Abstract

In this research, the condition for uniform lighting generated by array of LEDs with large view angle was studied. The luminous intensity distribution of LED is not monotone decreasing with view angle. A LED with freeform lens was designed as an example for analysis. In a system based on LEDs designed in house with a thickness of 20mm and rectangular arrangement, the condition for uniform lighting was derived and the analytical results demonstrated that the uniformity was not decreasing monotonously with the increasing of LED-to-LED spacing. The illuminance uniformities were calculated with Monte Carlo ray tracing simulations and the uniformity was found to increase with the increasing of certain LED-to-LED spacings anomalously. Another type of large view angle LED and different arrangements were discussed in addition. Both analysis and simulation results showed that the method is available for LED array lighting system design on the basis of large view angle LED..

© 2010 OSA

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References

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  1. M. G. Craford, “LEDs for solid state lighting and other emerging applications: status, trends, and challenges,” Proc. SPIE, Vol. 5941, 1–10 (2005).
  2. J. Kovác, J. Jakabovic, and M. Kytka, “Advanced light emitting devices for optoelectronic applications,” Proc. SPIE 7138, 71382A (2008).
    [CrossRef]
  3. E. F. Schubert, J.-K. Kim, H. Luo, and J.-Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69(12Issue 12), 3069–3099 (2006).
    [CrossRef]
  4. C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
    [CrossRef]
  5. I. Moreno and R. I. Tzonchev, “Effects on illuminance uniformity due to dilution on arrays of LEDs,” Proc. SPIE 5529, 268–275 (2004).
    [CrossRef]
  6. I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-10-2265 .
    [CrossRef] [PubMed]
  7. I. Moreno, “Configurations of LED arrays for uniform illuminance,” Proc. SPIE 5622, 713–718 (2004).
    [CrossRef]
  8. A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing Uniform Illuminance Systems by Surface-Tailored Lens and Configurations of LED Arrays,” J. Disp. Technol. 5(3), 94–103 (2009), http://www.opticsinfobase.org/jdt/abstract.cfm?URI=jdt-5-3-94 .
    [CrossRef]
  9. F. Zhao and J. F. Van Derlofske, “Side-emitting illuminators using LED sources,” Proc. SPIE 5186, 33–43 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  13. I. Moreno, C.-C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48(6), 1190–1197 (2009),
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    [CrossRef]
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  18. F. Chen, S. Liu, K. Wang, Z. Y. Liu, and X. B. Luo, “Free-form lenses for high illuminance quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
    [CrossRef]
  19. I. Moreno and C.-C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008),
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    [CrossRef] [PubMed]
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    [CrossRef]

2010 (1)

2009 (4)

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

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J.-P. M. G. Linnartz, and R. Rietman, “Uniform illuminance rendering using an array of LEDs: a signal processing perspective,” IEEE Trans. Signal Process. 57(3Issue 3), 1044–1057 (2009).
[CrossRef]

I. Moreno, C.-C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48(6), 1190–1197 (2009),
 http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-6-1190 .
[CrossRef]

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing Uniform Illuminance Systems by Surface-Tailored Lens and Configurations of LED Arrays,” J. Disp. Technol. 5(3), 94–103 (2009), http://www.opticsinfobase.org/jdt/abstract.cfm?URI=jdt-5-3-94 .
[CrossRef]

2008 (3)

J. Kovác, J. Jakabovic, and M. Kytka, “Advanced light emitting devices for optoelectronic applications,” Proc. SPIE 7138, 71382A (2008).
[CrossRef]

I. Moreno and C.-C. Sun, “LED array: Where does far-field begin?” Proc. SPIE 7058, 70580R (2008).
[CrossRef]

I. Moreno and C.-C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008),
 http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1808 .
[CrossRef] [PubMed]

2007 (2)

I. Moreno, C.-Y. Tsai, D. Bermúdez, and C.-C. Sun, “Simple function for luminous intensity distribution from LEDs,” Proc. SPIE 6670, 66700H (2007).
[CrossRef]

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

2006 (2)

2004 (2)

I. Moreno, “Configurations of LED arrays for uniform illuminance,” Proc. SPIE 5622, 713–718 (2004).
[CrossRef]

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

2003 (1)

F. Zhao and J. F. Van Derlofske, “Side-emitting illuminators using LED sources,” Proc. SPIE 5186, 33–43 (2003).
[CrossRef]

1964 (1)

1962 (1)

1936 (1)

W. A. Hendricks and K. W. Robey, “The sampling distribution of the coefficient of variation,” Ann. Math. Stat. 7(3), 129–132 (1936).
[CrossRef]

1916 (1)

C. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76 (1916).
[CrossRef]

Avendaño-Alejo, M.

