when the feed rate is 5 µm/r. However, an increasing number of micro grains were generated when the feed rate exceeded 10 µm/r. In particular, the entire DKDP surface was covered with micro grains when the feed rate reached 20 µm/r, as illustrated in Fig. 3(g). The formation mechanism of the DKDP surface micro grains is discussed in the next section.

When the feed rate was increased continuously to 30 µm/r, cracks and fracture pits were generated, implying that some of the DKDP surface material was removed in the fracture mode, as shown in Figs. 3(d) and (h).

AFM images of the DKDP surfaces created at various cutting depths are shown in Fig. 4. In this experiment, the feed rate and cutting direction were fixed at 10 µm/r and 0°, respectively, and the cutting depth was increased from 1 µm to 10 µm. It can be seen from these images that there is no significant difference in the surface morphology and roughness value of the surfaces created at different cutting depths. Even when the cutting depth reached 10 µm, a crack-free surface with a roughness of R_a = 1.45 nm could be achieved, as shown in Fig. 4(d).

 

Fig. 4. AFM images with a scanning size of 80 µm × 80 µm showing the morphologies of DKDP surfaces processed by SPDF with different cutting depths. (a) a_p = 1 µm; (b) a_p = 3 µm; (c) a_p = 6 µm; (d) a_p = 10 µm.

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In processing the DKDP crystal by SPDF, the maximal undeformed chip thickness (UCT_max) is determined by [18]:

In summary, the cutting depth has little influence on DKDP surface morphology and roughness. Therefore, in processing DKDP crystals using SPDF, a relatively large cutting depth can be selected to improve the machining efficiency.

Figure 5 illustrates the variation of DKDP surface morphology with the change of cutting direction. It can be seen from Figs. 5(a)–(d) that when the feed rate and cutting depth were respectively set to 15 µm/r and 2 µm, cutting along any of the four directions (i. e., 0°, 90°, 180°, 270°) can create a crack-free surface. Moreover, micro ripples parallel to the cutting edge projection is generated on each of the four surfaces. As the feed rate increased to 30 µm/r, the change of cutting direction led to completely different surface morphologies, as shown in Fig. 5(e)–(h). In the cases of cutting along 90° and 180° directions, crack-free surfaces generated by plastic flow can still be obtained. In comparison, cracks and pits caused by material fracture are formed on the machined surfaces as the cutting directions are 0° and 270°. This difference is attributed the anisotropic mechanical property of the crystal. As the feed rate further increased to 40 µm/r, a crack-free surface was created only under the condition of cutting along the 180° direction, while cutting along other three directions would result in the formation of surface cracks and pits, as illustrated in Fig. 5(i)–(l). Therefore, in processing a DKDP crystal by SPDF, 180° is the optimal cutting direction for achieving a crack-free surface.

 

Fig. 5. AFM images with a scanning size of 80 µm × 80 µm showing the morphologies of DKDP surfaces processed by SPDF under different cutting directions and feed rates. (a) θ = 0°, f = 15 µm/r; (b) θ = 90°, f = 15 µm/r; (c) θ = 180°, f = 15 µm/r; (d) θ = 270°, f = 15 µm/r; (e) θ = 0°, f = 30 µm/r; (f) θ = 90°, f = 30 µm/r; (g) θ = 180°, f = 30 µm/r; (h) θ = 270°, f = 30 µm/r; (i) θ = 0°, f = 40 µm/r; (j) θ = 90°, f = 40 µm/r; (k) θ = 180°, f = 40 µm/r; (l) θ = 270°, f = 40 µm/r.

