## Edge effect modeling and experiments on active lap processing |

Optics Express, Vol. 22, Issue 9, pp. 10761-10774 (2014)

http://dx.doi.org/10.1364/OE.22.010761

Acrobat PDF (2267 KB)

### Abstract

Edge effect is regarded as one of the most difficult technical issues for fabricating large primary mirrors, especially for large polishing tools. Computer controlled active lap (CCAL) uses a large size pad (e.g., 1/3 to 1/5 workpiece diameters) to grind and polish the primary mirror. Edge effect also exists in the CCAL process in our previous fabrication. In this paper the material removal rules when edge effects happen (i.e. edge tool influence functions (TIFs)) are obtained through experiments, which are carried out on a Φ1090-mm circular flat mirror with a 375-mm-diameter lap. Two methods are proposed to model the edge TIFs for CCAL. One is adopting the pressure distribution which is calculated based on the finite element analysis method. The other is building up a parametric equivalent pressure model to fit the removed material curve directly. Experimental results show that these two methods both effectively model the edge TIF of CCAL.

© 2014 Optical Society of America

## 1. Introduction

1. M. Johns, “The Giant Magellan Telescope (GMT),” Proc. SPIE **6986**, 698603 (2008). [CrossRef]

2. M. Cayrel, “E-ELT optomechanics: Overview,” Proc. SPIE **8444**, 84441X (2012). [CrossRef]

3. M. Clampin, “Status of the James Webb Space Telescope (JWST),” Proc. SPIE **7010**, 70100L (2008). [CrossRef]

4. R. A. Jones, “Computer-controlled optical surfacing with orbital tool motion,” Opt. Eng. **25**(6), 785–790 (1986). [CrossRef]

*et al.*[5

5. E. Luna-Aguilar, A. Cordero-Davila, J. Gonzalez Garcia, M. Nunez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sanchez, J. Cuautle-Cortez, and M. H. Pedrayes-Lopez, “Edge effects with Preston equation,” Proc. SPIE **4840**, 598–603 (2003). [CrossRef]

*et al*. [6

6. A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortés, and C. Robledo-Sánchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. **43**(6), 1250–1254 (2004). [CrossRef] [PubMed]

*et al.*[7

7. Y. Han, F. Wu, and Y. J. Wan, “Pressure distribution model in edge effect,” Proc. SPIE **7282**, 72822Q (2009). [CrossRef]

*et al.*[8

8. D. W. Kim, W. H. Park, S. W. Kim, and J. H. Burge, “Parametric modeling of edge effects for polishing tool influence functions,” Opt. Express **17**(7), 5656–5665 (2009). [CrossRef] [PubMed]

*et al.*[9

9. H. Hu, Y. Dai, X. Peng, and J. Wang, “Research on reducing the edge effect in magnetorheological finishing,” Appl. Opt. **50**(9), 1220–1226 (2011). [CrossRef] [PubMed]

*et al.*[10

10. D. D. Walker, G. Yu, H. Li, W. Messelink, R. Evans, and A. Beaucamp, “Edges in CNC polishing: from mirror-segments towards semiconductors, paper 1: edges on processing the global surface,” Opt. Express **20**(18), 19787–19798 (2012). [CrossRef] [PubMed]

*et al.*[11,12

12. H. Li, D. Walker, G. Yu, A. Sayle, W. Messelink, R. Evans, and A. Beaucamp, “Edge control in CNC polishing, paper 2: simulation and validation of tool influence functions on edges,” Opt. Express **21**(1), 370–381 (2013). [CrossRef] [PubMed]

12. H. Li, D. Walker, G. Yu, A. Sayle, W. Messelink, R. Evans, and A. Beaucamp, “Edge control in CNC polishing, paper 2: simulation and validation of tool influence functions on edges,” Opt. Express **21**(1), 370–381 (2013). [CrossRef] [PubMed]

13. L. Haitao, Z. Zhige, W. Fan, F. Bin, and W. Yongjian, “Study on active lap tool influence function in grinding 1.8 m primary mirror,” Appl. Opt. **52**(31), 7504–7511 (2013). [CrossRef] [PubMed]

6. A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortés, and C. Robledo-Sánchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. **43**(6), 1250–1254 (2004). [CrossRef] [PubMed]

*y*axis for better reading. The surface profile data were measured by a coordinates-measuring machine (CMM) with 1-μm accuracy.

