High-Gradient Optical Surface Figuring by IBF

In the above section, the measure data project distortion has been discussed. For high-gradient optical surface, this problem must be considered seriously when the surface error E(x, y, z) is projected from three-dimensional (3D) Cartesian frame to 2D frame, resulting in E(x, y) for the deconvolution operation. Furthermore, in order to maintain a constant removal function at every dwell point, the ion beam is usually held perpendicular to the surfaces. This means at least five axes and larger workspace is required. If wanting to figure high-slope surfaces with a linear threeaxis machine, which is definitely economical and more reliable compared with a five-axis system, some problems should be solved first. Thomas Haensel et al. (2008) and Dai Yifan et al. (2010) have discussed to figure strongly curved surfaces with a linear three-axis system in IBF.
In the section “Removal Function Modeling and Analyzing of IBF,” the changing rule of the removal function has been discussed with the incident angle based on theoretical and experimental investigation of the removal characteristics, which is used to figure high-gradient optical surface.

As shown in Fig. 30, a Gaussian ion beam bombards a high-slope surface z' = h(x', y') parallel to the optical axis. The actual removal rate at point B'(x', y') in the action region is Rθij when the beam dwells at point A'(x', y') with a removal function R(x, y). Since the dwell time is calculated in 2D plane (OXY), the nominal removal rate at B(x, y) is Rij, which is obviously unequal to the nominal one Rθij. The normalized removal rate is 

where θ is the incident angle. Rθij and Rij are defined as

So the actual removal rate becomes Rθij = KijRij. By compensating the change of coefficient Kij, according to the removal characteristics in IBF process, the original program developed for figuring flat surfaces is still  applicable to high-slope surfaces.
Substituting the normalized removal rate Kij into Eq. 19, which is the theoretical model of the removal function, it can be seen that the function at arbitrary incident angle can be gained when the removal function with perpendicular incident is obtained through experiments:

Based on Eq. 52, the curve of normalized peak removal rate is shown in Fig. 31 with the energy dispersion parameters determined. It can be verified through experiments with the ion beam scanning linearly and etching a sample along one of its generatrix. For example, the target distance is fixed and the constant scanning velocity is 1 mm/min. The aperture of the target surface is 21.3 mm, and the radius of curvature is 16 mm, which indicates the maximal incident angle is 41.7o. The actual material removal is shown in Fig. 32. For the purpose of comparison, the experimental curve of the removal rate is drawn in Fig. 31. It is consistent with the theoretical curve. Therefore it is reasonable to apply the theoretical model instead of a series of experiments to get the removal rate for high-slope samples of various apertures and various curvatures (Dai et al. 2010). A figuring experiment is done to testify this method with 5.7 mm (6σ) diameter ion beam on a linear three-axis IBF machine.

Figure 33a shows the original surface error map before figuring (101.9 nm PV and 13.1 nm RMS). According to the theoretical analysis and experiment, the distribution of the normalized removal rate Kij is given in Fig. 33b.

The first iteration aims to confirm the positioning precision and remove surface protuberances. It takes 4.2 min to reduce the figure error to 99.8 nm PV and 8.5 nm RMS with RMS convergence ratio 1.55. The residual error map shown in Fig. 34a indicates that the local protuberances at central and marginal regions have been removed, and the RMS error is reduced evidently. In this iteration, the ion beam positioning problem is met. Xuhui et al. (2011) have discussed how to solve it. Then in the next iteration, it takes only 4.8 min to reduce the figure error to 44.3 nm PV and 5.9 nm RMS, with RMS convergence ratio 1.44, comparable to that in flat surface figuring. The final figure error map is shown in Fig. 34b.

The total time consumed is 9 min, and the total convergence ratio reaches 2.24. Moreover, the figure error at the marginal region is successfully corrected without edge effect. This experiment proves the proposed method is excellent for figuring of high-slope surfaces.
Based on figure error compensation, the influence of varying removal function and projection distortion on the dwell-time solution is reduced when figuring a high-slope surface. Hence the five-axis figuring machine can be replaced by a three axis one with smaller workspace. And the program for figuring flat surfaces still works in high-slope cases. The limitation is that the maximal incident angle is required within about 60o. For higher-slope surfaces, the uniform gridding partition of surface error in 2D plane would lead to loss of local details including some highfrequency errors. To solve this problem, it will be with a smaller ion beam and subregion stitching method.

