Abstract

LED luminaires may deliver precise illumination patterns to control light pollution, comfort, visibility, and light utilization efficiency. Here, we provide simple equations to determine how the light distributes in the streets. In particular, we model the illuminance spatial distribution as a function of Cartesian coordinates on a floor, road, or street. The equations show explicit dependence on the luminary position (pole height and arm length), luminary angle (fixture tilt), and the angular intensity profile (radiation pattern) of the LED luminary. To achieve this, we propose two mathematical representations to model the sophisticated intensity profiles of LED luminaries. Furthermore, we model the light utilization efficiency, illumination uniformity, and veiling luminance of glare due to one or several LED streetlamps.

© 2014 Optical Society of America

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References

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  1. K. J. Gaston, “Sustainability: a green light for efficiency,” Nature 497, 560–561 (2013).
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  2. I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16, 1808–1819 (2008).
    [CrossRef]
  3. X. H. Lee, I. Moreno, and C. C. Sun, “High-performance LED street lighting using microlens arrays,” Opt. Express 21, 10612–10621 (2013).
    [CrossRef]
  4. Z. Feng, Y. Luo, and Y. Han, “Design of LED freeform optical system for road lighting with high luminance/illuminance ratio,” Opt. Express 18, 22020–22031 (2010).
    [CrossRef]
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  6. Y. C. Lo, K. T. Huang, X. H. Lee, and C. C. Sun, “Optical design of a butterfly lens for a street light based on a double-cluster LED,” Microelectron. Reliab. 52, 889–893 (2012).
  7. R. Wu, K. Li, P. Liu, Z. Zheng, H. Li, and X. Liu, “Conceptual design of dedicated road lighting for city park and housing estate,” Appl. Opt. 52, 5272–5278 (2013).
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  12. C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31, 2193–2195 (2006).
    [CrossRef]
  13. P. Raynham, “An examination of the fundamentals of road lighting for pedestrians and drivers,” Lighting Res. Technol. 36, 307–316 (2004).
    [CrossRef]
  14. W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).
  15. M. S. Rea, ed., “Roadway lighting,” in IESNA Lighting Handbook: Reference and Application, 9th ed. (IESNA, 2000), Chap. 22.
  16. J. A. Brons, J. D. Bullough, and M. S. Rea, “Outdoor site-lighting performance: a comprehensive and quantitative framework for assessing light pollution,” Lighting Res. Technol. 40, 201–224 (2008).
    [CrossRef]
  17. I. Moreno, “Illumination uniformity assessment based on human vision,” Opt. Lett. 35, 4030–4032 (2010).
    [CrossRef]
  18. J. J. Vos, “On the cause of disability glare and its dependence on glare angle, age and ocular pigmentation,” Clin. Exp. Optom. 86, 363–370 (2003).
    [CrossRef]
  19. A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).
  20. For example, ASAP ( http://www.breault.com/index.php ), and DIALux ( www.dial.de/DIAL/en/dialux.html ) software.

2013 (5)

2012 (2)

Y. C. Lo, K. T. Huang, X. H. Lee, and C. C. Sun, “Optical design of a butterfly lens for a street light based on a double-cluster LED,” Microelectron. Reliab. 52, 889–893 (2012).

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

2011 (1)

A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).

2010 (2)

2009 (1)

2008 (2)

I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16, 1808–1819 (2008).
[CrossRef]

J. A. Brons, J. D. Bullough, and M. S. Rea, “Outdoor site-lighting performance: a comprehensive and quantitative framework for assessing light pollution,” Lighting Res. Technol. 40, 201–224 (2008).
[CrossRef]

2006 (1)

2004 (1)

P. Raynham, “An examination of the fundamentals of road lighting for pedestrians and drivers,” Lighting Res. Technol. 36, 307–316 (2004).
[CrossRef]

2003 (1)

J. J. Vos, “On the cause of disability glare and its dependence on glare angle, age and ocular pigmentation,” Clin. Exp. Optom. 86, 363–370 (2003).
[CrossRef]

Brons, J. A.

J. A. Brons, J. D. Bullough, and M. S. Rea, “Outdoor site-lighting performance: a comprehensive and quantitative framework for assessing light pollution,” Lighting Res. Technol. 40, 201–224 (2008).
[CrossRef]

Bullough, J. D.

