1. INTRODUCTION

Large-aperture potassium dihydrogen phosphate (${{\rm KH}_2}{{\rm PO}_4}$, KDP) and its deuterated analog $({\rm K}{({{\rm D}_x}{{\rm H}_{1 - x}})_2}{{\rm PO}_4}$, DKDP) are used as electro-optic switches and frequency conversion crystals in high-power laser systems, such as the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory [1,2]. At present, single-point diamond turning (SPDT) is the most available fabrication technology for KDP or DKDP crystals, which shows water solubility, low fracture toughness (${\sim}{0.1}\;{{\rm MPa} \cdot {\rm m}^{0.5}}$), and high thermal expansion coefficient (${\sim}{4} \times {{10}^{ - 5}}\;{{\rm K}^{ - 1}}$) [3]. Producing finished crystal optics with a diameter-to-thickness ratio greater than 50:1 from large boules poses a variety of fabrication challenges, in which the transmittance wavefront root-mean-square gradient (GRMS) is a most difficult one. The GRMS values up to 20% above the specification of 11 nm/cm are acceptable at NIF for the tradeoff between performance and cost in the large-aperture KDP and DKDP manufacturing [4].

Transmittance wavefront characterizes the uniform thickness of the workpiece, which will affect the laser quality by reducing the energy concentration in a high-power laser facility. To ensure the laser performance, precisely controlled finishing of large-aperture crystal elements is very important for the stable achievement of transmittance wavefront GRMS specification. The uniform thickness of the workpiece is usually ascribed to holding quality during elements clamping, finishing parameters, and environmental factors. But good clamping is the first important task before cutting, especially for large-aperture optics with a diameter-to-thickness ratio above 50:1.

General mechanical clamping will cause damage to elements with low material and structural strength, and the local deformation caused by mechanical clamping will finally affect the processing accuracy. In this case, a vacuum chuck is used for good clamping. The air between the workpiece and chuck surface is pumped out, and the workpiece is holding on chuck with atmospheric pressure. The uniform elastic deformation of the whole workpiece requires high quality of vacuum chucking, which relies on the chuck material, vacuum hole (actuator) design, vacuum pressure, etc. All these factors should be carefully selected according to the material, shape, size, and application of the workpiece.

However, because the chucking research on KDP manufacturing is hard to find, the research on related areas has been taken as reference. A vacuum pin chuck was developed by A. Une to meet the demands of ArF excimer laser lithography. A 2-mm-wide static pressure seal, which extends beyond the wafer edge, makes it possible to almost completely flatten the wafer periphery. The new chuck can flatten the periphery of an 8-inch, 200-mm-bowed concave wafer to ${ \lt }{0.03}\;{\rm mm}$, and the residual bow is decreased to one-sixth of that possible with an ordinary ring-seal chuck [5–7]. The method of expanding the vacuum region by the wafer edge is worth using for reference. To improve the form accuracy of diamond-turned aluminum discs, Tani developed a plastic porous vacuum chuck with a ring groove air path for holding of magnetic aluminum substrates. The form accuracy of 3.5 inch aluminum disc is 0.1–0.3 µm [8]. Kang and Yang developed a segmented vacuum chuck as a bending system for stressed mirror polishing (SMP) of thin aspheric mirrors. The vacuum chuck is segmented into 12 pieces along its circumferential direction, and each is designed to allow small tilting with a hinge [9]. This is an interesting attempt to add vacuum hole units where holding is required.

In the literature mentioned above, the general idea of solving the local shortage of vacuum pressure can be used as a reference. However, there are great variations between different workpieces, such as processing requirement, material properties, application area, etc. As a transmission element, the KDP crystal requires not only a high precision of the surface form, but also a high precision of equal thickness. Therefore, it is necessary to design a vacuum chuck for a large-aperture KDP crystal according to the material characteristics, shape, size, processing requirements, etc.

