Abstract

We present an analytical model for the optical emission of a two-dimensional source in a flexible organic light-emitting diode formation with arbitrary curvature. The formulation rigorously produces closed-form analytical expressions which clearly relate the emission pattern and the device configuration, in particular, the radius of curvature. We investigate the optical properties of a prototype model through the resultant expressions, revealing that the bending induces a dramatic enhancement of emission to large angles, allowing for large viewing angle and reduced total internal reflection losses. These effects, shown to arise from geometrical considerations, demonstrate the unique advantages which curved flexible devices offer with respect to their planar counterparts. To the best of our knowledge, this is the first time that a rigorous analytical investigation of the optical characteristics of these novel devices is conducted. The resultant analytical formulae provide a robust basis for future analysis, as well as a set of design rules for efficient device engineering.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
    [CrossRef]
  2. G. Gu, P. E. Burrows, S. Venkatesh, S. R. Forrest, and M. E. Thompson, “Vacuum-deposited, nonpolymeric flexible organic light-emitting devices,” Opt. Lett. 22, 172–174 (1997).
    [CrossRef] [PubMed]
  3. N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996).
    [CrossRef]
  4. D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
    [CrossRef] [PubMed]
  5. B. Park, C. H. Park, M. Kim, and M. Han, “Polarized organic light-emitting device on a flexible giant birefringent optical reflecting polarizer substrate,” Opt. Express 17, 10136–10143 (2009).
    [CrossRef] [PubMed]
  6. T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
    [CrossRef] [PubMed]
  7. C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009).
    [CrossRef]
  8. C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. 11, 1870–1875 (2010).
    [CrossRef]
  9. B. Park and H. G. Jeon, “Spontaneous buckling in flexible organic light-emitting devices for enhanced light extraction,” Opt. Express 19, A1117–A1125 (2011).
    [CrossRef] [PubMed]
  10. H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
    [CrossRef]
  11. Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
    [CrossRef]
  12. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  13. P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. 31, 863–870 (1983).
    [CrossRef]
  14. W. Wasylkiwskyj, “Diffraction by a concave perfectly conducting circular cylinder,” IEEE Trans. Antennas Propag. 23, 480–492 (1975).
    [CrossRef]
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).
  16. A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. 46, 1388–1395 (2010).
    [CrossRef]
  17. A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. 35, 3366–3368 (2010).
    [CrossRef] [PubMed]
  18. P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. 31, 870–884 (1983).
    [CrossRef]
  19. A. Cherkassky, “Optimization of electromagnetic power absorption in a lossy circular cylinder,” Master’s thesis, Technion - Israel Institute of Technology (2006).
  20. M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
    [CrossRef]
  21. E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006).
    [CrossRef]
  22. T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
    [CrossRef]
  23. M. Ma, F. W. Mont, X. Yan, J. Cho, E. F. Schubert, G. B. Kim, and C. Sone, “Effects of the refractive index of the encapsulant on the light-extraction efficiency of light-emitting diodes,” Opt. Express 19, A1135–A1140 (2011).
    [CrossRef] [PubMed]
  24. S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
    [CrossRef]
  25. R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
    [CrossRef]
  26. S. Mladenovski, K. Neyts, D. Pavicic, A. Werner, and C. Rothe, “Exceptionally efficient organic light emitting devices using high refractive index substrates,” Opt. Express 17, 7562–7570 (2009).
    [CrossRef] [PubMed]
  27. K.-Y. Chen, Y.-T. Chang, Y.-H. Ho, H.-Y. Lin, J.-H. Lee, and M.-K. Wei, “Emitter apodization dependent angular luminance enhancement of microlens-array film attached organic light-emitting devices,” Opt. Express 18, 3238–3243 (2010).
    [CrossRef] [PubMed]
  28. Y. Sun and S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics 2, 483–487 (2008).
    [CrossRef]
  29. T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010).
    [CrossRef] [PubMed]

2011 (4)

