Abstract

Virtually all light-emitting diodes (LEDs) are encapsulated with a transparent epoxy or silicone gel. Here we analyze the optical efficiency of spherical encapsulants. We develop a quasi-radiometric equation for the light transmission efficiency, which incorporates some ideas of Monte Carlo ray tracing into the context of radiometry. The approach includes the extended source nature of the LED chip and the chip radiance distribution. The equation is an explicit function of the size and the refractive index of the package, and also of several chip parameters such as shape, size, radiance, and location inside the package. To illustrate the use of this equation, we analyze several packaging configurations of practical interest, for example, a hemispherical dome with multiple chips, a flat encapsulation as a special case of the spherical package, and approximate calculations of an encapsulant with a photonic crystal LED or a photonic quasi-crystal LED. These calculations are compared with Monte Carlo ray tracing, giving almost identical values.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
    [CrossRef]
  2. J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16, 21835-21842 (2008).
    [CrossRef] [PubMed]
  3. T. X. Lee, K. F. Gao, W. T. Chien, and C. C. Sun, “Light extraction analysis of GaN-based light-emitting diodes with surface texture and/or patterned substrate,” Opt. Express 15, 6670-6676 (2007).
    [CrossRef] [PubMed]
  4. H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
    [CrossRef]
  5. R. Zhang and S. Lee, “Wafer level LED packaging with integrated DRIE trenches for encapsulation,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).
  6. Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
    [CrossRef]
  7. E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
    [CrossRef]
  8. M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
    [CrossRef]
  9. N. T. Tran and F. G. Shi, “LED package design for high optical efficiency and low viewing angle,” in Microsystems Packaging, Assembly and Circuits Technology (IEEE, 2007), pp. 10-13.
    [CrossRef]
  10. H. Masui, N. N. Fellows, H. Sato, H. Asamizu, S. Nakamura, and S. P. DenBaars, “Direct evaluation of reflector effects on radiant flux from InGaN-based light-emitting diodes,” Appl. Opt. 46, 5974-5978 (2007).
    [CrossRef] [PubMed]
  11. W. Chi and N. George, “Light-emitting diode illumination design with a condensing sphere,” J. Opt. Soc. Am. A 23, 2295-2298 (2006).
    [CrossRef]
  12. Z. Liu, S. Liu, K. Wang, and X. Luo, “Effects of phosphor's location on LED packaging performance,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).
  13. Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
    [CrossRef]
  14. A. David, H. Benisty, and C. Weisbuch, “Optimization of light-diffracting photonic-crystals for high extraction efficiency LEDs,” J. Disp. Technol. 3, 133-148 (2007).
    [CrossRef]
  15. M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
    [CrossRef]
  16. H. Kikuta, S. Hino, and A. Maruyama, “Estimation method for the light extraction efficiency of light-emitting elements with a rigorous grating diffraction theory,” J. Opt. Soc. Am. A 23, 1207-1213 (2006).
    [CrossRef]
  17. H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
    [CrossRef]
  18. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).
  19. W. Cassarly, “Nonimaging optics: concentration and illumination,” in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. III.
  20. J. Jarominski, “Optical efficiency of LEDs with hemispherical lenses,” Appl. Opt. 21, 2461-2464 (1982).
    [CrossRef] [PubMed]
