Abstract

We propose a reverse functional design of modified Fresnel lens (MFL) with discontinuous refractive surfaces to achieve cost-effective high optical performance with thin lenses. The reverse-geometry design process was optimized to control the spatial illuminance distribution (SID) of light-emitting diodes (LEDs). Analysis results based on non-sequential ray-tracing simulations for flat SIDs indicated that the illuminance uniformity of LEDs with optimum MFL with different groove angles increased about 22 times, from 0.348 to 0.016, compared with the normalized standard deviation (NSD) of the general Fresnel lenses (GFL) with groove angles of 0°. In addition, the proposed method enhanced the color uniformity by reducing the circular yellow pattern. Tolerance analysis was carried out to determine tolerance limits for applying the optimum MFL in the assembly process. Finally, the feasibility of the reverse design process was verified by optical measurements of the optimum MFL.

© 2011 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[CrossRef]

2009

C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
[CrossRef] [PubMed]

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[CrossRef]

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[CrossRef] [PubMed]

2008

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16(26), 21835–21842 (2008).
[CrossRef] [PubMed]

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[CrossRef] [PubMed]

H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
[CrossRef] [PubMed]

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[CrossRef]

2007

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

2005

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[CrossRef] [PubMed]

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

2004

R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004).
[CrossRef]

2000

W. Tai and R. Schwarte, “Design of an aspherical lens to generate a homogenous irradiance for three-dimensional sensors with a light-emitting-diode source,” Appl. Opt. 39(31), 5801–5805 (2000).
[CrossRef]

1998

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[CrossRef]

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3428, 389–396 (1998).
[CrossRef]

Beck, J. V.

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[CrossRef]

Bergmans, J. W.

H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
[CrossRef] [PubMed]

Chen, F.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[CrossRef]

Chien, W.-T.

C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
[CrossRef] [PubMed]

Cho, J.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

Choi, M.

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[CrossRef] [PubMed]

Craford, M. G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Ding, Y.

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[CrossRef] [PubMed]

Gu, P. F.

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[CrossRef] [PubMed]

Harbers, G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Hsieh, C.-C.

C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
[CrossRef] [PubMed]

Kang, S.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[CrossRef]

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[CrossRef] [PubMed]

Kim, B.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[CrossRef]

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[CrossRef] [PubMed]

Kim, H.

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[CrossRef] [PubMed]

Kim, J.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[CrossRef]

Kim, J. K.

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16(26), 21835–21842 (2008).
[CrossRef] [PubMed]

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[CrossRef] [PubMed]

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

Koshel, R. J.

R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004).
[CrossRef]

Krames, M. R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Kudaev, S.

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

Lim, J.

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[CrossRef] [PubMed]

Linnartz, J. P.

H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
[CrossRef] [PubMed]

Liu, S.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[CrossRef]

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[CrossRef]

Liu, X.

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[CrossRef] [PubMed]

Liu, Z.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[CrossRef]

Liu, Z. Y.

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[CrossRef]

Lo, Y.-C.

C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
[CrossRef] [PubMed]

Luo, H.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

Luo, X.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[CrossRef]

Luo, X. B.

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[CrossRef]

Moreno, I.

C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
[CrossRef] [PubMed]

Mueller, G. O.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Mueller-Mach, R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Ohm, W.-S.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[CrossRef]

Park, Y.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

Parkyn, W. A.

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3428, 389–396 (1998).
[CrossRef]

Rietman, R.

H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
[CrossRef] [PubMed]

Schenk, T. C. W.

H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
[CrossRef] [PubMed]

Schreiber, P.

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

Schubert, E. F.

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16(26), 21835–21842 (2008).
[CrossRef] [PubMed]

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[CrossRef] [PubMed]

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

Schwarte, R.

W. Tai and R. Schwarte, “Design of an aspherical lens to generate a homogenous irradiance for three-dimensional sensors with a light-emitting-diode source,” Appl. Opt. 39(31), 5801–5805 (2000).
[CrossRef]

Shchekin, O. B.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Sone, C.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

Sun, C.-C.

C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
[CrossRef] [PubMed]

Tai, W.

W. Tai and R. Schwarte, “Design of an aspherical lens to generate a homogenous irradiance for three-dimensional sensors with a light-emitting-diode source,” Appl. Opt. 39(31), 5801–5805 (2000).
[CrossRef]

Wang, K.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[CrossRef]

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[CrossRef]

Woodbury, K. A.

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[CrossRef]

Yang, H.

H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
[CrossRef] [PubMed]

Zheng, Z. R.

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[CrossRef] [PubMed]

Zhou, L.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Appl. Opt.

W. Tai and R. Schwarte, “Design of an aspherical lens to generate a homogenous irradiance for three-dimensional sensors with a light-emitting-diode source,” Appl. Opt. 39(31), 5801–5805 (2000).
[CrossRef]

Appl. Phys. Lett.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[CrossRef]

J. Disp. Technol.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[CrossRef]

Meas. Sci. Technol.

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[CrossRef]

Opt. Eng.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[CrossRef]

Opt. Express

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[CrossRef] [PubMed]

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[CrossRef]

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16(26), 21835–21842 (2008).
[CrossRef] [PubMed]

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[CrossRef] [PubMed]

H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
[CrossRef] [PubMed]

C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
[CrossRef] [PubMed]

Proc. SPIE

R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004).
[CrossRef]

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3428, 389–396 (1998).
[CrossRef]

Science

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[CrossRef] [PubMed]

Other

B. Parkyna, and D. Pelkaa, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” in Nonimaging Optics and Efficient Illumination Systems III, R. Winston, P. Benítez, eds., Proc. SPIE 6338, 633808:1–7 (2006).

