## An analytical model for the illuminance distribution of a power LED

Optics Express, Vol. 16, Issue 26, pp. 21641-21646 (2008)

http://dx.doi.org/10.1364/OE.16.021641

Acrobat PDF (405 KB)

### Abstract

Light-emitting diodes (LEDs) will play a major role in future indoor illumination systems. In general, the generalized Lambertian pattern is widely used as the radiation pattern of a single LED. In this letter, we show that the illuminance distribution due to this Lambertian pattern, when projected onto a horizontal surface such as a floor, can be well approximated by a Gaussian function.

© 2008 Optical Society of America

## 1. Introduction

1. Lumileds, “LUXEON Power LEDs”, http://www.lumileds.com/products/luxeon/.

^{2}) is normally needed. An appealing feature of such a system with narrow beam LEDs is that it can provide localized, colorful, and dynamic lighting effects, especially because the intensity level of each LED can be easily changed.

2. J. M. Kahn and J. R. Barry, “Wireless Infrared Communications” Proc. IEEE , **85**, 265–298 (1997). [CrossRef]

3. I. Moreno, C.-Y. Tsai, D. Bermũdez, and C.-C. Sun, “Simple function for intensity distribution from LEDs”, Proc. SPIE , **6670**, 66700H-66700H-7 (2007). [CrossRef]

4. I. Moreno and U. Contreras, “Color distribution from multicolor LED arrays”, Opt. Express **15**, 3607–3618 (2007). [CrossRef] [PubMed]

5. L. Svilainis and V. Dumbrava, “LED Far Field Pattern Approximation Performance Study”, in *Prof. Int. Conf. on Information Technology Interfaces* (2007), pp. 645–649. [CrossRef]

6. Lumileds, “LUXEON LED Radiation Patterns:Light Distribution Patterns,” http://www.lumileds.com/technology/radiationpatterns.cfm.

2. J. M. Kahn and J. R. Barry, “Wireless Infrared Communications” Proc. IEEE , **85**, 265–298 (1997). [CrossRef]

## 2. Illuminance distribution

*r*is the distance between the LED and the illuminated location, the projection of

*r*onto the flat surface has length

*d*, and

*h*denotes the vertical distance between the LED and the flat surface. The polar angle of the location with respect to the LED is denoted by

*θ*, and the angle of light incidence on the location is clearly equal to

*θ*.

*d*or equivalently

*θ*. For convenience in describing the illuminance distribution on a flat surface at a distance

*h*, we write it as a function of

*d*, denoted by

*f*,

_{L}(d)### 2.1. Gaussian approximation

*f*. For instance, in [11], the two-dimensional (2D) Fourier transform is used as

_{L}(d)*F*= ∫∞ -∞∫∞ -∞

_{L}(u,v)*f*exp(-

_{L}(x,y)*j*2

*π*(

*ux*+

*vy*))dxdy, where

*f*is obtained by writing

_{L}(x,y)*f*into the 2D Cartesian coordinate system (

_{L}(d)*x,y*) through the relation

*d*

^{2}=

*x*

^{2}+

*y*

^{2}. The analytical form of

*F*for an integer

_{L}(u,v)*m*can be obtained as

*m*!! denotes the double factorial and

*K*

_{0}(·) is the modified Bessel function of the second kind. We can see that it is cumbersome to evaluate the values of

*F*for a large integer

_{L}(u,v)*m*. Moreover, to our best knowledge, there is in general no analytical expression of

*F*for a non-integer

_{L}(u,v)*m*. Therefore, we are particularly interested in the approximation models that can potentially bring convenience in the analysis of illumination effects by multiple LEDs. More particularly, in this paper, we propose a Gaussian approximation of Eq. (1).

*f*at

_{L}(d)*d*=0 is the largest, and decreases as d increases. Moreover, for an illumination effect, the human visual system tends to focus on the bright region rather than the background. Hence, we start from

*d*=0 and approximate the rate of decrease in

*f*.