B. Park and H. G. Jeon, “Spontaneous buckling in flexible organic light-emitting devices for enhanced light extraction,” Opt. Express 19, A1117–A1125 (2011).
[CrossRef] [PubMed]

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

M. Ma, F. W. Mont, X. Yan, J. Cho, E. F. Schubert, G. B. Kim, and C. Sone, “Effects of the refractive index of the encapsulant on the light-extraction efficiency of light-emitting diodes,” Opt. Express 19, A1135–A1140 (2011).
[CrossRef] [PubMed]

2010 (7)

R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
[CrossRef]

K.-Y. Chen, Y.-T. Chang, Y.-H. Ho, H.-Y. Lin, J.-H. Lee, and M.-K. Wei, “Emitter apodization dependent angular luminance enhancement of microlens-array film attached organic light-emitting devices,” Opt. Express 18, 3238–3243 (2010).
[CrossRef] [PubMed]

T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010).
[CrossRef] [PubMed]

A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. 46, 1388–1395 (2010).
[CrossRef]

A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. 35, 3366–3368 (2010).
[CrossRef] [PubMed]

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. 11, 1870–1875 (2010).
[CrossRef]

2009 (5)

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

B. Park, C. H. Park, M. Kim, and M. Han, “Polarized organic light-emitting device on a flexible giant birefringent optical reflecting polarizer substrate,” Opt. Express 17, 10136–10143 (2009).
[CrossRef] [PubMed]

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009).
[CrossRef]

S. Mladenovski, K. Neyts, D. Pavicic, A. Werner, and C. Rothe, “Exceptionally efficient organic light emitting devices using high refractive index substrates,” Opt. Express 17, 7562–7570 (2009).
[CrossRef] [PubMed]

2008 (2)

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
[CrossRef]

Y. Sun and S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics 2, 483–487 (2008).
[CrossRef]

2001 (1)

T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
[CrossRef]

1997 (1)

1996 (1)

N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996).
[CrossRef]

1992 (1)

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

1983 (2)

P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. 31, 870–884 (1983).
[CrossRef]

P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. 31, 863–870 (1983).
[CrossRef]

1975 (1)

W. Wasylkiwskyj, “Diffraction by a concave perfectly conducting circular cylinder,” IEEE Trans. Antennas Propag. 23, 480–492 (1975).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).

Aida, T.

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Bräuer, A. H.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

Brutting, W.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
[CrossRef]

Bull, S.

C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009).
[CrossRef]

Burrows, P. E.

Cao, Y.

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

Chang, Y.-T.

Chen, K.-Y.

Cherkassky, A.

A. Cherkassky, “Optimization of electromagnetic power absorption in a lossy circular cylinder,” Master’s thesis, Technion - Israel Institute of Technology (2006).

Chiang, C.-J.

C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. 11, 1870–1875 (2010).
[CrossRef]

C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009).
[CrossRef]

Cho, J.

Choi, W.

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Chun, Y.

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Colaneri, N.

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

Danz, N.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

Denton, G. J.

N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996).
[CrossRef]

Einziger, P.

P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. 31, 863–870 (1983).
[CrossRef]

P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. 31, 870–884 (1983).
[CrossRef]

Einziger, P. D.

A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. 35, 3366–3368 (2010).
[CrossRef] [PubMed]

A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. 46, 1388–1395 (2010).
[CrossRef]

Epstein, A.

A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. 35, 3366–3368 (2010).
[CrossRef] [PubMed]

A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. 46, 1388–1395 (2010).
[CrossRef]

Felsen, L.

P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. 31, 870–884 (1983).
[CrossRef]

P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. 31, 863–870 (1983).
[CrossRef]

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Flämmich, M.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

Forrest, S. R.

Y. Sun and S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics 2, 483–487 (2008).
[CrossRef]

G. Gu, P. E. Burrows, S. Venkatesh, S. R. Forrest, and M. E. Thompson, “Vacuum-deposited, nonpolymeric flexible organic light-emitting devices,” Opt. Lett. 22, 172–174 (1997).
[CrossRef] [PubMed]

Friend, R. H.

