## Non-sequential optimization technique for a computer controlled optical surfacing process using multiple tool influence functions

Optics Express, Vol. 17, Issue 24, pp. 21850-21866 (2009)

http://dx.doi.org/10.1364/OE.17.021850

Acrobat PDF (1334 KB)

### Abstract

Optical surfaces can be accurately figured by computer controlled optical surfacing (CCOS) that uses well characterized sub-diameter polishing tools driven by numerically controlled (NC) machines. The motion of the polishing tool is optimized to vary the dwell time of the polisher on the workpiece according to the desired removal and the calibrated tool influence function (TIF). Operating CCOS with small and very well characterized TIF achieves excellent performance, but it takes a long time. This overall polishing time can be reduced by performing sequential polishing runs that start with large tools and finish with smaller tools. In this paper we present a variation of this technique that uses a set of different size TIFs, but the optimization is performed globally – *i.e.* simultaneously optimizing the dwell times and tool shapes for the entire set of polishing runs. So the actual polishing runs will be sequential, but the optimization is comprehensive. As the optimization is modified from the classical method to the comprehensive non-sequential algorithm, the performance improvement is significant. For representative polishing runs we show figuring efficiency improvement from ~88% to ~98% in terms of residual RMS (root-mean-square) surface error and from ~47% to ~89% in terms of residual RMS slope error.

© 2009 OSA

## 1. Introduction

9. D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express **13**(3), 910–917 (
2005). [CrossRef] [PubMed]

11. J. H. Burge, S. Benjamin, D. Caywood, C. Noble, M. Novak, C. Oh, R. Parks, B. Smith, P. Su, M. Valente, and C. Zhao, “Fabrication and testing of 1.4-m convex off-axis aspheric optical surfaces,” in *Optical Manufacturing and Testing VIII*, J. H. Burge; O. W. Fähnle and R. Williamson, eds., Proc. SPIE **7426**, 74260L1–12 (2009).

14. D. D. Walker, A. P. Doel, R. G. Bingham, D. Brooks, A. M. King, G. Peggs, B. Hughes, S. Oldfield, C. Dorn, H. McAndrews, G. Dando, and D. Riley, “Design Study Report: The Primary and Secondary Mirrors for the Proposed Euro50 Telescope” (2002), http://www.zeeko.co.uk/papers/dl/New%20Study%20Report%20V%2026.pdf.

*a.k.a.*tool marks) suppression on these precision optical surfaces is important for maximum performance (

*i.e.*less scattering, well defined point spread function) of the optical systems [17]. Most of the recent large optical surfaces have been polished until the spatial frequencies of the surface errors satisfied a target structure function or power spectrum density (PSD) specification to quantify the target form accuracy as a function of spatial frequencies [17,18]. Thus, the improved CCOS technique must provide an efficient fabrication process for a mass-fabrication of precision optical surfaces while minimizing the mid-spatial frequency error.

*i.e.*ablation time as a function of position on the workpiece) of a tool influence function (TIF) is optimized as the major optimization parameter to achieve a target material removal (

*i.e.*target error map). The TIF represents instantaneous material removal for a tool with specific motion. Then a numerically controlled polishing machine executes the optimized dwell time map on the workpiece by altering the transverse speed of the tool [1–7

7. D. D. Walker, D. Brooks, A. King, R. Freeman, R. Morton, G. McCavana, and S. W. Kim, “The ‘Precessions’ tooling for polishing and figuring flat, spherical and aspheric surfaces,” Opt. Express **11**(8), 958–964 (
2003). [CrossRef] [PubMed]

*i.e.*collection of different TIFs) is provided also. Simulation results are presented to demonstrate the performance of the new technique in Section 4. Section 5 summarizes the implications.

