Abstract

We theoretically compare the excitation efficiency of waveguide and surface plasmon modes between quantum-mechanical and classical electromagnetic optical models of organic light-emitting diodes (OLEDs). A sophisticated optical model combining the two approaches is required to obtain an accurate calculation result and a comprehensive understanding of the micro-cavity effect in OLEDs. In the quantum-mechanical approach based on the Fermi’s golden rule, the mode expansion method is used to calculate the excitation efficiency. In the classical electromagnetic approach, the spectral power density calculated by the point dipole model is fitted by the summation of the Lorentzian line shape functions, which provide the excitation probability of each waveguide and surface plasmon modes. The mode coupling efficiencies on the basis of the two approaches are calculated in a bottom-emitting OLED when the position of a dipole emitter is varied. By comparing the calculation results, we confirm the equivalence of two approaches and obtain the better optical interpretation to the calculated excitation efficiency of waveguide and surface plasmon modes. The ratio of mode excitation efficiencies calculated by two approaches agrees well with each other except the contribution of the near-field absorption component.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

J. Kim, K. Kang, K.-Y. Kim, and J. Kim, “Origin of a sharp spectral peak near the critical angle in the spectral power density profile of top-emitting organic light-emitting diodes,” Jpn. J. Appl. Phys. 57(1), 012101 (2018).
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2016 (1)

K. Kang, Y. Lee, J. Kim, H. Lee, and B. Yang, “A generalized Fabry–Pérot formulation for optical modeling of organic light-emitting diodes considering the dipole orientation and light polarization,” IEEE Photonics J. 8(2), 1600519 (2016).
[Crossref]

2015 (4)

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

X. Wu, J. Liu, and G. He, “A highly conductive PEDOT:PSS film with the dipping treatment by hydroiodic acid as anode for organic light emitting diode,” Org. Electron. 22, 160–165 (2015).
[Crossref]

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

Y. R. Cho, H. S. Kim, Y.-J. Yu, and M. C. Suh, “Highly efficient organic light emitting diodes formed by solution processed red emitters with evaporated blue common layer structure,” Sci. Rep. 5(1), 15903 (2015).
[Crossref] [PubMed]

2014 (2)

2013 (2)

S. S. Jeong and J.-H. Ko, “Optical simulation study on the effect of diffusing substrate and pillow lenses on the outcoupling efficiency of organic light emitting diodes,” J. Opt. Soc. Korea 17(3), 269–274 (2013).
[Crossref]

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

2012 (4)

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B Condens. Matter Mater. Phys. 85(11), 115205 (2012).
[Crossref]

J.-H. Kim, S.-Y. Jung, and I.-K. Jeong, “Optical modeling for polarization-dependent optical power dissipation of thin-film organic solar cells at oblique incidence,” J. Opt. Soc. Korea 16(1), 6–12 (2012).
[Crossref]

A. K. Havarea, M. Cana, S. Demica, M. Kusb, and S. Icli, “The performance of OLEDs based on sorbitol doped PEDOT:PSS,” Synth. Met. 161(23–24), 2734–2738 (2012).
[Crossref]

M. Furno, M. C. Gather, B. Lüssem, and K. Leo, “Coupled plasmonic modes in organic planar microcavities,” Appl. Phys. Lett. 100(25), 253301 (2012).
[Crossref]

2011 (1)

2010 (3)

H. Cho, C. Yun, and S. Yoo, “Multilayer transparent electrode for organic light-emitting diodes: tuning its optical characteristics,” Opt. Express 18(4), 3404–3414 (2010).
[Crossref] [PubMed]

R. Meerheim, M. Furno, S. Hofmann, B. Lüssem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[Crossref]

L. Penninck, S. Mladenowski, and K. Neyts, “The effects of planar metallic interfaces on the radiation of nearby electrical dipoles,” J. Opt. 12(7), 075001 (2010).
[Crossref]

2008 (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(12), 123109 (2008).
[Crossref]

2007 (2)

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[Crossref]

