Abstract

This study presents a new method for achieving uniform illumination of an LED array by designing single LED illumination distribution (SLID) on the target plane. The SLID with the least coefficient of variation of the root mean squared error is calculated. For square LED arrays, the uniformity of illumination reaches 0.94. A secondary lens was designed to achieve the calculated SLID. Simulation results show that the uniformity of illumination reaches 0.92 when the distance–height ratio is 3.33. The calculated SLID can be used in different distance–height ratios. When the distance–height ratio is 5, the uniformity of illumination reaches 0.91. The calculation method can be used in different kinds of LED arrays. A feedback method of lens design is presented for a real direct-type backlight system.

© 2013 Optical Society of America

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  1. F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (1)

2011 (1)

2010 (2)

2009 (2)

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform illumination rendering using an array of LEDs: a signal processing perspective,” Proc. IEEE 57, 1044–1057 (2009).

A. J. W. Whang, Y. Y. Chen, and Y. T. Teng, “Designing uniform illumination systems by surface-tailored lens and configurations of LED arrays,” J. Display Technol. 5, 94–103 (2009).
[CrossRef]

2008 (1)

2006 (1)

2005 (1)

M. G. Craford, “LEDs for solid state lighting and other emerging applications: status, trends, and challenges,” Proc. SPIE 5941, 594101 (2005).
[CrossRef]

2004 (1)

I. Moreno and R. I. Tzonchev, “Effects on illumination uniformity due to dilution on arrays of LEDs,” Proc. SPIE 5529, 268–275 (2004).
[CrossRef]

2002 (1)

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Avendano-Alejo, M.

Bergmans, J. W. M.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform illumination rendering using an array of LEDs: a signal processing perspective,” Proc. IEEE 57, 1044–1057 (2009).

Bhat, J.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Chen, F.

Chen, Y. Y.

Collins, D.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Cook, L.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Craford, M. G.

M. G. Craford, “LEDs for solid state lighting and other emerging applications: status, trends, and challenges,” Proc. SPIE 5941, 594101 (2005).
[CrossRef]

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Ding, Y.

Feng, Z.

Fletcher, R.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Gan, Z.

Gardner, N.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Goetz, W.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Grillot, P.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Gu, P.

Han, Y.

Harbers, G.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Hu, R.

Keuper, M.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Khare, R.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Kim, A.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Krames, M.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Li, H.

Linnartz, J. P. M. G.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform illumination rendering using an array of LEDs: a signal processing perspective,” Proc. IEEE 57, 1044–1057 (2009).

Liu, S.

Liu, X.

Ludowise, M.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Luo, X.

Luo, Y.

Martin, P. S.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Misra, M.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Moreno, I.

I. Moreno, M. Avendano-Alejo, and R. I. Tzonchev, “Designing LED arrays for uniform near-field irradiance,” Appl. Opt. 45, 2265–2272 (2006).
[CrossRef]

I. Moreno and R. I. Tzonchev, “Effects on illumination uniformity due to dilution on arrays of LEDs,” Proc. SPIE 5529, 268–275 (2004).
[CrossRef]

Mueller, G.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Mueller-Mach, R.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Qin, Z.

Rietman, R.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform illumination rendering using an array of LEDs: a signal processing perspective,” Proc. IEEE 57, 1044–1057 (2009).

Rudaz, S.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Schenk, T. C. W.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform illumination rendering using an array of LEDs: a signal processing perspective,” Proc. IEEE 57, 1044–1057 (2009).

Shen, Y.-C.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Steigerwald, D.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Steranka, F. M.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Stockman, S.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Subramanya, S.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Teng, Y. T.

Trottier, T.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Tzonchev, R. I.

I. Moreno, M. Avendano-Alejo, and R. I. Tzonchev, “Designing LED arrays for uniform near-field irradiance,” Appl. Opt. 45, 2265–2272 (2006).
[CrossRef]

I. Moreno and R. I. Tzonchev, “Effects on illumination uniformity due to dilution on arrays of LEDs,” Proc. SPIE 5529, 268–275 (2004).
[CrossRef]

Wang, K.

Whang, A. J. W.

Wierer, J. J.

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Wu, B.

Wu, D.

Yang, H.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform illumination rendering using an array of LEDs: a signal processing perspective,” Proc. IEEE 57, 1044–1057 (2009).

Zheng, H.

Zheng, Z.

Zong, Q.