In our early work, a square aluminum vacuum chuck with a small vacuum hole was developed for KDP crystal cutting. Furthermore, the Finite Element Method (FEM) was used to analyze the anamorphosis of a ${320}\;{\rm mm} \times {320}\;{\rm mm} \times {14}\;{\rm mm}$ KDP crystal holding by the vacuum chuck, which is the major contributor to transmittance wavefront error. In contacting model, the calculated results show that the vacuum chuck parameters do have an effect on the anamorphosis [10]. According to the increasing demand of output energy of high-power laser systems, optics in large size is needed. Unfortunately, the edge effect (thickness difference or profile is obvious from edge to center) is getting worse in the fabrication of KDP crystal that is large in size (${ \gt }{380}\;{\rm mm} \times {380}\;{\rm mm}$) and of small thickness (${ \lt }{11}\;{\rm mm}$). Therefore, the edge effect is now considered the primary factor that leads to the local transmittance wavefront distortion, which will result in the disqualification of transmittance wavefront GRMS to meet the specification requirements (11 nm/cm).

In this paper, based on the theoretical analysis of suckhole configuration and the pressure distribution law under the vacuum chuck condition of a KDP crystal, the influence of the suckhole arrangement on transmittance wavefront GRMS is verified through fly-cutting experiments, which provides a theoretical and experimental basis for improving the accuracy of transmittance wavefront GRMS.

2. THEORETICAL ANALYSIS

As shown in Fig. 1(a), a precision ground single-point natural diamond with a big circular arc blade is used as the cutting tool in SPDT. The diamond tool with an ${\sim}{7}\;{\rm mm}$ nose radius is carried on the pose adjustment after the installation on the edge of a fly-cutting disk [see Fig. 1(b)]. The vacuum chuck surface was first cut to meet the requirement of the large-aperture KDP crystal finishing. The vacuum chuck is fly cut to generate a crystal mounting surface that will be parallel to the surface generated by the finish cuts on the crystal surface. The workpiece is held on the chuck by vacuum chucking, which is fed at a low and steady linear velocity after the fly-cutting disk reaches the rotating speed preset. Due to high-repeat running accuracy and high stiff machine, precision-sharpened diamond tool, precisely controlled environment temperature, and carefully controlled finishing schedule, a large-aperture KDP crystal is directly cut to the flat optical mirror surface, whose surface form accuracy is submicron and surface roughness is nanometer or subnanometer .

 

Fig. 1. (a) Schematic diagram of the SPDT; (b) photos of a fly-cutting machine.

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Fig. 2. Typical transmittance wavefront of the KDP crystal: (a) transmittance wavefront P-V $={0.47}\lambda $ ($\lambda ={632.8}\;{\rm nm}$); (b) surface form of the vacuum chuck (P-V $=\;\sim{1.5}\;\unicode{x00B5}{\rm m})$.

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The surface finish, surface figure, and transmittance wavefront of finished KDP crystal are measured using an optical surface profiler and large-aperture (such as 24 inch) interferometer. As mentioned, the measuring results of transmittance wavefront reflect the thickness distribution of the whole element. In the measuring results, the red area indicates that the site is thinner than other places, while the blue area indicates that the site is thicker.

Figure 2(a) is the typical transmittance wavefront of the ${390}\;{\rm mm} \times {390}\;{\rm mm}$ of KDP manufacturing. It is found that the edge is red, and it gradually turns to blue toward the middle of the element, which implies that the thickness of this KDP crystal gradually increases from the edge to the center. However, the local thickness inequality is unfavorable to the transmittance wavefront GRMS for large-aperture optics with a high diameter-to-thickness ratio. To ensure that there are no problems in the machine and processing environment, the surface profile of the ${460}\;{\rm mm} \times {445}\;{\rm mm}$ aluminum vacuum chuck used for the KDP holding was measured on the 24-inch interferometer, and the result is demonstrated in Fig. 2(b), in which a concave cutting surface can be observed. Ideally, the motion path of the machine worktable should be a straight line, and the fly-cutting surface should be an ideal plane. However, the surface of the processed vacuum chuck repeatedly exhibited the similar results, as shown in Fig. 2(b), which reflects the systematic motion error of the machine. Thus, this vacuum chuck surface condition is taken as the initial basis for the further cutting of the KDP crystal.