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3.2 Formation mechanism of DKDP surface in SPDF

To figure out the formation mechanism of DKDP surfaces processed by SPDF, the generation of surface microstructures including micro grains, ripples and cracks observed in the previous section was further analyzed. Firstly, GIXRD tests were performed to investigate the lattice structure of micro grains. In the GIXRD tests, the X-ray penetration depth is directly proportional to the incident angle i. Therefore, GIXRD can be used to characterize thin surface layers by setting a small incident angle [28]. Figure 6 illustrates the GIXRD patterns of the DKDP surface shown in Fig. 3(g), which is covered with a layer of micro grains. There are five diffraction peaks corresponding to the (101), (200), (112), (220), and (312) planes, respectively, for the grazing incident angle of i = 1°. In comparison, only three diffraction peaks were observed as the incident angle was increased to 2°, and the intensities of the diffraction peaks were lower. These results are in accordance with the diffraction patterns of the LMS layer reported by Hou et al. [16]. Therefore, the LMS layer reported in literature and micro grains discovered in this study should be the same in essence.

 

Fig. 6. GIXRD patterns of DKDP surface processed by SPDF with the cutting parameters of f = 20 µm/r, a_p = 2 µm and θ = 0°, under which condition the processed surface is covered with a layer of micro grains.

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By comparing the diffraction planes shown in Fig. 6 with the slip systems of the crystal, it can be found that the planes (101) and (112) are identical to the slip systems (101)1/2[111] and (112)1/2[111]. In addition, the (200) and (220) planes are parallel to the slip systems (100)1/2[111] and (110)1/2[111], respectively. Thus, it is reasonable to infer that the formation of micro grains on the DKDP surface is resulted from the slip motion of the crystal. During the SPDF process, the stress at the vicinity of the cutting tool edge exceeds 1 GPa [29,30]. Furthermore, the shear strength of slip systems is approximately 300–500 MPa [31]. Consequently, multiple slip planes are simultaneously activated in the cutting zone. These planes intersect with others to form a network of grain boundaries, resulting in the formation of micro grains.

To prove that the formation of micro grains is caused by the slip motion of the DKDP surface material, nano-indentation and nano-scratch experiments were performed to reveal the deformation behavior of the crystal. For the convenience of analysis, these experiments were carried out on the (001) plane of the DKDP crystal using a spherical indenter with a diameter of 20.7 µm. Figure 7(a) shows the SEM image of the indentation created under a normal force of 20 mN. At the bottom of the figure, the three groups of colored lines represent the theoretical intersections between the crystal surface (i.e., the (001) plane) and different slip systems. It can be seen that a large number of boundaries are generated in the indentation zone, and they coincide well with the expected intersection lines. In contrast, as the indenter slides on the DKDP surface along the [100] and [110] directions, the generated scratches are illustrated in Figs. 7(b) and (c), respectively. Similar to the phenomena observed in Fig. 3, the scratch zones are covered by micro grains with sizes ranging from tens to hundreds of nanometers. Moreover, as the scratch direction changes, the micro grains rotate accordingly and the boundaries always coincide with the expected intersection lines. In conclusion, these results validate that grain boundaries are created by the slip motion of DKDP surface material.

 

Fig. 7. SEM images of the indentation impression and scratch grooves on the (001) plane of DKDP crystal. (a) Impression created under a normal force of 20 mN; (b) Groove created by scratching along [100] direction with a normal force of 20 mN; (c) Groove created by scratching along [110] direction with a normal force of 20 mN.

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Next, the influence of the SDPF process on the formation of DKDP surface cracks is analyzed. According to Fig. 5, it can be found that the initially formed cracks (as indicated by the ellipses) are almost parallel to the projection of the cutting tool edge. Such phenomenon is more evident in the optical microscopic image of the DKDP surface illustrated in Fig. 8(a). To obtain more information about the geometry of the cracks, cross-sections of the cracks were obtained and observed using FIB-SEM, as shown in Fig. 8(b). It can be seen from this image that the crack propagates obliquely into the bulk, where the angle between the crystal surface and crack surface is approximately 10°.

 

Fig. 8. Microscopic morphology of DKDP surface cracks generated by SPDF. (a) Optical image showing the inclination of the cracks on the surface; (b) SEM image of the cross-section showing the crack propagation direction underneath the surface.