High Thermal Expansion and Crystal Optics Figuring by IBF

Unlike conventional methods, ion beam figuring process must be in the vacuum environment and generates high surface temperature on optical component because some part of the energy which is not transferred to component atom momentum heats the component. This section mainly talks about the thermal effects on optical component in the IBF process.
To know the temperature effect, the direct method is to measure the surface temperature by thermo-sensor, such as thermocouples, infrared camera, etc. (Gailly et al. 1999; Xuhui et al. 2012). The aim to discuss the temperature effect is to solve its influence to the optical component, especially to the high thermal expansion and crystal optics. The better method to solve this problem is to construct a thermal model to estimate the surface temperature (Xuhui et al. 2012; Nelson 2010).
For the thermal modeling of component, something should be known about the ion beam, such as ion beam power density distribution, ion beam power reached to component, absorbed energy (heat) by component, and so on. Only known to these problems, a useful thermal model may be set up to correctly estimate the component temperature and its distribution.
In ion beam, ion (Ar+) proportion is about 75–90 %; the other 25–10 % is the neutron-atom. When the ion beam runs from the outlet of ion source to the surface of component, its power will be lose partly because of resonance charge exchanging between the ions and neutron-atoms. So, the ion beam power P' that reached to the surface of the component is

where I is the ion beam electro-current, U is its voltage, P is the ion beam power generated by ion source, and η is the correcting factor of ion beam. Based on the ion beam correcting theory (Meinel et al. 1965), this correcting factor mainly lies on the section area of resonance charge exchange and ion velocity

where e, Ei, and Mi are the ion charge, ion energy, and ion mass, respectively; a and b are the ion constants; k is the Boltzmann constant; Po is the pressure of vacuum; d is the distance of ion source outlet to the surface of component; and σRCE is the section of resonance charge exchange:

The ion velocity is

Therefore, the ion beam power P' that reached to the surface of component can be gained theoretically by Eqs. 53–56.
The power that reached to the surface component is divided into two parts. One is so-called sputter power which is transferred to component atom momentum to make the atom escape the surface of component. The other is the main part absorbed by the component transferred heat to make the component surface temperature increase.
The absorbed heat may be calculated according to the rate of the component temperature increased:

where ꝺT/ꝺt is the rate of the component temperature increased and m and cp are the component mass and thermal capacity, respectively. It is therefore evident that the absorbed power would be estimated if it gained the rate of the component temperature increased. Another method to estimate the absorbed power is simulation based on the Monte Carlo method in the SRIM software which is an ion sputter simulation software (Shengyi and Xuhui 2010). With this method, it is also able to simulate the absorbed power P1:

where Q is the absorbed energy and Q1 is the sputter energy simulated. These two methods can validate each other to guarantee the model efficiency.
When the ion beam bombards the surface of component, a temperature gradient field is formed which may generate thermal stress. Here, a heat transfer model is constructed to analyze the temperature gradient distribution. Assuming that the radiated energy of ion source would be ignored, the component heat source is only ion beam which is a Gaussian function distribution as described in Fig. 5. The absorbed power may be described as

where r is the distance from the center of the Gaussian function as shown in Fig. 35. Equation 59 shows that the absorbed power is also a Gaussian distribution which may generate nonuniformed thermal stress on the component. It is important to construct a suitable heat transfer model to analyze the thermal stress of component which is the key step to analyze and control the component thermal stress magnitude and the stress distribution.

