Abstract

A novel approach named off-spindle-axis (OSA) spiral grinding for fabricating aspheric microlens array (AMLA) mold inserts for precision glass molding (PGM) is presented. In OSA spiral grinding, three translational motions of the grinding wheel are synchronized with the rotation of the workpiece to form a local spiral wheel path for individual lens-lets. With this approach, the form accuracy of lens-lets can be compensated within sub-micrometer by means of the on-machine measurement. The determination of wheel path and form error compensation via on-machine measurement are systematically studied. A tungsten carbide mold insert with four convex aspheric lens-lets is fabricated to evaluate the grinding performance. PGM experiments are performed to produce glass AMLA using the ground insert. The experimental results indicate that both the ground and molded AMLA with homogeneous quality are achieved. The form accuracy and surface roughness of both the mold insert and the molded AMLA were less than 0.3 µm in PV and 10 nm in Sa, respectively.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

W. L. Zhu, F. Duan, X. D. Zhang, Z. W. Zhu, and B. F. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
[Crossref]

2017 (4)

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12(1), 46–65 (2017).
[Crossref]

L. Zhang and W. Liu, “Precision glass molding: Toward an optimal fabrication of optical lenses,” Front. Mech. Eng. 12(1), 3–17 (2017).
[Crossref]

S. Yin, H. Jia, G. Zhang, F. Chen, and K. Zhu, “Review of small aspheric glass lens molding technologies,” Front. Mech. Eng. 12(1), 66–76 (2017).
[Crossref]

B. Guo and Q. Zhao, “Ultrasonic vibration assisted grinding of hard and brittle linear micro-structured surfaces,” Precis. Eng. 48(7179), 98–106 (2017).
[Crossref]

2016 (6)

Q. Zhang, S. To, Q. Zhao, and B. Guo, “Surface generation mechanism of WC/Co and RB-SiC/Si composites under high spindle speed grinding (HSSG),” Int. J. Refract. Met. Hard Mater. 56, 123–131 (2016).
[Crossref]

S. To, Z. Zhu, and H. Wang, “Virtual spindle based tool servo diamond turning of discontinuously structured microoptics arrays,” CIRP Ann. Manuf. Tech. 65(1), 475–478 (2016).
[Crossref]

F. Z. Fang, N. Zhang, and X. Zhang, “Precision injection molding of freeform optics,” Adv. Opt. Technol. 5(4), 303–324 (2016).

S. J. Zhang, S. To, Z. W. Zhu, and G. Q. Zhang, “A review of fly cutting applied to surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 103, 13–27 (2016).
[Crossref]

K. Pang, L. Song, F. Fang, Y. Zhang, and H. Zhang, “An imaging system with a large depth of field based on an overlapped micro-lens array,” CIRP Ann. Manuf. Tech. 65(1), 471–474 (2016).
[Crossref]

D. Xie, X. Chang, X. Shu, J. Wang, L. Mei, and S. Luo, “Replication of thermoplastic polymer spherical lens array using microforged molding technique,” Opt. Express 24(26), 30264–30274 (2016).
[Crossref] [PubMed]

2015 (4)

Z. Zhu, S. To, and S. Zhang, “Large-scale fabrication of micro-lens array by novel end-fly-cutting-servo diamond machining,” Opt. Express 23(16), 20593–20604 (2015).
[Crossref] [PubMed]

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. To, Z. Zhu, and W. Zeng, “Novel end-fly-cutting-servo system for deterministic generation of hierarchical micro–nanostructures,” CIRP Ann. Manuf. Tech. 64(1), 133–136 (2015).
[Crossref]

D. W. K. Neo, A. S. Kumar, and M. Rahman, “An automated Guilloche machining technique for the fabrication of polygonal Fresnel lens array,” Precis. Eng. 41, 55–62 (2015).
[Crossref]

2014 (2)

C. C. Chen, C. Y. Huang, W. J. Peng, and W. Y. Hsu, “Precision Glass Molding of High Filling Factor Micro Lens Arrays,” Adv. Mat. Res. 939(939), 349–354 (2014).
[Crossref]

E. Brinksmeier and L. Schönemann, “Generation of discontinuous microstructures by Diamond Micro Chiseling,” CIRP Ann. Manuf. Tech. 63(1), 49–52 (2014).
[Crossref]

2013 (6)

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. Manuf. Tech. 62(2), 823–846 (2013).
[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, 87693Q (2013).
[Crossref]

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]

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]

B. Guo, Q. Zhao, and M. J. Jackson, “Precision grinding of binderless ultrafine tungsten carbide (WC) microstructured surfaces,” Int. J. Adv. Manuf. Technol. 64, 727–735 (2013).
[Crossref]