K. Celebi, T. D. Heidel, and M. A. Baldo, “Simplified calculation of dipole energy transport in a multilayer stack using dyadic Green’s functions,” Opt. Express 15(4), 1762–1772 (2007).
[Crossref] [PubMed]

2006 (2)

J. F. Revelli, “Excitation of waveguide modes in organic light-emitting diode structures by classical dipole oscillators,” Appl. Opt. 45(27), 7151–7165 (2006).
[Crossref] [PubMed]

C.-L. Lin, T.-Y. Cho, C.-H. Chang, and C.-C. Wu, “Enhancing light outcoupling of organic light-emitting devices by locating emitters around the second antinode of the reflective metal,” Appl. Phys. Lett. 88(8), 081114 (2006).
[Crossref]

2005 (3)

C.-L. Lin, H.-W. Lin, and C.-C. Wu, “Examining microcavity organic light-emitting devices having two metal mirrors,” Appl. Phys. Lett. 87(2), 021101 (2005).
[Crossref]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6 (1), 3–9 (2005).
[Crossref]

J. F. Revelli, L. W. Tutt, and B. E. Kruschwitz, “Waveguide analysis of organic light-emitting diodes fabricated on surfaces with wavelength-scale periodic gratings,” Appl. Opt. 44(16), 3224–3237 (2005).
[Crossref] [PubMed]

2002 (2)

M. H. Lu and J. C. Sturm, “Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment,” J. Appl. Phys. 91(2), 595–604 (2002).
[Crossref]

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The Role of surface plasmons in organic light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8(2), 378–386 (2002).
[Crossref]

1998 (3)

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[Crossref]

K. A. Neyts, “Simulation of light emission from thin-film microcavities,” J. Opt. Soc. Am. A 15(4), 962–971 (1998).
[Crossref]

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).
[Crossref]

1997 (1)

1980 (1)

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B Condens. Matter 22(6), 3030–3038 (1980).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69(11), 681 (1946).

Asano, T.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[Crossref]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6 (1), 3–9 (2005).
[Crossref]

Bae, H. W.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Baldo, M. A.

Barnes, W. L.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The Role of surface plasmons in organic light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8(2), 378–386 (2002).
[Crossref]

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).
[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(12), 123109 (2008).
[Crossref]

Brütting, W.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Bulovic, V.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[Crossref]

Burrows, P. E.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[Crossref]

Callens, M. K.

Cana, M.

A. K. Havarea, M. Cana, S. Demica, M. Kusb, and S. Icli, “The performance of OLEDs based on sorbitol doped PEDOT:PSS,” Synth. Met. 161(23–24), 2734–2738 (2012).
[Crossref]

Celebi, K.

Chang, C.-H.

C.-L. Lin, T.-Y. Cho, C.-H. Chang, and C.-C. Wu, “Enhancing light outcoupling of organic light-emitting devices by locating emitters around the second antinode of the reflective metal,” Appl. Phys. Lett. 88(8), 081114 (2006).
[Crossref]

Cho, D.-H.

Cho, H.

Cho, T.-Y.

C.-L. Lin, T.-Y. Cho, C.-H. Chang, and C.-C. Wu, “Enhancing light outcoupling of organic light-emitting devices by locating emitters around the second antinode of the reflective metal,” Appl. Phys. Lett. 88(8), 081114 (2006).
[Crossref]

Cho, Y. R.

Y. R. Cho, H. S. Kim, Y.-J. Yu, and M. C. Suh, “Highly efficient organic light emitting diodes formed by solution processed red emitters with evaporated blue common layer structure,” Sci. Rep. 5(1), 15903 (2015).
[Crossref] [PubMed]

Chutinan, A.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6 (1), 3–9 (2005).
[Crossref]

de Groot, H.

Demica, S.

A. K. Havarea, M. Cana, S. Demica, M. Kusb, and S. Icli, “The performance of OLEDs based on sorbitol doped PEDOT:PSS,” Synth. Met. 161(23–24), 2734–2738 (2012).
[Crossref]

Forrest, S. R.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[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(12), 123109 (2008).
[Crossref]

Fuchs, C.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

Fujita, M.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[Crossref]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6 (1), 3–9 (2005).
[Crossref]

Furno, M.