_{L}(d)*d*is small compared to

*h*, i.e.,

*d*

^{2}≪

*h*

^{2},

*d*

^{2}+

*h*

^{2}≈

*h*

^{2}. Hence

*f*as a Gaussian function. The approximation error in

_{L}(d)*f*

^{′}

_{L}(

*d*) can be obtained as

*d*gets larger, since

*f*decreases quickly, especially when

_{L}(d)*m*is large, with the increase of

*d*(see Fig. 2).

*f*, denoted by

_{L}(d)*f*

_{g}(

*d*). Let

*σ*

^{2}is the variance and

*c*is a normalization factor. Thus, the derivative of

*f*with respect to

_{g}(d)*d*is

*f*(0)=

_{L}*f*(0), we get

_{g}*f*and

_{L}(d)*f*is illustrated in Fig. 2 for the case

_{g}(d)*h*=3 meter and for different

*m*. The illuminance at every

*d*is normalized by the value at d=0, i.e. the curves shown in Fig. 2 are actually

12. P. R. Boyce, *Human Factors in Lighting, Second Edition* (Taylar & Francis Inc, 2003). [CrossRef]

*m*is large, i.e., when the light from the LED is quite focused. The difference between

*f*and

_{L}(d)*f*is slightly larger for a smaller

_{g}(d)*m*, e.g., there is a 1 dB difference for

*m*=50 at

*d*=1.2m.

*f*and

_{L}(d)*f*can be explained as follows. Comparing Eq. (1) and Eq. (7), we observe that the approximation we make is actually

_{g}(d)*d*of interest is 0≤

*d*<

*h*. In this range,

*f*>

_{L}(d)*f*, as shown in Fig. 2. The difference between

_{g}(d)*f*and

_{L}(d)*f*, as can be seen from Eq. (8) as well as Fig. 2, increases with

_{g}(d)*d*, resulting a larger mismatch in the tail of the illuminance distribution.

*f̂*, with a slightly larger variance

_{g}(d)*f̂*provides a better fit of

_{g}(d)*f*, and yet has the benefit of a simpler expression than

_{L}(d)*f*. Equivalently from the Taylor expansion, see Eq. (8), the approximation error is now compensated by

_{g}(d)*f*, which might exist for a given range of

_{g}(d)*d*and certain criterion of optimality, is however beyond the scope of this paper.

*f*and

_{L}(d), f_{g}(d)*f̂*, are circularly symmetric. Therefore, we can easily obtain the equivalent expressions for these functions as

_{g}(d)*f*and

_{L}(x,y), f_{g}(x,y)*f̂*in the 2D Cartesian coordinate system. Henceforth, the 2D Fourier transform can be applied to these functions, resulting in

_{g}(x,y)*F*and

_{L}(u,v), F_{g}(u,v)*F̂*, respectively. For the Gaussian approximations,

_{g}(u,v)*F*and

_{g}(u,v)*F̂*, we can get the analytical expressions as

_{g}(u,v)*m*>0, no matter

*m*is an integer or a non-integer. In order to evaluate the performances of the Gaussian approximations in terms of Fourier transform, we present some numerical results in Fig. 3. Here, we again look at the numerical data on a logarithmic scale by evaluating

*m*such that we can numerically compute

*FL(u,v)*using Eq. (2). Furthermore, due to the symmetric property between

*u*and

*v*, and for the sake of convenience, we only show the values of the Fourier transform as a function of

*u*at

*v*=0. It can be seen that both

*F*and

_{g}(u,v)*F̂*give good approximations of

_{g}(u,v)*F*. The accuracy of the approximations is higher for a larger

_{L}(u,v)*m*, i.e. when a light beam is narrow. Furthermore,

*F̂*is closer to

_{g}(u,v)*F*when

_{L}(u,v)*F*is large, e.g.