  21. C. C. Sun, W. T. Chien, I. Moreno, C. C. Hsieh, and Y. C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17, 13918-13927 (2009).
    [CrossRef] [PubMed]
  22. 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]
  23. S. J. Lee, “Study of photon extraction efficiency in InGaN light/emitting diodes depending on chip structures and chip-mount schemes,” Opt. Eng. 45, 014601 (2006).
    [CrossRef]
  24. I. Moreno, “Spatial distribution of LED radiation,” Proc. SPIE 6342, 634216 (2006).
    [CrossRef]
  25. F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368-377 (1963).
    [CrossRef]
  26. E. Hecht, Optics (Addison-Wesley, 1998).
  27. 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]
  28. I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16, 1808-1819 (2008).
    [CrossRef] [PubMed]
  29. M. L. Wu, Y. C. Lee, S. P. Yang, P. S. Lee, and J. Y. Chang, “Azimuthally isotropic irradiance of GaN-based light-emitting diodes with GaN microlens arrays,” Opt. Express 17, 6148-6155 (2009).
    [CrossRef] [PubMed]
  30. Y. C. Lee, “Research on optical properties of micro/nano structures in light-emitting diodes and photovoltaics,” Ph.D. dissertation (National Central University, Taiwan, 2009).
  31. I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
    [CrossRef]
  32. H. Masui, S. Nakamura, and S. P. DenBaars, “Analytical light-ray tracing in two-dimensional objects for light-extraction problems in light-emitting diodes,” Appl. Opt. 47, 88-92 (2008).
    [CrossRef]
  33. J. Y. Kim, M. K. Kwon, J. P. Kim, and S. J. Park, “Enhanced light extraction from triangular GaN-based light-emitting diodes,” IEEE Photon. Technol. Lett. 19, 1865-1867 (2007).
    [CrossRef]
  34. G. Ouyang and K. Li, “Brightness increase in LED by recycling of light for projection applications,” Proc. SPIE 6489, 648903 (2007).
    [CrossRef]
  35. R. Leutz, L. Fu, and H. Ries, “Carambola optics for recycling of light,” Appl. Opt. 45, 2572-2575 (2006).
    [CrossRef] [PubMed]
  36. S. H. Jang and M. W. Shin, “Thermal analysis of LED arrays for automotive headlamp with a novel cooling system,” IEEE Trans. Device Mater. Reliab. 8, 561-564 (2008).
    [CrossRef]
  37. Z. Ma, X. Wang, D. Zhu, and S. Liu, “Thermal analysis and modeling of LED arrays integrated with an innovative liquid-cooling module,” in 6th International Conference on Electronic Packaging Technology (IEEE, 2005).
  38. W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
    [CrossRef]
  39. D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
    [CrossRef]
  40. J. J. Wierer Jr., A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photon. 3, 163 (2009).
    [CrossRef]
  41. F. Rahman, “Photonic crystal LEDs,” Opt. Photon. News 20(6), 24-29 (2009).
    [CrossRef]
  42. M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modeling, and optimization,” Proc. SPIE 6486, 64860R (2007).
    [CrossRef]
  43. PhlatLight LED data sheet, Luminus Devices Inc., www.luminus.com.
  44. Y.-J. Lee, S.-H. Kim, G.-H. Kim, Y.-H. Lee, S.-H. Cho, Y.-W. Song, Y. C. Kim, and Y. R. Do, “Far-field radiation of photonic crystal organic light-emitting diode,” Opt. Express 13, 5864-5870 (2005).
    [CrossRef] [PubMed]
  45. H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
    [CrossRef]
  46. A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
    [CrossRef]
  47. M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
    [CrossRef] [PubMed]
  48. I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48, 1190-1197 (2009).
    [CrossRef]
  49. S. C. Allen and A. J. Steckla, “A nearly ideal phosphor-converted white light-emitting diode,” Appl. Phys. Lett. 92, 143309 (2008).
    [CrossRef]
  50. F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
    [CrossRef]