S. Goel, X. Luo, P. Comley, R. L. Reuben, and A. Cox, “Brittle–ductile transition during diamond turning of single crystal silicon carbide,” Int. J. Mach. Tools Manuf. 65, 15–21 (2013).
[Crossref]

2011 (6)

M. P. Jahan, M. Rahman, and Y. S. Wong, “A review on the conventional and micro-electrodischarge machining of tungsten carbide,” Int. J. Mach. Tools Manuf. 51(12), 837–858 (2011).
[Crossref]

Z. Zhang, J. Yan, and T. Kuriyagawa, “Study on tool wear characteristics in diamond turning of reaction-bonded silicon carbide,” Int. J. Adv. Manuf. Technol. 57(1–4), 117–125 (2011).
[Crossref]

J. Xie, Y. W. Zhuo, and T. W. Tan, “Experimental study on fabrication and evaluation of micro pyramid-structured silicon surface using a V-tip of diamond grinding wheel,” Precis. Eng. 35(1), 173–182 (2011).
[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, 7927ON (2011).
[Crossref]

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]

A. Symmons and R. Pini, “Precision molded glass challenges plastic optics,” Laser Focus World 47(7), 42–45 (2011).

2010 (5)

S. Reichel and R. Biertümpfel, “Precision Molded Lens Arrays Made of Glass,” Optik-Photonik 5(2), 53–55 (2010).
[Crossref]

M. Malinauskas, A. Žukauskas, V. Purlys, K. Belazaras, A. Momot, D. Paipulas, R. Gadonas, A. Piskarskas, H. Gilbergs, A. Gaidukevičiūtė, I. Sakellari, M. Farsari, and S. Juodkazis, “Femtosecond laser polymerization of hybrid/integrated micro-optical elements and their characterization,” J. Opt. 12(12), 124010 (2010).
[Crossref]

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1(8), 124 (2010).
[Crossref] [PubMed]

K. Murphy, D. Burke, N. Devaney, and C. Dainty, “Experimental detection of optical vortices with a Shack-Hartmann wavefront sensor,” Opt. Express 18(15), 15448–15460 (2010).
[Crossref] [PubMed]

E. Brinksmeier, Y. Mutlugünes, F. Klocke, J. C. Aurich, P. Shore, and H. Ohmori, “Ultra-precision grinding,” CIRP Ann. Manuf. Tech. 59(2), 652–671 (2010).
[Crossref]

2009 (3)

E. Roy, B. Voisin, J. F. Gravel, R. Peytavi, D. Boudreau, and T. Veres, “Microlens array fabrication by enhanced thermal reflow process: Towards efficient collection of fluorescence light from microarrays,” Microelectron. Eng. 86(11), 2255–2261 (2009).
[Crossref]

G. E. Davis, J. W. Roblee, and A. R. Hedges, “Comparison of freeform manufacturing techniques in the production of monolithic lens arrays,” Proc. SPIE 7426, 742605 (2009).
[Crossref]

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 (2)

2007 (2)

Y. E. Tohme, “Grinding aspheric and freeform micro-optical molds,” Proc. SPIE 6462, 64620K (2007).
[Crossref]

Y. Yamamoto, H. Suzuki, T. Onishi, T. Okino, and T. Moriwaki, “Precision grinding of microarray lens molding die with 4-axes controlled microwheel,” Sci. Technol. Adv. Mater. 8(3), 173–176 (2007).
[Crossref]

2006 (2)

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

C. S. Lim, M. H. Hong, A. S. Kumar, M. Rahman, and X. D. Liu, “Fabrication of concave micro lens array using laser patterning and isotropic etching,” Int. J. Mach. Tools Manuf. 46(5), 552–558 (2006).
[Crossref]

2005 (3)

Q. Deng, C. Du, C. Wang, C. Zhou, and Y. Liu, “Microlens array for stacked laser diode beam collimation,” Proc. SPIE 5636, 666–670 (2005).
[Crossref]

C. Hall, M. Tricard, H. Murakoshi, Y. Yamamoto, K. Kuriyama, and H. Yoko, “New mold manufacturing techniques,” Proc. SPIE 5868, 58680V (2005).
[Crossref]

J. Seewig, “Linear and robust Gaussian regression filters,” J. Phys. Conf. Ser. 13(1), 254–257 (2005).
[Crossref]

2004 (2)

1994 (1)

D. L. Macfarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, and D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photonics Technol. Lett. 6(9), 1112–1114 (1994).
[Crossref]

1990 (1)

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

Aurich, J. C.

