Abstract

Fast/slow tool servo (FTS/STS) diamond turning is a very promising technique for the generation of micro-lens array (MLA). However, it is still a challenge to process MLA in large scale due to certain inherent limitations of this technique. In the present study, a novel ultra-precision diamond cutting method, as the end-fly-cutting-servo (EFCS) system, is adopted and investigated for large-scale generation of MLA. After a detailed discussion of the characteristic advantages for processing MLA, the optimal toolpath generation strategy for the EFCS is developed with consideration of the geometry and installation pose of the diamond tool. A typical aspheric MLA over a large area is experimentally fabricated, and the resulting form accuracy, surface micro-topography and machining efficiency are critically investigated. The result indicates that the MLA with homogeneous quality over the whole area is obtained. Besides, high machining efficiency, extremely small volume of control points for the toolpath, and optimal usage of system dynamics of the machine tool during the whole cutting can be simultaneously achieved.

© 2015 Optical Society of America

Full Article  |  PDF Article
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Zhanwen Sun, Suet To, Guoqing Zhang, and Shaojian Zhang
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2015 (2)

S. J. Zhang, S. To, G. Q. Zhang, and Z. W. Zhu, “A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 91, 34–42 (2015).
[Crossref]

Z. Zhu, S. To, and S. Zhang, “Theoretical and experimental investigation on the novel end-fly-cutting-servo diamond machining of hierarchical micro-nanostructures,” Int. J. Mach. Tools Manuf. 94, 15–25 (2015).
[Crossref]

2014 (4)

2013 (7)

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

X.-H. Lee, I. Moreno, and C.-C. Sun, “High-performance LED street lighting using microlens arrays,” Opt. Express 21(9), 10612–10621 (2013).
[Crossref] [PubMed]

B. McCall and T. S. Tkaczyk, “Rapid fabrication of miniature lens arrays by four-axis single point diamond machining,” Opt. Express 21(3), 3557–3572 (2013).
[Crossref] [PubMed]

C.-C. Chen, C.-Y. Huang, Y.-C. Cheng, and W.-Y. Hsu, “Ultra-precision diamond milling of aspheric microlens array,” Proc. SPIE 8769, 87693–87699 (2013).

S. J. Zhang, S. To, and H. T. Wang, “A theoretical and experimental investigation into five-DOF dynamic characteristics of an aerostatic bearing spindle in ultra-precision diamond turning,” Int. J. Mach. Tools Manuf. 71, 1–10 (2013).
[Crossref]

H. Zhang, L. Li, D. L. McCray, S. Scheiding, N. J. Naples, A. Gebhardt, S. Risse, R. Eberhardt, A. Tünnermann, and A. Y. Yi, “Development of a low cost high precision three-layer 3D artificial compound eye,” Opt. Express 21(19), 22232–22245 (2013).
[Crossref] [PubMed]

Z. Zhu, X. Zhou, D. Luo, and Q. Liu, “Development of pseudo-random diamond turning method for fabricating freeform optics with scattering homogenization,” Opt. Express 21(23), 28469–28482 (2013).
[Crossref] [PubMed]

2012 (4)

L. Li and A. Y. Yi, “Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera,” Appl. Opt. 51(12), 1843–1852 (2012).
[Crossref] [PubMed]

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

D. P. Yu, G. S. Hong, and Y. S. Wong, “Profile error compensation in fast tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 52(1), 13–23 (2012).
[Crossref]

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

2011 (4)

S. Scheiding, A. Y. Yi, A. Gebhardt, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Freeform manufacturing of a microoptical lens array on a steep curved substrate by use of a voice coil fast tool servo,” Opt. Express 19(24), 23938–23951 (2011).
[Crossref] [PubMed]

S. Scheiding, A. Y. Yi, A. Gebhardt, R. Loose, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Diamond milling or turning for the fabrication of micro lens arrays: comparing different diamond machining technologies,” Proc. SPIE 7927, 79270N (2011).
[Crossref]

