Ultra-precision grinding of monocrystalline silicon cylindrical mirror

As one of the important functional components in the optical diagnosis system, the processing accuracy of monocrystalline silicon cylindrical mirror directly affects the overall performance of the entire experimental device. In order to obtain high-precision monocrystalline silicon cylindrical optical elements, the coordinate transfer function models of the diamond wheel in the cylindrical surface parallel grinding process were established, and the forming process scheme was proposed based on the wheel dressing precision and element grinding precision. At the end, the experiment for a monocrystalline silicon cylindrical mirror with size of 440mm × 50mm was carried out. The PV value of form error after grinding was about 2.7μm, which proved the correctness of the transfer function and the forming process scheme of monocrystalline silicon cylindrical optical element.


Introduction
With the characteristics of hard, brittle, wear resistance, high mechanical strength, good chemical stability [1], as well as high thermal conductivity and low thermal expansion [2], the monocrystalline silicon is widely used as the mirror material in the optical diagnosis system of large precision physical experimental devices.The monocrystalline silicon cylindrical mirror with complex collimation line such as quadratic cylinder mirror is one of the important functional components in the optical diagnosis system such as synchrotron radiation systems, laser shaping devices, and X-ray telescope systems, whose processing accuracy directly affects the comprehensive performance of the entire experimental device [3].
With the development of the semiconductor industry, as an important substrate material in integrated circuits, the manufacturing technologies of monocrystalline silicon components are becoming increasingly mature.Rotary table grinding and in-feed grinding are the main two forming and thinning grinding technologies of plane monocrystalline silicon wafers [4].In the world a large number of special processing equipments and corresponding technologies have been developed for the processing of monocrystalline silicon wafers, such as G-500DS of GTI company in the United States [5] and VG401 MKII of Okmoto company in Japan [6].Kang Renke introduced the manufacturing process of large scale monocrystalline silicon wafer in integrated circuits and the research progress in recent years [7].Yang Shengrong established a precision grinding control system based on fuzzy-PID control according to the characteristics of wafer in-feed grinding, which obtained more stable grinding force and higher grinding efficiency [8].Jiang Haibo introduced the technology of thinning silicon wafer, and expounded the process of thinning and removing the damaged layer [9].However, it is difficult to ultra-precisely form cylindrical surface with complex collimation line using traditional rotary table grinding and in-feed grinding technologies.Furthermore, the depth of subsurface defects and surface roughness are also difficult to reduce, which restricts the improvement of subsequent polishing efficiency.The high-precision and high-stiffness grinding machine with multiaxis linkage is used for ultra-precision forming of cylindrical elements with complex collimation line based on the principle of motion copying, which has the advantages of high machining accuracy and easy automation [10].Yujie Niu investigated the influence of different process parameters on the surface quality of monocrystalline silicon and processed a 195mm diameter parabolic mirror of Si, whose PV value of form error was about 3.8μm and roughness was about 0.11μm [11].Sheng Wang grinded a larger size monocrystalline silicon biconical free-form optics by dressing arc-shaped diamond wheel precisely, and the profile error of optics was about 6.0μm [12].
For cylindrical optical components with complex collimation line such as elliptical cylindrical surface, hyperbolic cylindrical surface, and parabolic cylindrical surface, it's difficult to ultraprecision form by using the standard functions of a 3-axis CNC grinding machine.In order to realize the ultra-precision forming of monocrystalline silicon cylindrical mirror, based on the three-axis linkage ultra-precision grinding machine, the coordinate transfer function models of diamond wheel in parallel grinding using arc profile wheel and straight profile wheel were established in this paper.Then the advantages and disadvantages of grinding cylindrical components with different shapes of grinding wheels were analyzed, and the forming processing scheme of cylindrical surface was proposed.At the end, the form grinding experiment of a monocrystalline silicon cylindrical mirror with size of 440mm × 50mm was carried out.The PV value of component form error was about 2.7μm.

Modelling
Figure 1 is the schematic diagram of cylindrical surface grinding using arc profile diamond wheel on a three-axis linkage ultra-precision grinder.The generatrix of the cylindrical surface is perpendicular to the rotation axis of the grinding wheel.In the processing, the arc profile of the grinding wheel is tangent to the cylinder surface.And the feed direction is parallel to the generatrix of the cylindrical surface.Assume that the alignment equation of cylindrical surface is z=f (y), the coordinates of the cylindrical surface points are (x o , y o , z o ), and the coordinates of the wheel motion points are (x w , y w , z w ).According to the spatial position relationship between the grinding wheel and the components as shown in Figure 2, the coordinate transfer function of the grinding wheel motion points in the processing is analyzed as shown in Formula (1).Where, r a is the arc radius of the grinding wheel, r w is the total radius of the grinding wheel, and k is the slope at the machined point of cylindrical surface, as shown in Formula (2).
When the generatrix of cylindrical surface is parallel to the rotation axis of the grinding wheel, the cylindrical components can be formed by a straight section profile grinding wheel, as shown in Figure 3.During machining, the outer circle of the grinding wheel is tangent to the surface of the cylindrical component.The feed direction is perpendicular to the generatrix.Assume that the alignment equation of cylindrical surface is z=f (x), the coordinates of the cylindrical surface points are (x o , y o , z o ), and the coordinates of the wheel motion points are (x w , y w , z w ).According to the spatial position relationship between the grinding wheel and the components as shown in Figure 4, the coordinate transfer function of the grinding wheel motion points in the grinding process is analyzed as shown in Formula (3).Where, r w is the total radius of the grinding wheel, and k is the slope at the machined point of cylindrical surface, as shown in Formula (4).