E. Brinksmeier, Y. Mutlugünes, F. Klocke, J. C. Aurich, P. Shore, and H. Ohmori, “Ultra-precision grinding,” CIRP Ann. Manuf. Tech. 59(2), 652–671 (2010).
[Crossref]

Belazaras, K.

M. Malinauskas, A. Žukauskas, V. Purlys, K. Belazaras, A. Momot, D. Paipulas, R. Gadonas, A. Piskarskas, H. Gilbergs, A. Gaidukevičiūtė, I. Sakellari, M. Farsari, and S. Juodkazis, “Femtosecond laser polymerization of hybrid/integrated micro-optical elements and their characterization,” J. Opt. 12(12), 124010 (2010).
[Crossref]

Biertümpfel, R.

S. Reichel and R. Biertümpfel, “Precision Molded Lens Arrays Made of Glass,” Optik-Photonik 5(2), 53–55 (2010).
[Crossref]

Boor, Cd.

Cd. Boor, A Practical Guide to Splines (Springer-Verlag, 1978).

Boudreau, D.

E. Roy, B. Voisin, J. F. Gravel, R. Peytavi, D. Boudreau, and T. Veres, “Microlens array fabrication by enhanced thermal reflow process: Towards efficient collection of fluorescence light from microarrays,” Microelectron. Eng. 86(11), 2255–2261 (2009).
[Crossref]

Brinksmeier, E.

E. Brinksmeier and L. Schönemann, “Generation of discontinuous microstructures by Diamond Micro Chiseling,” CIRP Ann. Manuf. Tech. 63(1), 49–52 (2014).
[Crossref]

E. Brinksmeier, Y. Mutlugünes, F. Klocke, J. C. Aurich, P. Shore, and H. Ohmori, “Ultra-precision grinding,” CIRP Ann. Manuf. Tech. 59(2), 652–671 (2010).
[Crossref]

Burke, D.

Chang, X.

Chen, C. C.

C. C. Chen, C. Y. Huang, W. J. Peng, and W. Y. Hsu, “Precision Glass Molding of High Filling Factor Micro Lens Arrays,” Adv. Mat. Res. 939(939), 349–354 (2014).
[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, 87693Q (2013).
[Crossref]

Chen, F.

S. Yin, H. Jia, G. Zhang, F. Chen, and K. Zhu, “Review of small aspheric glass lens molding technologies,” Front. Mech. Eng. 12(1), 66–76 (2017).
[Crossref]

Chen, T.

D. L. Macfarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, and D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photonics Technol. Lett. 6(9), 1112–1114 (1994).
[Crossref]

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, 87693Q (2013).
[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]

Comley, P.

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D. L. Macfarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, and D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photonics Technol. Lett. 6(9), 1112–1114 (1994).
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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).
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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).
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G. E. Davis, J. W. Roblee, and A. R. Hedges, “Comparison of freeform manufacturing techniques in the production of monolithic lens arrays,” Proc. SPIE 7426, 742605 (2009).
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C. C. Chen, C. Y. Huang, W. J. Peng, and W. Y. Hsu, “Precision Glass Molding of High Filling Factor Micro Lens Arrays,” Adv. Mat. Res. 939(939), 349–354 (2014).
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C. C. Chen, C. Y. Huang, W. J. Peng, and W. Y. Hsu, “Precision Glass Molding of High Filling Factor Micro Lens Arrays,” Adv. Mat. Res. 939(939), 349–354 (2014).
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C. S. Lim, M. H. Hong, A. S. Kumar, M. Rahman, and X. D. Liu, “Fabrication of concave micro lens array using laser patterning and isotropic etching,” Int. J. Mach. Tools Manuf. 46(5), 552–558 (2006).
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K. H. Jeong, J. Kim, and L. P. Lee, “Biologically inspired artificial compound eyes,” Science 312(5773), 557–561 (2006).
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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).
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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).
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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).
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C. S. Lim, M. H. Hong, A. S. Kumar, M. Rahman, and X. D. Liu, “Fabrication of concave micro lens array using laser patterning and isotropic etching,” Int. J. Mach. Tools Manuf. 46(5), 552–558 (2006).
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F. Z. Fang and Y. C. Liu, “On minimum exit-burr in micro cutting,” J. Micromech. Microeng. 14(7), 984–988 (2004).
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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).
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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).
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Luo, X.