2009 (6)

M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
[CrossRef]

J. J. Wierer Jr., A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photon. 3, 163 (2009).
[CrossRef]

F. Rahman, “Photonic crystal LEDs,” Opt. Photon. News 20(6), 24-29 (2009).
[CrossRef]

I. Moreno, C. C. Sun, and R. Ivanov, “Far-field condition for light-emitting diode arrays,” Appl. Opt. 48, 1190-1197 (2009).
[CrossRef]

M. L. Wu, Y. C. Lee, S. P. Yang, P. S. Lee, and J. Y. Chang, “Azimuthally isotropic irradiance of GaN-based light-emitting diodes with GaN microlens arrays,” Opt. Express 17, 6148-6155 (2009).
[CrossRef] [PubMed]

C. C. Sun, W. T. Chien, I. Moreno, C. C. Hsieh, and Y. C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17, 13918-13927 (2009).
[CrossRef] [PubMed]

2008 (10)

H. Masui, S. Nakamura, and S. P. DenBaars, “Analytical light-ray tracing in two-dimensional objects for light-extraction problems in light-emitting diodes,” Appl. Opt. 47, 88-92 (2008).
[CrossRef]

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

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16, 21835-21842 (2008).
[CrossRef] [PubMed]

S. H. Jang and M. W. Shin, “Thermal analysis of LED arrays for automotive headlamp with a novel cooling system,” IEEE Trans. Device Mater. Reliab. 8, 561-564 (2008).
[CrossRef]

S. C. Allen and A. J. Steckla, “A nearly ideal phosphor-converted white light-emitting diode,” Appl. Phys. Lett. 92, 143309 (2008).
[CrossRef]

F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
[CrossRef]

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
[CrossRef]

Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
[CrossRef]

2007 (8)

A. David, H. Benisty, and C. Weisbuch, “Optimization of light-diffracting photonic-crystals for high extraction efficiency LEDs,” J. Disp. Technol. 3, 133-148 (2007).
[CrossRef]

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modeling, and optimization,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

J. Y. Kim, M. K. Kwon, J. P. Kim, and S. J. Park, “Enhanced light extraction from triangular GaN-based light-emitting diodes,” IEEE Photon. Technol. Lett. 19, 1865-1867 (2007).
[CrossRef]

G. Ouyang and K. Li, “Brightness increase in LED by recycling of light for projection applications,” Proc. SPIE 6489, 648903 (2007).
[CrossRef]

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

T. X. Lee, K. F. Gao, W. T. Chien, and C. C. Sun, “Light extraction analysis of GaN-based light-emitting diodes with surface texture and/or patterned substrate,” Opt. Express 15, 6670-6676 (2007).
[CrossRef] [PubMed]

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]

H. Masui, N. N. Fellows, H. Sato, H. Asamizu, S. Nakamura, and S. P. DenBaars, “Direct evaluation of reflector effects on radiant flux from InGaN-based light-emitting diodes,” Appl. Opt. 46, 5974-5978 (2007).
[CrossRef] [PubMed]

2006 (7)

R. Leutz, L. Fu, and H. Ries, “Carambola optics for recycling of light,” Appl. Opt. 45, 2572-2575 (2006).
[CrossRef] [PubMed]

H. Kikuta, S. Hino, and A. Maruyama, “Estimation method for the light extraction efficiency of light-emitting elements with a rigorous grating diffraction theory,” J. Opt. Soc. Am. A 23, 1207-1213 (2006).
[CrossRef]

W. Chi and N. George, “Light-emitting diode illumination design with a condensing sphere,” J. Opt. Soc. Am. A 23, 2295-2298 (2006).
[CrossRef]

S. J. Lee, “Study of photon extraction efficiency in InGaN light/emitting diodes depending on chip structures and chip-mount schemes,” Opt. Eng. 45, 014601 (2006).
[CrossRef]

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

E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
[CrossRef]

M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
[CrossRef]

2005 (3)

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
[CrossRef] [PubMed]

Y.-J. Lee, S.-H. Kim, G.-H. Kim, Y.-H. Lee, S.-H. Cho, Y.-W. Song, Y. C. Kim, and Y. R. Do, “Far-field radiation of photonic crystal organic light-emitting diode,” Opt. Express 13, 5864-5870 (2005).
[CrossRef] [PubMed]

2004 (1)

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 (1)

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

1993 (1)

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
[CrossRef]

1982 (1)

1963 (1)

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368-377 (1963).
[CrossRef]

Adibi, A.

M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
[CrossRef]

Alger, J. S.

M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
[CrossRef]

Allen, S. C.

S. C. Allen and A. J. Steckla, “A nearly ideal phosphor-converted white light-emitting diode,” Appl. Phys. Lett. 92, 143309 (2008).
[CrossRef]

Asamizu, H.

Asano, T.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
[CrossRef] [PubMed]

Bahadur, M.

M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
[CrossRef]

E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
[CrossRef]

Benisty, H.

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

A. David, H. Benisty, and C. Weisbuch, “Optimization of light-diffracting photonic-crystals for high extraction efficiency LEDs,” J. Disp. Technol. 3, 133-148 (2007).
[CrossRef]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

Benítez, P.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Bhat, J.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Caneau, C.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
[CrossRef]

Cassarly, W.

W. Cassarly, “Nonimaging optics: concentration and illumination,” in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. III.

Chang, J. Y.

Chao, C. H.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Charlton, M. D. B.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modeling, and optimization,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Chen, M. H.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Chi, J. Y.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Chi, W.

Chien, W. T.

Cho, S.-H.

Cho, Y. H.