D. P. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng. 21(9), 095011 (2011).
[Crossref]

D. P. Yu, Y. S. Wong, and G. S. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol. 57(1-4), 85–99 (2011).
[Crossref]

2010 (6)

C. Cheung, K. Hu, X. Jiang, and L. Kong, “Characterization of surface defects in fast tool servo machining of microlens array using a pattern recognition and analysis method,” Measurement 43(9), 1240–1249 (2010).
[Crossref]

C. Cheung, L. Kong, and M. Ren, “Measurement and characterization of ultra-precision freeform surfaces using an intrinsic surface feature-based method,” Meas. Sci. Technol. 21(11), 115109 (2010).
[Crossref]

J.-T. Wu, W.-Y. Chang, and S.-Y. Yang, “Fabrication of a nano/micro hybrid lens using gas-assisted hot embossing with an anodic aluminum oxide (AAO) template,” J. Micromech. Microeng. 20(7), 075023 (2010).
[Crossref]

J. Yan, Z. Zhang, T. Kuriyagawa, and H. Gonda, “Fabricating micro-structured surface by using single-crystalline diamond endmill,” Int. J. Adv. Manuf. Technol. 51(9-12), 957–964 (2010).
[Crossref]

B. McCall and T. S. Tkaczyk, “Fabrication of plastic microlens array for array microscopy by three-dimensional diamond micromilling,” Opt. Eng. 49(10), 103401 (2010).
[Crossref] [PubMed]

L. Li and A. Y. Yi, “Development of a 3D artificial compound eye,” Opt. Express 18(17), 18125–18137 (2010).
[Crossref] [PubMed]

2009 (1)

C. Huang, L. Li, and A. Y. Yi, “Design and fabrication of a micro Alvarez lens array with a variable focal length,” Microsyst. Technol. 15(4), 559–563 (2009).
[Crossref]

2008 (1)

2006 (3)

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

K.-H. Jeong, J. Kim, and L. P. Lee, “Biologically inspired artificial compound eyes,” Science 312(5773), 557–561 (2006).
[Crossref] [PubMed]

S. To, T. Kwok, C. Cheung, and W. Lee, “Study of ultra-precision diamond turning of a microlens array with a fast tool servo system,” Proc. SPIE 6149, 61490S (2006).
[Crossref]

2005 (1)

2003 (1)

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Araki, T.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Bennet, M.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Bian, H.

Brecher, C.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Chang, W.-Y.

J.-T. Wu, W.-Y. Chang, and S.-Y. Yang, “Fabrication of a nano/micro hybrid lens using gas-assisted hot embossing with an anodic aluminum oxide (AAO) template,” J. Micromech. Microeng. 20(7), 075023 (2010).
[Crossref]

Chen, C.-C.

C.-C. Chen, C.-Y. Huang, Y.-C. Cheng, and W.-Y. Hsu, “Ultra-precision diamond milling of aspheric microlens array,” Proc. SPIE 8769, 87693–87699 (2013).

Chen, F.

Cheng, Y.-C.

C.-C. Chen, C.-Y. Huang, Y.-C. Cheng, and W.-Y. Hsu, “Ultra-precision diamond milling of aspheric microlens array,” Proc. SPIE 8769, 87693–87699 (2013).

Cheung, C.

C. Cheung, L. Kong, and M. Ren, “Measurement and characterization of ultra-precision freeform surfaces using an intrinsic surface feature-based method,” Meas. Sci. Technol. 21(11), 115109 (2010).
[Crossref]

C. Cheung, K. Hu, X. Jiang, and L. Kong, “Characterization of surface defects in fast tool servo machining of microlens array using a pattern recognition and analysis method,” Measurement 43(9), 1240–1249 (2010).
[Crossref]

S. To, T. Kwok, C. Cheung, and W. Lee, “Study of ultra-precision diamond turning of a microlens array with a fast tool servo system,” Proc. SPIE 6149, 61490S (2006).
[Crossref]

Choi, K.-J.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Cölfen, H.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Crozier, K. B.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Deng, Z.

Du, G.

Eberhardt, R.

Fang, F. Z.