Barakat, R.

Bergmans, J. W. M.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J.-P. M. G. Linnartz, and R. Rietman, “Uniform illuminance rendering using an array of LEDs: a signal processing perspective,” IEEE Trans. Signal Process. 57(3Issue 3), 1044–1057 (2009).
[CrossRef]

Bermúdez, D.

I. Moreno, C.-Y. Tsai, D. Bermúdez, and C.-C. Sun, “Simple function for luminous intensity distribution from LEDs,” Proc. SPIE 6670, 66700H (2007).
[CrossRef]

Chen, F.

Chen, Y.-Y.

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing Uniform Illuminance Systems by Surface-Tailored Lens and Configurations of LED Arrays,” J. Disp. Technol. 5(3), 94–103 (2009), http://www.opticsinfobase.org/jdt/abstract.cfm?URI=jdt-5-3-94 .
[CrossRef]

Chien, W.-T.

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

Chung, S.-H.

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

Harris, J. L.

Hendricks, W. A.

W. A. Hendricks and K. W. Robey, “The sampling distribution of the coefficient of variation,” Ann. Math. Stat. 7(3), 129–132 (1936).
[CrossRef]

Hsieh, C.-T.

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

Ivanov, R.

Jakabovic, J.

J. Kovác, J. Jakabovic, and M. Kytka, “Advanced light emitting devices for optoelectronic applications,” Proc. SPIE 7138, 71382A (2008).
[CrossRef]

Kim, J.-K.

E. F. Schubert, J.-K. Kim, H. Luo, and J.-Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69(12Issue 12), 3069–3099 (2006).
[CrossRef]

Kovác, J.

J. Kovác, J. Jakabovic, and M. Kytka, “Advanced light emitting devices for optoelectronic applications,” Proc. SPIE 7138, 71382A (2008).
[CrossRef]

Kytka, M.

J. Kovác, J. Jakabovic, and M. Kytka, “Advanced light emitting devices for optoelectronic applications,” Proc. SPIE 7138, 71382A (2008).
[CrossRef]

Linnartz, J.-P. M. G.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J.-P. M. G. Linnartz, and R. Rietman, “Uniform illuminance rendering using an array of LEDs: a signal processing perspective,” IEEE Trans. Signal Process. 57(3Issue 3), 1044–1057 (2009).
[CrossRef]

Liu, S.

Liu, Z. Y.

Luo, H.

E. F. Schubert, J.-K. Kim, H. Luo, and J.-Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69(12Issue 12), 3069–3099 (2006).
[CrossRef]

Luo, X. B.

Moreno, I.

I. Moreno, C.-C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48(6), 1190–1197 (2009),
 http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-6-1190 .
[CrossRef]

I. Moreno and C.-C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008),
 http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1808 .
[CrossRef] [PubMed]

I. Moreno and C.-C. Sun, “LED array: Where does far-field begin?” Proc. SPIE 7058, 70580R (2008).
[CrossRef]

I. Moreno, C.-Y. Tsai, D. Bermúdez, and C.-C. Sun, “Simple function for luminous intensity distribution from LEDs,” Proc. SPIE 6670, 66700H (2007).
[CrossRef]

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. 45(10), 2265–2272 (2006), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-10-2265 .
[CrossRef] [PubMed]

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

I. Moreno, “Configurations of LED arrays for uniform illuminance,” Proc. SPIE 5622, 713–718 (2004).
[CrossRef]

Rietman, R.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J.-P. M. G. Linnartz, and R. Rietman, “Uniform illuminance rendering using an array of LEDs: a signal processing perspective,” IEEE Trans. Signal Process. 57(3Issue 3), 1044–1057 (2009).
[CrossRef]

Robey, K. W.