## 2. Theoretical background

### 2.1 Generation of the TIF library

9. D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express **13**(3), 910–917 (
2005). [CrossRef] [PubMed]

19. 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]

19. 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]

19. 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]

### 2.2 Dwell time map optimization using merit functions

*e.g.*measured errors on the optical surface). This optimization is also known as a de-convolution of the target removal map using a TIF. A TIF can be regarded as an impulse response of a tool with a given tool motion. In other words, a TIF represents the instantaneous material removal for a unit time at a location on the workpiece. The removal map (

*i.e.*accumulated TIFs over the whole workpiece) after the tool finishes its path on the workpiece can be expressed as

*x*,

_{workpiece}*y*are the coordinates on the workpiece,

_{workpiece}*x*,

_{TIF}*y*the coordinates on the TIF, and ** is the two dimensional convolution operator.

_{TIF}*e.g.*Fourier transform based algorithms, matrix-based least-squares algorithms) [20–23

23. H. Lee and M. Yang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. **40**(9), 1936–1943 (
2001). [CrossRef]

*i.e.*objective function) to search for the optimal solution. The merit function for the non-sequential optimization technique is presented in Section 3.4.

## 3. Non-sequential optimization technique using multiple TIFs

### 3.1 Conventional (i.e. sequential) vs. non-sequential optimization technique

*i.e.*sequential) CCOS optimization, a dwell time map for one TIF has been the major search space for the optimal solution. In other words, an optimization engine searches for the optimal dwell time values for a TIF on the workpiece, which gives the best residual error map. After the CCOS run is executed, another (or same) TIF is used for the next dwell time map optimization to attack the residual error map. This sequential process is repeated, usually using successively smaller TIFs until the target specification is achieved.

*i.e.*ignorable dwell time or removal) to be extracted from the TIF library during the optimization. However, the key difference of the non-sequential technique from the conventional one is not the number of utilized TIFs. The conventional case may use as many TIFs as the non-sequential case in sequential manner. The major improvement comes from considering all TIFs at the same time, so that the optimal combinations of TIFs are used in constructive manner to improve the performance of the CCOS process.

### 3.2 Non-sequential optimization engine using the gradient search method

*a.k.a.*steepest descent) method [24]. The method is known as one of the most simple and straight forward optimization technique which works in search spaces of any number of dimensions. This method presupposes that the gradient of the merit function space at a given point can be computed. It starts at a point, and moves to the next point by minimizing a figure of merit along the line extending from the initial point in the direction of the downhill gradient. This procedure is repeated as many times as required. Because the search space for the non-sequential optimization also has multiple dimensions (

*i.e.*many TIFs with various tool configuration parameters), the gradient descent method is suitable for our application.

*e.g.*5cm in diameter) in the TIF library to optimize an 8m diameter target removal map, the curvature of the figure of merit values along the 5cm TIF direction may be very shallow compared to the other reasonable size TIF (

*e.g.*20, 35, or 50cm in diameter) directions. Thus, including a 5cm TIF to the TIF library is inappropriate in this case. Limiting the total number of TIFs in the library improves computing efficiency. We do not rely on especially powerful computers for this work. Most of the optimization runs (including the case study runs in Section 4) are finished in 2-10 minutes on a regular desktop PC. Second, an improper perturbation step to calculate the local gradient may result in poor optimization performance. However, most of the search space dimensions are not a continuous space, but a discrete space depending on the given TIF library. For instance, there are only five available tool sizes (30, 40, 50, 60, and 70cm) in the reference TIF library in Table 3 (Appendix A). Although we carefully claim that the gradient descent method is suitable for this application, we still acknowledge the possibility of undesired optimization results for some special cases. For example, the TIFs are not orthogonal functions. Consequentially, the sequential application of TIFs for the optimization engine may not lead to the global minimum, but to a local minimum. However, we have not yet observed such cases in our trial optimization runs. Some actual optimization results using this optimization method are demonstrated in Section 4.

