Abstract

Microinjection molding is a mass production method to fabricate affordable optical components. However, the intense nature of this process often results in part deformation and uneven refractive index distribution. These two factors limit the precision of replicated optics. In order to understand the influences of injection molding on freeform optical devices, in this study, finite element method (FEM) was employed to investigate the miniature microinjection-molded Alvarez lenses. In addition, an innovative metrology setup was proposed to evaluate the optical wavefront patterns in the molded lenses using an interferometer-based wavefront measurement system. This measurement setup utilized an optical matching liquid to reduce or eliminate the lenses’ surface power such that the wavefront pattern with large deviation from the freeform lenses can be measured by a regular wavefront setup. The FEM simulation results were also used to explain the differences between the nominal and experimentally measured wavefront patterns of the microinjection-molded Alvarez lenses. In summary, the proposed method combining simulation and wavefront measurements is shown to be an effective approach for studying injection molding of freeform optics.

© 2014 Optical Society of America

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References

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2014

I. Sieber, A. Y. Yi, L. Li, E. Beckert, R. Steinkopf, and U. Gengenbach, “Design of freeform optics for an ophthalmological application,” Proc. SPIE 9131, 913108 (2014).
[CrossRef]

C. Y. Wang and P. J. Wang, “Analysis of optical properties in injection-molded and compression-molded optical lenses,” Appl. Opt. 53, 2523–2531 (2014).
[CrossRef]

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

2013

P. He, L. Li, J. Yu, W. Huang, Y. C. Yen, L. J. Lee, and A. Y. Yi, “Graphene-coated Si mold for precision glass optics molding,” Opt. Lett. 38, 2625–2628 (2013).
[CrossRef]

S. Barbero and J. Rubinstein, “Power-adjustable sphero-cylindrical refractor comprising two lenses,” Opt. Eng. 52, 063002 (2013).
[CrossRef]

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

L. Li, T. W. Raasch, and A. Y. Yi, “Simulation and measurement of optical aberrations of injection molded progressive addition lenses,” Appl. Opt. 52, 6022–6029 (2013).
[CrossRef]

L. Li and A. Y. Yi, “An affordable injection-molded precision hybrid glass–polymer achromatic lens,” Int. J. Adv. Manuf. Technol. 69, 1461–1467 (2013).

2012

2011

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

P. J. Smilie, B. S. Dutterer, J. L. Lineberger, M. A. Davies, and T. J. Suleski, “Freeform micromachining of an infrared Alvarez lens,” Proc. SPIE 7927, 79270K (2011).
[CrossRef]

2010

T. J. Tayag and B. L. Bachim, “Simulation of an interferometric computed tomography system for intraocular lenses,” Proc. SPIE 7791, 77910K (2010).
[CrossRef]

2009

2008

K. Park and W. Joo, “Numerical evaluation of a plastic lens by coupling injection molding analysis with optical simulation,” Jpn. J. Appl. Phys. 47, 8402–8407 (2008).
[CrossRef]

H. E. Lai and P. J. Wang, “Study of process parameters on optical qualities for injection-molded plastic lenses,” Appl. Opt. 47, 2017–2027 (2008).
[CrossRef]

2007

E. Savio, L. de Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. Manuf. Technol. 56, 810–835 (2007).
[CrossRef]

2006

S.-W. Kim and L.-S. Turng, “Three-dimensional numerical simulation of injection molding filling of optical lens and multiscale geometry using finite element method,” Polym. Eng. Sci. 46, 1263–1274 (2006).

A. N. Simonov, G. Vdovin, and M. C. Rombach, “Cubic optical elements for an accommodative intraocular lens,” Opt. Express 14, 7757–7775 (2006).
[CrossRef]

2005

2004

T. Bothe, W. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

2003

2002

1998

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

Z. Yun, Z. Liu, and Y. Li, “Aspheric surface testing with a liquid compensatory interferometer,” Opt. Eng. 37, 1364–1367 (1998).
[CrossRef]

1994

Bachim, B. L.

T. J. Tayag and B. L. Bachim, “Simulation of an interferometric computed tomography system for intraocular lenses,” Proc. SPIE 7791, 77910K (2010).
[CrossRef]

Barbero, S.

Beckert, E.