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B Condens. Matter Mater. Phys. 85(11), 115205 (2012).
[Crossref]

M. Furno, M. C. Gather, B. Lüssem, and K. Leo, “Coupled plasmonic modes in organic planar microcavities,” Appl. Phys. Lett. 100(25), 253301 (2012).
[Crossref]

R. Meerheim, M. Furno, S. Hofmann, B. Lüssem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[Crossref]

Garbuzov, D. Z.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[Crossref]

Gather, M. C.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

M. Furno, M. C. Gather, B. Lüssem, and K. Leo, “Coupled plasmonic modes in organic planar microcavities,” Appl. Phys. Lett. 100(25), 253301 (2012).
[Crossref]

Gu, G.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[Crossref]

Hall, D. G.

Havarea, A. K.

A. K. Havarea, M. Cana, S. Demica, M. Kusb, and S. Icli, “The performance of OLEDs based on sorbitol doped PEDOT:PSS,” Synth. Met. 161(23–24), 2734–2738 (2012).
[Crossref]

He, G.

X. Wu, J. Liu, and G. He, “A highly conductive PEDOT:PSS film with the dipping treatment by hydroiodic acid as anode for organic light emitting diode,” Org. Electron. 22, 160–165 (2015).
[Crossref]

Heidel, T. D.

Hobson, P. A.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The Role of surface plasmons in organic light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8(2), 378–386 (2002).
[Crossref]

Hofmann, S.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B Condens. Matter Mater. Phys. 85(11), 115205 (2012).
[Crossref]

S. Hofmann, M. Thomschke, B. Lüssem, and K. Leo, “Top-emitting organic light-emitting diodes,” Opt. Express 19(S6Suppl 6), A1250–A1264 (2011).
[Crossref] [PubMed]

R. Meerheim, M. Furno, S. Hofmann, B. Lüssem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[Crossref]

Icli, S.

A. K. Havarea, M. Cana, S. Demica, M. Kusb, and S. Icli, “The performance of OLEDs based on sorbitol doped PEDOT:PSS,” Synth. Met. 161(23–24), 2734–2738 (2012).
[Crossref]

Ishihara, K.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6 (1), 3–9 (2005).
[Crossref]

Jang, J.-H.

Jeong, I.-K.

Jeong, S. S.

Jeong, W.-I.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Jung, S.-Y.

Kang, K.

J. Kim, K. Kang, K.-Y. Kim, and J. Kim, “Origin of a sharp spectral peak near the critical angle in the spectral power density profile of top-emitting organic light-emitting diodes,” Jpn. J. Appl. Phys. 57(1), 012101 (2018).
[Crossref]

K. Kang, Y. Lee, J. Kim, H. Lee, and B. Yang, “A generalized Fabry–Pérot formulation for optical modeling of organic light-emitting diodes considering the dipole orientation and light polarization,” IEEE Photonics J. 8(2), 1600519 (2016).
[Crossref]

Khalfin, V. B.

V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[Crossref]

Kim, G. H.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Kim, H. S.

Y. R. Cho, H. S. Kim, Y.-J. Yu, and M. C. Suh, “Highly efficient organic light emitting diodes formed by solution processed red emitters with evaporated blue common layer structure,” Sci. Rep. 5(1), 15903 (2015).
[Crossref] [PubMed]

Kim, J.

J. Kim, K. Kang, K.-Y. Kim, and J. Kim, “Origin of a sharp spectral peak near the critical angle in the spectral power density profile of top-emitting organic light-emitting diodes,” Jpn. J. Appl. Phys. 57(1), 012101 (2018).
[Crossref]

J. Kim, K. Kang, K.-Y. Kim, and J. Kim, “Origin of a sharp spectral peak near the critical angle in the spectral power density profile of top-emitting organic light-emitting diodes,” Jpn. J. Appl. Phys. 57(1), 012101 (2018).
[Crossref]

K. Kang, Y. Lee, J. Kim, H. Lee, and B. Yang, “A generalized Fabry–Pérot formulation for optical modeling of organic light-emitting diodes considering the dipole orientation and light polarization,” IEEE Photonics J. 8(2), 1600519 (2016).
[Crossref]

Kim, J.-H.