J. A. Brons, J. D. Bullough, and M. S. Rea, “Outdoor site-lighting performance: a comprehensive and quantitative framework for assessing light pollution,” Lighting Res. Technol. 40, 201–224 (2008).
[CrossRef]

Cai, W.

Chen, H.-C.

Chen, W.

Chiu, H.-Y.

de la Rosa-Miranda, E.

I. Moreno, T. Saucedo-A, J. S. Pérez-Huerta, and E. de la Rosa-Miranda, “Designing LED street lighting,” Appl. Opt. (to be published).

Du, X.

Feng, Z.

Gardner, M.

M. Gardner, “Piet Hein’s Superellipse,” in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles (Scientific American, 1977), pp. 240–254.

Gaston, K. J.

K. J. Gaston, “Sustainability: a green light for efficiency,” Nature 497, 560–561 (2013).
[CrossRef]

Halonen, L.

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).

Han, Y.

Huang, K. T.

Y. C. Lo, K. T. Huang, X. H. Lee, and C. C. Sun, “Optical design of a butterfly lens for a street light based on a double-cluster LED,” Microelectron. Reliab. 52, 889–893 (2012).

Huang, S. M.

Ivanov, R.

Kufeoglu, S.

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

Lee, T. X.

Lee, X. H.

X. H. Lee, I. Moreno, and C. C. Sun, “High-performance LED street lighting using microlens arrays,” Opt. Express 21, 10612–10621 (2013).
[CrossRef]

Y. C. Lo, K. T. Huang, X. H. Lee, and C. C. Sun, “Optical design of a butterfly lens for a street light based on a double-cluster LED,” Microelectron. Reliab. 52, 889–893 (2012).

Lee, Y. L.

Lei, X.

Li, H.

Li, K.

Lin, J.-Y.

Liu, P.

Liu, X.

Lo, Y. C.

Y. C. Lo, K. T. Huang, X. H. Lee, and C. C. Sun, “Optical design of a butterfly lens for a street light based on a double-cluster LED,” Microelectron. Reliab. 52, 889–893 (2012).

Luo, W.

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

Luo, Y.

Ma, S. H.

Moreno, I.

Pellinen, T.

A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).

Pérez-Huerta, J. S.

I. Moreno, T. Saucedo-A, J. S. Pérez-Huerta, and E. de la Rosa-Miranda, “Designing LED street lighting,” Appl. Opt. (to be published).

Puolakka, M.

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).

Raynham, P.

P. Raynham, “An examination of the fundamentals of road lighting for pedestrians and drivers,” Lighting Res. Technol. 36, 307–316 (2004).
[CrossRef]

Rea, M. S.

J. A. Brons, J. D. Bullough, and M. S. Rea, “Outdoor site-lighting performance: a comprehensive and quantitative framework for assessing light pollution,” Lighting Res. Technol. 40, 201–224 (2008).
[CrossRef]

Saucedo-A, T.

I. Moreno, T. Saucedo-A, J. S. Pérez-Huerta, and E. de la Rosa-Miranda, “Designing LED street lighting,” Appl. Opt. (to be published).

Sun, C. C.

Sun, C.-C.

Valtonen, J.

A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).

Viikari, M.

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

Vos, J. J.

J. J. Vos, “On the cause of disability glare and its dependence on glare angle, age and ocular pigmentation,” Clin. Exp. Optom. 86, 363–370 (2003).
[CrossRef]

Wu, R.

Ylinen, A.

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

Ylinen, A. M.

A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).

Zheng, Z.

Appl. Opt. (3)

Clin. Exp. Optom. (1)

J. J. Vos, “On the cause of disability glare and its dependence on glare angle, age and ocular pigmentation,” Clin. Exp. Optom. 86, 363–370 (2003).
[CrossRef]

Light Eng. (1)

W. Luo, M. Puolakka, M. Viikari, S. Kufeoglu, A. Ylinen, and L. Halonen, “Lighting criteria for road lighting: a review,” Light Eng. 20, 64–74 (2012).