N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996).
[CrossRef]

Frischeisen, J.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
[CrossRef]

Fukushima, T.

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Furno, M.

R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
[CrossRef]

Gather, M. C.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

Greiner, M. T.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Gu, G.

Gustafsson, G.

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

Han, M.

Hata, K.

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Heeger, A. J.

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

Helander, M. G.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Ho, Y.-H.

Hofmann, S.

R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
[CrossRef]

Hudson, Z. M.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Jeon, H. G.

Kawano, K.

T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
[CrossRef]

Kee, I.

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Kim, G. B.

Kim, M.

Kim, S.

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Klauk, H.

T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010).
[CrossRef] [PubMed]

Klavetter, F.

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

Krummacher, B. C.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
[CrossRef]

Kwon, H.-J.

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Lee, J.-H.

Lee, S.

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Leo, K.

R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
[CrossRef]

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Lin, H.-Y.

Lindner, F.

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Liu, Z. W.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Lu, Z. H.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Lussem, B.

R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
[CrossRef]

Lüssem, B.

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Ma, M.

Maeda, H.

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Meerheim, R.

R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
[CrossRef]

Meerholz, K.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

Michaelis, D.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

Mladenovski, S.

Monkman, A.

C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. 11, 1870–1875 (2010).
[CrossRef]

C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009).
[CrossRef]

Mont, F. W.

Nakajima, H.

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Neyts, K.

Nowy, S.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
[CrossRef]

Park, B.

Park, C. H.

Pavicic, D.

Puzzo, D. P.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Qiu, J.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Reineke, D.

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Reinke, N. A.

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
[CrossRef]

Rothe, C.

Schubert, E. F.

Schwartz, G.

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Seidler, N.

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Sekitani, T.

T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010).
[CrossRef] [PubMed]

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Shim, H.

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

Someya, T.

T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010).
[CrossRef] [PubMed]

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Sone, C.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).

Sun, Y.

Y. Sun and S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics 2, 483–487 (2008).
[CrossRef]

Tessler, N.

A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. 46, 1388–1395 (2010).
[CrossRef]

A. Epstein, N. Tessler, and P. D. Einziger, “Analytical extraction of the recombination zone location in organic light-emitting diodes from emission pattern extrema,” Opt. Lett. 35, 3366–3368 (2010).
[CrossRef] [PubMed]

N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996).
[CrossRef]

Thompson, M. E.

Treacy, G. M.

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

Tsutsui, T.

T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
[CrossRef]

Tünnermann, A.

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

Venkatesh, S.

Walzer, K.

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Wang, S.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Wang, Z. B.

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Wasylkiwskyj, W.

W. Wasylkiwskyj, “Diffraction by a concave perfectly conducting circular cylinder,” IEEE Trans. Antennas Propag. 23, 480–492 (1975).
[CrossRef]

Wei, M.-K.

Werner, A.

Winscom, C.

C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. 11, 1870–1875 (2010).
[CrossRef]

C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009).
[CrossRef]

Yahiro, M.

T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
[CrossRef]

Yan, X.

Yokogawa, H.

T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
[CrossRef]

Yokoyama, M.

T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
[CrossRef]

Zschieschang, U.

T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010).
[CrossRef] [PubMed]

Adv. Mater. (1)

T. Tsutsui, M. Yahiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater. 13, 1149–1152 (2001).
[CrossRef]

Appl. Phys. Lett. (2)

R. Meerheim, M. Furno, S. Hofmann, B. Lussem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97, 253305 (2010).
[CrossRef]

H.-J. Kwon, H. Shim, S. Kim, W. Choi, Y. Chun, I. Kee, and S. Lee, “Mechanically and optically reliable folding structure with a hyperelastic material for seamless foldable displays,” Appl. Phys. Lett. 98, 151904 (2011).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Epstein, N. Tessler, and P. D. Einziger, “The impact of spectral and spatial exciton distributions on optical emission from thin-film weak-microcavity organic light-emitting diodes,” IEEE J. Quantum Electron. 46, 1388–1395 (2010).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