I. Sieber, A. Y. Yi, L. Li, E. Beckert, R. Steinkopf, and U. Gengenbach, “Design of freeform optics for an ophthalmological application,” Proc. SPIE 9131, 913108 (2014).
[CrossRef]

Bille, J.

Bothe, T.

T. Bothe, W. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Castro, J.

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

Davies, M. A.

P. J. Smilie, B. S. Dutterer, J. L. Lineberger, M. A. Davies, and T. J. Suleski, “Freeform micromachining of an infrared Alvarez lens,” Proc. SPIE 7927, 79270K (2011).
[CrossRef]

de Chiffre, L.

E. Savio, L. de Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. Manuf. Technol. 56, 810–835 (2007).
[CrossRef]

Dutterer, B. S.

P. J. Smilie, B. S. Dutterer, J. L. Lineberger, M. A. Davies, and T. J. Suleski, “Freeform micromachining of an infrared Alvarez lens,” Proc. SPIE 7927, 79270K (2011).
[CrossRef]

Evans, C.

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

Fang, F. Z.

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

Gengenbach, U.

I. Sieber, A. Y. Yi, L. Li, E. Beckert, R. Steinkopf, and U. Gengenbach, “Design of freeform optics for an ophthalmological application,” Proc. SPIE 9131, 913108 (2014).
[CrossRef]

Goelz, S.

Grimm, B.

He, P.

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

P. He, L. Li, J. Yu, W. Huang, Y. C. Yen, L. J. Lee, and A. Y. Yi, “Graphene-coated Si mold for precision glass optics molding,” Opt. Lett. 38, 2625–2628 (2013).
[CrossRef]

Huang, C.

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

Huang, H. X.

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

Huang, W.

Jeong, T. M.

Jiménez-Alfaro, I.

Joo, W.

K. Park and W. Joo, “Numerical evaluation of a plastic lens by coupling injection molding analysis with optical simulation,” Jpn. J. Appl. Phys. 47, 8402–8407 (2008).
[CrossRef]

Joo, W.-D.

Jung, M.-S.

Juptner, W.

T. Bothe, W. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Kim, S.-W.

S.-W. Kim and L.-S. Turng, “Three-dimensional numerical simulation of injection molding filling of optical lens and multiscale geometry using finite element method,” Polym. Eng. Sci. 46, 1263–1274 (2006).

Lai, H. E.

Lee, L. J.

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

P. He, L. Li, J. Yu, W. Huang, Y. C. Yen, L. J. Lee, and A. Y. Yi, “Graphene-coated Si mold for precision glass optics molding,” Opt. Lett. 38, 2625–2628 (2013).
[CrossRef]

Li, H.

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

Li, L.

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

I. Sieber, A. Y. Yi, L. Li, E. Beckert, R. Steinkopf, and U. Gengenbach, “Design of freeform optics for an ophthalmological application,” Proc. SPIE 9131, 913108 (2014).
[CrossRef]

L. Li and A. Y. Yi, “An affordable injection-molded precision hybrid glass–polymer achromatic lens,” Int. J. Adv. Manuf. Technol. 69, 1461–1467 (2013).

P. He, L. Li, J. Yu, W. Huang, Y. C. Yen, L. J. Lee, and A. Y. Yi, “Graphene-coated Si mold for precision glass optics molding,” Opt. Lett. 38, 2625–2628 (2013).
[CrossRef]

L. Li, T. W. Raasch, and A. Y. Yi, “Simulation and measurement of optical aberrations of injection molded progressive addition lenses,” Appl. Opt. 52, 6022–6029 (2013).
[CrossRef]

Li, W.

T. Bothe, W. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Li, Y.

Z. Yun, Z. Liu, and Y. Li, “Aspheric surface testing with a liquid compensatory interferometer,” Opt. Eng. 37, 1364–1367 (1998).
[CrossRef]

Liang, J.

Lineberger, J. L.

P. J. Smilie, B. S. Dutterer, J. L. Lineberger, M. A. Davies, and T. J. Suleski, “Freeform micromachining of an infrared Alvarez lens,” Proc. SPIE 7927, 79270K (2011).
[CrossRef]

Liu, Z.

Z. Yun, Z. Liu, and Y. Li, “Aspheric surface testing with a liquid compensatory interferometer,” Opt. Eng. 37, 1364–1367 (1998).
[CrossRef]

Marcos, S.

Menon, M.

Park, K.