## 2. Theoretical background

### 2.1 The Preston equation

*dz*is the integrated removal material during

*dt*time,

*k*is the Preston coefficient,

*P*is the pressure on the lap–mirror contact region, and

*V*is the magnitude of relative speed between the lap and mirror.

### 2.2 Methods to calculate the ring TIF

*ω*

_{1}, and the active lap will dwell at different radial positions with rotate speed

*ω*

_{2}. The values of

*ω*

_{1}and

*ω*

_{2}are at the same level, so the workpiece motion cannot be ignored when someone tries to calculate the TIFs for the active lap [13

13. L. Haitao, Z. Zhige, W. Fan, F. Bin, and W. Yongjian, “Study on active lap tool influence function in grinding 1.8 m primary mirror,” Appl. Opt. **52**(31), 7504–7511 (2013). [CrossRef] [PubMed]

15. D. W. Kim, S. W. Kim, and J. H. Burge, “Non-sequential optimization technique for a computer controlled optical surfacing process using multiple tool influence functions,” Opt. Express **17**(24), 21850–21866 (2009). [CrossRef] [PubMed]

*x*axis is the radial position on the workpiece, and its

*y*axis is the removal depth after one unit time.

*e*. We adopt the polar coordinates system in which the polar axis is along the line

*O*

_{1}

*O*

_{2}.

*O*

_{1}is the center of the workpiece, and

*O*

_{2}is the center of the lap. The edge ring TIF can be calculated by summing the removal material during one workpiece rotation. For example, the ring TIF on workpiece surface point

*A*is the sum of the removal material when

*A*moves from

*M*to

*N*along the arc

*MN*, which can be expressed aswhere

*ω*

_{1}/2π is the factor to scale the removed material in one unit time. According to our previous work [13

13. L. Haitao, Z. Zhige, W. Fan, F. Bin, and W. Yongjian, “Study on active lap tool influence function in grinding 1.8 m primary mirror,” Appl. Opt. **52**(31), 7504–7511 (2013). [CrossRef] [PubMed]

*φ*is the polar angle of the point

*N*. In Eq. (3), the lap–workpiece relative velocity

*V*is known [13

**52**(31), 7504–7511 (2013). [CrossRef] [PubMed]

*P*does not vary with the coordinates

*ρ*and

*θ*) when the active lap stays inside the mirror. Once the lap overhangs the mirror edge, this status will be changed, and the pressure distribution will no longer be uniform as well. To simplify the pressure model, we use a 1D equivalent pressure curve

*et al*. [8

8. D. W. Kim, W. H. Park, S. W. Kim, and J. H. Burge, “Parametric modeling of edge effects for polishing tool influence functions,” Opt. Express **17**(7), 5656–5665 (2009). [CrossRef] [PubMed]

### 2.3 The lap overhang ratio

*S*

_{lap}, is a dimensional irrelevant parameter that is used to describe the lap position status in edge figuring, as shown in Fig. 3.

## 3. Edge effect modeling

### 3.1 Parametric pressure model for edge ring TIFs

*et al.*[8

8. D. W. Kim, W. H. Park, S. W. Kim, and J. H. Burge, “Parametric modeling of edge effects for polishing tool influence functions,” Opt. Express **17**(7), 5656–5665 (2009). [CrossRef] [PubMed]

*κ*, is a function of the distance from the mirror edge,

*x*, and the lap overhang ratio,

*S*

_{lap}. It is a five-parameter function, as follows:where

*κ*

_{0}is the Preston coefficient when there is no overhang,

*f*

_{1}is the edge-side correction, and

*f*

_{2}is the workpiece-center-side correction.