H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
[CrossRef]

Choi, S. M.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Collins, D.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Craford, M. G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Danglot, J.

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

David, A.

J. J. Wierer Jr., A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photon. 3, 163 (2009).
[CrossRef]

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

A. David, H. Benisty, and C. Weisbuch, “Optimization of light-diffracting photonic-crystals for high extraction efficiency LEDs,” J. Disp. Technol. 3, 133-148 (2007).
[CrossRef]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

DeGroot, J.

E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
[CrossRef]

DenBaars, S. P.

Diana, F. S.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

Do, Y. R.

Eftekharian, A.

M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
[CrossRef]

Enoch, S.

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

Fellows, N. N.

Fletcher, R.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Fu, L.

Fujita, M.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
[CrossRef] [PubMed]

Gao, K. F.

George, N.

Gmitter, T. J.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
[CrossRef]

Harbers, G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

He, Y.

Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1998).

Hi, W.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Hino, S.

Holcomb, M. O.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Hong, C. H.

H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
[CrossRef]

Hong, S. Y.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Hsieh, C. C.

Hsu, Y. C.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Hsueh, H. T.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Hu, E.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

Hu, H. L.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Huang, S. B.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Ivanov, R.

Jang, S. H.

S. H. Jang and M. W. Shin, “Thermal analysis of LED arrays for automotive headlamp with a novel cooling system,” IEEE Trans. Device Mater. Reliab. 8, 561-564 (2008).
[CrossRef]

Jarominski, J.

Jeung, W. K.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Khorasani, S.

M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
[CrossRef]

Khoshnegar, M.

M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
[CrossRef]

Kikuta, H.

Kim, G.-H.

Kim, H. J.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Kim, J. K.

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16, 21835-21842 (2008).
[CrossRef] [PubMed]

F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
[CrossRef]

Kim, J. P.

J. Y. Kim, M. K. Kwon, J. P. Kim, and S. J. Park, “Enhanced light extraction from triangular GaN-based light-emitting diodes,” IEEE Photon. Technol. Lett. 19, 1865-1867 (2007).
[CrossRef]

Kim, J. Y.

J. Y. Kim, M. K. Kwon, J. P. Kim, and S. J. Park, “Enhanced light extraction from triangular GaN-based light-emitting diodes,” IEEE Photon. Technol. Lett. 19, 1865-1867 (2007).
[CrossRef]

Kim, S.-H.

Kim, Y. C.

Krames, M. R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Kuang, J. H.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Kuo, H. C.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Kwon, M. K.

J. Y. Kim, M. K. Kwon, J. P. Kim, and S. J. Park, “Enhanced light extraction from triangular GaN-based light-emitting diodes,” IEEE Photon. Technol. Lett. 19, 1865-1867 (2007).
[CrossRef]

Lai, C. F.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Lee, K. S.

H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
[CrossRef]

Lee, P. S.

Lee, S.

R. Zhang and S. Lee, “Wafer level LED packaging with integrated DRIE trenches for encapsulation,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

Lee, S. J.

S. J. Lee, “Study of photon extraction efficiency in InGaN light/emitting diodes depending on chip structures and chip-mount schemes,” Opt. Eng. 45, 014601 (2006).
[CrossRef]

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Lee, T.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modeling, and optimization,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Lee, T. X.

T. X. Lee, K. F. Gao, W. T. Chien, and C. C. Sun, “Light extraction analysis of GaN-based light-emitting diodes with surface texture and/or patterned substrate,” Opt. Express 15, 6670-6676 (2007).
[CrossRef] [PubMed]

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]

Lee, Y. C.

M. L. Wu, Y. C. Lee, S. P. Yang, P. S. Lee, and J. Y. Chang, “Azimuthally isotropic irradiance of GaN-based light-emitting diodes with GaN microlens arrays,” Opt. Express 17, 6148-6155 (2009).
[CrossRef] [PubMed]

Y. C. Lee, “Research on optical properties of micro/nano structures in light-emitting diodes and photovoltaics,” Ph.D. dissertation (National Central University, Taiwan, 2009).

Lee, Y.-H.

Lee, Y.-J.

Leutz, R.

Li, K.

G. Ouyang and K. Li, “Brightness increase in LED by recycling of light for projection applications,” Proc. SPIE 6489, 648903 (2007).
[CrossRef]

Lin, C. Y.