*e.g.*maximum acceleration), the minimum dwell time is set by the CCPM specification. The optimization engine evaluates each TIF to achieve the target removal map for all possible TIF locations on the workpiece. For each trial, the change in the total figure of merit,

*FOM*in Section 3.4, is recorded to determine the steepest descent case as follows. Using the TIFs with their own dwell time maps for each perturbation case, the expected removal maps are calculated using Eq. (1). The difference between the total expected removal map (

_{total}*i.e.*sum of all expected removal maps from each TIF) and the target removal map is the residual error map. This residual error map is used to evaluate the

*FOM*. After all TIFs (

_{total}*i.e.*dimensions of the search space) have been tried, the optimization engine updates the dwell time maps with the optimal trial, which recorded the steepest improvements in

*FOM*.

_{total}*FOM*reaches the specification or does not decrease anymore (

_{total}*i.e.*saturated). The current dwell time maps for each TIF become the optimization result. If these conditions are not met, more TIFs are fed into the TIF library. The TIFs which were hardly used are extracted from the TIF library. By performing more rounds of optimization using the updated TIF library, the optimal TIF set with their dwell time maps is determined eventually.

### 3.3 TIF library

*e.g.*computing power, time), only reasonable TIFs need to be generated and saved in the library. A square tool, a circular tool, and a sector tool (

*e.g.*TIF #7 in Fig. 6 , Appendix A) with orbital or spin tool motions may create a sufficient tool shape search space (

*i.e.*TIF library) for most cases. Also, the shop does not need to have a large inventory for all tools in the library. Only some optimal tool sets need to be made and maintained.

*i.e.*workpiece RPM = 0). The ring TIF is the axisymmetric removal profile when the workpiece also rotates, and is calculated using the relative speed between the tool motion and the workpiece rotation. The ring TIF looks like a ring (

*i.e.*donut) on the workpiece. The ring TIF shape is a function of radial position of the TIF center,

*ρ*, on the workpiece. The ring TIF radial profiles in Fig. 6 are only displayed for

*ρ*= 50, 150, and 250cm. The full TIF library includes the ring TIFs for all

*ρ*values on the workpiece. These two different types of TIFs can be selected depending on the relative speed between the tool and workpiece. If the workpiece motion is slow compared to the tool motion, the static TIFs are used because their shapes do not change significantly by the workpiece motion. However, if the workpiece rotates quickly, then the ring TIFs, which incorporate the workpiece motion effect, are used.

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

### 3.4 Merit functions for the non-sequential optimization technique

*C*is the weighting factors for

_{1-6}*FOM*. Each

_{1-6}*FOM*is defined as where the surface integral limit

_{i}*M*represents the error map surface.

*M +*and

*M-*are the error map areas with positive and negative residual error values, respectively. The six weighting factors can be adjusted depending on a specific purpose of a CCOS run as a design parameter.

*FOM*and

_{1}*FOM*.

_{2}*FOM*is the RMS of the positive error map, where the final surface is still higher than the target surface.

_{1}*FOM*is the RMS of the negative error map, where the final surface is lower than the target surface. Because the polishing process can only remove material from the workpiece, the surface often needs to be kept higher than the target surface to a certain extent during the polishing process. This can be achieved by increasing the weighting factor

_{2}*C*for

_{2}*FOM*. At the final polishing run to finish the project, both

_{2}*FOM*and

_{1}*FOM*may need to be minimized with the same weightings (

_{2}*C*=

_{1}*C*) to minimize the conventional RMS of the error map.

_{2}*i.e. FOM*&

_{3}*FOM*in Eq. (5) and (6)] and the RMS deviation of the surface curvature map [

_{4}*FOM*&

_{5}*FOM*in Eq. (7) and (8)] are used to quantify the mid-spatial frequency error and localized small errors. The approaches using Fourier transform or PSD based figure of merits were excluded due to their computing power requirements. In contrast, the differential calculations in

_{6}*FOM*,

_{3}*FOM*,

_{4}*FOM*and

_{5}*FOM*can be easily done for a numerical data set (

_{6}*e.g.*matrix for a pixelized error map) in most computing language platforms, such as MATLAB

^{TM}.