W. A. Hendricks and K. W. Robey, “The sampling distribution of the coefficient of variation,” Ann. Math. Stat. 7(3), 129–132 (1936).
[CrossRef]

Schenk, T. C. W.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J.-P. M. G. Linnartz, and R. Rietman, “Uniform illuminance rendering using an array of LEDs: a signal processing perspective,” IEEE Trans. Signal Process. 57(3Issue 3), 1044–1057 (2009).
[CrossRef]

Schubert, E. F.

E. F. Schubert, J.-K. Kim, H. Luo, and J.-Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69(12Issue 12), 3069–3099 (2006).
[CrossRef]

Sparrow, C.

C. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76 (1916).
[CrossRef]

Sun, C.-C.

I. Moreno, C.-C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48(6), 1190–1197 (2009),
 http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-6-1190 .
[CrossRef]

I. Moreno and C.-C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16(3), 1808–1819 (2008),
 http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1808 .
[CrossRef] [PubMed]

I. Moreno and C.-C. Sun, “LED array: Where does far-field begin?” Proc. SPIE 7058, 70580R (2008).
[CrossRef]

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

I. Moreno, C.-Y. Tsai, D. Bermúdez, and C.-C. Sun, “Simple function for luminous intensity distribution from LEDs,” Proc. SPIE 6670, 66700H (2007).
[CrossRef]

Teng, Y.-T.

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing Uniform Illuminance Systems by Surface-Tailored Lens and Configurations of LED Arrays,” J. Disp. Technol. 5(3), 94–103 (2009), http://www.opticsinfobase.org/jdt/abstract.cfm?URI=jdt-5-3-94 .
[CrossRef]

Tsai, C.-Y.

I. Moreno, C.-Y. Tsai, D. Bermúdez, and C.-C. Sun, “Simple function for luminous intensity distribution from LEDs,” Proc. SPIE 6670, 66700H (2007).
[CrossRef]

Tzonchev, R. I.

Van Derlofske, J. F.

F. Zhao and J. F. Van Derlofske, “Side-emitting illuminators using LED sources,” Proc. SPIE 5186, 33–43 (2003).
[CrossRef]

Wang, K.

Whang, A. J.-W.

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing Uniform Illuminance Systems by Surface-Tailored Lens and Configurations of LED Arrays,” J. Disp. Technol. 5(3), 94–103 (2009), http://www.opticsinfobase.org/jdt/abstract.cfm?URI=jdt-5-3-94 .
[CrossRef]

Xi, J.-Q.

E. F. Schubert, J.-K. Kim, H. Luo, and J.-Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69(12Issue 12), 3069–3099 (2006).
[CrossRef]

Yang, H.

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J.-P. M. G. Linnartz, and R. Rietman, “Uniform illuminance rendering using an array of LEDs: a signal processing perspective,” IEEE Trans. Signal Process. 57(3Issue 3), 1044–1057 (2009).
[CrossRef]

Yang, T.-H.

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

Zhao, F.

F. Zhao and J. F. Van Derlofske, “Side-emitting illuminators using LED sources,” Proc. SPIE 5186, 33–43 (2003).
[CrossRef]

Ann. Math. Stat. (1)

W. A. Hendricks and K. W. Robey, “The sampling distribution of the coefficient of variation,” Ann. Math. Stat. 7(3), 129–132 (1936).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

C. Sparrow, “On spectroscopic resolving power,” Astrophys. J. 44, 76 (1916).
[CrossRef]

IEEE Trans. Signal Process. (1)

H. Yang, J. W. M. Bergmans, T. C. W. Schenk, J.-P. M. G. Linnartz, and R. Rietman, “Uniform illuminance rendering using an array of LEDs: a signal processing perspective,” IEEE Trans. Signal Process. 57(3Issue 3), 1044–1057 (2009).
[CrossRef]

J. Disp. Technol. (1)

A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing Uniform Illuminance Systems by Surface-Tailored Lens and Configurations of LED Arrays,” J. Disp. Technol. 5(3), 94–103 (2009), http://www.opticsinfobase.org/jdt/abstract.cfm?URI=jdt-5-3-94 .
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