Kim, J.-J.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Kim, J.-W.

Kim, K.-H.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Kim, K.-Y.

J. Kim, K. Kang, K.-Y. Kim, and J. Kim, “Origin of a sharp spectral peak near the critical angle in the spectral power density profile of top-emitting organic light-emitting diodes,” Jpn. J. Appl. Phys. 57(1), 012101 (2018).
[Crossref]

Kim, S.-Y.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Ko, J.-H.

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(12), 123109 (2008).
[Crossref]

Kruschwitz, B. E.

Kusb, M.

A. K. Havarea, M. Cana, S. Demica, M. Kusb, and S. Icli, “The performance of OLEDs based on sorbitol doped PEDOT:PSS,” Synth. Met. 161(23–24), 2734–2738 (2012).
[Crossref]

Kwon, J. H.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Lampande, R.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Lee, H.

K. Kang, Y. Lee, J. Kim, H. Lee, and B. Yang, “A generalized Fabry–Pérot formulation for optical modeling of organic light-emitting diodes considering the dipole orientation and light polarization,” IEEE Photonics J. 8(2), 1600519 (2016).
[Crossref]

Lee, J.-H.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Lee, J.-I.

Lee, Y.

K. Kang, Y. Lee, J. Kim, H. Lee, and B. Yang, “A generalized Fabry–Pérot formulation for optical modeling of organic light-emitting diodes considering the dipole orientation and light polarization,” IEEE Photonics J. 8(2), 1600519 (2016).
[Crossref]

Lee, Y. K.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Leo, K.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

M. Furno, M. C. Gather, B. Lüssem, and K. Leo, “Coupled plasmonic modes in organic planar microcavities,” Appl. Phys. Lett. 100(25), 253301 (2012).
[Crossref]

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B Condens. Matter Mater. Phys. 85(11), 115205 (2012).
[Crossref]

S. Hofmann, M. Thomschke, B. Lüssem, and K. Leo, “Top-emitting organic light-emitting diodes,” Opt. Express 19(S6Suppl 6), A1250–A1264 (2011).
[Crossref] [PubMed]

R. Meerheim, M. Furno, S. Hofmann, B. Lüssem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[Crossref]

Lin, C.-L.

C.-L. Lin, T.-Y. Cho, C.-H. Chang, and C.-C. Wu, “Enhancing light outcoupling of organic light-emitting devices by locating emitters around the second antinode of the reflective metal,” Appl. Phys. Lett. 88(8), 081114 (2006).
[Crossref]

C.-L. Lin, H.-W. Lin, and C.-C. Wu, “Examining microcavity organic light-emitting devices having two metal mirrors,” Appl. Phys. Lett. 87(2), 021101 (2005).
[Crossref]

Lin, H.-W.

C.-L. Lin, H.-W. Lin, and C.-C. Wu, “Examining microcavity organic light-emitting devices having two metal mirrors,” Appl. Phys. Lett. 87(2), 021101 (2005).
[Crossref]

Liu, J.

X. Wu, J. Liu, and G. He, “A highly conductive PEDOT:PSS film with the dipping treatment by hydroiodic acid as anode for organic light emitting diode,” Org. Electron. 22, 160–165 (2015).
[Crossref]

Lu, M. H.

M. H. Lu and J. C. Sturm, “Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment,” J. Appl. Phys. 91(2), 595–604 (2002).
[Crossref]

Lukosz, W.

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B Condens. Matter 22(6), 3030–3038 (1980).
[Crossref]

Lüssem, B.