Appl. Opt. (1)

J. Display Technol. (1)

Opt. Express (5)

Phys. Status Solidi A (1)

F. M. Steranka, J. Bhat, D. Collins, L. Cook, M. G. Craford, R. Fletcher, N. Gardner, P. Grillot, W. Goetz, M. Keuper, R. Khare, A. Kim, M. Krames, G. Harbers, M. Ludowise, P. S. Martin, M. Misra, G. Mueller, R. Mueller-Mach, S. Rudaz, Y.-C. Shen, D. Steigerwald, S. Stockman, S. Subramanya, T. Trottier, and J. J. Wierer, “High power LEDs technology status and market applications,” Phys. Status Solidi A 194, 380–388 (2002).
[CrossRef]

Proc. IEEE (1)

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J. P. M. G. Linnartz, and R. Rietman, “Uniform illumination rendering using an array of LEDs: a signal processing perspective,” Proc. IEEE 57, 1044–1057 (2009).

Proc. SPIE (2)

M. G. Craford, “LEDs for solid state lighting and other emerging applications: status, trends, and challenges,” Proc. SPIE 5941, 594101 (2005).
[CrossRef]

I. Moreno and R. I. Tzonchev, “Effects on illumination uniformity due to dilution on arrays of LEDs,” Proc. SPIE 5529, 268–275 (2004).
[CrossRef]

Other (1)

S. Liu and X. Luo, LED Packaging for Lighting Applications: Design, Manufacturing, and Testing (Wiley, 2011).

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Figures (12)

Fig. 1.
Fig. 1.

Schematic illustration of LED uniform illumination. d represents the distance between two adjacent LEDs, R represents the radius of SLID, h represents the distance between the source and the target plane.

Fig. 2.
Fig. 2.

Schematic illustration of a 16-LED square array where d represents the distance between two adjacent LEDs, R represents the radius of the SLID.

Fig. 3.
Fig. 3.

Relative variance and uniformity of illumination versus order of polynomial m for different types of basis functions.

Fig. 4.
Fig. 4.

SLIDs approximated by a polynomial with basis functions gk(r)=cos((2k+1)πr/2R), gk(r)=rk, and gk(r)=Jk(r), when the polynomial order is 6, 6, and 17.

Fig. 5.
Fig. 5.

Schematic of corresponding relationship of energy between the LED light source and the calculated SLID.

Fig. 6.
Fig. 6.

(a) Lens designed to achieve the SLID when DHR=3.33. (b) Simulated illumination distribution of a 4×4 LED array produced by the lens in (a), DHR=3.33, U=0.92.

Fig. 7.
Fig. 7.

(a) Lens designed to achieve the SLID when DHR=5. (b) Simulated illumination distribution of a 4×4 LED array produced by the lens in (a), DHR=5, U=0.91.

Fig. 8.
Fig. 8.

Schematic illustration of triangular LED array with 23 LEDs.

Fig. 9.
Fig. 9.

SLID approximated by a polynomial with basis functions gk(r)=cos((2k+1)πr/2R) when the polynomial order is 8.

Fig. 10.
Fig. 10.

(a) Lens designed to achieve the SLID of a triangular LED array when DHR=3.33. (b) Simulated illumination distribution of the triangular LED array in Fig. 8, produced by the lens in (a), DHR=3.33, U=0.94.

Fig. 11.
Fig. 11.

(a) Lens design without feedback for SLID described by Eq. (10), DHR=3.33. (b) Simulated illumination distribution of a 4×4 LED array produced by the lens in (a), DHR=3.33, U=0.86.

Fig. 12.
Fig. 12.

(a) Lens design with feedback For SLID described by Eq. (10), DHR=3.33. (b) Simulated illumination distribution of a 4×4 LED array produced by the lens in (a), DHR=3.33, U=0.91.

Equations (12)

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Eu(x,y)=E[(x2+y2)1/2]+E{[x2+(yd)2]1/2}+E{[(xd)2+y2]1/2}+E{[(xd)2+(yd)2]1/2},
E(r){0=0(0<rd),(r<0,rd).
D=0d0d[Eu(x,y)E0]2dxdy,
CV(RMSE)=[D/(d2E02)]1/2.
E(r)=a0·g0(r)+a1·g1(r)+a2·g2(r)+(0<rd),
D/ak=0,k=0,1,2,.
E(r)=a0·g0(r)+a1·g1(r)++am1·gm1(r)(0<rd).
D/a0=0,D/a1=0,,D/am1=0.
E(r)=E0E0·E(rd·d).
E(r)=0.8245×cos(πr/2R)0.1176×cos(3πr/2R),0.0013×cos(5πr/2R)+0.0166×cos(7πr/2R),+0.0057×cos(9πr/2R)+0.0037×cos(11πr/2R).
0θsouI0cos(θ)sin(θ)dθ=K0rSLIDE(r)rdr,
K=0π2I0cos(θ)sin(θ)dθ/0RE(r)rdr.

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