## Abstract

Edge mis-figure is regarded as one of the most difficult technical issues for manufacturing the segments of extremely large telescopes, which can dominate key aspects of performance. A novel edge-control technique has been developed, based on *‘Precessions’* polishing technique and for which accurate and stable edge tool influence functions (TIFs) are crucial. In the first paper in this series [D. Walker Opt. Express **20**, 19787–19798 (2012)], multiple parameters were experimentally optimized using an extended set of experiments. The first purpose of this new work is to ‘short circuit’ this procedure through modeling. This also gives the prospect of optimizing *local* (as distinct from global) polishing for edge mis-figure, now under separate development. This paper presents a model that can predict edge TIFs based on surface-speed profiles and pressure distributions over the polishing spot at the edge of the part, the latter calculated by finite element analysis and verified by direct force measurement. This paper also presents a hybrid-measurement method for edge TIFs to verify the simulation results. Experimental and simulation results show good agreement.

© 2013 OSA

## 1. Introduction

The next generation of optical/infrared telescope will see primary apertures greater than 20m, increasing photon collecting area and angular resolution [2]. With technology limitations, 8.4m seems to be the practical maximum diameter for monolithic mirrors, transportation being a key issue [3, 4]. Nelson and Malacara et al. pioneered large telescopes by mirror segmentation [5], avoiding manufacturing and shipping a large monolithic primary mirror. Segmented mirrors have successfully been applied in Keck I&II, HET, SALT and GTC telescopes [6, 7].

Manufacturing segmented mirrors presents its own challenges. Edge mis-figure is regarded as one of the most difficult technical issues, which can dominate the performance of segmented-mirror telescopes. This is because the integrated edge-length in segmented optical system greatly exceeds the equivalent monolithic system. For example, there are nearly *4km* of edges in the E-ELT’s primary mirror, and edge mis-figure can be the limiting factor in stray-light and IR-emissivity performance. The traditional technique pioneered by Keck is to oversize the segment during polishing and figuring, after which the segment is cut hexagonal [8]. This introduces global distortion, which is rectified using the slow process of ion figuring.

As we pointed out in Paper 1 [1], other applications for fine edge control include pupil and image slicer optics, datum straight edges in various applications including wafer-steppers. There are other requirements in the semiconductor industry, for example specialist semiconductors that have been diced to final size, and post-polishing is needed to achieve a uniform thickness of a deposited layer.

In Paper 1, we demonstrated that effective edge-control on hexagonal parts could be achieved whilst polishing the global surface, using developments of the *‘Precessions’* process [9, 10]. Now, when a spherical compressible polishing-bonnet significantly overhangs the edge of the part, it will naturally roll-down the edge. The methodology described was therefore based on the tool-lift method, where the bonnet is progressively raised or lowered (‘Z offset’), delivering polishing spots (‘tool influence functions’ or ‘TIFs’) of variable size. As the tool-path approaches the edge of the part, tool-lift enables the spot to be controlled so that it never overlaps the precise edge at all, or more normally, it overlaps by a small and controlled amount. In Paper 1, the process variables then comprised:

- 1. Z-offset (and hence spot-size) at each incremental distance from the edge of the part
- 2. Dwell-time at each incremental distance from the edge of the part

These were empirically optimized using repeated trials and measuring the resulting edge-profiles. Due to the number of possible combinations of parameters (Z-offset, overhang, precess angle, feed-rate, raster-spacing), this proved a tedious and time-consuming process. Whilst amply demonstrating the effectiveness of the method, it does not lend itself directly to process-automation.

This second paper considers how the process can be ruggedized and automated by modeling edge TIFs from basic mechanical data, and then using these edge TIFs to predict the edge-removal. This immediately enables ‘edge control’ experiments to be performed in software rather than in the optics shop, and lends itself to full numerical optimization. The detailed morphology of the TIFs of variable size and edge-overlap then becomes of paramount importance.

This simulation predicts the edge TIFs by i) modeling the surface speed distribution across the TIF, and ii) the pressure distribution at the edge of the part by finite element analysis (FEA). The latter, verified by force measurement, as described in Section 3. To verify the simulation results of the edge TIFs, a hybrid-measurement method to obtain the experimental edge TIFs is presented in Section 4, using both simultaneous phase interferometry and profilometry. The simulation and measurement results show good agreement, and led to a preliminary model which can predict the edge removal profile, with edge-control result shown in Section 5.

## 2. The strategy of practical edge control-Tool lift

The strategy we adopt is to use comparatively large polishing spots over the bulk surface to give high volumetric removal rates in pre-polishing and early form-correction polishing, followed by smaller spot-sizes. The tool-lift method [1] then enables the edge-profiles to be optimized at each stage, as described in detail in Paper 1 [1] and summarized in Fig. 1 below.