*P*(

_{E}*ρ*,

*e*), and put it into Eq. (5) to calculate the edge ring TIF. Considering that the active lap is working like the spin tool motion, the workpiece-center-side correction,

*f*

_{2}, is not necessary. So, when we build up

*P*(

_{E}*ρ*,

*e*), only the edge-side correction,

*f*

_{1}, is considered. Also, the variables

*ρ*and

*e*were replaced by

*x*and

*S*

_{lap}in

*P*(

_{E}*ρ*,

*e*) for convenience. The basic form of the parametric equivalent pressure curve iswhere

*p*is the pressure scale factor, which scales the whole pressure to a reasonable level. The parameter

_{s}*ε*represents how fast the pressure increases in edge area. The basic idea of the parametric equivalent pressure curve is shown in Fig. 4.

*f*in Eq. (7) is used to describe the nonlinear pressure distribution in edge effects. It is a second-order function of variable

*x*, and its influence width is equal to

*W*

_{lap}·

*α*. The maximum value of this function can be controlled by the parameter

*β*.where Θ(x) is the step function: 1 for

*x*≥ 0 and 0 for

*x*< 0. Consider that the width of edge effects may be different at each overhang ratio, so the parameter

*α*is a function of

*S*

_{lap},

*α*,

*β*, and

*ε*) are determined by using the measured edge ring TIF data to fit Eq. (5), and the least square method is used to solve this problem. This will be discussed in Section 4.

*α*= –2

*S*

_{lap}+ 0.8,

*β*= 3, and

*ε*= 0.2. The example shows that both width and slope of the nonlinear pressure curve are different at each overhang ratio in this model.

### 3.2 FEA model considering a soft layer for pressure analysis

**17**(7), 5656–5665 (2009). [CrossRef] [PubMed]

*E*, is much smaller than that of the mirror,

_{s}*E*. The soft layer is about 2 mm thick and with 1-mm mesh size in our model. The deformable to deformable separable surface contact is used in the model. Two symmetric contact pairs are created for lap–soft layer and soft layer–mirror contact. The FEA model is shown in Fig. 6.Fixed constraint is applied on the workpiece bottom, and only the gravity load is considered. The mesh size is 6 mm for the workpiece part and the lap and 1.25 mm for the grinding layer. The FEA modeling and analyzing are carried out on ANSYS Workbench 14.5.

_{m}*E*, is essential to this model. Different

_{s}*E*will produce different pressure distribution on the workpiece surface, as shown in Fig. 7.The width of the nonlinear pressure area is narrow, and the slope of the pressure along the radial line is high when

_{s}*E*is close to

_{s}*E*; otherwise, the width and slope are wide and low when

_{m}*E*is smaller than

_{s}*E*.

_{m}*E*which satisfies the condition that theoretical edge ring TIFs coincide with the measured TIFs may exist. But the proper

_{s}*E*is hard to find out. Our approach to solve this is to use one group of measured TIF data with the same overhang ratio to fit the

_{s}*E*and then use this

_{s}*E*to analyze the pressure distribution for different overhang ratios. After that, put the pressure in Eq. (3) to calculate the edge ring TIFs for those overhang ratios. Finally, we compared these TIFs with the measured data from experiments to verify

_{s}*E*. This will be presented in Section 4.

_{s}## 4. Edge ring TIF experiments and model fitting

### 4.1 Experiment setup and measured TIFs

### 4.2 Parametric pressure model fitting

*P*[see Eq. (7)]. There are five parameters that need to be fitted: parameter

_{E}*α*for each overhang ratio (i.e., for

*S*

_{lap}= 0.1, 0.2, and 0.3),

*β*, and

*ε*. We use the least square method to solve this problem. The target curves are the edge ring TIFs for

*S*

_{lap}= 0.1 and 0.2 and the average edge ring TIF for

*S*

_{lap}= 0.3. The fitted results are

*α*

_{0.1}= 1.559,

*α*

_{0.2}= 1.101,

*α*

_{0.3}= 0.540,

*β*= 18.511, and

*ε*= 0.642, where

*α*

_{0.1}represents

*α*for

*S*

_{lap}= 0.1 and so on. The fitted parametric pressure curves are shown in Fig. 10.The curves are normalized by the pressure when the active lap is inside the mirror. Edge ring TIFs predicted by the fitted parametric pressure model are shown in Fig. 11, where the solid lines represent the predicted TIFs and the dashed lines represent the experimental TIFs. The unit of the TIFs is μm/min.

