Abstract

Light-emitting diodes (LEDs) come in many varieties and with a wide range of radiation patterns. We propose a general, simple but accurate analytic representation for the radiation pattern of the light emitted from an LED. To accurately render both the angular intensity distribution and the irradiance spatial pattern, a simple phenomenological model takes into account the emitting surfaces (chip, chip array, or phosphor surface), and the light redirected by both the reflecting cup and the encapsulating lens. Mathematically, the pattern is described as the sum of a maximum of two or three Gaussian or cosine-power functions. The resulting equation is widely applicable for any kind of LED of practical interest. We accurately model a wide variety of radiation patterns from several world-class manufacturers.

© 2008 Optical Society of America

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  5. H. Zerfhau-Dreihöfer, U. Haack, T. Weber, and D. Wendt, "Light source modeling for automotive lighting devices," Proc. SPIE 4775, 58 (2002).
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  13. P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, "Method for analysing luminous intensity of light-emitting diodes," Meas. Sci. Technol. 18, 223-229 (2007).
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    [CrossRef]
  18. C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, "Enhancement of light extraction of GaN-based light-emitting diodes with a microstructure array," Opt. Eng. 43, 1700-1701 (2004).
    [CrossRef]
  19. A. Estrada-Hernandez, L. P. Gonzalez-Galvan, H. Zarate-Hernandez, R. Cardoso, and E. Rosas, "Luminous flux and correlated color temperature determination for LED sources," SPIE 6422, 64220O (2007).
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  20. R. Young, "Measuring light emission from LEDs," SPIE 6355, 63550H (2006).
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  25. S. Preuss, D. Potthoff, T. Preuss, and K. Lischka, "LED encapsulation- a new approach of rear light design," SPIE 6198, 61980I (2006).
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  26. S. C. Allen and A. J. Steckl, "ELiXIR—Solid-State Luminaire With Enhanced Light Extraction by Internal Reflection," J. Display Technol. 3, 155-159 (2007).
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  27. I. Moreno and U. Contreras, "Color distribution of multicolor LED arrays," Opt. Express 15, 3607-3618 (2007).
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  28. I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, "Designing light-emitting diode arrays for uniform near-field irradiance," Appl. Opt. 45, 2265-2272 (2006).
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  29. S. Tryka, "Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane," Opt. Express 15, 3777-3790 (2007)
    [CrossRef] [PubMed]
  30. S. Tryka, "Spherical object in radiation field from a point source," Opt. Express 12, 512-517 (2004).
    [CrossRef] [PubMed]
  31. J. L. Balenzategui and A. Marti, "Design of hemispherical cavities for LED-based illumination devices," Appl. Phys. B 82, 75-80 (2006).
    [CrossRef]
  32. A. L. Dubovikov, S. S. Repin, and S. N. Natarovskii, "Features of the use of LEDs in artificial-vision systems," J. Opt. Technol. 72, 40-42 (2005).
    [CrossRef]
  33. T. Matsumoto, N. Inoue, and M. Suzuki, "Optimum arrangement of LEDs in base station of optical wireless LANs," SPIE 6134, 613403 (2006).
    [CrossRef]
  34. C. G. Lee, C. S. Park, J. H. Kim, and D. H. Kim, "Experimental verification of optical wireless communication link using high-brightness illumination light-emitting diodes," Opt. Eng. 46, 125005 (2007).
    [CrossRef]
  35. J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265-298 (1997).
    [CrossRef]
  36. D. W. K. Wong and G. Chen, "Illumination design of a white-light-emitting diode wireless transmission system," Opt. Eng. 46, 085002 (2007).
    [CrossRef]
  37. M. Akanegawa, Y. Tanaka, and M. Nakagawa, "Basic study on traffic information system using LED traffic lights," IEEE Transactions on Intelligent Transportation Systems 2, 197-203 (2001).
    [CrossRef]
  38. I. Moreno, J. Muñoz, and R. Ivanov, "Uniform illumination of distant targets using a spherical light-emitting diode array," Opt. Eng. 46, 033001 (2007).
    [CrossRef]
  39. I. Moreno, "Color tunable hybrid lamp: LED-incandescent and LED-fluorescent," SPIE 6422, 64220N (2007).
    [CrossRef]
  40. Y. Tu, S. Jin, Y. Wang, and L. Dou, "Color uniformity and data simulation in High-Power RGB LED modules using different LED-chips arrays," SPIE 6828, 682816 (2007).
    [CrossRef]
  41. C. Deller, G. Smith, and J. Franklin, "Colour mixing LEDs with short microsphere doped acrylic rods," Opt. Express 12, 3327-3333 (2004).
    [CrossRef] [PubMed]
  42. C. A. Deller, J. B. Franklin, and G. B. Smith, "Lighting simulations using smoothed LED profiles compared with measured profiles," SPIE 6337, 63370X (2006).
    [CrossRef]
  43. M. A. Volkova, S. V. Zlatina, S. N. Natarovskii, O. N. Nemkova, T. F. Selezneva, N. B. Skobeleva, D. N. Frolov, L. M. Kogan, and B. P. Papchenko, "Prospects of using LEDs in the illuminating systems of microscopes," J. Opt. Technol. 72, 186-190 (2005).
    [CrossRef]
  44. C. Ye, Y. Liu, and F. Yu, "New illumination patterns in microscopes" SPIE 6033, 60330L (2005).
    [CrossRef]
  45. J. Xu, Z. Xiao, and T. Lin, "The design of microscope field illumination system based on LED" SPIE 6841, 68410U (2007).
    [CrossRef]
  46. C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, "Brightness management in a direct LED backlight for LCD TVs," J. Soc. Inf. Disp. In press (2007).
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    [CrossRef]
  48. J. W. Whang and Y. T. Teng, "Uniform illumination system with desired emitting angle," SID Conf. Rec. Int. Display Res. Conf., 100-103 (2006).
  49. Y. Uchida and T. Taguchi, "Lighting theory and luminous characteristics of white light-emitting diodes," Opt. Eng. 44, 124003 (2005).
    [CrossRef]
  50. L. Svilainis and V. Dumbrava, "LED Far Field Pattern Approximation Performance Study," Information Technology Interfaces, 2007. ITI 2007. 29th International Conference on, 645-649 (2007).
  51. M. Bennahmias, E. Arik, K. Yu, D. Voloshenko, K. Chua, R. Pradhan, T. Forrester, and T. Jannson, "Modeling of non-Lambertian sources in lighting applications," SPIE 6669, 66691A (2007).
    [CrossRef]
  52. A. L. Fischer, "LEDs and displays: Analytical method for computing color patterns in LEDs," Photonics Spectra 41, 87-88 (June 2007).
  53. K. Man and I. Ashdown, "Accurate colorimetric feedback for RGB LED clusters," SPIE 6337, 633702 (2006).
    [CrossRef]