P. Einziger and L. Felsen, “Rigorous asymptotic analysis of transmission through a curved dielectric slab,” IEEE Trans. Antennas Propag. 31, 863–870 (1983).
[CrossRef]

W. Wasylkiwskyj, “Diffraction by a concave perfectly conducting circular cylinder,” IEEE Trans. Antennas Propag. 23, 480–492 (1975).
[CrossRef]

P. Einziger and L. Felsen, “Ray analysis of two-dimensional radomes,” IEEE Trans. Antennas Propag. 31, 870–884 (1983).
[CrossRef]

J. Appl. Phys. (1)

S. Nowy, B. C. Krummacher, J. Frischeisen, N. A. Reinke, and W. Brutting, “Light extraction and optical loss mechanisms in organic light-emitting diodes: Influence of the emitter quantum efficiency,” J. Appl. Phys. 104, 123109 (2008).
[CrossRef]

Nat. Mater. (2)

T. Sekitani, U. Zschieschang, H. Klauk, and T. Someya, “Flexible organic transistors and circuits with extreme bending stability,” Nat. Mater. 9, 1015–1022 (2010).
[CrossRef] [PubMed]

T. Sekitani, H. Nakajima, H. Maeda, T. Fukushima, T. Aida, K. Hata, and T. Someya, “Stretchable active-matrix organic light-emitting diode display using printable elastic conductors,” Nat. Mater. 8, 494–499 (2009).
[CrossRef] [PubMed]

Nat. Photonics (2)

Z. B. Wang, M. G. Helander, J. Qiu, D. P. Puzzo, M. T. Greiner, Z. M. Hudson, S. Wang, Z. W. Liu, and Z. H. Lu, “Unlocking the full potential of organic light-emitting diodes on flexible plastic,” Nat. Photonics 5, 753–757 (2011).
[CrossRef]

Y. Sun and S. R. Forrest, “Enhanced light out-coupling of organic light-emitting devices using embedded low-index grids,” Nat. Photonics 2, 483–487 (2008).
[CrossRef]

Nature (3)

G. Gustafsson, Y. Cao, G. M. Treacy, F. Klavetter, N. Colaneri, and A. J. Heeger, “Flexible light-emitting diodes made from soluble conducting polymers,” Nature 357, 477–479 (1992).
[CrossRef]

N. Tessler, G. J. Denton, and R. H. Friend, “Lasing from conjugated-polymer microcavities,” Nature 382, 695–697 (1996).
[CrossRef]

D. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459, 234–238 (2009).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

Org. Electron. (3)

M. Flämmich, M. C. Gather, N. Danz, D. Michaelis, A. H. Bräuer, K. Meerholz, and A. Tünnermann, “Orientation of emissive dipoles in OLEDs: Quantitative in situ analysis,” Org. Electron. 11, 1039–1046 (2010).
[CrossRef]

C.-J. Chiang, C. Winscom, S. Bull, and A. Monkman, “Mechanical modeling of flexible OLED devices,” Org. Electron. 10, 1268–1274 (2009).
[CrossRef]

C.-J. Chiang, C. Winscom, and A. Monkman, “Electroluminescence characterization of FOLED devices under two type of external stresses caused by bending,” Org. Electron. 11, 1870–1875 (2010).
[CrossRef]

Other (4)

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

E. F. Schubert, Light-Emitting Diodes (Cambridge University Press, 2006).
[CrossRef]

A. Cherkassky, “Optimization of electromagnetic power absorption in a lossy circular cylinder,” Master’s thesis, Technion - Israel Institute of Technology (2006).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions : with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1970).

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 (4)

Fig. 1
Fig. 1

Two-dimensional configuration for the curved FOLED model. The source location is denoted by ρ⃗′ (red) and the observation point by ρ⃗ (blue). Only outbound azimuthal waves are taken into account by placing a ”perfect azimuthal absorber” at the azimuthal boundaries of the curved FOLED [13].