Discussion
When cylindrical surface element is processed by arc profile grinding wheel, it is necessary to install a special and complex wheel dresser on the grinder to shape and sharpen the surface of the wheel [13].As shown in Figure 5, during the dressing process of arc diamond wheel, it also needs to swing back and forth with high accuracy arc trajectory while the dressing wheel rotating at high speed.As crucial parameters affecting wheel motion points and machining accuracy according Formula (1), the arc radius and total radius of diamond wheel are two key parameters for the dressing process.In addition, due to the contour copying effect in the grinding process, there will be a large amount of small-scale waveness on the surface of the processed components due to the copying of the arc contour, and the period and amplitude of the waveness will increase with the increase of the step distance.This waveness will seriously reduce the efficiency of subsequent polishing process.The only way to reduce the waveness amplitude is to decrease the step distance, but this will multiply the grinding time.
Compared with the arc profile grinding wheel, the straight profile wheel is easier to dress and can also obtain a higher profile precision because it only needs rotational motion of the dressing wheel and simpler requirements for dresser as shown in Figure 6 [13].According to the coordinate transfer function of Formula (1) and Formula (3), the arc radius and the total radius of the grinding wheel need to be strictly controlled when grinding with the arc profile grinding wheel.When grinding components with the straight profile grinding wheel, only the total radius of the grinding wheel will affect the final coordinate accuracy.So it's easy to get higher shape accuracy when grinding cylindrical components with the straight profile grinding wheel.When grinding cylindrical components with a straight profile grinding wheel, as long as the step distance does not exceed the contour width, the linear profile will be completely copied on the surface of the component, thus greatly reducing the amplitude of smallscale waveness on the surface of the component and improving the processing quality of the component.

Grinding experiment
In order to verify the correctness of the coordinate transfer function model, grinding experiment of cylindrical optical element was carried out on an ultra-precision three axis CNC grinding machine, whose position accuracies of the three linear axes were within 2μm.According to the parameters of elliptical cylindrical optical element as shown in Table 1, the 3D shape of the cylindrical surface was calculated as shown in Figure 7. It's easier to get higher shape accuracy and surface quality by using straight profile grinding wheel, whose parameters were shown in Table 2.After on-machine dressing, the profile error of grinding wheel was about 1μm, and radial run-out error was about 0.8μm.According to Formula (3), combined with the surface point coordinates of cylindrical optical element and the total radius of the grinding wheel, the coordinates of the grinding wheel movement points matrix during the grinding process were calculated, as shown in Figure 8.After integrating the grinding wheel motion points according to the raster processing path and outputting them as a numerical control program, the monocrystalline silicon cylindrical element was grinded according to the process parameters as shown in Table 3.The processing was shown in Figure 9, and the processed element was shown in Figure 10.By on-machine measurement using chromatic confocal displacement sensor with measuring accuracy of 0.05μm, the form error of element was measured and the PV value was about 2.7μm, as shown in Figure 11.The profile along the generatrix of cylindrical surface was measured by profilometer, as shown in Figure 12, and the maximum PV value of small scale waviness was about 0.3μm.The experimental results indicated that ultra-precision forming of cylindrical optical element with complex collimation line cloud be obtained by the above coordinate transfer function and manufacturing method using straight profile diamond wheel.

Conclusions
In order to form monocrystalline silicon cylindrical optical element with complex collimation line high-precisely, based on the principle of parallel grinding, the coordinate transfer function models using arc profile grinding wheel and straight profile grinding wheel were established.Then the advantages and disadvantages of those two grinding methods were compared and analyzed.It's not only achieving higher dressing accuracy of wheel, but also achieving lighter surface waviness and higher accuracy of element when using straight profile grinding wheel.And then the forming process scheme was proposed.At the end, the ultra-precision grinding experiment of a large-scale monocrystalline silicon elliptical cylindrical component was carried out.According to the coordinate transfer function with straight profile grinding wheel, the coordinates of grinding wheel were calculated and the element was processed.The surface form error PV value of the component was about 2.7μm, which proved the correctness of the transfer function and realized the ultra-precision forming of monocrystalline silicon cylindrical optical element.

Figure 1 .
Figure 1.Grinding diagram by arc profile diamond wheel.

Figure 2 .
Figure 2. Position relationship between arc profile diamond wheel and element.

Figure 3 .
Figure 3. Grinding diagram by straight profile diamond wheel.

Figure 4 .
Figure 4. Position relationship between straight profile diamond wheel and element.

Figure 5 .
Figure 5. Dressing arc diamond wheel by cup dressing wheel.

Figure 6 .
Figure 6.Dressing straight profile diamond wheel by cup dressing wheel.

Figure 11 .
Figure 11.Form error of monocrystalline silicon cylindrical element.

Figure 12 .
Figure 12.Small scale waviness along the generatrix of cylindrical surface.

Table 2 .
The parameters of diamond wheel.

Table 3 .
The processing parameters.