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]

Lin, Y. C.

Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
[CrossRef]

Lin, Y. K.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Liu, S.

Z. Liu, S. Liu, K. Wang, and X. Luo, “Effects of phosphor's location on LED packaging performance,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

Z. Ma, X. Wang, D. Zhu, and S. Liu, “Thermal analysis and modeling of LED arrays integrated with an innovative liquid-cooling module,” in 6th International Conference on Electronic Packaging Technology (IEEE, 2005).

Liu, Z.

Z. Liu, S. Liu, K. Wang, and X. Luo, “Effects of phosphor's location on LED packaging performance,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

Lo, Y. C.

Lu, T. C.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Ludowise, M. J.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Luo, X.

Z. Liu, S. Liu, K. Wang, and X. Luo, “Effects of phosphor's location on LED packaging performance,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

Ma, Z.

Z. Ma, X. Wang, D. Zhu, and S. Liu, “Thermal analysis and modeling of LED arrays integrated with an innovative liquid-cooling module,” in 6th International Conference on Electronic Packaging Technology (IEEE, 2005).

Martin, P. S.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Maruyama, A.

Masui, H.

Megens, M. M.

J. J. Wierer Jr., A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photon. 3, 163 (2009).
[CrossRef]

Meier, C.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

Miñano, J. C.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Mont, F. W.

F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
[CrossRef]

Moreno, I.

Mueller, G. O.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Mueller-Mach, R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Nakamura, S.

Nicodemus, F. E.

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368-377 (1963).
[CrossRef]

Noda, S.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
[CrossRef] [PubMed]

Norris, A. W.

E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
[CrossRef]

M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
[CrossRef]

Ouyang, G.

G. Ouyang and K. Li, “Brightness increase in LED by recycling of light for projection applications,” Proc. SPIE 6489, 648903 (2007).
[CrossRef]

Park, K. Y.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Park, S. J.

J. Y. Kim, M. K. Kwon, J. P. Kim, and S. J. Park, “Enhanced light extraction from triangular GaN-based light-emitting diodes,” IEEE Photon. Technol. Lett. 19, 1865-1867 (2007).
[CrossRef]

Pottage, J. M.

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

Rahman, F.

F. Rahman, “Photonic crystal LEDs,” Opt. Photon. News 20(6), 24-29 (2009).
[CrossRef]

Ries, H.

Rudaz, S. L.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Ryu, J. H.

H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
[CrossRef]

Sato, H.

Scherer, A.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
[CrossRef]

Schnitzer, I.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
[CrossRef]

Schubert, E. F.

F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
[CrossRef]

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16, 21835-21842 (2008).
[CrossRef] [PubMed]

Schubert, M. F.

F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
[CrossRef]

Sharma, R.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

Shchekin, O. B.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Shi, F. G.

Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
[CrossRef]

N. T. Tran and F. G. Shi, “LED package design for high optical efficiency and low viewing angle,” in Microsystems Packaging, Assembly and Circuits Technology (IEEE, 2007), pp. 10-13.
[CrossRef]

Shin, M. W.

S. H. Jang and M. W. Shin, “Thermal analysis of LED arrays for automotive headlamp with a novel cooling system,” IEEE Trans. Device Mater. Reliab. 8, 561-564 (2008).
[CrossRef]

Shin, S. H.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Siegel, R. W.

F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
[CrossRef]

Sodagar, M.

M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
[CrossRef]

Song, Y.-W.

Steckla, A. J.

S. C. Allen and A. J. Steckla, “A nearly ideal phosphor-converted white light-emitting diode,” Appl. Phys. Lett. 92, 143309 (2008).
[CrossRef]

Steigerwald, D.

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

Su, Y. I.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Sun, C. C.

Takahashi, S.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
[CrossRef] [PubMed]

Talneau, A.

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

Tanaka, Y.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
[CrossRef] [PubMed]

Tran, N.

Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
[CrossRef]

Tran, N. T.

N. T. Tran and F. G. Shi, “LED package design for high optical efficiency and low viewing angle,” in Microsystems Packaging, Assembly and Circuits Technology (IEEE, 2007), pp. 10-13.
[CrossRef]

Tsai, C. C.

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

Vanlathem, E.

E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
[CrossRef]

Wang, H.