Fig. 2
Fig. 2

(a) Physical configuration of the prototype FOLED specified in Table 2.4. (b) For very large radius of curvature with respect to the device dimensions, the prototype PPOLED is received. A polar coordinate system (ρ̃,θ) is defined as in [16] (note the different origin).

Fig. 3
Fig. 3

Emission patterns of (a) the prototype PPOLED and (b)–(g) the prototype FOLED with radii of curvature varying from (b) R = 2000μm to (g) R = 50μm. The emission patterns are plotted for two source-cathode separations of z′ = 20nm (red dashed line) and z′ = 140nm (blue solid line). For the z′ = 140nm case, the local maximum angle is denoted in the legend of each subplot. (h) Illustration of the main curvature effects, emphasizing the difference between the planar (dotted lines and captions) and the curved (solid lines and captions) devices. The red lines demonstrate the behavior of a ray which departs the anode/substrate interface in an angle below the critical angle for the substrate/air interface, θ c , and the blue lines follow a ray with an angle of departure above θ c . Normals to the interfaces at the point of incidence are denoted by thin dotted lines. The geometrical ray-optics interpretation indicates that the angle of incidence, and hence the local reflection coefficients, at the substrate/air interface are always smaller for the curved FOLED than for the PPOLED.

Fig. 4
Fig. 4

Outcoupling efficiency enhancement (with respect to the corresponding PPOLED) as a function of radius of curvature for the prototype FOLED, with emission zones located at z′ = 20nm (red circles) or z′ = 140nm (blue × symbols), along with the 2D substrate escape efficiency enhancement (green solid line, calculated according to Eqs. (24) and (22)).

Tables (3)

Tables Icon

Table 1 Wave Equation and Boundary Conditions for the 1D and 2D Cylindrical Green Functions

Tables Icon

Table 2 Recursive Relations of the Cylindrical 1D Green Function

Tables Icon

Table 3 Geometrical and Electrical Properties of a Prototype FOLED, Corresponding to Fig. 1 *

Equations (24)