Fratzl, P.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Gan, S. W.

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

Gao, W.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Gebhardt, A.

Gonda, H.

J. Yan, Z. Zhang, T. Kuriyagawa, and H. Gonda, “Fabricating micro-structured surface by using single-crystalline diamond endmill,” Int. J. Adv. Manuf. Technol. 51(9-12), 957–964 (2010).
[Crossref]

Hong, G.

D. P. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng. 21(9), 095011 (2011).
[Crossref]

Hong, G. S.

D. P. Yu, G. S. Hong, and Y. S. Wong, “Profile error compensation in fast tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 52(1), 13–23 (2012).
[Crossref]

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

D. P. Yu, Y. S. Wong, and G. S. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol. 57(1-4), 85–99 (2011).
[Crossref]

Hou, X.

Hsu, W.-Y.

C.-C. Chen, C.-Y. Huang, Y.-C. Cheng, and W.-Y. Hsu, “Ultra-precision diamond milling of aspheric microlens array,” Proc. SPIE 8769, 87693–87699 (2013).

Hu, K.

C. Cheung, K. Hu, X. Jiang, and L. Kong, “Characterization of surface defects in fast tool servo machining of microlens array using a pattern recognition and analysis method,” Measurement 43(9), 1240–1249 (2010).
[Crossref]

Hu, X. T.

Huang, C.

C. Huang, L. Li, and A. Y. Yi, “Design and fabrication of a micro Alvarez lens array with a variable focal length,” Microsyst. Technol. 15(4), 559–563 (2009).
[Crossref]

Huang, C.-Y.

C.-C. Chen, C.-Y. Huang, Y.-C. Cheng, and W.-Y. Hsu, “Ultra-precision diamond milling of aspheric microlens array,” Proc. SPIE 8769, 87693–87699 (2013).

Huang, Y.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Jeong, K.-H.

K.-H. Jeong, J. Kim, and L. P. Lee, “Biologically inspired artificial compound eyes,” Science 312(5773), 557–561 (2006).
[Crossref] [PubMed]

Jiang, X.

C. Cheung, K. Hu, X. Jiang, and L. Kong, “Characterization of surface defects in fast tool servo machining of microlens array using a pattern recognition and analysis method,” Measurement 43(9), 1240–1249 (2010).
[Crossref]

Jung, I.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Kim, J.

K.-H. Jeong, J. Kim, and L. P. Lee, “Biologically inspired artificial compound eyes,” Science 312(5773), 557–561 (2006).
[Crossref] [PubMed]

Kim, R.-H.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Kiyono, S.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Kommareddy, K. P.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Kong, L.

C. Cheung, K. Hu, X. Jiang, and L. Kong, “Characterization of surface defects in fast tool servo machining of microlens array using a pattern recognition and analysis method,” Measurement 43(9), 1240–1249 (2010).
[Crossref]

C. Cheung, L. Kong, and M. Ren, “Measurement and characterization of ultra-precision freeform surfaces using an intrinsic surface feature-based method,” Meas. Sci. Technol. 21(11), 115109 (2010).
[Crossref]

Kumar, A. S.

D. W. K. Neo, A. S. Kumar, and M. Rahman, “A novel surface analytical model for cutting linearization error in fast tool/slow slide servo diamond turning,” Precis. Eng. 38(4), 849–860 (2014).
[Crossref]

Kuriyagawa, T.

J. Yan, Z. Zhang, T. Kuriyagawa, and H. Gonda, “Fabricating micro-structured surface by using single-crystalline diamond endmill,” Int. J. Adv. Manuf. Technol. 51(9-12), 957–964 (2010).
[Crossref]

Kwok, T.

S. To, T. Kwok, C. Cheung, and W. Lee, “Study of ultra-precision diamond turning of a microlens array with a fast tool servo system,” Proc. SPIE 6149, 61490S (2006).
[Crossref]

Lange, S.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Lee, K.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Lee, L. P.