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

Opt. Express (2)

Proc. SPIE (7)

I. Moreno, C.-Y. Tsai, D. Bermúdez, and C.-C. Sun, “Simple function for luminous intensity distribution from LEDs,” Proc. SPIE 6670, 66700H (2007).
[CrossRef]

I. Moreno and C.-C. Sun, “LED array: Where does far-field begin?” Proc. SPIE 7058, 70580R (2008).
[CrossRef]

F. Zhao and J. F. Van Derlofske, “Side-emitting illuminators using LED sources,” Proc. SPIE 5186, 33–43 (2003).
[CrossRef]

I. Moreno, “Configurations of LED arrays for uniform illuminance,” Proc. SPIE 5622, 713–718 (2004).
[CrossRef]

J. Kovác, J. Jakabovic, and M. Kytka, “Advanced light emitting devices for optoelectronic applications,” Proc. SPIE 7138, 71382A (2008).
[CrossRef]

C.-C. Sun, I. Moreno, S.-H. Chung, W.-T. Chien, C.-T. Hsieh, and T.-H. Yang, “Direct LED backlight for large area LCD TVs: brightness analysis,” Proc. SPIE 6669, 666909 (2007).
[CrossRef]

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

Rep. Prog. Phys. (1)

E. F. Schubert, J.-K. Kim, H. Luo, and J.-Q. Xi, “Solid-state lighting—a benevolent technology,” Rep. Prog. Phys. 69(12Issue 12), 3069–3099 (2006).
[CrossRef]

Other (1)

M. G. Craford, “LEDs for solid state lighting and other emerging applications: status, trends, and challenges,” Proc. SPIE, Vol. 5941, 1–10 (2005).

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Figures (19)

Fig. 1
Fig. 1

The illuminance of the point (x,y) is generated by 4 LEDs in 2 × 2 array. Point (0,0) corresponds to the central point of the region enclosed by the 4 LEDs. Parameters to calculate the illuminance generated by the ith (i = 1~4) LED are indicated in the figure.

Fig. 2
Fig. 2

Freeform lens that can produce a circular light pattern with a radius of 60mm at a plane 20mm away

Fig. 3
Fig. 3

Simulated illuminance distribution produced by the freeform lens in Fig. 2. Different color in the left sub-figure represents different illuminance value. The relationship between color and illuminance value is shown in the right sub-figure. The right sub-figure also shows how much luminous flux accumulates at each illuminance value.

Fig. 4
Fig. 4

Luminous intensity distribution curve attached to the circular light pattern in Fig. 3

Fig. 5
Fig. 5

Fitting result of the NMDD produced by freeform lens where P1 = 0.7221, P2 = 1.2212, P3 = 0.0527, P4 = −0.3454, P5 = 1.4540, P6 = 0.1791, P7 = 0.3086, P8 = 1.4767, P9 = 0.5920 for Eq. (8). Parameters are all in radians although this figure is drawn in degrees

Fig. 6
Fig. 6

Model of a LAL system with a 4 × 4 LED array where the illuminance distribution in the gray region will be derived. Black dots states the positions of LEDs. A white dot indicates the origin

Fig. 7
Fig. 7

The functional image of f ( d ) = 2 E / x 2 | x = 0 , y = 0 , z = 20

Fig. 8
Fig. 8

Illuminance distributions in the direction of x axis of the gray region in Fig. 6 where (a) dz1 = 28.90mm, (b) dz2 = 37.90mm, (c) dz3 = 65.12mm, (d) dz4 = 83.78mm

Fig. 9
Fig. 9

Illuminance distributions in the direction of x axis of the gray region in Fig. 6 where (a) dp1 = 32.44mm, (b) dp2 = 59.11mm, (c) dp3 = 72.33mm

Fig. 10
Fig. 10

Simulated illuminance distributions corresponding to zero points (a) dz1 = 28.90mm, CV(RMSE) = 0.014 (b) dz2 = 37.90mm, CV(RMSE) = 0.032 (c) dz3 = 65.12mm, CV(RMSE) = 0.080 (d) dz4 = 83.78mm, CV(RMSE) = 0.147. Black dots are used to state positions of LEDs.