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

J m e = m e I 0 δ ( φ φ ) δ ( ρ ρ ) ρ z ^
E t ( ρ , ρ ; ω ) = j k Z J s G ( ρ , ρ ) M s G ( ρ , ρ ) ρ , H t ( ρ , ρ ; ω ) = J s G ( ρ , ρ ) ρ + j k Y M s G ( ρ , ρ )
G ( ρ , ρ ) = 1 π ρ 0 g ( ν ) ( ρ , ρ ) cos [ ν ( φ φ ) ] d ν
g N + 1 ( ρ , ρ ) = n = 2 N + 1 ( 1 + Γ n 1 ) n = 1 N + 1 H n , n ( 2 ) H n , n 1 ( 2 ) h n , n ( 1 ) h n , n ( 2 ) h n , n 1 ( 1 ) h n , n 1 ( 2 ) k N + 1 ( h N + 1 , N + 1 ( 1 ) h N + 1 , N + 1 ( 2 ) ) l = 0 1 l 1 = 1 ( w 1 , 0 w 1 , 1 ) l s 1 = 0 l 1 + 1 p 1 = 0 s N 1 = 0 l N 1 + 1 p N 1 = 0 K ( 1 , N , l q , s q , p q ) s ^ 1 = 0 l ^ 1 + 1 p ^ 1 = 0 s ^ M = 0 l ^ M + 1 p ^ M = 0 K ^ ( M , 1 , l ^ q , s q , p ^ q ) n = 2 N [ l ˜ n n = 1 ( Γ n 1 w n , n w n , n 1 ) l ˜ n n + 1 s ˜ n n = 0 l ˜ n n + 1 p ˜ n n = 0 s ˜ n + 1 n = 0 l ˜ n + 1 n + 1 p ˜ n + 1 n = 0 s ˜ N 1 n = 0 l ˜ N 1 n + 1 p ˜ N 1 n = 0 K ( n , N , l ˜ q n , s ˜ q n , p ˜ q n ) ]
S ρ ( φ φ ) = lim ρ 8 P N + 1 d ω p ( ω ) G N + 1 ( ρ , ρ ) [ ρ G N + 1 ( ρ , ρ ) ] *
H ν ( 1 ) ( Ω ) [ ( 2 / π ) 2 Ω 2 ν 2 ] 1 / 4 { exp { j [ ( Ω 2 ν 2 ) 1 / 2 ν ( π 2 arcsin ν Ω ) ] j π 4 } ν < | Ω | exp { [ ( ν 2 Ω 2 ) 1 / 2 ν arccosh ν Ω ] j π 2 } ν > | Ω |
w l , m = ( w ^ l , m ) 1 { e 2 j Ω l , m ( cos α l , m + α l , m sin α l , m ) ν < | Ω l , m | e j ( π ν π 2 ) ν > | Ω l , m |
± ( φ φ ) + n = 1 N + 1 ( α n , n α n , n 1 ) + 2 n = 1 N ( α n , n α n , n 1 ) ( n + l n + p = 2 n l ˜ n p ) + 2 n = M 1 ( α n , n + 1 α n , n ) ( l ^ n + 1 ) + ( l ^ ( M + 1 ) + 1 ) ( 2 α ( M + 1 ) , M π ) = 0
G N ± 1 ± ( ρ , ρ ) = 1 2 π l = 0 1 l 1 = 1 s 1 = 0 l 1 + 1 p 1 = 0 s N 1 = 0 l N 1 + 1 p N 1 = 0 s ^ 1 = 0 l ^ 1 + 1 p ^ 1 = 0 s ^ M = 0 l ^ M + 1 p ^ M = 0 l ˜ 2 2 = 1 s ˜ 2 2 = 0 l ˜ 2 2 + 1 p ˜ 2 2 = 0 s ˜ N 1 2 = 0 l ˜ N 1 2 + 1 p ˜ N 1 2 = 0 l ˜ N N = 1 s ˜ N N = 0 l ˜ N N + 1 p ˜ N N = 0 s ˜ N 1 N = 0 l ˜ N 1 N + 1 p ˜ N 1 N = 0 { 2 π Q l q , l ^ q , l , l ˜ q p ( ν ) n = 2 N + 1 ( 1 + Γ n 1 ) n = 1 N + 1 H n , n ( 2 ) H n , n 1 ( 2 ) h n , n ( 1 ) h n , n ( 2 ) h n , n 1 ( 1 ) h n , n 1 ( 2 ) k N + 1 ( h N + 1 , N + 1 ( 1 ) h N + 1 , N + 1 ( 2 ) ) ( w 1 , 0 w 1 , 1 ) l ( Γ 1 w 2 , 2 w 2 , 1 ) l ˜ 2 2 + 1 ( Γ N 1 w N , N w N , N 1 ) l ˜ N N + 1 K ( 2 , N , l ˜ q 2 , s ˜ q 2 , p ˜ q 2 ) K ( N , N , l ˜ q N , s ˜ q N , p ˜ q N ) K ( 1 , N , l q , s q , p q ) K ^ ( M , 1 , l ^ q , s ^ q , p ^ q ) e ± j ν ( φ φ ) } ν = ν s ± ( l q , l ^ q , l , l ˜ q p )
