Abstract

Light extraction from two-dimensional objects is discussed. Analytical calculations in terms of three different parameters have been applied to equiangular polygons to trace light rays during multiple reflections in a polygon. Based on the result that there are a finite number of incident angles in a polygon for a light ray, it was found that the triangle has the least chance to trap light rays among the polygons. The discussion has been extended to parallelograms, which have an advantage in light extraction to rectangles. Placement of a possible light source in polygons is discussed.

© 2007 Optical Society of America

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References

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  • Publication
  1. W. N. Carr, "Photometric figures of merit for semiconductor luminescent sources operating in spontaneous mode," Infrared Phys. 6, 1-19 (1966).
    [Crossref]
  2. 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]
  3. R. Krames, M. Ochiai-Holcomb, G. E. Höfler, C. Carter-Coman, E. I. Chen, I.-H. Tan, P. Grillot, N. F. Gardner, H. C. Chui, J.-W. Huang, S. A. Stockman, F. A. Kish, and M. G. Craford, "High-power truncated-inverted-pyramid (AlxGa1−x)0.5In0.5P/GaP light-emitting diodes exhibiting >50% external quantum efficiency," Appl. Phys. Lett. 75, 2365-2367 (1999).
    [Crossref]
  4. U. Strauss, H.-J. Lugauer, A. Weimar, J. Baur, G. Brüderl, D. Eisert, F. Kühn, U. Zehnder, and V. Härle, "Progress of InGaN light emitting diodes on SiC," Phys. Status Solidi C 0, 276-279 (2002).
    [Crossref]
  5. T. Fujii, A. David, Y. Gao, M. Iza, S. P. DenBaars, E. L. Hu, C. Weisbuch, and S. Nakamura, "Cone-shaped surface GaN-based light-emitting diodes," Phys. Status Solidi C 2, 2836-2840 (2005).
    [Crossref]
  6. Y. Narukawa, J. Narita, T. Sakamoto, K. Deguchi, T. Yamada, and T. Mukai, "Ultra-high efficiency white light emitting diodes," Jpn. J. Appl. Phys. , Part 2 45, L1084-L1086 (2006).
    [Crossref]
  7. A. David, T. Fujii, B. Moran, S. Nakamura, S. P. DenBaars, C. Weisbuch, and H. Benisty, "Photonic crystal laser lift-off GaN light-emitting diodes," Appl. Phys. Lett. 88, 133514 (2006).
    [Crossref]
  8. E. F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge Press, 2006), p. 151.
  9. 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]
  10. Standing-wave modes are a consequence of propagating light interfering with itself due to the total internal reflection. Discussions on waveguides can be found in C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989), pp. 366-372.
  11. Light can be trapped at higher bounce mode as well. In this case the incident angle is smaller than that of the lowest bounce mode (m = 1), so that light is more likely to exit than the lowest mode.
  12. J.-H. Liang, T. Maruyama, Y. Ogawa, S. Kobayashi, J. Sonoda, H. Urae, S. Tomita, Y. Tomioka, S. Kon, and Y. Nakano, "High-power high-efficiency superluminescent diodes with J-shaped ridge waveguide structure," in Proceedings of 14th International Conference on Indium Phosphide and Related Materials Conference (IEEE, 2002), pp. 119-122.
    [PubMed]

2007 (1)

2006 (3)

Y. Narukawa, J. Narita, T. Sakamoto, K. Deguchi, T. Yamada, and T. Mukai, "Ultra-high efficiency white light emitting diodes," Jpn. J. Appl. Phys. , Part 2 45, L1084-L1086 (2006).
[Crossref]

A. David, T. Fujii, B. Moran, S. Nakamura, S. P. DenBaars, C. Weisbuch, and H. Benisty, "Photonic crystal laser lift-off GaN light-emitting diodes," Appl. Phys. Lett. 88, 133514 (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]

2005 (1)

T. Fujii, A. David, Y. Gao, M. Iza, S. P. DenBaars, E. L. Hu, C. Weisbuch, and S. Nakamura, "Cone-shaped surface GaN-based light-emitting diodes," Phys. Status Solidi C 2, 2836-2840 (2005).
[Crossref]

2002 (1)

U. Strauss, H.-J. Lugauer, A. Weimar, J. Baur, G. Brüderl, D. Eisert, F. Kühn, U. Zehnder, and V. Härle, "Progress of InGaN light emitting diodes on SiC," Phys. Status Solidi C 0, 276-279 (2002).
[Crossref]

1999 (1)

R. Krames, M. Ochiai-Holcomb, G. E. Höfler, C. Carter-Coman, E. I. Chen, I.-H. Tan, P. Grillot, N. F. Gardner, H. C. Chui, J.-W. Huang, S. A. Stockman, F. A. Kish, and M. G. Craford, "High-power truncated-inverted-pyramid (AlxGa1−x)0.5In0.5P/GaP light-emitting diodes exhibiting >50% external quantum efficiency," Appl. Phys. Lett. 75, 2365-2367 (1999).
[Crossref]