*FOM*combines the functions

_{total}*FOM*with appropriate weighting coefficients depending on the purpose of a CCOS run, and provides a good criterion to optimize a CCOS run using a TIF library. For instance, if large

_{1-6}*C*and

_{3}*C*values were entered, the optimization engine would try to minimize the slope errors on the final workpiece. By minimizing

_{4}*FOM*, the non-sequential optimization engine prevents the unwanted mid-spatial frequency error and localized small errors, while it achieves a small RMS of the residual error map.

_{total}## 4. Performance

### 4.1 High figuring efficiency

*FE*) is defined by

*FE*= 88%, since those two TIFs were not utilized in a constructive manner.

*FE*= 98.4%) was accomplished. The two removal profiles from both TIFs (green and brown dotted lines in Case 1.2, Fig. 3) matched well, so that the total removal (blue solid lines in Case 1.2, Fig. 3) is almost a constant (

*i.e.*piston) removal profile. The residual error (red solid line in Case 1.2, Fig. 3) shows flat profile, which is much improved over Case 1.1.

*i.e.*initial profile) in Case 1.3 and 1.4, Fig. 3. The TIF from 50cm square tool with orbital tool motion was given as a common primary TIF.

*FE*= 91.7%) with hard-to-correct bumpy features on the residual error profile.

*FE*= 96.8%.

### 4.2 Mid-spatial frequency error suppression with high time-efficiency

*i.e.*tool marks) was evaluated in a two-dimensional simulation of polishing the 1.6m New Solar Telescope (NST) primary mirror [27]. A 1.6m optical surface map with 701nm RMS of irregular errors was simulated as shown in Fig. 4 . The target specification for the residual error map was set as <20nm RMS, the NST primary final optical surface specification [27].

*i.e.*tool marks) on the finished optical surface.

^{−1}in the PSD graph) which were relatively smaller than the TIF size. In contrast, for the Case 2.2, almost 99.5% of the form error volume was removed using the smallest TIF. However, it caused significant mid-spatial frequency error on the final optical surface. This is easily observed by comparing the initial and final PSD graphs in Case 2.2, Fig. 5. Even though the low-spatial frequency errors (<5m

^{−1}) were removed, there was a significant generation of mid-spatial frequency error (5-30m

^{−1}). As a result, the final RMS slope error was 0.277arcsec which was the worst among three cases in Table 2.

*i.e.*no increase from the initial PSD) in the mid-spatial frequency range (5- 30m

^{−1}) during the polishing process. This also means that the figures of merit in Section 3.4 were effectively representing the errors in terms of the spatial frequencies in the course of the optimization. The final surface had 0.057arcsec RMS slope variation and 10nm RMS surface irregularity, which meets the target specification. About 99.6% of the initial error volume was removed. This demonstrates that the non-sequential optimization technique successfully balanced between various size TIFs by selecting the large TIFs for most of the error volume and the small TIFs only for the localized small errors. The final surface error map is shown in Case 2.3, Fig. 5.

## 5. Concluding remarks

*i.e.*short polishing time) of the CCOS process using the new technique was clearly demonstrated. The CCOS aided with this new optimization technique enables mass fabrication processes for high quality optical surfaces, and will meaningfully contribute to the materialization of the next generation optical systems, such as Laser Inertial Fusion Engine [28

28. A. Heller, “Safe and sustainable energy with LIFE” (2009), https://str.llnl.gov/AprMay09/pdfs/05.09.02.pdf.

## Appendix B. Analytical solution for the dwell time map

*i.e.*impulse response) should not be a function of workpiece coordinates

*x*and

_{workpiece}*y*[29]. The TIF should be same everywhere on the workpiece.