### 4.3 FEA model fitting

*E*, in Fig. 6. The average experimental edge ring TIFs with 0.3 overhang ratio are used as the target in the searching. Figure 13 shows the comparison between the experimental TIF and theoretical TIFs with different scale

_{s}*E*.From the figure we can find out (1) the theoretical TIF when

_{s}*E*equal or close to

_{s}*E*is far away from the experimental TIF and (2) the best fitting

_{m}*E*exists between 0.0001

_{s}*E*and 0.001

_{m}*E*. Through the searching we found that the best fitting modulus for this case is about 20.78 MPa, which is approximately 0.0003

_{m}*E*. More FEA models are built up with this Young’s modulus soft layer, and the pressure distributions are analyzed. The results are shown in Fig. 14.The overhang ratio varies from 0.05 to 0.4 at regular intervals of 0.05.

_{m}## 5. Model comparison and application

### 5.1 Comparison and discussion

**17**(7), 5656–5665 (2009). [CrossRef] [PubMed]

### 5.2 Application in simulation process

*y*axis value is equal to 1. The parametric pressure model can fit the target profile very well, and the maximum fit residual error is only 0.4%. Because the number of FEA results is limited, FEA modeling results are not as smooth as the parametric modeling results, but they also fit the target profile reasonable well, and the maximum fit residual error is 2.6%. The skin model cannot fit well at the edge area (i.e., the right side), and the maximum fit residual error is 12%.

## 6. Concluding remarks

## Acknowledgments

## References and links

1. | M. Johns, “The Giant Magellan Telescope (GMT),” Proc. SPIE |

2. | M. Cayrel, “E-ELT optomechanics: Overview,” Proc. SPIE |

3. | M. Clampin, “Status of the James Webb Space Telescope (JWST),” Proc. SPIE |

4. | R. A. Jones, “Computer-controlled optical surfacing with orbital tool motion,” Opt. Eng. |

5. | E. Luna-Aguilar, A. Cordero-Davila, J. Gonzalez Garcia, M. Nunez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sanchez, J. Cuautle-Cortez, and M. H. Pedrayes-Lopez, “Edge effects with Preston equation,” Proc. SPIE |

6. | A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortés, and C. Robledo-Sánchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. |

7. | Y. Han, F. Wu, and Y. J. Wan, “Pressure distribution model in edge effect,” Proc. SPIE |

8. | D. W. Kim, W. H. Park, S. W. Kim, and J. H. Burge, “Parametric modeling of edge effects for polishing tool influence functions,” Opt. Express |

9. | H. Hu, Y. Dai, X. Peng, and J. Wang, “Research on reducing the edge effect in magnetorheological finishing,” Appl. Opt. |

10. | D. D. Walker, G. Yu, H. Li, W. Messelink, R. Evans, and A. Beaucamp, “Edges in CNC polishing: from mirror-segments towards semiconductors, paper 1: edges on processing the global surface,” Opt. Express |

11. | H. Li, G. Yu, D. Walker, and R. Evans, “Modelling and measurement of polishing tool influence functions for edge control,” J. Eur. Opt. Soc. Rap. Publ. |

12. | H. Li, D. Walker, G. Yu, A. Sayle, W. Messelink, R. Evans, and A. Beaucamp, “Edge control in CNC polishing, paper 2: simulation and validation of tool influence functions on edges,” Opt. Express |

13. | L. Haitao, Z. Zhige, W. Fan, F. Bin, and W. Yongjian, “Study on active lap tool influence function in grinding 1.8 m primary mirror,” Appl. Opt. |

14. | F. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. |

15. | D. W. Kim, S. W. Kim, and J. H. Burge, “Non-sequential optimization technique for a computer controlled optical surfacing process using multiple tool influence functions,” Opt. Express |