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To better understand how these cracks are initiated by SPDF, a finite element (FE) simulation model was established using LS-DYNA software to investigate the stress state in the cutting zone, as shown in Fig. 9(a). In this model, the simulation parameters, such as cutting velocity and tool geometry, were set according to the actual experimental conditions listed in Table 1. Because the rigidity of diamond is much larger than that of DKDP crystal, the cutting tool was simplified to a rigid body, and a self-developed material model was utilized to simulate the deformation behavior of DKDP crystal [29]. The details of the material model are described in the Appendix. Figure 9(b) shows the first principal stress in the cutting zone when the UCT is 100 nm. The simulation result indicates that the substrate material in front of the cutting edge is in a compressive state, whereas the newly formed surface just behind the cutting edge is subjected to a tensile stress with the maximal value of 176.6 MPa. Moreover, the orientation of the first principal stress shown in Fig. 9(c) demonstrates that the angle between the maximal tensile stress (the longest red line) and vertical direction is 12°, implying that the stress direction is almost perpendicular to the crack surface observed in Fig. 8(b). Thus, it can be concluded that the cracks on the DKDP surface are initiated by the tensile stress formed behind the cutting edge.

 

Fig. 9. Simulation on the cutting of DKDP crystal by FE method. (a) Simulation model; (b) Distribution of the first principal stress; (c) Direction of the first principal stress.

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Based on the above analysis, a comprehensive understanding of the DKDP surface formation process can be established, as illustrated in Fig. 10. Depending on the difference in UCT and the resulting variation in the material removal mode, the whole cutting area as shown in Fig. 10(a), can be roughly divided into three regions: the elastic recovery region (region I), the plastic slip region (region II) and the plastic-fracture concomitant region (region III).

 

Fig. 10. Schematics of the formation of DKDP surface processed by SPDF. (a) Material deformation mode varies with the increase of UCT; (b) Elastic recovery is the dominant deformation mode in region I; (c) Plastic chip flow in region II is realized by slip and GBS, resulting in the formation of micro grains and ripples; (d) Plastic flow and brittle fracture co-exist in region III, leading to the formation of micro grains, ripples and cracks.

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In region I, the UCT is so small as the generated stress is insufficient to cause massive slip motions. Therefore, neither micro grain nor chip is formed in this region, and the deformation of DKDP crystal is dominated by elastic recovery, as shown in Fig. 10(b). According to the minimum chip thickness (MCT) theory and the relation L_I = (0.2–0.3) ·ρ·R /f [32], it can be calculated that the width of region I L_I is in the range of 13–19 µm as f = 5 µm/r, and decreases to 6.5–9.5 µm as f increases to 10 µm/r. This result can explain the surface phenomena observed in Fig. 3, where a smooth surface is generated when f is 5 µm/r, while relatively rough surfaces with micro grains are formed when f exceeds 10 µm/r.

With a continuous increase of UCT, the material removal mode of DKDP crystal is transformed to plastic flow, thus entering the plastic slip region, as illustrated in Fig. 10(c). In this region, the stress generated in front of the cutting edge is sufficiently large to induce massive slip motions. Consequently, several slip planes are simultaneously activated. The intersections of these planes convert the material from a single-crystalline to a poly-crystalline structure, just as the phenomena observed in Fig. 8. Such variations could change the mechanical property of the local material, enabling the crystal to achieve large plastic deformation through grain boundary sliding (GBS) [33], which is an important plastic deformation mechanism for materials whose homologous temperature T/T_m is above 0.4, where T and T_m are the material and melt temperatures, respectively (for DKDP crystal at room temperature, T/T_m is approximately 0.56). Therefore, in the subsequent cutting process, the micro grains in front of the cutting edge could slide under the effect of shear stress to form chip. It is notable that the maximal shear stress is distributed in the primary shear zone; therefore, the micro grains should also slide along the shear plane direction. Consequently, the chip separation point moves upward gradually, resulting in an increase of the residual height, as shown in Fig. 10(c-iii). However, this state is unstable because the contact between the tool edge and ascending surface will increase the stress near the cutting edge and produce more severe slip motions. As the shear stress in the vicinity of the cutting edge exceeds the critical value for sliding, the chip separation point will be drawn back to its original height position, as shown in Fig. 10(c-iv), leading to a sudden decrease of the residual height. The repetition of these processes generates micro ripples on the finished surface, as shown in Fig. 10(c-v).