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B Condens. Matter Mater. Phys. 85(11), 115205 (2012).
[Crossref]

M. Furno, M. C. Gather, B. Lüssem, and K. Leo, “Coupled plasmonic modes in organic planar microcavities,” Appl. Phys. Lett. 100(25), 253301 (2012).
[Crossref]

S. Hofmann, M. Thomschke, B. Lüssem, and K. Leo, “Top-emitting organic light-emitting diodes,” Opt. Express 19(S6Suppl 6), A1250–A1264 (2011).
[Crossref] [PubMed]

R. Meerheim, M. Furno, S. Hofmann, B. Lüssem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[Crossref]

Marsman, H.

Mayr, C.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Meerheim, R.

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B Condens. Matter Mater. Phys. 85(11), 115205 (2012).
[Crossref]

R. Meerheim, M. Furno, S. Hofmann, B. Lüssem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[Crossref]

Mladenowski, S.

L. Penninck, S. Mladenowski, and K. Neyts, “The effects of planar metallic interfaces on the radiation of nearby electrical dipoles,” J. Opt. 12(7), 075001 (2010).
[Crossref]

Moon, C.-K.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Moon, J.-H.

Neyts, K.

Neyts, K. A.

Noda, S.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[Crossref]

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6 (1), 3–9 (2005).
[Crossref]

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(12), 123109 (2008).
[Crossref]

Oh, M.-C.

Park, M. J.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Park, Y.-S.

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Peeters, P.

Penninck, L.

Pode, R.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69(11), 681 (1946).

Reineke, S.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

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(12), 123109 (2008).
[Crossref]

Revelli, J. F.

Sage, I.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The Role of surface plasmons in organic light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8(2), 378–386 (2002).
[Crossref]

Scholz, R.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

Shin, J.-W.

Son, Y. H.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Song, W. J.

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

Sturm, J. C.

M. H. Lu and J. C. Sturm, “Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment,” J. Appl. Phys. 91(2), 595–604 (2002).
[Crossref]

Suh, M. C.

Y. R. Cho, H. S. Kim, Y.-J. Yu, and M. C. Suh, “Highly efficient organic light emitting diodes formed by solution processed red emitters with evaporated blue common layer structure,” Sci. Rep. 5(1), 15903 (2015).
[Crossref] [PubMed]

Sullivan, K. G.

ter Meulen, J. M.

Thomschke, M.

Tutt, L. W.

Wasey, J. A. E.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The Role of surface plasmons in organic light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8(2), 378–386 (2002).
[Crossref]

Wieczorek, M.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

Will, P.-A.

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

Wu, C.-C.

C.-L. Lin, T.-Y. Cho, C.-H. Chang, and C.-C. Wu, “Enhancing light outcoupling of organic light-emitting devices by locating emitters around the second antinode of the reflective metal,” Appl. Phys. Lett. 88(8), 081114 (2006).
[Crossref]

C.-L. Lin, H.-W. Lin, and C.-C. Wu, “Examining microcavity organic light-emitting devices having two metal mirrors,” Appl. Phys. Lett. 87(2), 021101 (2005).
[Crossref]

Wu, X.

X. Wu, J. Liu, and G. He, “A highly conductive PEDOT:PSS film with the dipping treatment by hydroiodic acid as anode for organic light emitting diode,” Org. Electron. 22, 160–165 (2015).
[Crossref]

Yang, B.

K. Kang, Y. Lee, J. Kim, H. Lee, and B. Yang, “A generalized Fabry–Pérot formulation for optical modeling of organic light-emitting diodes considering the dipole orientation and light polarization,” IEEE Photonics J. 8(2), 1600519 (2016).
[Crossref]

Yoo, S.

Yu, Y.-J.

Y. R. Cho, H. S. Kim, Y.-J. Yu, and M. C. Suh, “Highly efficient organic light emitting diodes formed by solution processed red emitters with evaporated blue common layer structure,” Sci. Rep. 5(1), 15903 (2015).
[Crossref] [PubMed]

Yun, C.