16. | H. Liu, Z. Zeng, F. Wu, B. Fan, and Y. Wan, “Improvement of active lap in the grinding of a 1.8m honeycomb primary mirror,” AOMMAT, in press. |

**OCIS Codes**

(220.0220) Optical design and fabrication : Optical design and fabrication

(220.4610) Optical design and fabrication : Optical fabrication

(220.5450) Optical design and fabrication : Polishing

**ToC Category:**

Geometric Optics

**History**

Original Manuscript: January 6, 2014

Revised Manuscript: March 27, 2014

Manuscript Accepted: April 3, 2014

Published: April 28, 2014

**Citation**

Haitao Liu, Fan Wu, Zhige Zeng, Bin Fan, and Yongjian Wan, "Edge effect modeling and experiments on active lap processing," Opt. Express **22**, 10761-10774 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10761

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### References

- M. Johns, “The Giant Magellan Telescope (GMT),” Proc. SPIE 6986, 698603 (2008). [CrossRef]
- M. Cayrel, “E-ELT optomechanics: Overview,” Proc. SPIE 8444, 84441X (2012). [CrossRef]
- M. Clampin, “Status of the James Webb Space Telescope (JWST),” Proc. SPIE 7010, 70100L (2008). [CrossRef]
- R. A. Jones, “Computer-controlled optical surfacing with orbital tool motion,” Opt. Eng. 25(6), 785–790 (1986). [CrossRef]
- E. Luna-Aguilar, A. Cordero-Davila, J. Gonzalez Garcia, M. Nunez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sanchez, J. Cuautle-Cortez, M. H. Pedrayes-Lopez, “Edge effects with Preston equation,” Proc. SPIE 4840, 598–603 (2003). [CrossRef]
- A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortés, C. Robledo-Sánchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. 43(6), 1250–1254 (2004). [CrossRef] [PubMed]
- Y. Han, F. Wu, Y. J. Wan, “Pressure distribution model in edge effect,” Proc. SPIE 7282, 72822Q (2009). [CrossRef]
- D. W. Kim, W. H. Park, S. W. Kim, J. H. Burge, “Parametric modeling of edge effects for polishing tool influence functions,” Opt. Express 17(7), 5656–5665 (2009). [CrossRef] [PubMed]
- H. Hu, Y. Dai, X. Peng, J. Wang, “Research on reducing the edge effect in magnetorheological finishing,” Appl. Opt. 50(9), 1220–1226 (2011). [CrossRef] [PubMed]
- D. D. Walker, G. Yu, H. Li, W. Messelink, R. Evans, A. Beaucamp, “Edges in CNC polishing: from mirror-segments towards semiconductors, paper 1: edges on processing the global surface,” Opt. Express 20(18), 19787–19798 (2012). [CrossRef] [PubMed]
- H. Li, G. Yu, D. Walker, R. Evans, “Modelling and measurement of polishing tool influence functions for edge control,” J. Eur. Opt. Soc. Rap. Publ. 6, 110480 (2011).
- H. Li, D. Walker, G. Yu, A. Sayle, W. Messelink, R. Evans, A. Beaucamp, “Edge control in CNC polishing, paper 2: simulation and validation of tool influence functions on edges,” Opt. Express 21(1), 370–381 (2013). [CrossRef] [PubMed]
- L. Haitao, Z. Zhige, W. Fan, F. Bin, W. Yongjian, “Study on active lap tool influence function in grinding 1.8 m primary mirror,” Appl. Opt. 52(31), 7504–7511 (2013). [CrossRef] [PubMed]
- F. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 9, 214–256 (1927).
- D. W. Kim, S. W. Kim, J. H. Burge, “Non-sequential optimization technique for a computer controlled optical surfacing process using multiple tool influence functions,” Opt. Express 17(24), 21850–21866 (2009). [CrossRef] [PubMed]
- H. Liu, Z. Zeng, F. Wu, B. Fan, Y. Wan, “Improvement of active lap in the grinding of a 1.8m honeycomb primary mirror,” AOMMAT, in press.

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