2007

P. Benitez and J. C. Miñano, "The future of illumination design," Opt. Photonics News 18, 20-25 (2007).
[CrossRef]

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, "Method for analysing luminous intensity of light-emitting diodes," Meas. Sci. Technol. 18, 223-229 (2007).
[CrossRef]

W. T. Chien, C. C. Sun, and I. Moreno, "Precise optical model of multi-chip white LEDs," Opt. Express 15, 7572-7577 (2007).
[CrossRef] [PubMed]

A. Estrada-Hernandez, L. P. Gonzalez-Galvan, H. Zarate-Hernandez, R. Cardoso, and E. Rosas, "Luminous flux and correlated color temperature determination for LED sources," SPIE 6422, 64220O (2007).
[CrossRef]

S. C. Allen and A. J. Steckl, "ELiXIR—Solid-State Luminaire With Enhanced Light Extraction by Internal Reflection," J. Display Technol. 3, 155-159 (2007).
[CrossRef]

I. Moreno and U. Contreras, "Color distribution of multicolor LED arrays," Opt. Express 15, 3607-3618 (2007).
[CrossRef] [PubMed]

S. Tryka, "Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane," Opt. Express 15, 3777-3790 (2007)
[CrossRef] [PubMed]

C. G. Lee, C. S. Park, J. H. Kim, and D. H. Kim, "Experimental verification of optical wireless communication link using high-brightness illumination light-emitting diodes," Opt. Eng. 46, 125005 (2007).
[CrossRef]

D. W. K. Wong and G. Chen, "Illumination design of a white-light-emitting diode wireless transmission system," Opt. Eng. 46, 085002 (2007).
[CrossRef]