## 3. The modeling of edge TIFs

#### 3.1 Theoretical background

The theoretical basis of prediction of material removal in optical surface polishing was presented by Preston in 1927 [11], as follows:

where:- $\Delta h(x,\text{y})$- Removal in uni
^{t time at point$(x,y)$} *k*- Preston coefficient, related to the work piece material, polishing-tool, polishing slurry and temperature of work environment_{$\nu (x,y)$}- Instantaneous relative surface velocity of polish pad at point$(x,y)$_{$p(x,y)$}- Instantaneous pressure of the polishing pad at point$(x,y)$

Define the average removal value of surface materials$R(x,y)$ in unit time T as the tool influence function, i.e.:

The removal function *R(x,y)* can be determined by Eq. (4), if the surface velocity distribution *v(x,y*) and the pressure distribution *p(x,y)* can be obtained.

In the ‘*Precession*’ polishing technique, when the spot extends beyond the edge of the part, the pressure distribution between tool and part is complex. Wagner and Shannon, (1974), used the force equation in conjunction with the torque equation for static equilibrium [12]. This model, however, presents an important problem. Whenever the tool centre is near the edge of the part, the minimum pressure can become negative, which means that this model is no longer valid. Jones, (1986), suggested a linear pressure distribution model in 1986 [13]. Cordero-Davila et al, (2004), developed this approach further using a non-linear high pressure distribution near the edge of the part; however, they did not report the model’s validity by experimental results [14]. Kim (2009) established a parametric modeling of edge TIFs based on ‘Preston equation’, which is able to predict edge TIFs [15]. This model was based on the presumption of linear pressure distribution for the hard tool. However, the cause of edge effect in flexible bonnet tool is different.

In the work presented here, the surface velocity distribution is obtained according to the geometry of the precess tool-motion. The pressure distribution over the polishing spot at the edge of the part is calculated by means of finite element analysis (FEA), the FEA results being verified by direct force measurement.

#### 3.2 Modeling of surface velocity distribution v (x,y)

A sketch of the movement of a precessed bonnet is shown in Fig. 2 . The velocity relation of any point in the polishing contact zone is shown in Fig. 3 .where:

_{$P(x,y)$}is any point in polishing contact zone;_{${\omega}_{0}$}is the angular velocity around axis of tool;_{$\omega $}is the angular velocity around a normal work piece;_{$O$}is the centre of polishing spot;- $\rho $ is the process angle;
*d*is the compression value of tool (Z-offset);_{$R$}is the radius of curvature of the tool

According to the geometry of precess in the polishing area, the velocity component distribution ${v}^{\text{'}}{}_{p1}(x,y)$can be expressed as:

#### 3.3 FEA model for pressure distribution and results

The pressure on the work piece is caused by elastic deformation of the tool. To simplify the problem, a thin layer of polishing cloth is considered to be a second order effect and is neglected in the modelling. A nominal 100mm x 100mm square, 10mm thick, Zerodur part were chosen for the model. The bonnet material was natural rubber (BS-1154: 2003). The material properties for the modeling are listed in Table 1 . The R160mm tool, 2.8mm Z-offset and 10° precess angle is chosen for the modeling.

The FEA of the bonnet tool polishing model is defined as a two-body contact. A contact pair is created between the surface of the tool and the polishing surface of the part. During the polishing process, the back surface of the part is fixed on the support system. Thus, all degree of freedom (DOF) of the back surface are constrained with zero displacements. The top surface of the tool is fixed on the polishing machine. The bonnet tool is depressed by the Z-offset to deliver a spot-size along the Z-direction. Thus, the top surface of the tool is constrained with zero displacements along X-axis and Y-axis and −2.8mm displacement along Z-axis. The elements and restraints of the FEA model are shown in Fig. 5
. The pressure distribution simulation result is shown in Fig. 6
, for which the maximum pressure on the Zerodur part is 1.17 x 10^{5} Pa for the case considered.

Figure 7 shows that pressure distribution at the edge of the part with different overhangs (5mm, 15mm, 20mm, 25mm), which can be seen that, when the spot beyond the edge of the part, the pressure applied on the edge becomes extremely high.

After the surface velocity distribution *v(x,y)* and the pressure distribution *p(x,y)* have been obtained, the tool influence function *R(x,y)* can be modeled according to Eq. (4) in Section 3.1. As a demonstration of the modeling, the edge TIFs for different overhang (5mm, 15mm, 20mm, 25mm) were simulated using MatLab code. According to the Preston equation, the absolute material removal is also determined by the Preston coefficient k, which is a constant related to the part material, polishing liquid and temperature. To simplify the modeling result, the magnitude of the TIF has been normalized (scaling of TIF from −1 to 0) in the simulation. The modeling results with experimental results are shown together in Section 4.

#### 3.4 Experimental verification for FEA results of the pressure distribution

The total force *f* applied on the part should be the same as the integral of the pressure distribution *p(x,y)* over the part contact area A, which can be described as:

*f*

_{R160}can be calculated according to Eq. (8), which is:To verify the pressure distribution

*p*(x,y) simulation results, the force

*f*exerted on the part has been measured on the polishing machine. OpTIC Glyndwr has developed a device for force measurement, using three Omega (SN: LCM201) standard load cells whose accuracy is ± 1.0% (linearity, hysteresis and repeatability combined). The sketch of the set-up of the force measurement is shown in Fig. 8 .