Q l q , l ^ q , l , l ˜ q p ( ν ) = n = 1 N + 1 [ ( Ω n , n cos α n , n ) 1 ( Ω n , n 1 cos α n , n 1 ) 1 ] + 2 n = 1 N ( n + l n + p = 2 n l ˜ n p ) [ ( Ω n , n cos α n , n ) 1 ( Ω n , n 1 cos α n , n 1 ) 1 ] + 2 n = M 1 ( l ^ n + 1 ) [ ( Ω n , n + 1 cos α n , n + 1 ) 1 ( Ω n , n cos α n , n ) 1 ] + 2 ( l ^ ( M + 1 ) + 1 ) ( Ω ( M + 1 ) , M cos α ( M + 1 ) , M ) 1
α n + 1 , n + 1 α n + 1 , n Δ a n + 1 , n a n tan α n + 1 , n
| φ φ | + ( α 4 , 4 α 4 , 3 ) + [ 2 ( l 3 + 1 ) + 1 ] ( α 3 , 3 α 3 , 2 ) = 0
G 4 ( ρ , ρ ) 1 2 π ρ ρ l 1 = 1 l max s 1 = 0 l 1 + 1 s 2 = 0 l 2 + 1 { e j ν | φ φ | 2 j k 4 cos α 4 , 4 t I S ( ν ) t D R ( ν ) K W M ( l 1 , s 1 , s 2 ) ( ν ) } ν = ν s ( l 3 )
k n L n , m , l = Ω n , m ( cos α n , m + α n , m sin α n , m ) Ω n , l ( cos α n , l + α n , l sin α n , l )
t I S ( ν ) = 1 Γ ^ 1 exp { 2 j k 1 L 1 , 0 , 1 } = l = 0 1 K I S ( l ) ( ν )
t D R ( ν ) = n = 2 4 ( 1 + Γ n 1 ) n = 1 4 cos α n , n cos α n , n 1 exp { j k n L n , n , n 1 }
K W M ( l 1 , s 1 , s 2 ) ( ν ) = 1 Q l 3 ( ν ) ( l 1 + 1 s 1 ) ( l 2 + 1 s 2 ) [ Γ ^ 1 Γ 1 ] s 1 [ Γ ^ 1 ( 1 Γ 1 2 ) Γ 2 ] s 2 [ Γ ^ 1 ( 1 Γ 1 2 ) ( 1 Γ 2 2 ) Γ 3 ] l 3 + 1 exp { 2 j [ ( l 1 + 1 ) k 1 L 1 , 1 , 1 + ( l 2 + 1 ) k 2 L 2 , 2 , 1 + ( l 3 + 1 ) k 3 L 3 , 3 , 2 ] }
S ρ ( | φ φ | ) P 4 π ρ l 1 = 1 3 s 1 = 0 l 1 + 1 s 2 = 0 l 2 + 1 l 1 ¯ = 1 3 s 1 ¯ = 0 l 1 ¯ + 1 l = 0 1 l ¯ = 0 1 { | t D R ( ν ) | 2 k 4 ρ cos α 4 , 4 K I S ( l ) ( ν ) K W M ( l 1 , s 1 , s 2 ) ( ν ) [ K I S ( l ¯ ) ( ν ) K W M ( l 1 ¯ , s 1 ¯ , s 2 ¯ ) ( ν ) ] * exp { 1 2 ( k 4 L c ) 2 [ 2 ( l l ¯ ) { k 1 L 1 , 0 , 1 } + 2 ( l 1 l 1 ¯ ) { k 1 L 1 , 1 , 1 } + 2 ( l 2 l 2 ¯ ) { k 2 L 2 , 2 , 1 } ] 2 } } ν = ν s ( l 3 = l 3 ¯ )
| φ φ | = θ + arcsin ( k 4 a 3 k 3 a 2 sin θ ) arcsin ( k 4 k 3 sin θ )
| φ φ | view ( R ) = π / 2 + arcsin ( k 4 a 3 k 3 a 2 ) arcsin ( k 4 k 3 )
a 3 a 2 = k 3 k 4 R opt k 4 k 3 k 4 d 3 | φ φ | view ( R opt ) = π arcsin ( k 4 k 3 )
η esc , 3 D PPOLED = 1 cos θ c = 1 1 ( k 4 k 3 ) 2 , η esc , 2 D PPOLED = 2 θ c π = 2 π arcsin ( k 4 k 3 )
sin α c = sin ( α 3 , 2 ) c = a 3 a 2 sin θ c = k 4 a 3 k 3 a 2
η esc , 3 D FOLED ( R ) = 1 cos α c = 1 1 ( k 4 a 3 k 3 a 2 ) 2 , η esc , 2 D FOLED ( R ) = 2 α c π = 2 π arcsin ( k 4 a 3 k 3 a 2 )

Metrics