_{workpiece}*i.e.*spatially invariant TIF) [20,22

22. M. Negishi, M. Ando, M. Takimoto, A. Deguchi, and N. Nakamura, “Studies on super-smooth polishing (2nd report),” J. Jpn. Soc. Precis. Eng. **62**, 408–412 (
1996). [CrossRef]

*TIF*(

*x*) with

_{TIF,}y_{TIF,}x_{workpiece,}y_{workpiece}*TIF*(

*x*) in Eq. (1), the analytical solution for the dwell time map can be calculated using the property of Fourier transform as below. (The 2D convolution operator can be changed to the multiplication operator.)

_{TIF,}y_{TIF}*FF*is the 2D Fourier transform. Then, using the inverse Fourier transform, the dwell time map is

*FF*is the inverse 2D Fourier transform. The removal map is replaced with the target removal map, which is the ideal goal. This is an analytical and ideal dwell time map solution, which gives the perfect removal map.

^{−1}**17**(7), 5656–5665 (
2009). [CrossRef] [PubMed]

*e.g.*square tool case) and the workpiece rotates, the orientation of the square TIF also rotates with respect to the workpiece. Thus, no general solution to the dwell time map in Eq. (1) exists analytically.

## Acknowledgments

## References and links

1. | R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra , 34–39 ( 1963). |

2. | R. Aspden, R. McDonough, and F. R. Nitchie Jr., “Computer assisted optical surfacing,” Appl. Opt. |

3. | R. E. Wagner and R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. |

4. | R. A. Jones, “Computer-controlled polishing of telescope mirror segments,” Opt. Eng. |

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

6. | J. R. Johnson, and E. Waluschka, “Optical fabrication-process modeling-analysis tool box,” in |

7. | D. D. Walker, D. Brooks, A. King, R. Freeman, R. Morton, G. McCavana, and S. W. Kim, “The ‘Precessions’ tooling for polishing and figuring flat, spherical and aspheric surfaces,” Opt. Express |

8. | H. M. Pollicove, E. M. Fess, and J. M. Schoen, “Deterministic manufacturing processes for precision optical surfaces,” in |

9. | D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express |

10. | S. D. Jacobs, “International innovations in optical finishing,” in Current Developments in |

11. | J. H. Burge, S. Benjamin, D. Caywood, C. Noble, M. Novak, C. Oh, R. Parks, B. Smith, P. Su, M. Valente, and C. Zhao, “Fabrication and testing of 1.4-m convex off-axis aspheric optical surfaces,” in |

12. | M. Johns, “The Giant Magellan Telescope (GMT),” in |

13. | J. Nelson, and G. H. Sanders, “The status of the Thirty Meter Telescope project,” in |

14. | D. D. Walker, A. P. Doel, R. G. Bingham, D. Brooks, A. M. King, G. Peggs, B. Hughes, S. Oldfield, C. Dorn, H. McAndrews, G. Dando, and D. Riley, “Design Study Report: The Primary and Secondary Mirrors for the Proposed Euro50 Telescope” (2002), http://www.zeeko.co.uk/papers/dl/New%20Study%20Report%20V%2026.pdf. |

15. | T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, and D. Walker, “The Euro50 Extremely Large Telescope,” in |

16. | A. Ardeberg, T. Andersen, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, “From Euro50 towards a European ELT,” in |

17. | R. E. Parks, “Specifications: Figure and Finish are not enough,” in |

18. | J. M. Hill, “Optical Design, Error Budget and Specifications for the Columbus Project Telescope,” in |

19. | 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 |

20. | A. P. Bogodanov, “Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool,” Sov. J. Opt. Technol. |

21. | C. L. Carnal, C. M. Egert, and K. W. Hylton, “Advanced matrix-based algorithms for ion beam milling of optical components,” in |

22. | M. Negishi, M. Ando, M. Takimoto, A. Deguchi, and N. Nakamura, “Studies on super-smooth polishing (2nd report),” J. Jpn. Soc. Precis. Eng. |