K.-H. Jeong, J. Kim, and L. P. Lee, “Biologically inspired artificial compound eyes,” Science 312(5773), 557–561 (2006).
[Crossref] [PubMed]

Lee, S.-W.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Lee, W.

S. To, T. Kwok, C. Cheung, and W. Lee, “Study of ultra-precision diamond turning of a microlens array with a fast tool servo system,” Proc. SPIE 6149, 61490S (2006).
[Crossref]

Lee, X.-H.

Li, L.

R. Sun, Y. Li, and L. Li, “Rapid method for fabricating polymeric biconvex parabolic lenslets,” Opt. Lett. 39(18), 5391–5394 (2014).
[Crossref]

H. Zhang, L. Li, D. L. McCray, S. Scheiding, N. J. Naples, A. Gebhardt, S. Risse, R. Eberhardt, A. Tünnermann, and A. Y. Yi, “Development of a low cost high precision three-layer 3D artificial compound eye,” Opt. Express 21(19), 22232–22245 (2013).
[Crossref] [PubMed]

L. Li and A. Y. Yi, “Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera,” Appl. Opt. 51(12), 1843–1852 (2012).
[Crossref] [PubMed]

S. Scheiding, A. Y. Yi, A. Gebhardt, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Freeform manufacturing of a microoptical lens array on a steep curved substrate by use of a voice coil fast tool servo,” Opt. Express 19(24), 23938–23951 (2011).
[Crossref] [PubMed]

S. Scheiding, A. Y. Yi, A. Gebhardt, R. Loose, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Diamond milling or turning for the fabrication of micro lens arrays: comparing different diamond machining technologies,” Proc. SPIE 7927, 79270N (2011).
[Crossref]

L. Li and A. Y. Yi, “Development of a 3D artificial compound eye,” Opt. Express 18(17), 18125–18137 (2010).
[Crossref] [PubMed]

C. Huang, L. Li, and A. Y. Yi, “Design and fabrication of a micro Alvarez lens array with a variable focal length,” Microsyst. Technol. 15(4), 559–563 (2009).
[Crossref]

A. Y. Yi and L. Li, “Design and fabrication of a microlens array by use of a slow tool servo,” Opt. Lett. 30(13), 1707–1709 (2005).
[Crossref] [PubMed]

Li, R.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Li, Y.

Liu, Q.

Liu, Z.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Loose, R.

S. Scheiding, A. Y. Yi, A. Gebhardt, R. Loose, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Diamond milling or turning for the fabrication of micro lens arrays: comparing different diamond machining technologies,” Proc. SPIE 7927, 79270N (2011).
[Crossref]

Lu, C.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Luo, D.

Malyarchuk, V.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Manjubala, I.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Masic, A.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

McCall, B.

B. McCall and T. S. Tkaczyk, “Rapid fabrication of miniature lens arrays by four-axis single point diamond machining,” Opt. Express 21(3), 3557–3572 (2013).
[Crossref] [PubMed]

B. McCall and T. S. Tkaczyk, “Fabrication of plastic microlens array for array microscopy by three-dimensional diamond micromilling,” Opt. Eng. 49(10), 103401 (2010).
[Crossref] [PubMed]

McCray, D. L.

Merz, M.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Moreno, I.

Naples, N. J.

Neo, D. W. K.

D. W. K. Neo, A. S. Kumar, and M. Rahman, “A novel surface analytical model for cutting linearization error in fast tool/slow slide servo diamond turning,” Precis. Eng. 38(4), 849–860 (2014).
[Crossref]

Niehaus, F.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Okazaki, Y.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Park, H.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Park, S. B.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Rahman, M.

D. W. K. Neo, A. S. Kumar, and M. Rahman, “A novel surface analytical model for cutting linearization error in fast tool/slow slide servo diamond turning,” Precis. Eng. 38(4), 849–860 (2014).
[Crossref]

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

Ren, M.

C. Cheung, L. Kong, and M. Ren, “Measurement and characterization of ultra-precision freeform surfaces using an intrinsic surface feature-based method,” Meas. Sci. Technol. 21(11), 115109 (2010).
[Crossref]

Risse, S.