Y. Honguh, Y. Takashima, H. Mihara, and M. Kitamura, “Illumination apparatus and a projection type image display apparatus using it,” US Patent 6264332.

I. Moreno, “Spatial distribution of LED radiation,” in The International Optical Design Conference, G.Gregory, J. Howard, J. Koshel, eds., Proc. SPIE 6342, 634216:1–7 (2006).

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Figures (14)

Fig. 1
Fig. 1

Forward-geometry design based on ray matrices and special illuminance distributions (SIDs) by ray-tracing simulations of (a) a general Fresnel lens (GFL) with a normalized standard deviation (NSD) of 0.348 by groove angles of 0° and (b) a modified Fresnel lens (MFL) with a NSD of 0.164 by equal optimum groove angles.

Fig. 2
Fig. 2

Reverse functional design process of an optimum MFL with different groove angles for a real LED module by compensation of the input energy.

Fig. 3
Fig. 3

Schematic of functional relationship for MFL with different groove angles based on a discrete energy distribution for the reverse-geometric design.

Fig. 4
Fig. 4

Construction concept of groove facets (a) with equal groove angle in general optimized MFL of Ref [11]. and (b) with different groove angles in functional designed MFL based on a discrete energy distribution with the reverse-geometric relationship.

Fig. 5
Fig. 5

Results of (a) initial MFL with different groove angles modeling for the ray-tracing simulation, (b) spatial illuminance distribution image for the point source, and (c) relative illuminance distribution of the x-axis with a NSD of 0.011.

Fig. 6
Fig. 6

Results of (a) the final MFL with different groove angles modeling for a ray-tracing simulation, (b) SID image for the initial MFL, (c) SID image for the final compensated MFL, and (d) relative illuminance distributions of the x-axis with a NSD of 0.073 before optimization and with a NSD of 0.016 after optimization for a real LED module.

Fig. 7
Fig. 7

Comparison of geometry for the initial and final compensated MFL with different groove angles.

Fig. 8
Fig. 8

(a) Schematic for analysis of an extended source, (b) analysis of a SID by the initial MFL with different groove angles for discrete sources, and (c) analysis of a SID by the final compensated MFL for discrete sources.

Fig. 9
Fig. 9

Comparison of (a) the color distribution by a spherical lens, (b) the color distribution by an optimal MFL with a real extended white LED source module, and (c) the yellow–blue ratio (YBR) of each lens.

Fig. 10
Fig. 10

Analysis of illuminance uniformity variation for the decenter tolerance between the LED module and the designed MFL (e.g., tolerance range for the decenter was ± 50 µm for an illuminance uniformity of 96%).

Fig. 11
Fig. 11

Analysis of illuminance uniformity variation for the defocus tolerance between the LED module and the designed MFL (e.g., tolerance range for the defocus was ± 100 µm for a viewing angle deviation of ± 2° and an illuminance uniformity of 96%).

Fig. 12
Fig. 12

Analysis of illuminance uniformity variation for the tilt tolerance between the LED module and the designed MFL (e.g., tolerance range for tilt was ± 0.35° for an illuminance uniformity of 96%).

Fig. 13
Fig. 13

The main UV-imprinting process and images of the fabricated original mold and the fabricated MFL.

Fig. 14
Fig. 14

Comparison of (a) the color distribution image with the NSD of 0.218 by a spherical lens, (b) the color distribution image with the NSD of 0.039 using the fabricated MFL with a real, extended white LED source module, and (c) the YBR of each lens.

Equations (7)

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s o u r c e I ( θ ) d Ω = t arg e t E ( R ) d A ,
φ i n p u t ( i ) = θ i 1 θ i 2 π cos θ sin θ d θ = π R 2 m , ( i = 1 , 2 , , m , θ 0 = 0 ) ,
n ' O i = n I i + ( n ' O i N i n I i N i ) N i , where   O i = ( t i x P i x , 0 , t i z P i z ) ,           I i = ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) , 0 , P i z s h ) ,         N i = ( d P i z , 0 , d P i x ) ,
d P i z d P i x = ( ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) ) ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) ) 2 + ( P i z s h ) 2 ) n ( ( t i x P i x ) ( t i x P i x ) 2 + ( t i z P i z ) 2 ) n ' ( ( P i z s h ) ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) ) 2 + ( P i z s h ) 2 ) n ( ( t i z P i z ) ( t i x P i x ) 2 + ( t i z P i z ) 2 ) n ' ,
E ( Z j ( x ) | n 1 ) d e v i a t i o n = 1 E a v g i = 1 m ( E ( t i x ; Z j ( x ) | n 1 ) E a v g ) 2 m ,               ( i = 1 , 2 , , m ) , ( j = 0 , 1 , , N 1 ) , ( n = 1 , 2 , )
φ i n p u t ( i ) | n = w ( i ) | n × φ i n p u t ( i ) | n 1 1 m i = 1 m w ( i ) | n × φ i n p u t ( i ) | n 1 , ( i = 1 , 2 , , m ) , ( n = 1 , 2 , ) , where   w ( i ) | n = ( E ( t i x ; Z j ( x ) | n 1 ) 1 1 m i = 1 m ( E ( t i x ; Z j ( x ) | n 1 ) 1 , ( j = 0 , 1 , , N 1 ) ,
P i x | n = s tan ( θ i | n ) + h tan ( sin 1 ( 1 n sin ( θ i | n ) ) ) , ( i = 1 , 2 , , m ) , ( n = 1 , 2 , ) where   θ i | n = 1 2 cos 1 ( cos ( 2 θ i 1 | n 2 φ i n p u t ( i ) | n π ) ) , θ 0 | n = θ 0 = 0 ,

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