S. Goel, X. Luo, P. Comley, R. L. Reuben, and A. Cox, “Brittle–ductile transition during diamond turning of single crystal silicon carbide,” Int. J. Mach. Tools Manuf. 65, 15–21 (2013).
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H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1(8), 124 (2010).
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D. L. Macfarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, and D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photonics Technol. Lett. 6(9), 1112–1114 (1994).
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M. Malinauskas, A. Žukauskas, V. Purlys, K. Belazaras, A. Momot, D. Paipulas, R. Gadonas, A. Piskarskas, H. Gilbergs, A. Gaidukevičiūtė, I. Sakellari, M. Farsari, and S. Juodkazis, “Femtosecond laser polymerization of hybrid/integrated micro-optical elements and their characterization,” J. Opt. 12(12), 124010 (2010).
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Mei, L.

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D. W. K. Neo, A. S. Kumar, and M. Rahman, “An automated Guilloche machining technique for the fabrication of polygonal Fresnel lens array,” Precis. Eng. 41, 55–62 (2015).
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K. Pang, L. Song, F. Fang, Y. Zhang, and H. Zhang, “An imaging system with a large depth of field based on an overlapped micro-lens array,” CIRP Ann. Manuf. Tech. 65(1), 471–474 (2016).
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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).
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Peng, W. J.

C. C. Chen, C. Y. Huang, W. J. Peng, and W. Y. Hsu, “Precision Glass Molding of High Filling Factor Micro Lens Arrays,” Adv. Mat. Res. 939(939), 349–354 (2014).
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M. P. Jahan, M. Rahman, and Y. S. Wong, “A review on the conventional and micro-electrodischarge machining of tungsten carbide,” Int. J. Mach. Tools Manuf. 51(12), 837–858 (2011).
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C. S. Lim, M. H. Hong, A. S. Kumar, M. Rahman, and X. D. Liu, “Fabrication of concave micro lens array using laser patterning and isotropic etching,” Int. J. Mach. Tools Manuf. 46(5), 552–558 (2006).
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S. Goel, X. Luo, P. Comley, R. L. Reuben, and A. Cox, “Brittle–ductile transition during diamond turning of single crystal silicon carbide,” Int. J. Mach. Tools Manuf. 65, 15–21 (2013).
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Risse, S.

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, 7927ON (2011).
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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).
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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).
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E. Roy, B. Voisin, J. F. Gravel, R. Peytavi, D. Boudreau, and T. Veres, “Microlens array fabrication by enhanced thermal reflow process: Towards efficient collection of fluorescence light from microarrays,” Microelectron. Eng. 86(11), 2255–2261 (2009).
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M. Malinauskas, A. Žukauskas, V. Purlys, K. Belazaras, A. Momot, D. Paipulas, R. Gadonas, A. Piskarskas, H. Gilbergs, A. Gaidukevičiūtė, I. Sakellari, M. Farsari, and S. Juodkazis, “Femtosecond laser polymerization of hybrid/integrated micro-optical elements and their characterization,” J. Opt. 12(12), 124010 (2010).
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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, 7927ON (2011).
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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).
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E. Brinksmeier and L. Schönemann, “Generation of discontinuous microstructures by Diamond Micro Chiseling,” CIRP Ann. Manuf. Tech. 63(1), 49–52 (2014).
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E. Brinksmeier, Y. Mutlugünes, F. Klocke, J. C. Aurich, P. Shore, and H. Ohmori, “Ultra-precision grinding,” CIRP Ann. Manuf. Tech. 59(2), 652–671 (2010).
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Shu, X.

Song, L.

K. Pang, L. Song, F. Fang, Y. Zhang, and H. Zhang, “An imaging system with a large depth of field based on an overlapped micro-lens array,” CIRP Ann. Manuf. Tech. 65(1), 471–474 (2016).
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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).
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D. L. Macfarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, and D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photonics Technol. Lett. 6(9), 1112–1114 (1994).
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S. J. Zhang, S. To, Z. W. Zhu, and G. Q. Zhang, “A review of fly cutting applied to surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 103, 13–27 (2016).
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S. To, Z. Zhu, and H. Wang, “Virtual spindle based tool servo diamond turning of discontinuously structured microoptics arrays,” CIRP Ann. Manuf. Tech. 65(1), 475–478 (2016).
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Q. Zhang, S. To, Q. Zhao, and B. Guo, “Surface generation mechanism of WC/Co and RB-SiC/Si composites under high spindle speed grinding (HSSG),” Int. J. Refract. Met. Hard Mater. 56, 123–131 (2016).
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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).
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S. To, Z. Zhu, and W. Zeng, “Novel end-fly-cutting-servo system for deterministic generation of hierarchical micro–nanostructures,” CIRP Ann. Manuf. Tech. 64(1), 133–136 (2015).
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Z. Zhu, S. To, and S. Zhang, “Large-scale fabrication of micro-lens array by novel end-fly-cutting-servo diamond machining,” Opt. Express 23(16), 20593–20604 (2015).
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C. Hall, M. Tricard, H. Murakoshi, Y. Yamamoto, K. Kuriyama, and H. Yoko, “New mold manufacturing techniques,” Proc. SPIE 5868, 58680V (2005).
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Tünnermann, A.