Fig. 11
Fig. 11

Simulated illuminance distributions corresponding to peak points (a) dp1 = 32.44mm, CV(RMSE) = 0.052 (b) dp2 = 59.11mm, CV(RMSE) = 0.107 (c) dp3 = 72.33mm, CV(RMSE) = 0.256. Black dots are used to state positions of LEDs.

Fig. 12
Fig. 12

Luminous intensity distribution curve of LUXEON® Batwing LED from Philips Corporation

Fig. 13
Fig. 13

Simulated illuminance distribution on a 20mm-height plane generated by a LUXEON® Batwing LED from Philips Corporation

Fig. 14
Fig. 14

The functional image of f ( d ) = 2 E / x 2 | x = 0 , y = 0 , z = 20 for a LUXEON® Batwing LED array

Fig. 15
Fig. 15

Simulated illuminance distributions corresponding to zero points (a) dz1 = 11.29mm, CV(RMSE) = 0.012 (b) dz2 = 16.66mm, CV(RMSE) = 0.016 (c) dz3 = 20.66mm, CV(RMSE) = 0.139. Black dots are used to state positions of LEDs.

Fig. 16
Fig. 16

Simulated illuminance distributions corresponding to peak points (a) dp1 = 13.41mm, CV(RMSE) = 0.028 (b) dp2 = 20.66mm, CV(RMSE) = 0.087. Black dots are used to state positions of LEDs.

Fig. 17
Fig. 17

Model of a LAL system with a triangular array formed up by 15 LEDs. The gray region is a repeated mesh in the system. Black dots states the positions of LEDs. A white dot indicates the origin

Fig. 18
Fig. 18

The functional image of f ( d ) = 2 E / x 2 | x = 0 , y = 0 , z = 20 in the analysis of triangular arrangement

Fig. 19
Fig. 19

Flow chart to clarify the method of designing a uniform LAL system with large view angle LEDs

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

C V ( R M S E ) = R M S E / x ¯ o r σ / x ¯
I ( θ ) = I 0 cos m θ
E ( r , θ ) = I ( θ ) cos θ / r 2
E ( r , θ ) = i = 1 4 E i ( r i , θ i )
E ( x , y ) = I 0 z m + 1 { [ ( x d 2 ) 2 + ( y d 2 ) 2 + z 2 ] m + 3 2 + [ ( x d 2 ) 2 + ( y + d 2 ) 2 + z 2 ] m + 3 2 + [ ( x + d 2 ) 2 + ( y d 2 ) 2 + z 2 ] m + 3 2 + [ ( x + d 2 ) 2 + ( y + d 2 ) 2 + z 2 ] m + 3 2 }
d = 4 m + 3 z
I ( θ ) = i c 1 i cos ( | θ | c 2 i ) c 3 i
I ( θ ) = i g 1 i exp [ ( ln 2 ) ( | θ | g 2 i g 3 i ) 2 ]
I ( θ ) = P 1 exp [ ( ln 2 ) ( | θ | P 2 P 3 ) 2 ] + P 4 exp [ ( ln 2 ) ( | θ | P 5 P 6 ) 2 ] + P 7 exp [ ( ln 2 ) ( | θ | P 8 P 9 ) 2 ]
E ( r , θ ) = i = 1 16 E i ( r i , θ i ) = i = 1 16 I ( θ i ) cos θ i r i 2
E ( x , y ) = i = 1 16 E [ ( x x i ) 2 + ( y y i ) 2 + z 2 , arc tan ( ( x x i ) 2 + ( y y i ) 2 z ) ]
I ( θ ) = P 1 exp [ ( ln 2 ) ( θ P 3 ) 2 ] cos h ( P 2 180 θ π ) + P 4 exp [ ( ln 2 ) ( | θ | P 5 P 6 ) 2 ]
E ( x , y ) = i = 1 15 E [ ( x x i ) 2 + ( y y i ) 2 + z 2 , arc tan ( ( x x i ) 2 + ( y y i ) 2 z ) ]

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