23. | H. Lee and M. Yang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. |

24. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

25. | C. Bob, Crawford, Don Loomis, Norm Schenck, and Bill Anderson, Optical Engineering and Fabrication Facility, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, (personal communication, 2008). |

26. | D. W. Kim, W. H. Park, S. W. Kim, and J. H. Burge, “Edge tool influence function library using the parametric edge model for computer controlled optical surfacing,” in |

27. | P. R. Goode, C. J. Denker, L. I. Didkovsky, J. R. Kuhn, and H. Wang, “1.6 m Solar Telescope in Big Bear – the NST,” J. Korean Astron. Soc. |

28. | A. Heller, “Safe and sustainable energy with LIFE” (2009), https://str.llnl.gov/AprMay09/pdfs/05.09.02.pdf. |

29. | H. H. Barrett, and K. J. Myers, |

30. | D. W. Kim, and S. W. Kim, “Novel simulation technique for efficient fabrication of 2m class hexagonal segments for extremely large telescope primary mirrors,” in |

**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:**

Optical Design and Fabrication

**History**

Original Manuscript: August 28, 2009

Revised Manuscript: October 30, 2009

Manuscript Accepted: November 6, 2009

Published: November 13, 2009

**Citation**

Dae Wook Kim, Sug-Whan Kim, and James H. Burge, "Non-sequential optimization technique for a computer controlled optical surfacing process using multiple tool influence functions," Opt. Express **17**, 21850-21866 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21850