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, 7927ON (2011).
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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).
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Veres, T.

E. Roy, B. Voisin, J. F. Gravel, R. Peytavi, D. Boudreau, and T. Veres, “Microlens array fabrication by enhanced thermal reflow process: Towards efficient collection of fluorescence light from microarrays,” Microelectron. Eng. 86(11), 2255–2261 (2009).
[Crossref]

Voisin, B.

E. Roy, B. Voisin, J. F. Gravel, R. Peytavi, D. Boudreau, and T. Veres, “Microlens array fabrication by enhanced thermal reflow process: Towards efficient collection of fluorescence light from microarrays,” Microelectron. Eng. 86(11), 2255–2261 (2009).
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Wang, C.

Q. Deng, C. Du, C. Wang, C. Zhou, and Y. Liu, “Microlens array for stacked laser diode beam collimation,” Proc. SPIE 5636, 666–670 (2005).
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Wang, H.

S. To, Z. Zhu, and H. Wang, “Virtual spindle based tool servo diamond turning of discontinuously structured microoptics arrays,” CIRP Ann. Manuf. Tech. 65(1), 475–478 (2016).
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Wang, J.

Wang, X.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12(1), 46–65 (2017).
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Weckenmann, A.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. Manuf. Tech. 62(2), 823–846 (2013).
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Wong, Y. S.

M. P. Jahan, M. Rahman, and Y. S. Wong, “A review on the conventional and micro-electrodischarge machining of tungsten carbide,” Int. J. Mach. Tools Manuf. 51(12), 837–858 (2011).
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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).
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Xie, D.

Xie, J.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12(1), 46–65 (2017).
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J. Xie, Y. W. Zhuo, and T. W. Tan, “Experimental study on fabrication and evaluation of micro pyramid-structured silicon surface using a V-tip of diamond grinding wheel,” Precis. Eng. 35(1), 173–182 (2011).
[Crossref]

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).
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Yamamoto, Y.

Y. Yamamoto, H. Suzuki, T. Onishi, T. Okino, and T. Moriwaki, “Precision grinding of microarray lens molding die with 4-axes controlled microwheel,” Sci. Technol. Adv. Mater. 8(3), 173–176 (2007).
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C. Hall, M. Tricard, H. Murakoshi, Y. Yamamoto, K. Kuriyama, and H. Yoko, “New mold manufacturing techniques,” Proc. SPIE 5868, 58680V (2005).
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Z. Zhang, J. Yan, and T. Kuriyagawa, “Study on tool wear characteristics in diamond turning of reaction-bonded silicon carbide,” Int. J. Adv. Manuf. Technol. 57(1–4), 117–125 (2011).
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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, 7927ON (2011).
[Crossref]

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).
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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).
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S. Yin, H. Jia, G. Zhang, F. Chen, and K. Zhu, “Review of small aspheric glass lens molding technologies,” Front. Mech. Eng. 12(1), 66–76 (2017).
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Yoko, H.

C. Hall, M. Tricard, H. Murakoshi, Y. Yamamoto, K. Kuriyama, and H. Yoko, “New mold manufacturing techniques,” Proc. SPIE 5868, 58680V (2005).
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Zeng, W.

S. To, Z. Zhu, and W. Zeng, “Novel end-fly-cutting-servo system for deterministic generation of hierarchical micro–nanostructures,” CIRP Ann. Manuf. Tech. 64(1), 133–136 (2015).
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Zhang, G.

S. Yin, H. Jia, G. Zhang, F. Chen, and K. Zhu, “Review of small aspheric glass lens molding technologies,” Front. Mech. Eng. 12(1), 66–76 (2017).
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Zhang, G. Q.

S. J. Zhang, S. To, Z. W. Zhu, and G. Q. Zhang, “A review of fly cutting applied to surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 103, 13–27 (2016).
[Crossref]

Zhang, G. X.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. Manuf. Tech. 62(2), 823–846 (2013).
[Crossref]

Zhang, H.

K. Pang, L. Song, F. Fang, Y. Zhang, and H. Zhang, “An imaging system with a large depth of field based on an overlapped micro-lens array,” CIRP Ann. Manuf. Tech. 65(1), 471–474 (2016).
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Zhang, L.

L. Zhang and W. Liu, “Precision glass molding: Toward an optimal fabrication of optical lenses,” Front. Mech. Eng. 12(1), 3–17 (2017).
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Zhang, N.