I. Moreno, J. Muñoz, and R. Ivanov, "Uniform illumination of distant targets using a spherical light-emitting diode array," Opt. Eng. 46, 033001 (2007).
[CrossRef]

I. Moreno, "Color tunable hybrid lamp: LED-incandescent and LED-fluorescent," SPIE 6422, 64220N (2007).
[CrossRef]

Y. Tu, S. Jin, Y. Wang, and L. Dou, "Color uniformity and data simulation in High-Power RGB LED modules using different LED-chips arrays," SPIE 6828, 682816 (2007).
[CrossRef]

J. Xu, Z. Xiao, and T. Lin, "The design of microscope field illumination system based on LED" SPIE 6841, 68410U (2007).
[CrossRef]

C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, "Brightness management in a direct LED backlight for LCD TVs," J. Soc. Inf. Disp. In press (2007).

G. Harbers, S. J. Bierhuizen, and M. R. Krames, "Performance of high power light emitting diodes in display illumination applications" J. Display Technol. 3, 98-109 (2007).
[CrossRef]

M. Bennahmias, E. Arik, K. Yu, D. Voloshenko, K. Chua, R. Pradhan, T. Forrester, and T. Jannson, "Modeling of non-Lambertian sources in lighting applications," SPIE 6669, 66691A (2007).
[CrossRef]

A. L. Fischer, "LEDs and displays: Analytical method for computing color patterns in LEDs," Photonics Spectra 41, 87-88 (June 2007).

2006

K. Man and I. Ashdown, "Accurate colorimetric feedback for RGB LED clusters," SPIE 6337, 633702 (2006).
[CrossRef]

C. A. Deller, J. B. Franklin, and G. B. Smith, "Lighting simulations using smoothed LED profiles compared with measured profiles," SPIE 6337, 63370X (2006).
[CrossRef]

T. Matsumoto, N. Inoue, and M. Suzuki, "Optimum arrangement of LEDs in base station of optical wireless LANs," SPIE 6134, 613403 (2006).
[CrossRef]

J. L. Balenzategui and A. Marti, "Design of hemispherical cavities for LED-based illumination devices," Appl. Phys. B 82, 75-80 (2006).
[CrossRef]

I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, "Designing light-emitting diode arrays for uniform near-field irradiance," Appl. Opt. 45, 2265-2272 (2006).
[CrossRef] [PubMed]

I. Moreno, "Spatial distribution of LED radiation," Proc. SPIE 6342, 634216 (2006).
[CrossRef]

S. Preuss, D. Potthoff, T. Preuss, and K. Lischka, "LED encapsulation- a new approach of rear light design," SPIE 6198, 61980I (2006).
[CrossRef]

R. Young, "Measuring light emission from LEDs," SPIE 6355, 63550H (2006).
[CrossRef]

L. Fu, R. Leutz, and H. Ries, "Physical modeling of filament light sources," J. Appl. Phys. 100, 103528 (2006).
[CrossRef]

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

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] [PubMed]

2005

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

A. L. Dubovikov, S. S. Repin, and S. N. Natarovskii, "Features of the use of LEDs in artificial-vision systems," J. Opt. Technol. 72, 40-42 (2005).
[CrossRef]

M. A. Volkova, S. V. Zlatina, S. N. Natarovskii, O. N. Nemkova, T. F. Selezneva, N. B. Skobeleva, D. N. Frolov, L. M. Kogan, and B. P. Papchenko, "Prospects of using LEDs in the illuminating systems of microscopes," J. Opt. Technol. 72, 186-190 (2005).
[CrossRef]

C. Ye, Y. Liu, and F. Yu, "New illumination patterns in microscopes" SPIE 6033, 60330L (2005).
[CrossRef]

Y. Uchida and T. Taguchi, "Lighting theory and luminous characteristics of white light-emitting diodes," Opt. Eng. 44, 124003 (2005).
[CrossRef]

2004

S. Tryka, "Spherical object in radiation field from a point source," Opt. Express 12, 512-517 (2004).
[CrossRef] [PubMed]

C. Deller, G. Smith, and J. Franklin, "Colour mixing LEDs with short microsphere doped acrylic rods," Opt. Express 12, 3327-3333 (2004).
[CrossRef] [PubMed]