K. Park and W. Joo, “Numerical evaluation of a plastic lens by coupling injection molding analysis with optical simulation,” Jpn. J. Appl. Phys. 47, 8402–8407 (2008).
[CrossRef]

Pril, W.

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

Raasch, T. W.

Rombach, M. C.

Rosielle, N.

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

Rubinstein, J.

S. Barbero and J. Rubinstein, “Power-adjustable sphero-cylindrical refractor comprising two lenses,” Opt. Eng. 52, 063002 (2013).
[CrossRef]

Savio, E.

E. Savio, L. de Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. Manuf. Technol. 56, 810–835 (2007).
[CrossRef]

Schellekens, P.

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

Schmitt, R.

E. Savio, L. de Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. Manuf. Technol. 56, 810–835 (2007).
[CrossRef]

Sieber, I.

I. Sieber, A. Y. Yi, L. Li, E. Beckert, R. Steinkopf, and U. Gengenbach, “Design of freeform optics for an ophthalmological application,” Proc. SPIE 9131, 913108 (2014).
[CrossRef]

Simonov, A. N.

Smilie, P. J.

P. J. Smilie, B. S. Dutterer, J. L. Lineberger, M. A. Davies, and T. J. Suleski, “Freeform micromachining of an infrared Alvarez lens,” Proc. SPIE 7927, 79270K (2011).
[CrossRef]

Steinkopf, R.

I. Sieber, A. Y. Yi, L. Li, E. Beckert, R. Steinkopf, and U. Gengenbach, “Design of freeform optics for an ophthalmological application,” Proc. SPIE 9131, 913108 (2014).
[CrossRef]

Su, L.

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

Suhara, H.

Suleski, T. J.

P. J. Smilie, B. S. Dutterer, J. L. Lineberger, M. A. Davies, and T. J. Suleski, “Freeform micromachining of an infrared Alvarez lens,” Proc. SPIE 7927, 79270K (2011).
[CrossRef]

Tayag, T. J.

T. J. Tayag and B. L. Bachim, “Simulation of an interferometric computed tomography system for intraocular lenses,” Proc. SPIE 7791, 77910K (2010).
[CrossRef]

Turng, L.-S.

S.-W. Kim and L.-S. Turng, “Three-dimensional numerical simulation of injection molding filling of optical lens and multiscale geometry using finite element method,” Polym. Eng. Sci. 46, 1263–1274 (2006).

Vdovin, G.

Vermeulen, H.

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

Vermeulen, M.

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

von Kopylow, C.

T. Bothe, W. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Wang, C. Y.

Wang, P. J.

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. Technol. 62, 823–846 (2013).
[CrossRef]

Wetzels, S.

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

Yang, C.

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

Yen, Y. C.

Yi, A. Y.

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

I. Sieber, A. Y. Yi, L. Li, E. Beckert, R. Steinkopf, and U. Gengenbach, “Design of freeform optics for an ophthalmological application,” Proc. SPIE 9131, 913108 (2014).
[CrossRef]

L. Li and A. Y. Yi, “An affordable injection-molded precision hybrid glass–polymer achromatic lens,” Int. J. Adv. Manuf. Technol. 69, 1461–1467 (2013).

P. He, L. Li, J. Yu, W. Huang, Y. C. Yen, L. J. Lee, and A. Y. Yi, “Graphene-coated Si mold for precision glass optics molding,” Opt. Lett. 38, 2625–2628 (2013).
[CrossRef]

L. Li, T. W. Raasch, and A. Y. Yi, “Simulation and measurement of optical aberrations of injection molded progressive addition lenses,” Appl. Opt. 52, 6022–6029 (2013).
[CrossRef]

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

Yoon, G.

Yu, J.

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

P. He, L. Li, J. Yu, W. Huang, Y. C. Yen, L. J. Lee, and A. Y. Yi, “Graphene-coated Si mold for precision glass optics molding,” Opt. Lett. 38, 2625–2628 (2013).
[CrossRef]

Yun, Z.

Z. Yun, Z. Liu, and Y. Li, “Aspheric surface testing with a liquid compensatory interferometer,” Opt. Eng. 37, 1364–1367 (1998).
[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. Technol. 62, 823–846 (2013).
[CrossRef]

Zhang, X. D.

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

Adv. Polym. Technol.