Adv. Funct. Mater. (1)

S.-Y. Kim, W.-I. Jeong, C. Mayr, Y.-S. Park, K.-H. Kim, J.-H. Lee, C.-K. Moon, W. Brütting, and J.-J. Kim, “Organic light-emitting diodes with 30% external quantum efficiency based on a horizontally oriented emitter,” Adv. Funct. Mater. 23(31), 3896–3900 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

R. Meerheim, M. Furno, S. Hofmann, B. Lüssem, and K. Leo, “Quantification of energy loss mechanisms in organic light-emitting diodes,” Appl. Phys. Lett. 97(25), 253305 (2010).
[Crossref]

M. Furno, M. C. Gather, B. Lüssem, and K. Leo, “Coupled plasmonic modes in organic planar microcavities,” Appl. Phys. Lett. 100(25), 253301 (2012).
[Crossref]

C.-L. Lin, H.-W. Lin, and C.-C. Wu, “Examining microcavity organic light-emitting devices having two metal mirrors,” Appl. Phys. Lett. 87(2), 021101 (2005).
[Crossref]

C.-L. Lin, T.-Y. Cho, C.-H. Chang, and C.-C. Wu, “Enhancing light outcoupling of organic light-emitting devices by locating emitters around the second antinode of the reflective metal,” Appl. Phys. Lett. 88(8), 081114 (2006).
[Crossref]

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

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The Role of surface plasmons in organic light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8(2), 378–386 (2002).
[Crossref]

IEEE Photonics J. (1)

K. Kang, Y. Lee, J. Kim, H. Lee, and B. Yang, “A generalized Fabry–Pérot formulation for optical modeling of organic light-emitting diodes considering the dipole orientation and light polarization,” IEEE Photonics J. 8(2), 1600519 (2016).
[Crossref]

J. Appl. Phys. (2)

M. H. Lu and J. C. Sturm, “Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment,” J. Appl. Phys. 91(2), 595–604 (2002).
[Crossref]

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(12), 123109 (2008).
[Crossref]

J. Mod. Opt. (1)

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).
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L. Penninck, S. Mladenowski, and K. Neyts, “The effects of planar metallic interfaces on the radiation of nearby electrical dipoles,” J. Opt. 12(7), 075001 (2010).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Opt. Soc. Korea (2)

Jpn. J. Appl. Phys. (1)

J. Kim, K. Kang, K.-Y. Kim, and J. Kim, “Origin of a sharp spectral peak near the critical angle in the spectral power density profile of top-emitting organic light-emitting diodes,” Jpn. J. Appl. Phys. 57(1), 012101 (2018).
[Crossref]

Nat. Photonics (1)

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1(8), 449–458 (2007).
[Crossref]

Opt. Express (5)

Org. Electron. (3)

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6 (1), 3–9 (2005).
[Crossref]

M. J. Park, Y. H. Son, G. H. Kim, R. Lampande, H. W. Bae, R. Pode, Y. K. Lee, W. J. Song, and J. H. Kwon, “Device performances of third order micro-cavity green top-emitting organic light emitting,” Org. Electron. 26, 458–463 (2015).
[Crossref]

X. Wu, J. Liu, and G. He, “A highly conductive PEDOT:PSS film with the dipping treatment by hydroiodic acid as anode for organic light emitting diode,” Org. Electron. 22, 160–165 (2015).
[Crossref]

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69(11), 681 (1946).

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W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B Condens. Matter 22(6), 3030–3038 (1980).
[Crossref]

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V. Bulović, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak microcavity effects in organic light-emitting devices,” Phys. Rev. B Condens. Matter Mater. Phys. 58(7), 3730–3740 (1998).
[Crossref]

C. Fuchs, P.-A. Will, M. Wieczorek, M. C. Gather, S. Hofmann, S. Reineke, K. Leo, and R. Scholz, “Enhanced light emission from top-emitting organic light-emitting diodes by optimizing surface plasmon polariton losses,” Phys. Rev. B Condens. Matter Mater. Phys. 92(24), 245306 (2015).
[Crossref]