Rogers, J. A.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Scheiding, S.

Si, J.

Song, Y. M.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Sun, C.-C.

Sun, R.

Tkaczyk, T. S.

B. McCall and T. S. Tkaczyk, “Rapid fabrication of miniature lens arrays by four-axis single point diamond machining,” Opt. Express 21(3), 3557–3572 (2013).
[Crossref] [PubMed]

B. McCall and T. S. Tkaczyk, “Fabrication of plastic microlens array for array microscopy by three-dimensional diamond micromilling,” Opt. Eng. 49(10), 103401 (2010).
[Crossref] [PubMed]

To, S.

S. J. Zhang, S. To, G. Q. Zhang, and Z. W. Zhu, “A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 91, 34–42 (2015).
[Crossref]

Z. Zhu, S. To, and S. Zhang, “Theoretical and experimental investigation on the novel end-fly-cutting-servo diamond machining of hierarchical micro-nanostructures,” Int. J. Mach. Tools Manuf. 94, 15–25 (2015).
[Crossref]

S. J. Zhang, S. To, and H. T. Wang, “A theoretical and experimental investigation into five-DOF dynamic characteristics of an aerostatic bearing spindle in ultra-precision diamond turning,” Int. J. Mach. Tools Manuf. 71, 1–10 (2013).
[Crossref]

S. To, T. Kwok, C. Cheung, and W. Lee, “Study of ultra-precision diamond turning of a microlens array with a fast tool servo system,” Proc. SPIE 6149, 61490S (2006).
[Crossref]

Tong, S.

Tünnermann, A.

Wagermaier, W.

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Wang, H. T.

S. J. Zhang, S. To, and H. T. Wang, “A theoretical and experimental investigation into five-DOF dynamic characteristics of an aerostatic bearing spindle in ultra-precision diamond turning,” Int. J. Mach. Tools Manuf. 71, 1–10 (2013).
[Crossref]

Weck, M.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Wenzel, C.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Winterschladen, M.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Wong, Y.

D. P. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng. 21(9), 095011 (2011).
[Crossref]

Wong, Y. S.

D. P. Yu, G. S. Hong, and Y. S. Wong, “Profile error compensation in fast tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 52(1), 13–23 (2012).
[Crossref]

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

D. P. Yu, Y. S. Wong, and G. S. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol. 57(1-4), 85–99 (2011).
[Crossref]

Wu, J.-T.

J.-T. Wu, W.-Y. Chang, and S.-Y. Yang, “Fabrication of a nano/micro hybrid lens using gas-assisted hot embossing with an anodic aluminum oxide (AAO) template,” J. Micromech. Microeng. 20(7), 075023 (2010).
[Crossref]

Xiao, J.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Xie, Y.

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Yamanaka, M.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Yan, J.

J. Yan, Z. Zhang, T. Kuriyagawa, and H. Gonda, “Fabricating micro-structured surface by using single-crystalline diamond endmill,” Int. J. Adv. Manuf. Technol. 51(9-12), 957–964 (2010).
[Crossref]

Yang, Q.

Yang, S.-Y.

J.-T. Wu, W.-Y. Chang, and S.-Y. Yang, “Fabrication of a nano/micro hybrid lens using gas-assisted hot embossing with an anodic aluminum oxide (AAO) template,” J. Micromech. Microeng. 20(7), 075023 (2010).
[Crossref]

Yao, J.

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

Yi, A. Y.

Yu, D. P.

D. P. Yu, G. S. Hong, and Y. S. Wong, “Profile error compensation in fast tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 52(1), 13–23 (2012).
[Crossref]

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

D. P. Yu, Y. S. Wong, and G. S. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol. 57(1-4), 85–99 (2011).
[Crossref]

D. P. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng. 21(9), 095011 (2011).
[Crossref]

Zhang, G. Q.

S. J. Zhang, S. To, G. Q. Zhang, and Z. W. Zhu, “A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 91, 34–42 (2015).
[Crossref]

Zhang, H.

Zhang, S.