_{L}(u,v)### 2.2. Impact of diffuse light

13. V. Jungnickel, V. Pohl, S. Nönnig, and C. V. Helmolt, “A Physical Model of theWireless Infrared Communication Channel,” IEEE J. Select. Areas Commun. **20**, 631–640 (2002). [CrossRef]

## 3. Concluding remarks

## References and links

1. | Lumileds, “LUXEON Power LEDs”, http://www.lumileds.com/products/luxeon/. |

2. | J. M. Kahn and J. R. Barry, “Wireless Infrared Communications” Proc. IEEE , |

3. | I. Moreno, C.-Y. Tsai, D. Bermũdez, and C.-C. Sun, “Simple function for intensity distribution from LEDs”, Proc. SPIE , |

4. | I. Moreno and U. Contreras, “Color distribution from multicolor LED arrays”, Opt. Express |

5. | L. Svilainis and V. Dumbrava, “LED Far Field Pattern Approximation Performance Study”, in |

6. | Lumileds, “LUXEON LED Radiation Patterns:Light Distribution Patterns,” http://www.lumileds.com/technology/radiationpatterns.cfm. |

7. | R. Otte, L. P. de Jong, and A. H. M. van Roermund, |

8. | Lumileds, “LUXEON for Flashlight Applications,” http://www.lumileds.com/pdfs/DR02.PDF. |

9. | Faren Srl, “FHS Lens Series,” http://www.fraen.com/pdf/FHS Lens Series Datasheet.pdf. |

10. | Marubeni, “Fully Sealable APOLLO Lens for LUXEON,” http://www.led-spot.com/data/APOLLO.pdf. |

11. | 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,” to appear in IEEE Trans. Signal Processing, 2009. |

12. | P. R. Boyce, |

13. | V. Jungnickel, V. Pohl, S. Nönnig, and C. V. Helmolt, “A Physical Model of theWireless Infrared Communication Channel,” IEEE J. Select. Areas Commun. |

**OCIS Codes**

(150.2950) Machine vision : Illumination

(230.3670) Optical devices : Light-emitting diodes

**ToC Category:**

Optical Devices

**History**

Original Manuscript: July 30, 2008

Revised Manuscript: November 28, 2008

Manuscript Accepted: December 4, 2008

Published: December 16, 2008

**Citation**

Hongming Yang, Jan W. M. Bergmans, Tim C. W. Schenk, Jean-Paul M. G. Linnartz, and Ronald Rietman, "An analytical model for the illuminance distribution of a power LED," Opt. Express **16**, 21641-21646 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-26-21641

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### References

- Lumileds, "LUXEON Power LEDs," http://www.lumileds.com/products/luxeon/.
- J. M. Kahn and J. R. Barry, "Wireless Infrared Communications," Proc. IEEE 85, 265-298 (1997). [CrossRef]
- I. Moreno, C.-Y. Tsai, D. Berm˜udez and C.-C. Sun, "Simple function for intensity distribution from LEDs," Proc. SPIE 6670, 66700H-66700H-7 (2007). [CrossRef]
- I. Moreno and U. Contreras, "Color distribution from multicolor LED arrays," Opt. Express 15, 3607-3618 (2007). [CrossRef] [PubMed]
- L. Svilainis and V. Dumbrava, "LED Far Field Pattern Approximation Performance Study," in Prof. Int. Conf. on Information Technology Interfaces (2007), pp. 645-649. [CrossRef]
- Lumileds, "LUXEON LED Radiation Patterns:Light Distribution Patterns," http://www.lumileds.com/technology/radiationpatterns.cfm
- R. Otte, L. P. de Jong, and A. H. M. van Roermund, Low-Power Wireless Infrared Communications (Kluwer Academic Publishers, 1999), Chap. 3.
- Lumileds, "LUXEON for Flashlight Applications," http://www.lumileds.com/pdfs/DR02.PDF.
- Faren Srl, "FHS Lens Series," http://www.fraen.com/pdf/FHS Lens Series Datasheet.pdf.
- Marubeni, "Fully Sealable APOLLO Lens for LUXEON," http://www.led-spot.com/data/APOLLO.pdf.
- 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," to appear in IEEE Trans. Signal Processing, 2009.
- P. R. Boyce, Human Factors in Lighting, Second Edition (Taylar & Francis Inc, 2003). [CrossRef]
- V. Jungnickel, V. Pohl, S. Nonnig and C. V. Helmolt, "A Physical Model of theWireless Infrared Communication Channel," IEEE J. Select. Areas Commun. 20, 631-640 (2002). [CrossRef]

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