Y. Narukawa, "White-light LEDs," Opt. Photon. News 15, 24-29 (2004).

C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, "Enhancement of light extraction of GaN-based light-emitting diodes with a microstructure array," Opt. Eng. 43, 1700-1701 (2004).
[CrossRef]

2002

M. S. Kaminski, K. J. Garcia, M. A. Stevenson, M. Frate, and R. J. Koshel, "Advanced Topics in Source Modeling," Proc. SPIE 4775, 46 (2002).
[CrossRef]

H. Zerfhau-Dreihöfer, U. Haack, T. Weber, and D. Wendt, "Light source modeling for automotive lighting devices," Proc. SPIE 4775, 58 (2002).
[CrossRef]

M. Jongewaard, "Guide to selecting the appropriate type of light source model," Proc. SPIE 4775, 86-98 (2002).
[CrossRef]

2001

M. Akanegawa, Y. Tanaka, and M. Nakagawa, "Basic study on traffic information system using LED traffic lights," IEEE Transactions on Intelligent Transportation Systems 2, 197-203 (2001).
[CrossRef]

1999

1997

J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265-298 (1997).
[CrossRef]

1996

M. W. Siegel and R. D. Stock, "General near-zone light source model and its application to computer-automated reflector design," Opt. Eng. 35, 2661-2679 (1996).
[CrossRef]

R. D. Stock and M. W. Siegel, "Orientation invariant light source parameters," Opt. Eng. 35, 2651 (1996).
[CrossRef]

1993

I. Ashdown, "Near-field photometry: a new approach," J. Illum. Eng. Soc. 22, 163-180 (1993).

Appl. Opt.

Appl. Phys. B

J. L. Balenzategui and A. Marti, "Design of hemispherical cavities for LED-based illumination devices," Appl. Phys. B 82, 75-80 (2006).
[CrossRef]

IEEE Transactions on Intelligent Transportation Systems

M. Akanegawa, Y. Tanaka, and M. Nakagawa, "Basic study on traffic information system using LED traffic lights," IEEE Transactions on Intelligent Transportation Systems 2, 197-203 (2001).
[CrossRef]

J. Appl. Phys.

L. Fu, R. Leutz, and H. Ries, "Physical modeling of filament light sources," J. Appl. Phys. 100, 103528 (2006).
[CrossRef]

J. Display Technol.

J. Illum. Eng. Soc.

I. Ashdown, "Near-field photometry: a new approach," J. Illum. Eng. Soc. 22, 163-180 (1993).

J. Opt. Technol.

J. Soc. Inf. Disp.

C. C. Sun, I. Moreno, S. H. Chung, W. T. Chien, C. T. Hsieh, and T. H. Yang, "Brightness management in a direct LED backlight for LCD TVs," J. Soc. Inf. Disp. In press (2007).

Meas. Sci. Technol.

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, "Method for analysing luminous intensity of light-emitting diodes," Meas. Sci. Technol. 18, 223-229 (2007).
[CrossRef]

Opt. Eng.

M. W. Siegel and R. D. Stock, "General near-zone light source model and its application to computer-automated reflector design," Opt. Eng. 35, 2661-2679 (1996).
[CrossRef]

R. D. Stock and M. W. Siegel, "Orientation invariant light source parameters," Opt. Eng. 35, 2651 (1996).
[CrossRef]

C. G. Lee, C. S. Park, J. H. Kim, and D. H. Kim, "Experimental verification of optical wireless communication link using high-brightness illumination light-emitting diodes," Opt. Eng. 46, 125005 (2007).
[CrossRef]

C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, "Enhancement of light extraction of GaN-based light-emitting diodes with a microstructure array," Opt. Eng. 43, 1700-1701 (2004).
[CrossRef]

Y. Uchida and T. Taguchi, "Lighting theory and luminous characteristics of white light-emitting diodes," Opt. Eng. 44, 124003 (2005).
[CrossRef]

I. Moreno, J. Muñoz, and R. Ivanov, "Uniform illumination of distant targets using a spherical light-emitting diode array," Opt. Eng. 46, 033001 (2007).
[CrossRef]

D. W. K. Wong and G. Chen, "Illumination design of a white-light-emitting diode wireless transmission system," Opt. Eng. 46, 085002 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photon. News

Y. Narukawa, "White-light LEDs," Opt. Photon. News 15, 24-29 (2004).

Optics & Photonics News

P. Benitez and J. C. Miñano, "The future of illumination design," Opt. Photonics News 18, 20-25 (2007).
[CrossRef]

Photonics Spectra

A. L. Fischer, "LEDs and displays: Analytical method for computing color patterns in LEDs," Photonics Spectra 41, 87-88 (June 2007).