Z. Zhu, S. To, and S. Zhang, “Theoretical and experimental investigation on the novel end-fly-cutting-servo diamond machining of hierarchical micro-nanostructures,” Int. J. Mach. Tools Manuf. 94, 15–25 (2015).
[Crossref]

Zhang, S. J.

S. J. Zhang, S. To, G. Q. Zhang, and Z. W. Zhu, “A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 91, 34–42 (2015).
[Crossref]

S. J. Zhang, S. To, and H. T. Wang, “A theoretical and experimental investigation into five-DOF dynamic characteristics of an aerostatic bearing spindle in ultra-precision diamond turning,” Int. J. Mach. Tools Manuf. 71, 1–10 (2013).
[Crossref]

Zhang, X. D.

Zhang, Z.

J. Yan, Z. Zhang, T. Kuriyagawa, and H. Gonda, “Fabricating micro-structured surface by using single-crystalline diamond endmill,” Int. J. Adv. Manuf. Technol. 51(9-12), 957–964 (2010).
[Crossref]

Zhou, X.

Zhu, Z.

Z. Zhu, S. To, and S. Zhang, “Theoretical and experimental investigation on the novel end-fly-cutting-servo diamond machining of hierarchical micro-nanostructures,” Int. J. Mach. Tools Manuf. 94, 15–25 (2015).
[Crossref]

Z. Zhu, X. Zhou, D. Luo, and Q. Liu, “Development of pseudo-random diamond turning method for fabricating freeform optics with scattering homogenization,” Opt. Express 21(23), 28469–28482 (2013).
[Crossref] [PubMed]

Zhu, Z. W.

S. J. Zhang, S. To, G. Q. Zhang, and Z. W. Zhu, “A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 91, 34–42 (2015).
[Crossref]

Appl. Opt. (1)

CIRP Ann. Manuf. Tech. (1)

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. Manuf. Tech. 55(1), 547–550 (2006).
[Crossref]

Int. J. Adv. Manuf. Technol. (3)

D. P. Yu, Y. S. Wong, and G. S. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol. 57(1-4), 85–99 (2011).
[Crossref]

D. P. Yu, S. W. Gan, Y. S. Wong, G. S. Hong, M. Rahman, and J. Yao, “Optimized tool path generation for fast tool servo diamond turning of micro-structured surfaces,” Int. J. Adv. Manuf. Technol. 63(9-12), 1137–1152 (2012).
[Crossref]

J. Yan, Z. Zhang, T. Kuriyagawa, and H. Gonda, “Fabricating micro-structured surface by using single-crystalline diamond endmill,” Int. J. Adv. Manuf. Technol. 51(9-12), 957–964 (2010).
[Crossref]

Int. J. Mach. Tools Manuf. (4)

D. P. Yu, G. S. Hong, and Y. S. Wong, “Profile error compensation in fast tool servo diamond turning of micro-structured surfaces,” Int. J. Mach. Tools Manuf. 52(1), 13–23 (2012).
[Crossref]

S. J. Zhang, S. To, and H. T. Wang, “A theoretical and experimental investigation into five-DOF dynamic characteristics of an aerostatic bearing spindle in ultra-precision diamond turning,” Int. J. Mach. Tools Manuf. 71, 1–10 (2013).
[Crossref]

S. J. Zhang, S. To, G. Q. Zhang, and Z. W. Zhu, “A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 91, 34–42 (2015).
[Crossref]

Z. Zhu, S. To, and S. Zhang, “Theoretical and experimental investigation on the novel end-fly-cutting-servo diamond machining of hierarchical micro-nanostructures,” Int. J. Mach. Tools Manuf. 94, 15–25 (2015).
[Crossref]

J. Micromech. Microeng. (2)

D. P. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng. 21(9), 095011 (2011).
[Crossref]

J.-T. Wu, W.-Y. Chang, and S.-Y. Yang, “Fabrication of a nano/micro hybrid lens using gas-assisted hot embossing with an anodic aluminum oxide (AAO) template,” J. Micromech. Microeng. 20(7), 075023 (2010).
[Crossref]