Lighting Res. Technol. (2)

J. A. Brons, J. D. Bullough, and M. S. Rea, “Outdoor site-lighting performance: a comprehensive and quantitative framework for assessing light pollution,” Lighting Res. Technol. 40, 201–224 (2008).
[CrossRef]

P. Raynham, “An examination of the fundamentals of road lighting for pedestrians and drivers,” Lighting Res. Technol. 36, 307–316 (2004).
[CrossRef]

Microelectron. Reliab. (1)

Y. C. Lo, K. T. Huang, X. H. Lee, and C. C. Sun, “Optical design of a butterfly lens for a street light based on a double-cluster LED,” Microelectron. Reliab. 52, 889–893 (2012).

Nature (1)

K. J. Gaston, “Sustainability: a green light for efficiency,” Nature 497, 560–561 (2013).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Road Mater. Pavement (1)

A. M. Ylinen, T. Pellinen, J. Valtonen, M. Puolakka, and L. Halonen, “Investigation of pavement light reflection characteristics,” Road Mater. Pavement 12, 587–614 (2011).

Other (4)

For example, ASAP ( http://www.breault.com/index.php ), and DIALux ( www.dial.de/DIAL/en/dialux.html ) software.

M. S. Rea, ed., “Roadway lighting,” in IESNA Lighting Handbook: Reference and Application, 9th ed. (IESNA, 2000), Chap. 22.

I. Moreno, T. Saucedo-A, J. S. Pérez-Huerta, and E. de la Rosa-Miranda, “Designing LED street lighting,” Appl. Opt. (to be published).

M. Gardner, “Piet Hein’s Superellipse,” in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles (Scientific American, 1977), pp. 240–254.

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Figures (13)

Fig. 1.
Fig. 1.

How LED luminaries distribute light is usually answered by Monte Carlo ray-tracing and radiosity-based software. However, a mathematical model may be able to handle rapid calculations and become an additional and complementary design tool.

Fig. 2.
Fig. 2.

Graphic representations of the angular radiation pattern of LED luminaries: (a) in polar coordinates and (b) in Cartesian coordinates. Note that the same radiation pattern is shown in sub-figures (a) and (b).

Fig. 3.
Fig. 3.

Modeled radiation patterns of two commercial LED luminaries. (a) Pattern from CREE, and modeled with Eq. (1). (b) Pattern from OSRAM, and modeled with Eq. (2). Plot also shows the curve reported in manufacturers’ datasheets, for comparison.

Fig. 4.
Fig. 4.

Modeled radiation pattern of the LED streetlight of BBE LEDs. Intensity curve through two perpendicular azimuthal directions is modeled with Eq. (2). (a) Pattern along the horizontal plane and (b) is the pattern across the vertical plane. Plot also shows the curve reported in manufacturers’ datasheets, for comparison.

Fig. 5.
Fig. 5.

Azimuthal variation modeled by a super-ellipse in polar coordinates.

Fig. 6.
Fig. 6.

Three-dimensional radiation pattern in spherical coordinates (θ,ϕ), modeled with Eqs. (5) and (7). (a) shows the sharp profile of the BBE LED lamp, i.e., the 3D pattern of Fig. 4. Inset shows a spherical coordinate system for reference. Luminary is located at 0 in the z direction. (b) shows the smooth profile of an OSRAM LED lamp. The inset shows the 2D pattern in the XZ and YZ planes.

Fig. 7.
Fig. 7.

Schematic for calculating illuminance on the road. (a) Schematic diagram of street for calculating the illuminance at each point P on the road due to a street lamp with radiation pattern I(θ,ϕ). (b) Two-dimensional scheme showing: arm length d, height of luminary h, and the luminary angle σ.

Fig. 8.
Fig. 8.

Illuminance distribution on the street, modeled with Eq. (15) for different lamp tilts. Sub-figures (a)–(d) show light patterns for different luminary inclinations: σ=0°, 10°, 15°, and 25°, respectively. In each sub-figure, the left side shows a gray-level illuminance picture of area 10m×25m, and a color plot is shown at the right, where the area enclosed by the red dotted rectangle is 10m×25m. Parameters are: h=6m, d=1.5m, and intensity pattern of Fig. 6(a).

Fig. 9.
Fig. 9.

Illuminance distribution on the street, modeled with Eq. (15). Sub-figures (a)–(d) show light distributions for tilts σ=0°, 10°, 15°, and 25°. In each sub-figure, the left side shows a gray-level illuminance picture of area 10×30m, and a color plot is shown at the right, where the red dotted area is 10m×30m. Parameters are: h=6m, d=1.5m, and the intensity distribution of Fig. 6(b).