As shown in Fig. 35, assuming that the ion beam bombards the center of the component along Z direction, the component heat transfer equation based on the Fourier heat transfer formulated in the cylinder coordinate frame is

where ρ, cp and k are the material density, thermal capacity, and thermal conductivity, respectively. T is the interior temperature field distribution which is the function of radial coordinate r, axial coordinate z, and heated time t.
Using the above temperature model, it can be analyzed and modeled that the temperature field and its temperature stress field of actual optical component figuring by ion beam. The experimental parameters are ion beam voltage 700 eV and its electro-current 60 mA, the diameter of ion beam 10 mm with ion diaphragm, and the vacuum 2.1 x 10-2 Pa. The BK7 experimental component is Ф 40 mm with thickness 10 mm. The reason to select a small component is that it is convenient to simulate the machining process. The actual component surface contour is shown in Fig. 36a, and its figured scanning is raster with 2 mm raster pitch shown in Fig. 36b.

Based on the data of actual surface contour, a simulation is done to get the temperature field and its temperature stress field by the above temperature model. The simulated results are shown in Fig. 37. In this figure, it can be seen that the machined component temperature field and its corresponding stress field are not uniform. It is known that the high-temperature gradient field on these materials will generate large thermal stress in the component, which apparently distorts its surface, and when the thermal stress is larger than the material mechanic stress limit of component, the component generates crack or break. The reasons to generate thermal stress are the rapid change of the component’s temperature and high-temperature gradient. So, the troublesome and important problem is to select a suitable machining technique to control the temperature change rate and make temperature distribution uniformity in the IBF process.

A suitable machining technique method to make the component’s temperature more even is to select a reasonable figuring method which can reduce the temperature gradient distribution, such as by designing a low-pass spatial frequency filter to process the component surface data measured by Zygo interferometer to only hold the component’s low-frequency part and its middle- and high-frequency parts to be filtered. By this way, computer simulation results about the related temperature field and its thermal stress field shown in Fig. 38. Comparing Figs. 37 and 38, it is seen that the filtered IBF can gain the more even temperature field and thermal stress distribution. A final simulated result shows that the component maximum temperature is decreased about 10 % and its thermal stress is decreased about 22 %.
Based on the above analysis, an ion beam figuring flow is set up for high thermal expansion and crystal optics as shown in Fig. 39.

Supersmooth Surface Figuring and Micro-roughness Evolution

The supersmooth surfaces of optical component is very precisely figured plane, spherical, and aspherical surfaces with the accuracies in depth down to the sub-nanometer level over the entire spatial wavelength range. The kinds of optics are very important for advanced DUVL, EUVL, and synchrotron. According to the Mare´chal condition, the wave front of a diffraction-limited imaging system must achieve a deviation of λ/14 in the exit pupil (λ ¼ operating wavelength). However, requirements for a lithographic system are even more demanding. For nowadays 193 nm and 13.5 nm systems, the expected residual wave front error of each optical element amounts to at least λ/20 or below: 0.20–0.28 nm RMS and 0.11–0.20 nm RMS, respectively (Bruning 2007). Similar or even smaller values hold for spatial frequencies higher than 1/1 mm as outlined elsewhere. The roughness of EUVL substrates is described with two different areas of spatial frequency:

MSFR = mid-spatial-frequency roughness between 1/mm and 1/μm and
HSFR = high-spatial-frequency roughness ranging from 1/μm to 50/μm.

It is proved that the ion beam figuring is a very useful tool to figure supersmooth optical surface. For example, supersmooth optical surface figure by IBF in Carl Zeiss, serial spherewith a diameter of 178mm, rms = 0.13 nm, serial asphere with a diameter of 260 mm, rms = 0.19 nm (Weiser 2009). In China, our research group has also figured supersmooth optical surface shown in Fig. 40 using our designed IBF machine.
With ion beam to figure optical component, the micro-roughness is a regardful problem. Ion beam sputtering or ion beam erosion of surfaces can generate a diversity of surface topographies. Typically, during ion beam sputtering, the surface of the solid is far from equilibrium and a variety of atomistic surface processes and mechanisms become effective. It is the complex interplay of these processes that either tends to roughen (e.g., by curvature-dependent sputtering) or smoothen (e.g., by surface diffusion or viscous flow of surface atoms) the surface, which, finally, can result in the spontaneous formation of patterns (Frost et al. 2009).