F. Z. Fang, N. Zhang, and X. Zhang, “Precision injection molding of freeform optics,” Adv. Opt. Technol. 5(4), 303–324 (2016).

Zhang, Q.

Q. Zhang, S. To, Q. Zhao, and B. Guo, “Surface generation mechanism of WC/Co and RB-SiC/Si composites under high spindle speed grinding (HSSG),” Int. J. Refract. Met. Hard Mater. 56, 123–131 (2016).
[Crossref]

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]

Z. Zhu, S. To, and S. Zhang, “Large-scale fabrication of micro-lens array by novel end-fly-cutting-servo diamond machining,” Opt. Express 23(16), 20593–20604 (2015).
[Crossref] [PubMed]

Zhang, S. J.

S. J. Zhang, S. To, Z. W. Zhu, and G. Q. Zhang, “A review of fly cutting applied to surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 103, 13–27 (2016).
[Crossref]

Zhang, X.

F. Z. Fang, N. Zhang, and X. Zhang, “Precision injection molding of freeform optics,” Adv. Opt. Technol. 5(4), 303–324 (2016).

Zhang, X. D.

W. L. Zhu, F. Duan, X. D. Zhang, Z. W. Zhu, and B. F. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
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F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. Manuf. Tech. 62(2), 823–846 (2013).
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F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express 16(10), 7323–7329 (2008).
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Zhang, Y.

K. Pang, L. Song, F. Fang, Y. Zhang, and H. Zhang, “An imaging system with a large depth of field based on an overlapped micro-lens array,” CIRP Ann. Manuf. Tech. 65(1), 471–474 (2016).
[Crossref]

Zhang, Z.

Z. Zhang, J. Yan, and T. Kuriyagawa, “Study on tool wear characteristics in diamond turning of reaction-bonded silicon carbide,” Int. J. Adv. Manuf. Technol. 57(1–4), 117–125 (2011).
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Zhao, Q.

B. Guo and Q. Zhao, “Ultrasonic vibration assisted grinding of hard and brittle linear micro-structured surfaces,” Precis. Eng. 48(7179), 98–106 (2017).
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Q. Zhang, S. To, Q. Zhao, and B. Guo, “Surface generation mechanism of WC/Co and RB-SiC/Si composites under high spindle speed grinding (HSSG),” Int. J. Refract. Met. Hard Mater. 56, 123–131 (2016).
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B. Guo, Q. Zhao, and M. J. Jackson, “Precision grinding of binderless ultrafine tungsten carbide (WC) microstructured surfaces,” Int. J. Adv. Manuf. Technol. 64, 727–735 (2013).
[Crossref]

Zhou, C.

Q. Deng, C. Du, C. Wang, C. Zhou, and Y. Liu, “Microlens array for stacked laser diode beam collimation,” Proc. SPIE 5636, 666–670 (2005).
[Crossref]

Zhou, T.

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12(1), 46–65 (2017).
[Crossref]

Zhu, K.

S. Yin, H. Jia, G. Zhang, F. Chen, and K. Zhu, “Review of small aspheric glass lens molding technologies,” Front. Mech. Eng. 12(1), 66–76 (2017).
[Crossref]

Zhu, W. L.

W. L. Zhu, F. Duan, X. D. Zhang, Z. W. Zhu, and B. F. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
[Crossref]

Zhu, Z.

S. To, Z. Zhu, and H. Wang, “Virtual spindle based tool servo diamond turning of discontinuously structured microoptics arrays,” CIRP Ann. Manuf. Tech. 65(1), 475–478 (2016).
[Crossref]

S. To, Z. Zhu, and W. Zeng, “Novel end-fly-cutting-servo system for deterministic generation of hierarchical micro–nanostructures,” CIRP Ann. Manuf. Tech. 64(1), 133–136 (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]

Z. Zhu, S. To, and S. Zhang, “Large-scale fabrication of micro-lens array by novel end-fly-cutting-servo diamond machining,” Opt. Express 23(16), 20593–20604 (2015).
[Crossref] [PubMed]

Zhu, Z. W.

W. L. Zhu, F. Duan, X. D. Zhang, Z. W. Zhu, and B. F. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
[Crossref]

S. J. Zhang, S. To, Z. W. Zhu, and G. Q. Zhang, “A review of fly cutting applied to surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 103, 13–27 (2016).
[Crossref]

Zhuo, Y. W.

J. Xie, Y. W. Zhuo, and T. W. Tan, “Experimental study on fabrication and evaluation of micro pyramid-structured silicon surface using a V-tip of diamond grinding wheel,” Precis. Eng. 35(1), 173–182 (2011).
[Crossref]

Žukauskas, A.