The force measurements for R160mm tool were carried out on the machine. Figure 9 shows the force exerted by a R160mm bonnet applied to the part with different Z-offsets. It can be seen that the force on the part is 15.87KgF, when the Z-offset is 2.8mm for R160mm tool (the simulation result is 16.43kg). The simulation error is 3.6% for this tool according to the force measurement results.

## 4. The measurement of edge TIFs

When the polishing spot from a bonnet projects beyond the edge of the part, the bonnet material at some level wraps around the edge. The local edge-removal of the TIFs then becomes abnormally high. If not managed, this will turn the edge down. Moreover, the resulting slopes can be beyond the measurement-range of full-aperture interferometry. To obtain the full TIFs data at the edge, we have therefore developed a measurement method using both 3D interferometer and 2D Profilometer data [10]. The sketch of this method is shown in Fig. 10 . Firstly, the depth data of certain points of the edge were measured by individual 2D scanning. Five 2D scans were then chosen for interpolating the boundary of each TIF, as shown in Fig. 10(a). After the boundary of the TIF was obtained, the 3D topology of the TIF at the edge of the part was then interpolated, as shown in Fig. 10(b).

To verify the simulation results, four TIFs at the edge of the part were interpolated, for which the parameters were the same as in the simulation (R160mm bonnet; precess angle: 10°; off-set: 2.8mm, overhang: 5mm, 15mm, 20mm, 25mm). The stitched results are shown in Fig. 11 . Approximately 2mm of the edge-data is lost in the interferometer field, which was recovered with the profilometer.

The 2D comparison of modeling and experimental edge TIFs results is shown in Fig. 12 . For simplification of the comparison, experimental results have been normalized (by scaling the magnitude of removal from 0 to 1). This shows that the shape of the simulated edge-TIFs is in reasonable agreement with measurement. The residual error is attributed to the contribution of errors in the pressure distribution calculated by FEA, and to the measurement error.

## 5. First application of edge modeling result

To predict and optimize the profile of the edge of the part, a simple model has been developed, which effectively co-adds the multiple edge TIFs in the edge zone. An experiment to verify this modeling result was carried out on a 200mm across corners, hexagonal part, using the same process parameters. Figure 13
shows the comparison of modeling and experimental results (edge to edge). It shows a reasonable agreement. Note that in Fig. 13 of this paper, the edge-modeling results are based on *modeling* the edge TIFs. In contrast, the preliminary results presented in Fig. 9 of Paper 1 were based on the *empirically measured* TIFs in the vicinity of edges.

By this means, the tool lift parameters were optimized and the whole process was demonstrated on a 200mm across corners, hexagonal witness part. The final result is shown in Fig. 14
, from which it can be seen that the PVq (95%) for each of the edges within the 10mm wide trapezoid zone, as previously defined [1], are approximately 100nm and no edge turn down. The dominant edge-defect remaining in the PV number is the turning-up of the corners. This is because, for a defined maximum spot overlap on the edges, the overlap on the corners is less, simply from geometry. Therefore we are currently developing a *local* edge rectification method using a small tool, as a final process step. The details of the method and will be published in due course.

## 6. Conclusion

This paper has reported on the most recent phase of our edge control development, driven by the E-ELT mirror segments, but looking ahead to wider applications. The *‘Precessions’* polishing technique has proved highly competent in addressing aspheres, but raises the challenge of edge-roll when the flexible bonnet overhangs the edge of the part. We have summarized our strategy, as reported in detail in Paper 1, based on shrinking the TIFs in the vicinity of edges using tool-lift. We have then pointed out the limitations of using empirical TIF data in an automated segment serial-production context. This has led to the improved technique reported here, based on modeling TIFs using Preston’s equation. The input data comprised the computed speed-distributions over the polishing spot delivered by a precessed bonnet, combined with the pressure distributions exerted by the bonnet calculated using finite element analysis. The latter has been confirmed by direct force-measurement.

To verify the edge TIFs simulation results, a hybrid-measurement method for edge-TIFs has been described, in which the interferometry and profilometry results are stitched together. The simulation and measurement results showed good agreement. Using these simulated edge TIFs, we have established a model which can predict the edge removal profile. By this means, the edge control parameters have been successfully optimized, for the first time *without* recourse to empirical TIF data. The method opens up the possibility of locally correcting edges using a tool-path constrained to the edge-zone; the subject of current work.

Overall, the work reported in this paper has provided a sound basis for automating optimization of the process for controlling edges on mirror-segments and in other applications.

## Acknowledgments

We gratefully acknowledge financial support under an RCUK Basic Technology Translation Grant, ‘Ultra Precision Surfaces: A New Paradigm’, through an R&D project funded under the EPSRC Integrated Knowledge Centre in Ultra Precision and Structured Surfaces, and through an STFC IPS grant. Particular thanks are also due to Glyndŵr University, and to the Vice Chancellor Prof. Mike Scott, in regard to their substantial commitment and financial support of the segment project. We also wish to thank Zeeko Ltd for assistance in many ways, not least developing the code for edge-control in polishing. H. Li’s PhD studentship has been sponsored by UCL and OpTIC, and W. Messelink through a UCL Impact studentship.

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