Fig. 10.
Fig. 10.

Comparison of our model with ray-tracing simulation. (a), (b) Illuminance distribution on the street, of a luminary tilted by 0° and 15°, respectively. Ray-tracing is performed using ASAP software. The similarity between the ray-trace and the model calculation is high: NCC=98.5% in (a) and 98.1% in (b). (c) Cross sections of radiation pattern analyzed [3], modeled using Eq. (7).

Fig. 11.
Fig. 11.

Illumination efficiency calculated with Eq. (18). (a) Illumination efficiency as a function of luminary tilt σ, for different luminary positions d. This calculation is for the luminary [3], modeled in Fig. 10. (b) Both the modeled illuminance distribution and the integrating area A=14m×30m. Calculation is carried for h=10m.

Fig. 12.
Fig. 12.

Calculation of the illumination uniformity Uo, with luminaries in zig–zag configuration. (a) Illumination uniformity as a function of both luminary tilt σ and pole distance s. (b) Illuminance distribution for the three points marked in (a). The red-dotted rectangle encloses the area for the Uo calculation, as tens of m2. The modeling is carried for h=6m, d=0m, and the intensity pattern of Fig. 6(a).

Fig. 13.
Fig. 13.

Veiling luminance (glare) calculation. (a) Schematic of the geometry for calculating the veiling luminance at the observer position, standing on a point P(x,y) on the road due to one luminary. (b) Veiling luminance, calculated by Eq. (23), at each observer’s position (x,y), due to three luminaries in the field-of-view.

Equations (26)

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I(θ)=i=13gi1exp(gi2(|θ|gi3)2),
I(θ)=UG(θ)+(1U)G(θp)exp[g5(|θ|θp)2],
G(θ)=g1g2exp[g3(|θ|g4)2],
U={1if|θ|<θp0if|θ|θp}.
I(θ,ϕ)=i=13gi1(ϕ)exp{gi2(ϕ)[|θ|gi3(ϕ)]2},
gik(ϕ)=gxikgyik(gxiksinϕ)mik+(gyikcosϕ)mikmik,
I(θ,ϕ)=UG(θ)+(1U)G[θp(ϕ)]exp{g5(ϕ)[|θ|θp(ϕ)]2},
g5(ϕ)=gx5gy5(gx5sinϕ)2+(gy5cosϕ)22,
U={1if|θ|<θp(ϕ)0if|θ|θp(ϕ)},
θp(ϕ)=arctan[tanθpxtanθpy(tanθpxsinϕ)m+(tanθpycosϕ)mm],
E=I(θ,ϕ)cosθsr2,
E(x)=I(θ)h(x2+h2)3/2,
E(x)=I(θσ)h[(xd)2+h2]3/2,
E(x,y)=I(θ,ϕ)h(x2+y2+h2)3/2,
E(x,y)=I(θ,ϕ)h[(xd)2+y2+h2]3/2,
θ=arcos{hcosσ+(xd)sinσ[(xd)2+y2+h2]0.5},
ϕ=arcsin{y{[(xd)cosσhsinσ]2+y2}0.5}.
η=ΦstreetΦluminary=EdAIdΩ=E(x,y)dxdyI(θ,ϕ)sinθdθdϕ,
LP=Φout-streetΦluminary=ΦluminaryΦstreetΦluminary=1η.
U0=EminEave,
Lv=10Evθv2+1.5θv,
Lv=10Icosθvr2(θv2+1.5θv).
Lv(x,y)=10I(θ,ϕ)|y|(θv2+1.5θv)1[(xd)2+y2+(hho)2]3/2.
θv=arcos{|y|[(xd)2+y2+(hho)2]1/2},
g11=0.203,g12=0.0780,g13=55.99g21=0.579,g22=0.0067,g23=58.06g31=0.318,g32=0.0009,g33=38.17
gx11=0.035,gy11=0.4610,g21=2.984,gx31=43,gy31=61.81gx12=0.035,gy12=0.4738,g22=11.15,gx32=43,gy32=60.80g13=0.1693,g23=38.78,g33=19.57m11=m12=8,m31=m32=2.

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