Proc. IEEE

J. M. Kahn and J. R. Barry, "Wireless infrared communications," Proc. IEEE 85, 265-298 (1997).
[CrossRef]

Proc. SPIE

M. S. Kaminski, K. J. Garcia, M. A. Stevenson, M. Frate, and R. J. Koshel, "Advanced Topics in Source Modeling," Proc. SPIE 4775, 46 (2002).
[CrossRef]

H. Zerfhau-Dreihöfer, U. Haack, T. Weber, and D. Wendt, "Light source modeling for automotive lighting devices," Proc. SPIE 4775, 58 (2002).
[CrossRef]

M. Jongewaard, "Guide to selecting the appropriate type of light source model," Proc. SPIE 4775, 86-98 (2002).
[CrossRef]

I. Moreno, "Spatial distribution of LED radiation," Proc. SPIE 6342, 634216 (2006).
[CrossRef]

Rep. Prog. Phys.

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

Science

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

SPIE

A. Estrada-Hernandez, L. P. Gonzalez-Galvan, H. Zarate-Hernandez, R. Cardoso, and E. Rosas, "Luminous flux and correlated color temperature determination for LED sources," SPIE 6422, 64220O (2007).
[CrossRef]

R. Young, "Measuring light emission from LEDs," SPIE 6355, 63550H (2006).
[CrossRef]

T. Matsumoto, N. Inoue, and M. Suzuki, "Optimum arrangement of LEDs in base station of optical wireless LANs," SPIE 6134, 613403 (2006).
[CrossRef]

K. Man and I. Ashdown, "Accurate colorimetric feedback for RGB LED clusters," SPIE 6337, 633702 (2006).
[CrossRef]

C. A. Deller, J. B. Franklin, and G. B. Smith, "Lighting simulations using smoothed LED profiles compared with measured profiles," SPIE 6337, 63370X (2006).
[CrossRef]

S. Preuss, D. Potthoff, T. Preuss, and K. Lischka, "LED encapsulation- a new approach of rear light design," SPIE 6198, 61980I (2006).
[CrossRef]

I. Moreno, "Color tunable hybrid lamp: LED-incandescent and LED-fluorescent," SPIE 6422, 64220N (2007).
[CrossRef]

Y. Tu, S. Jin, Y. Wang, and L. Dou, "Color uniformity and data simulation in High-Power RGB LED modules using different LED-chips arrays," SPIE 6828, 682816 (2007).
[CrossRef]

M. Bennahmias, E. Arik, K. Yu, D. Voloshenko, K. Chua, R. Pradhan, T. Forrester, and T. Jannson, "Modeling of non-Lambertian sources in lighting applications," SPIE 6669, 66691A (2007).
[CrossRef]

C. Ye, Y. Liu, and F. Yu, "New illumination patterns in microscopes" SPIE 6033, 60330L (2005).
[CrossRef]

J. Xu, Z. Xiao, and T. Lin, "The design of microscope field illumination system based on LED" SPIE 6841, 68410U (2007).
[CrossRef]

Other

J. W. Whang and Y. T. Teng, "Uniform illumination system with desired emitting angle," SID Conf. Rec. Int. Display Res. Conf., 100-103 (2006).

L. Svilainis and V. Dumbrava, "LED Far Field Pattern Approximation Performance Study," Information Technology Interfaces, 2007. ITI 2007. 29th International Conference on, 645-649 (2007).

C. Daniel and F. S. Wood, Fitting equations to data: computer analysis of multifactor data (2nd ed.), (Wiley, New York, 1999).

D. R. Jenkins and H. Monch, "Source Imaging Goniometer Method of Light Source Characterization for Accurate Projection System Design," SID Symposium Digest 31, 862-865 (2000).
[CrossRef]

A. Zukauskas, M. S. Schur, R. Gaska, Introduction to Solid State Lighting (Wiley-Interscience, NY, 2002).

W. J. Cassarly, "LED modelling: pros and cons of common methods," Photon. Tech Briefs IIa-2a (April 2002), special supplement to NASA Tech Briefs.