H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
[CrossRef]

Wang, J.-T.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Wang, K.

Z. Liu, S. Liu, K. Wang, and X. Luo, “Effects of phosphor's location on LED packaging performance,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

Wang, S. C.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Wang, X.

Z. Ma, X. Wang, D. Zhu, and S. Liu, “Thermal analysis and modeling of LED arrays integrated with an innovative liquid-cooling module,” in 6th International Conference on Electronic Packaging Technology (IEEE, 2005).

Weisbuch, C.

A. David, H. Benisty, and C. Weisbuch, “Optimization of light-diffracting photonic-crystals for high extraction efficiency LEDs,” J. Disp. Technol. 3, 133-148 (2007).
[CrossRef]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

Wierer, J. J.

J. J. Wierer Jr., A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photon. 3, 163 (2009).
[CrossRef]

Windiate, C. C.

M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
[CrossRef]

Winston, R.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Wu, M. L.

Yablonovitch, E.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
[CrossRef]

Yang, S. P.

Yang, T. H.

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]

Yeh, W. Y.

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

Yi, S.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Yoon, Y. B.

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

Yoshitake, M.

E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
[CrossRef]

Zarisfi, A.

M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
[CrossRef]

Zhang, R.

R. Zhang and S. Lee, “Wafer level LED packaging with integrated DRIE trenches for encapsulation,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

Zhou, L.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Zhou, Y.

Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
[CrossRef]

Zhu, D.

Z. Ma, X. Wang, D. Zhu, and S. Liu, “Thermal analysis and modeling of LED arrays integrated with an innovative liquid-cooling module,” in 6th International Conference on Electronic Packaging Technology (IEEE, 2005).

Zoorob, M. E.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modeling, and optimization,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Am. J. Phys. (1)

F. E. Nicodemus, “Radiance,” Am. J. Phys. 31, 368-377 (1963).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (3)

S. C. Allen and A. J. Steckla, “A nearly ideal phosphor-converted white light-emitting diode,” Appl. Phys. Lett. 92, 143309 (2008).
[CrossRef]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, “Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction,” Appl. Phys. Lett. 87, 101107(2005).
[CrossRef]

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Benisty, J. Danglot, A. Talneau, S. Enoch, J. M. Pottage, and A. David, “Investigation of extracting photonic crystal lattices for guided modes of GaAs-based heterostructures,” IEEE J. Quantum Electron. 44, 777-789 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. Steigerwald, J. Bhat, D. Collins, R. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with solid state lighting technology,” IEEE J. Sel. Top. Quantum Electron. 8, 310-320 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. Y. Kim, M. K. Kwon, J. P. Kim, and S. J. Park, “Enhanced light extraction from triangular GaN-based light-emitting diodes,” IEEE Photon. Technol. Lett. 19, 1865-1867 (2007).
[CrossRef]

H. Wang, K. S. Lee, J. H. Ryu, C. H. Hong, and Y. H. Cho, “Active packaging method for light-emitting diode lamps with photosensitive epoxy resins,” IEEE Photon. Technol. Lett. 20, 87-89 (2008).
[CrossRef]

IEEE Trans. Advanced Packaging (1)

Y. Zhou, N. Tran, Y. C. Lin, Y. He, and F. G. Shi, “One-component, low-temperature, and fast cure epoxy encapsulant with high refractive index for LED applications,” IEEE Trans. Advanced Packaging 31, 484-489 (2008).
[CrossRef]

IEEE Trans. Device Mater. Reliab. (1)

S. H. Jang and M. W. Shin, “Thermal analysis of LED arrays for automotive headlamp with a novel cooling system,” IEEE Trans. Device Mater. Reliab. 8, 561-564 (2008).
[CrossRef]

IEEE Trans. Electron Devices (1)

Y. C. Hsu, Y. K. Lin, M. H. Chen, C. C. Tsai, J. H. Kuang, S. B. Huang, H. L. Hu, Y. I. Su, and W. Hi, “Failure mechanisms associated with lens shape of high-power LED modules in aging test,” IEEE Trans. Electron Devices 55, 689-694(2008).
[CrossRef]

J. Appl. Phys. (1)

F. W. Mont, J. K. Kim, M. F. Schubert, E. F. Schubert, and R. W. Siegel, “High-refractive-index TiO2-nanoparticle-loaded encapsulants for light-emitting diodes,” J. Appl. Phys. 103, 083120 (2008).
[CrossRef]