1966 (1)

W. N. Carr, "Photometric figures of merit for semiconductor luminescent sources operating in spontaneous mode," Infrared Phys. 6, 1-19 (1966).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

R. Krames, M. Ochiai-Holcomb, G. E. Höfler, C. Carter-Coman, E. I. Chen, I.-H. Tan, P. Grillot, N. F. Gardner, H. C. Chui, J.-W. Huang, S. A. Stockman, F. A. Kish, and M. G. Craford, "High-power truncated-inverted-pyramid (AlxGa1−x)0.5In0.5P/GaP light-emitting diodes exhibiting >50% external quantum efficiency," Appl. Phys. Lett. 75, 2365-2367 (1999).
[Crossref]

A. David, T. Fujii, B. Moran, S. Nakamura, S. P. DenBaars, C. Weisbuch, and H. Benisty, "Photonic crystal laser lift-off GaN light-emitting diodes," Appl. Phys. Lett. 88, 133514 (2006).
[Crossref]

Infrared Phys. (1)

W. N. Carr, "Photometric figures of merit for semiconductor luminescent sources operating in spontaneous mode," Infrared Phys. 6, 1-19 (1966).
[Crossref]

Jpn. J. Appl. Phys. (1)

Y. Narukawa, J. Narita, T. Sakamoto, K. Deguchi, T. Yamada, and T. Mukai, "Ultra-high efficiency white light emitting diodes," Jpn. J. Appl. Phys. , Part 2 45, L1084-L1086 (2006).
[Crossref]

Opt. Eng. (1)

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]

Phys. Status Solidi C (2)

U. Strauss, H.-J. Lugauer, A. Weimar, J. Baur, G. Brüderl, D. Eisert, F. Kühn, U. Zehnder, and V. Härle, "Progress of InGaN light emitting diodes on SiC," Phys. Status Solidi C 0, 276-279 (2002).
[Crossref]

T. Fujii, A. David, Y. Gao, M. Iza, S. P. DenBaars, E. L. Hu, C. Weisbuch, and S. Nakamura, "Cone-shaped surface GaN-based light-emitting diodes," Phys. Status Solidi C 2, 2836-2840 (2005).
[Crossref]

Other (4)

Standing-wave modes are a consequence of propagating light interfering with itself due to the total internal reflection. Discussions on waveguides can be found in C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989), pp. 366-372.

Light can be trapped at higher bounce mode as well. In this case the incident angle is smaller than that of the lowest bounce mode (m = 1), so that light is more likely to exit than the lowest mode.

J.-H. Liang, T. Maruyama, Y. Ogawa, S. Kobayashi, J. Sonoda, H. Urae, S. Tomita, Y. Tomioka, S. Kon, and Y. Nakano, "High-power high-efficiency superluminescent diodes with J-shaped ridge waveguide structure," in Proceedings of 14th International Conference on Indium Phosphide and Related Materials Conference (IEEE, 2002), pp. 119-122.
[PubMed]

E. F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge Press, 2006), p. 151.

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Figures (5)

Fig. 1
Fig. 1 Definition of bounce mode m in the octagon as an example.

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Fig. 2
Fig. 2 Derivation of the master equation. The sum of the three angles is π.

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Fig. 3
Fig. 3 Relationship between the maximum angle of incidence among two successive reflections in the lowest mode, θ max , and the number of sides of the polygon, n ( 10 < n < ∞ has been omitted because of the obviousness). On the vertical axis on the right, critical angles of some of the material combinations are shown.

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Fig. 4
Fig. 4 Extraction cones in a square. The shaded area is in the two cones, therefore a light ray emitted from this area will exit the square at the first incidence upon the side.

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Fig. 5
Fig. 5 Isosceles triangle with a vertical angle α. The two successive reflections fall in the condition described in Eq. (11). Note the sequent reflection (not indicated by the light-ray line) would direct the ray toward the bottom, resulting in a bounce mode with m = 1 .

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Equations (10)

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

θ 1 + θ 2 + 2 π m / n = π ,
θ 2 + θ 3 + 2 π m / n = π ,
θ 3 = θ 1 .
θ 3 = θ 1 + 2 π ( m 1 − m 2 ) / n .
θ 4 = θ 2 + 2 π ( m 2 − m 3 ) / n ,
θ max = π ( 1 2 − 1 n ) .
( π 2 − θ 1 ) + θ 2 = π n − α ,
θ 1 − θ 2 = α .
θ 1 + θ 2 = α .
θ 1 − [ ( f − 1 ) α ] < α ,

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