M. Malinauskas, A. Žukauskas, V. Purlys, K. Belazaras, A. Momot, D. Paipulas, R. Gadonas, A. Piskarskas, H. Gilbergs, A. Gaidukevičiūtė, I. Sakellari, M. Farsari, and S. Juodkazis, “Femtosecond laser polymerization of hybrid/integrated micro-optical elements and their characterization,” J. Opt. 12(12), 124010 (2010).
[Crossref]

Adv. Mat. Res. (1)

C. C. Chen, C. Y. Huang, W. J. Peng, and W. Y. Hsu, “Precision Glass Molding of High Filling Factor Micro Lens Arrays,” Adv. Mat. Res. 939(939), 349–354 (2014).
[Crossref]

Adv. Opt. Technol. (1)

F. Z. Fang, N. Zhang, and X. Zhang, “Precision injection molding of freeform optics,” Adv. Opt. Technol. 5(4), 303–324 (2016).

CIRP Ann. Manuf. Tech. (6)

S. To, Z. Zhu, and W. Zeng, “Novel end-fly-cutting-servo system for deterministic generation of hierarchical micro–nanostructures,” CIRP Ann. Manuf. Tech. 64(1), 133–136 (2015).
[Crossref]

E. Brinksmeier and L. Schönemann, “Generation of discontinuous microstructures by Diamond Micro Chiseling,” CIRP Ann. Manuf. Tech. 63(1), 49–52 (2014).
[Crossref]

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. Manuf. Tech. 62(2), 823–846 (2013).
[Crossref]

K. Pang, L. Song, F. Fang, Y. Zhang, and H. Zhang, “An imaging system with a large depth of field based on an overlapped micro-lens array,” CIRP Ann. Manuf. Tech. 65(1), 471–474 (2016).
[Crossref]

E. Brinksmeier, Y. Mutlugünes, F. Klocke, J. C. Aurich, P. Shore, and H. Ohmori, “Ultra-precision grinding,” CIRP Ann. Manuf. Tech. 59(2), 652–671 (2010).
[Crossref]

S. To, Z. Zhu, and H. Wang, “Virtual spindle based tool servo diamond turning of discontinuously structured microoptics arrays,” CIRP Ann. Manuf. Tech. 65(1), 475–478 (2016).
[Crossref]

Front. Mech. Eng. (3)

T. Zhou, X. Liu, Z. Liang, Y. Liu, J. Xie, and X. Wang, “Recent advancements in optical microstructure fabrication through glass molding process,” Front. Mech. Eng. 12(1), 46–65 (2017).
[Crossref]

L. Zhang and W. Liu, “Precision glass molding: Toward an optimal fabrication of optical lenses,” Front. Mech. Eng. 12(1), 3–17 (2017).
[Crossref]

S. Yin, H. Jia, G. Zhang, F. Chen, and K. Zhu, “Review of small aspheric glass lens molding technologies,” Front. Mech. Eng. 12(1), 66–76 (2017).
[Crossref]

IEEE Photonics Technol. Lett. (1)

D. L. Macfarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, and D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photonics Technol. Lett. 6(9), 1112–1114 (1994).
[Crossref]

Int. J. Adv. Manuf. Technol. (2)

Z. Zhang, J. Yan, and T. Kuriyagawa, “Study on tool wear characteristics in diamond turning of reaction-bonded silicon carbide,” Int. J. Adv. Manuf. Technol. 57(1–4), 117–125 (2011).
[Crossref]

B. Guo, Q. Zhao, and M. J. Jackson, “Precision grinding of binderless ultrafine tungsten carbide (WC) microstructured surfaces,” Int. J. Adv. Manuf. Technol. 64, 727–735 (2013).
[Crossref]

Int. J. Mach. Tools Manuf. (6)

S. Goel, X. Luo, P. Comley, R. L. Reuben, and A. Cox, “Brittle–ductile transition during diamond turning of single crystal silicon carbide,” Int. J. Mach. Tools Manuf. 65, 15–21 (2013).
[Crossref]

M. P. Jahan, M. Rahman, and Y. S. Wong, “A review on the conventional and micro-electrodischarge machining of tungsten carbide,” Int. J. Mach. Tools Manuf. 51(12), 837–858 (2011).
[Crossref]

C. S. Lim, M. H. Hong, A. S. Kumar, M. Rahman, and X. D. Liu, “Fabrication of concave micro lens array using laser patterning and isotropic etching,” Int. J. Mach. Tools Manuf. 46(5), 552–558 (2006).
[Crossref]