Meas. Sci. Technol. (1)

C. Cheung, L. Kong, and M. Ren, “Measurement and characterization of ultra-precision freeform surfaces using an intrinsic surface feature-based method,” Meas. Sci. Technol. 21(11), 115109 (2010).
[Crossref]

Measurement (1)

C. Cheung, K. Hu, X. Jiang, and L. Kong, “Characterization of surface defects in fast tool servo machining of microlens array using a pattern recognition and analysis method,” Measurement 43(9), 1240–1249 (2010).
[Crossref]

Microsyst. Technol. (1)

C. Huang, L. Li, and A. Y. Yi, “Design and fabrication of a micro Alvarez lens array with a variable focal length,” Microsyst. Technol. 15(4), 559–563 (2009).
[Crossref]

Nat. Commun. (1)

K. Lee, W. Wagermaier, A. Masic, K. P. Kommareddy, M. Bennet, I. Manjubala, S.-W. Lee, S. B. Park, H. Cölfen, and P. Fratzl, “Self-assembly of amorphous calcium carbonate microlens arrays,” Nat. Commun. 3, 725 (2012).
[Crossref] [PubMed]

Nature (1)

Y. M. Song, Y. Xie, V. Malyarchuk, J. Xiao, I. Jung, K.-J. Choi, Z. Liu, H. Park, C. Lu, R.-H. Kim, R. Li, K. B. Crozier, Y. Huang, and J. A. Rogers, “Digital cameras with designs inspired by the arthropod eye,” Nature 497(7447), 95–99 (2013).
[Crossref] [PubMed]

Opt. Eng. (1)

B. McCall and T. S. Tkaczyk, “Fabrication of plastic microlens array for array microscopy by three-dimensional diamond micromilling,” Opt. Eng. 49(10), 103401 (2010).
[Crossref] [PubMed]

Opt. Express (8)

S. Scheiding, A. Y. Yi, A. Gebhardt, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Freeform manufacturing of a microoptical lens array on a steep curved substrate by use of a voice coil fast tool servo,” Opt. Express 19(24), 23938–23951 (2011).
[Crossref] [PubMed]

S. Tong, H. Bian, Q. Yang, F. Chen, Z. Deng, J. Si, and X. Hou, “Large-scale high quality glass microlens arrays fabricated by laser enhanced wet etching,” Opt. Express 22(23), 29283–29291 (2014).
[PubMed]

B. McCall and T. S. Tkaczyk, “Rapid fabrication of miniature lens arrays by four-axis single point diamond machining,” Opt. Express 21(3), 3557–3572 (2013).
[Crossref] [PubMed]

L. Li and A. Y. Yi, “Development of a 3D artificial compound eye,” Opt. Express 18(17), 18125–18137 (2010).
[Crossref] [PubMed]

X.-H. Lee, I. Moreno, and C.-C. Sun, “High-performance LED street lighting using microlens arrays,” Opt. Express 21(9), 10612–10621 (2013).
[Crossref] [PubMed]

H. Zhang, L. Li, D. L. McCray, S. Scheiding, N. J. Naples, A. Gebhardt, S. Risse, R. Eberhardt, A. Tünnermann, and A. Y. Yi, “Development of a low cost high precision three-layer 3D artificial compound eye,” Opt. Express 21(19), 22232–22245 (2013).
[Crossref] [PubMed]

Z. Zhu, X. Zhou, D. Luo, and Q. Liu, “Development of pseudo-random diamond turning method for fabricating freeform optics with scattering homogenization,” Opt. Express 21(23), 28469–28482 (2013).
[Crossref] [PubMed]

F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008).
[Crossref] [PubMed]

Opt. Lett. (3)

Precis. Eng. (2)

D. W. K. Neo, A. S. Kumar, and M. Rahman, “A novel surface analytical model for cutting linearization error in fast tool/slow slide servo diamond turning,” Precis. Eng. 38(4), 849–860 (2014).
[Crossref]

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Proc. SPIE (3)