Figure 10(d) illustrates the formation of DKDP surface in region III, where the UCT is larger than the BDT depth of the crystal. In this region, plastic flow realized by GBS is still the major material removal mode, so the generated surface morphology contains microstructures like those of region II, such as micro grains and ripples. In addition, because of the increase in UCT, the tensile stress formed behind the cutting edge exceeds the fracture stress of the crystal. Consequently, cracks and fracture pits are generated in this region.

3.3 Laser-induced damage performance of DKDP surface

Figure 11 shows the laser-induced damage probability curves of DKDP surfaces generated under different SDPF conditions. For each curve, a damage probability of 10% indicates that one of the ten tested sites was damaged. Analogously, if all sites were damaged, a damage probability of 100% was assigned. A direct comparison of these curves indicates that the influence of the cutting depth on the damage performance of DKDP surface is insignificant, whereas the variation of the feed rate and cutting direction can change the damage performance significantly.

 

Fig. 11. Laser-induced damage probability curves of DKDP surfaces generated under different fly-cutting conditions.

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Moreover, through a comprehensive comparison of the damage test results and the surface morphologies shown in Fig. 3, the influence of the microstructure on the damage performance of DKDP surface can be obtained. For instance, by comparing curve (a) and curve (b), it can be known that the damage thresholds of the surfaces shown in Figs. 3(a) and (b) have no significant difference. This result indicates that the formation of the micro ripples parallel to the cutting edge projection has little impact on the laser-induced damage performance of DKDP surface. Besides, curve (d) shows that the damage threshold of the surface shown in Fig. 3(d) is in the range of 7–10 J/cm², which is much lower than that of the crack-free surfaces. Thus, it can be concluded that the cracks will degrade the damage performance of DKDP surface dramatically. Compared to the surface with micro ripples or cracks, the damage thresholds of the surfaces covered with micro grains vary over a wider range of 9–21 J/cm², as shown by curve (c). Because the upper segment of this curve almost coincide with curve (a), the formation of micro grains is unlikely to degrade the damage performance of DKDP surface. Otherwise, all sites will be damaged at lower fluences, and the generated probability curve could shift leftward. As for the occurrence of the two damages at the fluences about 10 J/cm², a possible reason is that a few micro cracks, which are hardly detected by AFM, have already been generated in this cutting condition.

In summary, the laser-induced damage performance of DKDP surface is determined by the microstructures generated during the SPDF process. Moreover, the formation of microstructures is closely related to the selection of cutting parameters. Therefore, understanding the process-structure-performance relationship in SPDF process is the key to improving the laser-induced damage performance of DKDP surface.

4. Conclusions

In this study, the influence of SPDF parameters on the microstructure and laser-induced damage performance of DKDP surface has been investigated. The formation mechanisms of the different microstructures were revealed, based on which a comprehensive understanding of the surface formation process was established for the first time. The main conclusions are as follows:

Appendix

In a previous research, a material model for KDP crystal had been established [29]. This model was adopted to study the deformation behavior of DKDP, considering the similarity of these two crystals. Within the elastic range, the stress–strain relationship of DKDP is described as:

(2)$$\left[ {\begin{array}{{c}} {{\sigma_x}}\\ {{\sigma_y}}\\ {{\sigma_z}}\\ {{\tau_{xy}}}\\ {{\tau_{yz}}}\\ {{\tau_{zx}}} \end{array}} \right] = C\left[ {\begin{array}{{c}} {{\varepsilon_x}}\\ {{\varepsilon_y}}\\ {{\varepsilon_z}}\\ {{\gamma_{xy}}}\\ {{\gamma_{yz}}}\\ {{\gamma_{zx}}} \end{array}} \right] = \left[ {\begin{array}{{cccccc}} {{C_{11}}}&{{C_{12}}}&{{C_{13}}}&0&0&0\\ {{C_{12}}}&{{C_{11}}}&{{C_{13}}}&0&0&0\\ {{C_{13}}}&{{C_{13}}}&{{C_{33}}}&0&0&0\\ 0&0&0&{{C_{44}}}&0&0\\ 0&0&0&0&{{C_{44}}}&0\\ 0&0&0&0&0&{{C_{66}}} \end{array}} \right]\left[ {\begin{array}{{c}} {{\varepsilon_x}}\\ {{\varepsilon_y}}\\ {{\varepsilon_z}}\\ {{\gamma_{xy}}}\\ {{\gamma_{yz}}}\\ {{\gamma_{zx}}} \end{array}} \right],$$