W. L. Zhu, F. Duan, X. D. Zhang, Z. W. Zhu, and B. F. Ju, “A new diamond machining approach for extendable fabrication of micro-freeform lens array,” Int. J. Mach. Tools Manuf. 124, 134–148 (2018).
[Crossref]

S. J. Zhang, S. To, Z. W. Zhu, and G. Q. Zhang, “A review of fly cutting applied to surface generation in ultra-precision machining,” Int. J. Mach. Tools Manuf. 103, 13–27 (2016).
[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]

Int. J. Refract. Met. Hard Mater. (1)

Q. Zhang, S. To, Q. Zhao, and B. Guo, “Surface generation mechanism of WC/Co and RB-SiC/Si composites under high spindle speed grinding (HSSG),” Int. J. Refract. Met. Hard Mater. 56, 123–131 (2016).
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J. Micromech. Microeng. (1)

F. Z. Fang and Y. C. Liu, “On minimum exit-burr in micro cutting,” J. Micromech. Microeng. 14(7), 984–988 (2004).
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J. Opt. (1)

M. Malinauskas, A. Žukauskas, V. Purlys, K. Belazaras, A. Momot, D. Paipulas, R. Gadonas, A. Piskarskas, H. Gilbergs, A. Gaidukevičiūtė, I. Sakellari, M. Farsari, and S. Juodkazis, “Femtosecond laser polymerization of hybrid/integrated micro-optical elements and their characterization,” J. Opt. 12(12), 124010 (2010).
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A. Symmons and R. Pini, “Precision molded glass challenges plastic optics,” Laser Focus World 47(7), 42–45 (2011).

Meas. Sci. Technol. (1)

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

Microelectron. Eng. (1)

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[Crossref]

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H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1(8), 124 (2010).
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[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Schematic of configuration of the OSA spiral grinding system.
Fig. 2
Fig. 2 Schematic of the basic principle of OSA spiral grinding of AMLA. (a) Initialization, (b) When the work spindle rotated with an angle of θj. (c) Spiral grinding.
Fig. 3
Fig. 3 Schematic of the on-machine measurement principle.
Fig. 4
Fig. 4 Schematic of the generation of new tool path for compensation grinding.
Fig. 5
Fig. 5 View of the grinding experiment for four-lens-let convex AMLA mould insert.
Fig. 6
Fig. 6 (a) Features of the designed AMLA mold insert, (b) profile and slope characteristic of single lens-let, (c) the rough grinding wheel path for #1 lens-let, and (d) the xm-, ym-, and zm-component of the rough grinding wheel path for #1 lens-let.
Fig. 7
Fig. 7 Photographs of the (a) AMLA mold insert and (b) molded glass AMLA.
Fig. 8
Fig. 8 Characteristics of the ground AMLA mold insert and molded glass AMLA, (a) the 3D structure of the mold insert, (b) the corresponding profiles of mold insert along the cross-hair directions, (c) the 3D structure of the molded glass AMLA, and (d) the corresponding profiles of molded glass AMLA along the cross-hair directions,
Fig. 9
Fig. 9 Profile error curves of the #1 lens-let obtained from WECS (a) after finish grinding without compensation, (b) after the first compensation grinding, (c) after the second compensation grinding and (d) obtained from Form Talysurf after second compensated grinding.
Fig. 10
Fig. 10 Comparison of the profile errors of (a) the ground lens-lets and (b) molded lens-lets.
Fig. 11
Fig. 11 Surface micro-topography of different regions in (a) the 1# lens-let of mold insert and in (b) the 1# lens-let of molded glass.

Tables (4)

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Table 1 Grinding Conditions for fabrication of the microlens array mold insert

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Table 2 Parameters of the convex AMLA

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Table 3 Profile errors of the ground and molded lens-lets

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Table 4 Surface roughness (Sa) of the ground and molded AMLA

Equations (8)

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{ x m A j =ρcos( θ j + θ 0 ) y m A j =ρsin( θ j + θ 0 ) z m A j =0
{ x i P j = f 2π θ j y i P j =0 z i P j =ASP( f 2π θ j )
ASP( r )= r 2 R b + R b ( 1+k ) r 2 + i=2 n A 2i r 2i
{ x i C j = x i P j + r w sinβ y i C j =0 z i C j = x i P j + r w cosβ+ R w
{ x m C j = x m A j + x i C j =ρcos( θ j + θ 0 )+ f 2π θ j + r w sinβ y m C j = y m A j + y i C j =ρsin( θ j + θ 0 ) z m C j = z m A j + z i C j =ASP( f 2π θ j )+ r w cosβ+ R w
F=D+ δ z D [ 0,0,1 ]
M=F R p n F
C''=C δ n P n P

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