Sort: Year | Journal | Reset

### References

- R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra , 34–39 (1963).
- R. Aspden, R. McDonough, and F. R. Nitchie., “Computer assisted optical surfacing,” Appl. Opt. 11(12), 2739–2747 (1972). [CrossRef] [PubMed]
- R. E. Wagner and R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. 13(7), 1683–1689 (1974). [CrossRef] [PubMed]
- R. A. Jones, “Computer-controlled polishing of telescope mirror segments,” Opt. Eng. 22, 236–240 (1983).
- R. A. Jones, “Computer-controlled optical surfacing with orbital tool motion,” Opt. Eng. 25, 785–790 (1986).
- J. R. Johnson, and E. Waluschka, “Optical fabrication-process modeling-analysis tool box,” in Manufacturing and metrology tooling for the Solar-A Soft X-Ray Telescope, W.R. Sigman, L.V. Burns, C.G. Hull-Allen, A.F. Slomba, and R.G. Kusha, eds., Proc. SPIE 1333, 106–117 (1990).
- D. D. Walker, D. Brooks, A. King, R. Freeman, R. Morton, G. McCavana, and S. W. Kim, “The ‘Precessions’ tooling for polishing and figuring flat, spherical and aspheric surfaces,” Opt. Express 11(8), 958–964 (2003). [CrossRef] [PubMed]
- H. M. Pollicove, E. M. Fess, and J. M. Schoen, “Deterministic manufacturing processes for precision optical surfaces,” in Window and Dome Technologies VIII, R. W. Tustison, eds., Proc. SPIE 5078, 90–96 (2003).
- D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express 13(3), 910–917 (2005). [CrossRef] [PubMed]
- S. D. Jacobs, “International innovations in optical finishing,” in Current Developments in Lens Design and Optical Engineering V, P.Z. Mouroulis, W.J. Smith, and R.B. Johnson, eds., Proc. SPIE 5523, 264–272 (2004).
- J. H. Burge, S. Benjamin, D. Caywood, C. Noble, M. Novak, C. Oh, R. Parks, B. Smith, P. Su, M. Valente, and C. Zhao, “Fabrication and testing of 1.4-m convex off-axis aspheric optical surfaces,” in Optical Manufacturing and Testing VIII, J. H. Burge; O. W. Fähnle and R. Williamson, eds., Proc. SPIE 7426, 74260L1–12 (2009).
- M. Johns, “The Giant Magellan Telescope (GMT),” in Extremely Large Telescopes: Which Wavelengths? T. E. Andersen, eds., Proc. SPIE 6986, 698603 1–12 (2008).
- J. Nelson, and G. H. Sanders, “The status of the Thirty Meter Telescope project,” in Ground-based and Airborne Telescopes II, L. M. Stepp and R. Gilmozzi, eds., Proc. SPIE 7012, 70121A1–18 (2008).
- D. D. Walker, A. P. Doel, R. G. Bingham, D. Brooks, A. M. King, G. Peggs, B. Hughes, S. Oldfield, C. Dorn, H. McAndrews, G. Dando, and D. Riley, “Design Study Report: The Primary and Secondary Mirrors for the Proposed Euro50 Telescope” (2002), http://www.zeeko.co.uk/papers/dl/New%20Study%20Report%20V%2026.pdf .
- T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, and D. Walker, “The Euro50 Extremely Large Telescope,” in Future Giant Telescopes, J.R.P. Angel and R. Gilmozzi, eds., Proc. SPIE 4840, 214–225 (2003).
- A. Ardeberg, T. Andersen, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, “From Euro50 towards a European ELT,” in Ground-based and Airborne Telescopes, L. M. Stepp, eds., Proc. SPIE 6267, 626725 1–10 (2006).
- R. E. Parks, “Specifications: Figure and Finish are not enough,” in An optical Believe It or Not: Key Lessons Learned, M. A. Kahan, eds., Proc. SPIE 7071, 70710B1–9 (2008).
- J. M. Hill, “Optical Design, Error Budget and Specifications for the Columbus Project Telescope,” in Advanced Technology Optical Telescopes IV, L. D. Barr, eds., Proc. SPIE 1236, 86–107 (1990).
- 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]
- A. P. Bogodanov, “Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool,” Sov. J. Opt. Technol. 52, 409–413 (1985).
- C. L. Carnal, C. M. Egert, and K. W. Hylton, “Advanced matrix-based algorithms for ion beam milling of optical components,” in Current Developments in Optical Design and Optical Engineering II, R. E. Fischer and W. J. Smith, eds., Proc. SPIE 1752, 54–62 (1992).
- M. Negishi, M. Ando, M. Takimoto, A. Deguchi, and N. Nakamura, “Studies on super-smooth polishing (2nd report),” J. Jpn. Soc. Precis. Eng. 62, 408–412 (1996). [CrossRef]
- H. Lee and M. Yang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. 40(9), 1936–1943 (2001). [CrossRef]
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical recipes in C (Cambridge, 1988)
- C. Bob, Crawford, Don Loomis, Norm Schenck, and Bill Anderson, Optical Engineering and Fabrication Facility, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, (personal communication, 2008).
- D. W. Kim, W. H. Park, S. W. Kim, and J. H. Burge, “Edge tool influence function library using the parametric edge model for computer controlled optical surfacing,” in Optical Manufacturing and Testing VIII, J. H. Burge; O. W. Fähnle and R. Williamson, Proc. SPIE 7426, 74260G1–12 (2009).
- P. R. Goode, C. J. Denker, L. I. Didkovsky, J. R. Kuhn, and H. Wang, “1.6 m Solar Telescope in Big Bear – the NST,” J. Korean Astron. Soc. 35, 1–8 (2002).
- A. Heller, “Safe and sustainable energy with LIFE” (2009), https://str.llnl.gov/AprMay09/pdfs/05.09.02.pdf .
- H. H. Barrett, and K. J. Myers, Foundations of Image Science (Wiley, 2004)
- D. W. Kim, and S. W. Kim, “Novel simulation technique for efficient fabrication of 2m class hexagonal segments for extremely large telescope primary mirrors,” in Optical Design and Testing II, Y. Wang, Z. Weng, S. Ye and J. M. Sasian, eds., Proc. SPIE 5638, 48–59 (2005).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.