Application brief AB20-5, "Secondary Optics Design Considerations for SuperFlux LEDs" Philips Lumileds.

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Figures (3)

Fig. 1.
Fig. 1.

Three radiation patterns of some major manufacturers. (a) and (b) NSPW345CS from Nichia. (c) and (d) XLamp® XR-E LED from Cree. (e) and (f) LUXEON® Rebel from Lumileds Philips. (a), (c), and (e) are the modeled three-dimensional radiation patterns. (b), (d), and (f) are a comparison between modeled and manufacturer reported radiation pattern. UB-Upper bound, LB-Lower bound, and XX and YY are two perpendicular azimuthal directions.

Fig. 2.
Fig. 2.

Other radiation patterns of some major manufacturers. (a) and (b) LUXEON® K2 from Lumileds Philips. (c) and (d) EdiPower Emitter 3-RD-01-H0001 LED from Edison. (e) and (f) FIREFLY® Hyper-Bright LB-V19G LED from OSRAM. (a), (c), and (e) are the modeled three-dimensional radiation patterns. (b), (d), and (f) are a comparison between modeled and manufacturer reported radiation pattern. UB-Upper bound, LB-Lower bound, and V and H are the vertical and horizontal directions.

Fig. 3.
Fig. 3.

Other radiation patterns of some major manufacturers. (a) and (b) LUXEON® Side Emitter from Lumileds Philips. (c) and (d) LUXEON® Batwing from Lumileds Philips. (e) and (f) Z-Power Side Emitter LED from Seoul Semiconductor. (a), (c), and (e) are the modeled three-dimensional radiation patterns. (b), (d), and (f) are a comparison between modeled and manufacturer reported radiation pattern. UB-Upper bound, LB-Lower bound.

Equations (14)

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I ( θ ) = i g 1 i exp ( ln 2 ( θ g 2 i g 3 i ) 2 ) ,
I ( θ ) = i c 1 i cos ( θ c 2 i ) c 3 i .
I ( θ , ϕ ) = i g 1 i exp [ ( ln 2 ) ( θ g 2 i ) 2 ( cos 2 ϕ ( g 3 i ) 2 + sin 2 ϕ ( g 4 i ) 2 ) ] ,
I ( θ , ϕ ) = i c 1 i cos [ θ ( c 2 i cos 2 ϕ + c 3 i sin 2 ϕ ) ] c 4 i .
I ( θ , ϕ ) A = [ Φ 0 4 π I ( θ , ϕ ) d Ω ] I ( θ , ϕ ) .
RMS = 1 M i j [ I ( θ i , ϕ j ) m - I ( θ i , ϕ j ) e ] 2 .
NCC = i j I ( θ i , ϕ j ) m I ¯ m I ( θ i , ϕ j ) e I ¯ e i j [ I ( θ i , ϕ j ) m I ¯ m ] 2 i j [ I ( θ i , ϕ j ) e I ¯ e ] 2 ,
I ( θ ) = g 1 1 [ exp ( ln 2 ( θ + g 2 1 g 3 1 ) 2 ) + exp ( ln 2 ( θ g 2 1 g 3 1 ) 2 ) ] + g 1 2 exp ( ln 2 ( θ g 2 2 g 3 2 ) 2 )
= h 1 1 exp [ ln 2 ( θ h 3 1 ) 2 ] cosh ( h 2 1 θ ) + g 1 2 exp ( ln 2 ( θ g 2 2 g 3 2 ) 2 ) ,
E ( r , θ , ϕ ) = I ( θ , ϕ ) cos θ r 2 ,
E ( x , y , z ) = z ( x 2 + y 2 + z 2 ) 3 2 I [ θ ( x , y , z ) , ϕ ( x , y ) ] ,
θ ( x , y , z ) = arctan [ x 2 + y 2 z ] ,
ϕ ( x , y ) = arctan ( y x ) .
E ( x , y , z ) A = 120 z ( x 2 + y 2 + z 2 ) 3 2 i = 1 2 g 1 i exp { ln 2 [ arctan ( z 1 x 2 + y 2 ) g 2 i g 3 i ] 2 } ,

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