As shown in Fig. 3, the yellow area is the chuck base, in which the chuck holes are uniformly arranged vertically to the mounting surface. The interval is 20 mm. In ideal conditions, vacuum pressure distributes uniformly over the KDP holding surface, and a perfect contact between the chuck mounting surface and KDP should be created after the elastic deformation. But the vacuum pressure is nonuniform in the actual fly-cutting process, and its distribution is assumed. The raw tape is used to seal around the edge of the square element, but it is impossible to achieve absolute sealing. This will result in less pressure at the edge compared with the center, so that the edge area of the KDP crystal is upwarping during processing. When the worktable of the machine repeats the motion path in fly-cutting, the peripheral depth of the KDP crystal cut is greater than the central depth of the cut. As a result, the peripheral thickness of a finished KDP crystal is thinner than the central thickness after elastic recovery.

 

Fig. 3. Schematic diagram of the principle of a KDP crystal fly-cutting, holding under nonuniform vacuum pressure.

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3. SIMULATION

For in-depth understanding and analysis of the edge effect in the KDP crystal fly-cutting, the FEM was adopted to theoretical simulate the workpiece holding by the vacuum chuck.

Based on the analysis of the edge effect formation, deformation of the crystal elements caused by the chucking process is further calculated, and the results can provide evidence for decreasing such error. In previous work, vacuum-chucking processes have been analyzed by many researchers using structure analysis methods [11–13]. However, it can be seen from Fig. 4 that the KDP chucking is a complicated fluid–solid interaction problem that is hard to be solved accurately using the conventional single-field analysis method.

 

Fig. 4. Flowchart of the fluid-structure analysis method for studying the vacuum-chucking process.

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Fig. 5. Simulation results of the vacuum-chucking process: (a) pressure distribution in the gas film generated by the original chuck; (b) pressure distribution in the gas film generated by the optimized chuck; and (c) displacement of the KDP top surface.

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In this paper, a fluid-structure coupled analysis method is proposed to study the KDP deformation during the chucking process, and the flowchart of such method is presented in Fig. 5. First, the gas film between the vacuum chuck and the undeformed KDP surface is analyzed using the Finite Volume Method and Computational Fluid Dynamics software FLUENT. In this simulation, the boundary of the gas film is determined using the concaved chuck surface (edge of the chuck is 1.5 µm higher than its central position), as well as the undeformed KDP surface (flat), and the film is composed of a total of 265,344 brick elements. Then, the obtained pressure distribution in the gas film is exerted on the KDP surface, and the deformation of the crystal element is calculated by FEM using ANSYS. As the structure analysis is completed, the boundary of the gas film is updated according to the deformation of the crystal surface, and the new pressure distribution result will be utilized in the subsequent structure analysis. Such a fluid-structure analysis will be repeated until the change of the KDP deformation is less than 0.05 µm between two adjacent calculations.

According to this method, deformations of KDP elements absorbed by the original vacuum chuck, as well as a structure-optimized vacuum chuck, are analyzed. In the first case, the vacuum holes distribute uniformly on the whole chuck surface with the spacing of 20 mm, while the optimized chuck has an additional row (column) of holes inserted between the two outermost rows (columns) of the original one. Figures 5(a) and 5(b) compare the pressure distribution in the gas film formed by the original chuck and the optimized one. In both cases, pressure in the central part of the gas film equals the vacuum pressure (${-}{0.08}\;{\rm MPa}$), and it increases gradually to atmospheric pressure (0 MPa) at the margin. In comparison, the width of the pressure transition area of the optimized chuck (about 15 mm) is smaller than that of the original one (about 25 mm). Therefore, it can be concluded that inserting a row (column) of holes at the margin of the chuck can improve the chucking quality of the KDP crystal. Just because of the difference in pressure distribution, deformations of the crystals chucked by these two vacuum chucks also behave differently, as shown in Fig. 5(c). One can find that the deformation of the top surface of the crystal chucked by the optimized chuck is much closer to the profile of the chuck surface, which means the edge effect can be decreased effectively by using the structure-optimized vacuum chuck.