J. Disp. Technol. (2)

A. David, H. Benisty, and C. Weisbuch, “Optimization of light-diffracting photonic-crystals for high extraction efficiency LEDs,” J. Disp. Technol. 3, 133-148 (2007).
[CrossRef]

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nat. Photon. (1)

J. J. Wierer Jr., A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photon. 3, 163 (2009).
[CrossRef]

Opt. Eng. (2)

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]

S. J. Lee, “Study of photon extraction efficiency in InGaN light/emitting diodes depending on chip structures and chip-mount schemes,” Opt. Eng. 45, 014601 (2006).
[CrossRef]

Opt. Express (7)

Opt. Photon. News (1)

F. Rahman, “Photonic crystal LEDs,” Opt. Photon. News 20(6), 24-29 (2009).
[CrossRef]

Proc. SPIE (6)

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modeling, and optimization,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

G. Ouyang and K. Li, “Brightness increase in LED by recycling of light for projection applications,” Proc. SPIE 6489, 648903 (2007).
[CrossRef]

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

M. Khoshnegar, A. Eftekharian, M. Sodagar, S. Khorasani, and A. Adibi, “Quantum well design and diffraction efficiency of quantum well light emitting diode,” Proc. SPIE 7223, 722311 (2009).
[CrossRef]

E. Vanlathem, A. W. Norris, M. Bahadur, J. DeGroot, and M. Yoshitake, “Novel silicone materials for LED packaging and optoelectronics devices,” Proc. SPIE 6192, 619202 (2006).
[CrossRef]

M. Bahadur, A. W. Norris, A. Zarisfi, J. S. Alger, and C. C. Windiate, “Silicone materials for LED packaging,” Proc. SPIE 6337, 63370F (2006).
[CrossRef]

Science (1)

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296 (2005).
[CrossRef] [PubMed]

Other (11)

Z. Ma, X. Wang, D. Zhu, and S. Liu, “Thermal analysis and modeling of LED arrays integrated with an innovative liquid-cooling module,” in 6th International Conference on Electronic Packaging Technology (IEEE, 2005).

W. K. Jeung, S. H. Shin, S. Y. Hong, S. M. Choi, S. Yi, Y. B. Yoon, H. J. Kim, S. J. Lee, and K. Y. Park, “Silicon-based multi-chip LED package,” in Electronic Components and Technology Conference 2007 (IEEE, 2007), pp. 722-727.
[CrossRef]

H. T. Hsueh, J.-T. Wang, C. H. Chao, W. Y. Yeh, J. Y. Chi, C. F. Lai, H. C. Kuo, T. C. Lu, and S. C. Wang, “Azimuthal anisotropy of light extraction from photonic crystal light-emitting diodes,” in Conference on Lasers and Electro-Optics 2007(Optical Society of America, 2007), paper ThF1_4.
[CrossRef]

PhlatLight LED data sheet, Luminus Devices Inc., www.luminus.com.

Y. C. Lee, “Research on optical properties of micro/nano structures in light-emitting diodes and photovoltaics,” Ph.D. dissertation (National Central University, Taiwan, 2009).

N. T. Tran and F. G. Shi, “LED package design for high optical efficiency and low viewing angle,” in Microsystems Packaging, Assembly and Circuits Technology (IEEE, 2007), pp. 10-13.
[CrossRef]

Z. Liu, S. Liu, K. Wang, and X. Luo, “Effects of phosphor's location on LED packaging performance,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

R. Zhang and S. Lee, “Wafer level LED packaging with integrated DRIE trenches for encapsulation,” in International Conference on Electronic Packaging Technology and High Density Packaging (IEEE, 2008).

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

W. Cassarly, “Nonimaging optics: concentration and illumination,” in OSA Handbook of Optics, 2nd ed. (McGraw-Hill, 2001), Vol. III.

E. Hecht, Optics (Addison-Wesley, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Geometry of an LED with a spherical package of radius R.

Fig. 2
Fig. 2

Geometry of an LED with a flat package.

Fig. 3
Fig. 3

Results obtained for a square central chip inside a spherical package. In (a) and (b)  n = 1.5 ; in (b) and (c)  R = 10 a .