C. Yang, L. Su, C. Huang, H. X. Huang, J. Castro, and A. Y. Yi, “Effect of packing pressure on refractive index variation in injection molding of precision plastic optical lens,” Adv. Polym. Technol. 30, 51–61 (2011).
[CrossRef]

Appl. Opt.

CIRP Ann. Manuf. Technol.

E. Savio, L. de Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. Manuf. Technol. 56, 810–835 (2007).
[CrossRef]

P. Schellekens, N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril, “Design for precision: current status and trends,” CIRP Ann. Manuf. Technol. 47, 557–586 (1998).
[CrossRef]

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

Int. J. Adv. Manuf. Technol.

L. Li and A. Y. Yi, “An affordable injection-molded precision hybrid glass–polymer achromatic lens,” Int. J. Adv. Manuf. Technol. 69, 1461–1467 (2013).

J. Opt. Soc. Am. A

J. Opt. Soc. Korea

Jpn. J. Appl. Phys.

K. Park and W. Joo, “Numerical evaluation of a plastic lens by coupling injection molding analysis with optical simulation,” Jpn. J. Appl. Phys. 47, 8402–8407 (2008).
[CrossRef]

Manuf. Lett.

P. He, L. Li, H. Li, J. Yu, L. J. Lee, and A. Y. Yi, “Compression molding of glass freeform optics using diamond machined silicon mold,” Manuf. Lett. 2, 17–20 (2014).
[CrossRef]

Opt. Eng.

Z. Yun, Z. Liu, and Y. Li, “Aspheric surface testing with a liquid compensatory interferometer,” Opt. Eng. 37, 1364–1367 (1998).
[CrossRef]

S. Barbero and J. Rubinstein, “Power-adjustable sphero-cylindrical refractor comprising two lenses,” Opt. Eng. 52, 063002 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Polym. Eng. Sci.

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P. J. Smilie, B. S. Dutterer, J. L. Lineberger, M. A. Davies, and T. J. Suleski, “Freeform micromachining of an infrared Alvarez lens,” Proc. SPIE 7927, 79270K (2011).
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[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Top freeform surface of Alvarez lens.

Fig. 2.
Fig. 2.

(a) Finished ultraprecision diamond-machined mold for injection molding. (b) 3D model of the molded Alvarez lens with small straight pins and flats designed for assembly.

Fig. 3.
Fig. 3.

Injection-molded Alvarez lens.

Fig. 4.
Fig. 4.

FEM mesh model of the Alvarez lens. Node (i,1) is numbered along the vertical line at coordinates (xi,yi) at the bottom surface, and Node (i,N) is the node along the vertical line at coordinates (xi,yi) at its top surface.

Fig. 5.
Fig. 5.

Interferometry setup for measuring wavefront pattern of microinjection-molded Alvarez lens immersed in a wet cell. 1, He–Ne laser light source; 2, optical pin hole; 3, collimation lens; 4, beam splitter A; 5, flat mirror A; 6, flat mirror B; 7, Alvarez lens immersed in a wet cell; 8, beam splitter B; 9, CCD camera; 10, PZT stage.

Fig. 6.
Fig. 6.

(a) Simulated and (b) measured bottom surface deformations of the microinjection-molded Alvarez lens.

Fig. 7.
Fig. 7.

Simulation plot of the deformed microinjection-molded Alvarez lens with a magnification scale of 50×.

Fig. 8.
Fig. 8.

(a) Simulated and (b) measured wavefront pattern describing refractive index distribution of the microinjection-molded Alvarez lenses.

Fig. 9.
Fig. 9.

(a) Nominal and (b) measured wavefront pattern describing surface power distribution of the microinjection-molded Alvarez lenses. (c) Difference between the simulated and measured results.

Fig. 10.
Fig. 10.

Optical path change on the freeform surface.

Tables (1)

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Table 1. Microinjection Molding Conditions

Equations (9)

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z=a1x2y+a2y3,
ti=z(i,N)z(i,1),
ti=z(i,N)z(i,1),
ni=(j=1Nn(i,j))/N,
Δni=nij=1NVnj/NV,
CF=nlensnliquidnlensnair,
OPi=ni×ti+nliquid×(Tchamberti),
ΔOPi=OPij=1NVOPj/NV.
Wair(xi,yi)=CF×Wliquid(xi,yi),

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