4. EXPERIMENT

Based on the theoretical analysis and FEM simulation results, an optimized aluminum vacuum chuck was developed, as shown in Fig. 6(b). It has chuck holes 2.5 mm in diameter and non-uniform hole distribution, with a 20 mm interval at the center and 10 mm interval near the edge. Comparatively, the original chuck has a uniform hole distribution at an interval of 20 mm, as shown in Fig. 6(a). The optimized vacuum chuck was installed on the worktable of the same machine, and the same finishing parameters were set: the depth of cut, 0.5 µm; spindle speed, 290 rpm; and feeding speed, 20 µm/r. The flat surfaces of ${390}\;{\rm mm} \times {390}\;{\rm mm} \times {10}\;{\rm mm}$ KDP crystals were processed. After two to three transmittance wavefront measurements were carried out after each cutting, the transmission wavefront GRMS was calculated.

 

Fig. 6. Photos of the peripheral area of the vacuum chuck: (a) original aluminum vacuum chuck; (b) optimized aluminum vacuum chuck.

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5. RESULTS AND DISCUSSION

Figures 7(a) and 7(b) present the transmittance wavefront results of the KDP elements finishing with the original and optimized vacuum chuck, respectively. There is a thickness difference of 244.9 nm in $X$ and 165.8 nm in $Y$ before optimization, and 98.7 nm in $X$ and 95.0 nm in $Y$ after optimization. Correspondingly, the transmittance wavefront GRMS decreases from 25.50 nm/cm to 8.15 nm/cm. Meanwhile, the edge effect after optimization is obviously suppressed compared to the original, as shown in Figs. 7(c) and 7(d). The peripheral area of measuring results after optimization as pointed by white arrows is obviously different from the central area, compared to the original one. Thus, the new vacuum chuck shows a good chucking quality on the edge effect in the fly-cutting of KDP crystals.

 

Fig. 7. Thickness difference and transmittance wavefront GRMS of the KDP crystal fly-cutting: (a) thickness difference 244.9 nm in $X$ and 165.8 nm in $Y$ (original); (b) thickness difference 98.7 nm in $X$ and 95 nm in $Y$ (optimized); (c) 8.15 nm/cm (original); and (d) 25.50 nm/cm (optimized).

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The batch processing results show that (see Fig. 8) the edge warping is suppressed, and the difference in the actual depth of the cut between the edge and center is reduced from 150.6–280.1 nm to 59.6–122.3 nm, indicating a prominent improvement regarding the thickness uniformity of a finished KDP crystal. Meanwhile, the accuracy of the transmittance wavefront and transmittance wavefront GRMS can be improved by some 25% (from 10.9 to13.9nm/cm to 8.1–11.3 nm/cm (see Fig. 8). Furthermore, the experimental results validate the theoretical analysis and simulation results.

 

Fig. 8. Statistical results of thickness difference and transmittance wavefront GRMS of the KDP crystal holding with an original and optimized vacuum chuck.

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6. CONCLUSION

Based on the theoretical analysis of the fly-cutting process of an element with vacuum chucking, it has been found that the reduced vacuum pressure on the edge will lead to the thickness difference between the edge and the center of the element. Based on the fluid-structure coupled FEM simulation method, results show that the increase of chuck holes at edge can improve the uniformity of pressure distribution. Thus, an optimized aluminum vacuum chuck was developed, which can effectively improve the vacuum-chucking quality and suppress the edge effect in the SPDT (an average thickness difference of the KDP crystal was reduced to ${\sim}{50}\% $). The accuracy of transmittance wavefront GRMS of a large-aperture KDP crystal is improved by 25% with the optimized chuck, which is helpful to meet the specification of high-power laser system. This method provides a reference for improving the accuracy of transmittance wavefront in ultra-precision machining of optics with a large-aperture and diameter-to-thickness ratio.