where C is the elastic matrix, [σ_x, σ_y, σ_z, τ_xy, τ_yz, τ_zx]^T and [ε_x, ε_y, ε_z, γ_xy, γ_yz, γ_zx]^T are engineering stress and engineering strain respectively.

In the plastic range, an ellipsoid yield surface proposed by Vodenitcharova and Zhang [34] was used to describe the pressure-dependent characteristic of the crystal. The yield function is expressed as follows:

(3)$$f({\sigma _m},{J_2}) = \frac{{3\sigma _m^2}}{{R_1^2}} + \frac{{2J_2^{\prime}}}{{R_2^2}} - 1 = 0,$$

where R₁ and R₂ are the plastic parameters of the crystal, σ_m and J₂^’ are the hydrostatic pressure and second deviatoric stress invariant, respectively.

Moreover, the flow rule used for describing the relationship between the stress and strain increment is written as:

(4)$$d\varepsilon _{ij}^p = \lambda \frac{{\partial f({\sigma _m},J_2^{\prime})}}{{\partial {\sigma _{ij}}}} = \lambda \frac{{\partial f({\sigma _m},J_2^{\prime})}}{{\partial {\sigma _m}}} \cdot \frac{{\partial {\sigma _m}}}{{\partial {\sigma _{ij}}}} + \lambda \frac{{\partial f({\sigma _m},J_2^{\prime})}}{{\partial {J_2}}} \cdot \frac{{\partial J_2^{\prime}}}{{\partial {\sigma _{ij}}}},$$

where dε_ij^p is the plastic strain increment and λ is the plastic multiplier, which is determined by:

(5)$$\lambda = \frac{{{{\left( {\frac{{\partial f({\sigma_m},{J_2})}}{{\partial {\sigma_{ij}}}}} \right)}^T}Cd\varepsilon _{ij}^T}}{{{{\left( {\frac{{\partial f({\sigma_m},{J_2})}}{{\partial {\sigma_{ij}}}}} \right)}^T}C\left( {\frac{{\partial f({\sigma_m},{J_2})}}{{\partial {\sigma_{ij}}}}} \right)}}.$$

By converting these formulas to FE code and integrating them into LS-DYNA solver, the material model can be invoked in the subsequent simulation research.

In this model, there are eight material parameters in total, including six elastic constants C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, C₆₆, and two plastic parameters R₁, R₂. The values of the elastic constants of DKDP crystal could be found in [35], while the plastic parameters were determined by the nano-indentation experiment and simulation. By adjusting these two parameters in the model and comparing the load-displacement curves from simulation and experiment, the plastic parameters were determined as R₁= 1900MPa and R₂= 400 MPa. Figure 12 compares the load-displacement curves from simulation and experiment, where the maximum indentation depth is 500 nm. It can be seen that the simulated curve coincides well with the experiment result, indicating that the obtained parameters are accurate enough to describe the plastic property of DKDP crystal.

 

Fig. 12. Comparison of the load-displacement curves from experiment and simulation. The solid line represents the experiment curve generated by a spherical indenter with a diameter of 20.7 µm, while the circles show the simulated curve under the same condition. Both results were performed on the THG plane of DKDP crystal.

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Finally, all the material parameters of DKDP crystal were determined, and the results are listed in Table 2.

Table 2. Material parameters of DKDP crystal used in the cutting simulation

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Funding

Laser Fusion Research Center, China Academy of Engineering Physics (LFRC-PD2020-004); National Natural Science Foundation of China (61905227).

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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