Fig. 4
Fig. 4

Results for the four identical chips in the central plane of a spherical encapsulant as a function of the total emitting area in both the analytic method and the ray-trace simulation. In this case z = 0 , n = 1.5 , and R = 10 a .

Fig. 5
Fig. 5

Efficiency by varying the encapsulant radius for the chip with lateral faces. For this geometry n = 1.5 , z = 0 , and a = 10 c .

Fig. 6
Fig. 6

Efficiency by varying the size of the LED chip for several LED radiation patterns. Results obtained for a square central chip inside a spherical package with n = 1.5 and z = 0 ; see inset of Fig. 3a.

Fig. 7
Fig. 7

Flat package. (a) Efficiency by varying the index of refraction of package for several LED radiation patterns (see insets of Fig. 6) with n out = 1 and n in = n . (b) LTE of directional LEDs for several half-angles θ 1 / 2 .

Tables (1)

Tables Icon

Table 1 Coefficients of Eqs. (20, 21) Used in Calculations of Figs. 6, 7

Equations (24)

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

η = ϕ out ϕ in .
ϕ in = S in 2 π L in d Ω d A ,
ϕ out = n out 2 n in 2 S out Ω out L in τ d Ω d A ,
d ϕ in = L in d A = L in ( θ p , φ p , S ) cos θ p d S .
d ϕ out = L ( θ p , φ p , S ) τ ( α in ) cos θ p sin θ p d θ p d φ p d S .
ϕ out = S in 0 2 π 0 π / 2 L in ( θ p , φ p , S ) τ ( α in ) cos θ p sin θ p d θ p d φ p d S ,
η = S in 0 2 π 0 π / 2 L in ( θ p , φ p ; d ) τ ( α in ) cos θ p sin θ p d θ p d φ p d S S in 0 2 π 0 π / 2 L in ( θ p , φ p , d ) cos θ p sin θ p d θ p d φ p d S ,
α in ( R , θ p , φ p ; d ) = ar c cos ( r r | r | | r | ) = ar c cos ( r r ^ | r | ) ,
r ^ = ( sin φ p cos θ p sin φ p sin θ p cos φ p ) , d = ( x y z ) ,
r = d + r r ^ , r = [ | d r ^ | + ( d r ^ ) 2 + ( R 2 | d | 2 ) ] .
τ ( α in ) = T ( α in ) U ( α in ) ,
T ( α in ) = ( n out cos α out n in cos α in ) ( t P 2 + t S 2 ) 2 .
t P = 2 n in cos α in n in cos α out + n out cos α in ,
t S = 2 n in cos α in n in cos α in + n out cos α out ,
U ( α in ) = { 0 i f α in α c in 1 i f α in < α c in ,
α c in = arc sin ( n out n in ) .
η = S in 0 2 π 0 π / 2 I in ( θ p , φ p ) f ( S ) τ ( α in ) sin θ p d θ p d φ p d S S in 0 2 π 0 π / 2 I in ( θ p , φ p ) f ( S ) sin θ p d θ p d φ p d S ,
ϕ out = n out 2 n in 2 S in 0 2 π 0 π / 2 L in ( θ p , φ p , S ) T ( θ ) cos θ sin θ d θ d φ d S .
ϕ out = n out 2 n in 2 S in 0 2 π 0 π / 2 I in ( θ p , φ p ) f ( S ) T ( θ ) ( cos θ cos θ p ) sin θ d θ d φ d S ,
η = n out 2 n in 2 0 π / 2 0 π / 2 I ( θ p , φ ) T ( θ ) ( cos θ cos θ p ) sin θ d θ d φ 0 π / 2 0 π / 2 I ( θ , φ ) sin θ d θ d φ .
η = 2 n out 2 n in 2 0 π / 2 T ( θ ) cos θ sin θ d θ = 2 n out 2 n in 2 T ¯ .
I ( θ p ) = m g m , 1 exp ( ln 2 ( | θ p | g m , 2 g m , 3 ) 2 ) .
I ( θ p , φ p ) = m G ( φ p ) m , 1 exp ( ln 2 ( | θ p | G ( φ p ) m , 2 G ( φ p ) m , 3 ) 2 ) ,
G ( φ p ) m n = g 1 m n cos 2 ( N φ p ) + g 2 m n sin 2 ( N φ p ) .

Metrics