S. To, T. Kwok, C. Cheung, and W. Lee, “Study of ultra-precision diamond turning of a microlens array with a fast tool servo system,” Proc. SPIE 6149, 61490S (2006).
[Crossref]

S. Scheiding, A. Y. Yi, A. Gebhardt, R. Loose, L. Li, S. Risse, R. Eberhardt, and A. Tünnermann, “Diamond milling or turning for the fabrication of micro lens arrays: comparing different diamond machining technologies,” Proc. SPIE 7927, 79270N (2011).
[Crossref]

C.-C. Chen, C.-Y. Huang, Y.-C. Cheng, and W.-Y. Hsu, “Ultra-precision diamond milling of aspheric microlens array,” Proc. SPIE 8769, 87693–87699 (2013).

Science (1)

K.-H. Jeong, J. Kim, and L. P. Lee, “Biologically inspired artificial compound eyes,” Science 312(5773), 557–561 (2006).
[Crossref] [PubMed]

Other (1)

S. To and Z. Zhu, Novel end-fly-cutting-servo system for deterministic generation of hierarchical micro-nanostructures (CIRP Ann. Manuf. Tech., 2015).

Cited By

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Figures (8)

Fig. 1
Fig. 1 Configuration of the EFCS System, (a) hardware configuration, (b) the horizontal cutting and (c) the vertical cutting.
Fig. 2
Fig. 2 Schematic of the cutting system, (a) the relative positions between the tool and the basic cell, and (b) the local spherical coordinate system of the tool. Note: the angle α here is negative.
Fig. 3
Fig. 3 EFCS system configuration for cutting experiments.
Fig. 4
Fig. 4 Features of (a) the designed MLA, and (b) the corresponding optimal toolpath.
Fig. 5
Fig. 5 (a) Photography of the workpiece, and microscope diagram of the structures in (b) Zone A, (c) Zone B and (d) Zone C.
Fig. 6
Fig. 6 Characteristics of the machined MLA, (a) the 3-D structure of a large area, (b) the 2D profile of the cross-section, (c) an extracted square area, and (d) the corresponding profiles along the cross-hair directions.
Fig. 7
Fig. 7 Details of the machined lenslet, (a) the 3D structure, (b) and (c), the profiles and the machining errors along the cross-hair directions.
Fig. 8
Fig. 8 Surface micro-topography of a lenslet

Tables (2)

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Table 1 Cutting parameters and the tool geometry

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Table 2 Parameters of the micro-asphere array

Equations (10)

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{ x T = R T cosθcosα y T = R T sinθsin γ 0 + R T cosθsinα z T = R T sinθcos γ 0 , θ[ θ min , θ max ]
{ T i = ( x T θ , y T θ , z T θ ) | θ= θ i θ i = θ min +(i1) ( θ max θ min ) N 0 ,i=1,2,3( N 0 +1)
φ k,l =2πk+ 2πl N s
x axis (k,l) = φ k,l f x 2π R d
[ x i (k,l) y i (k,l) z i (k,l) ]=[ cos( φ k,l +α ) sin( φ k,l +α ) x axis (k,l) x axis (k,l) cos( φ k,l +α ) sin( φ k,l +α ) cos( φ k,l +α ) x axis (k,l) sin( φ k,l +α ) 0 0 1 ][ x i T y i T z i T ]
V i (k,l) = ( f W x , f W y ,1) | x= x i (k,l) ,y= y i (k,l)
P m (k,l) := arg P i T min{| V i (k,l) T i |,i}
z (k,l) =d+ z m T f w ( x m (k,l) , y m (k,l) )
[ x CLP (k,l) y CLP (k,l) z CLP (k,l) ]=[ x axis (k,l) + R d cos( φ k,l +α ) R d sin( φ k,l +α ) f w ( x m (k,l) , y m (k,l) ) z m T ]
z(x,y)= sC R o 2 4+4 1(1+k) C 2 R o 2 - sC ρ 2 (x,y) 4+4 1(1+k) C 2 ρ 2 (x,y)

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