M. Furno, R. Meerheim, S. Hofmann, B. Lüssem, and K. Leo, “Efficiency and rate of spontaneous emission in organic electroluminescent devices,” Phys. Rev. B Condens. Matter Mater. Phys. 85(11), 115205 (2012).
[Crossref]

Sci. Rep. (1)

Y. R. Cho, H. S. Kim, Y.-J. Yu, and M. C. Suh, “Highly efficient organic light emitting diodes formed by solution processed red emitters with evaporated blue common layer structure,” Sci. Rep. 5(1), 15903 (2015).
[Crossref] [PubMed]

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A. K. Havarea, M. Cana, S. Demica, M. Kusb, and S. Icli, “The performance of OLEDs based on sorbitol doped PEDOT:PSS,” Synth. Met. 161(23–24), 2734–2738 (2012).
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S. L. Chuang, Physics of Photonic Devices, 2nd. ed (Wiley, 2009).

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Figures (8)

Fig. 1
Fig. 1 Device structure of a bottom-emitting OLED along with the corresponding layer thickness and refractive index. The Alq3 layer works as both the electron-transport layer and EML. The dipole emitter with the wavelength of 520 nm is assumed to be isotropic and have the δ-distributed emission zone. In the inset, the configurations of the wave vectors in the EML and the definition of the in-plane component of the normalized wave vector u are shown.
Fig. 2
Fig. 2 Calculated dispersion curve of (a) TE-polarized and (b) TM-polarized waveguide or surface plasmon modes. The short-dotted line in orange color indicates the dispersion curve of light in vacuum.
Fig. 3
Fig. 3 Spatial distribution of the normalized electric field intensity of waveguide and surface plasmon modes in the bottom-emitting OLED. The purple dashed lines indicate three dipole emitter positions of A (50 nm), B (124 nm), and C (198 nm).
Fig. 4
Fig. 4 The calculated spectral power densities at three emitter positions for (a) TE and (b) TM polarizations. Five spectral peaks in the waveguide or surface plasmon modes are fitted to the summation of the respective Lorentzian line shape function.
Fig. 5
Fig. 5 Calculated mode coupling ratio of five waveguide or surface plasmon modes between the quantum-mechanical mode expansion and electromagnetic point dipole methods at various dipole emitter positions for (a) TE and (b) TM polarizations.
Fig. 6
Fig. 6 (a) Device structure of the thin bottom-emitting OLED, whose layer structure and refractive index is the same as that of the thick OLED in Fig. 1 except that the thickness of the Alq3 layer is reduced from 342 to 40 nm. (b) Calculation results of the normalized electric field intensity of waveguide and surface plasmon modes in the thin bottom-emitting OLED. The TE0, TM0, and TM1 modes of the thin OLED are matched with the TE1, TM1, and TM2 modes of the thick OLED in Fig. 3, respectively. The purple dashed lines indicate three dipole emitter positions of D (10 nm), E (20 nm), and F (30 nm).
Fig. 7
Fig. 7 The calculated spectral power densities of the thin bottom-emitting OLED at three emitter positions for (a) TE and (b) TM polarizations. Three spectral peaks in the waveguide or surface plasmon modes are fitted to the respective Lorentzian line shape function.
Fig. 8
Fig. 8 Comparison of the calculated mode coupling ratio of two TM-polarized confined modes between the quantum-mechanical mode expansion and electromagnetic point dipole methods at various dipole emitter positions in the thin bottom-emitting OLED. The mode coupling ratio for TE polarization is assumed to be 100% and not shown here because there is only one TE waveguide mode in the thin bottom-emitting OLED.

Tables (14)

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Table 1 Calculated effective mode index, phase index, group index, and photonic density of states of five confined modes in the bottom-emitting OLED shown in Fig. 1.

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Table 2 Calculated electric field intensity of five confined modes at three dipole positions

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Table 3 Calculated relative excitation efficiencies based on the quantum-mechanical mode expansion method

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Table 4 The values of the phase index and extinction coefficient of five confined modes obtained by the fitting to the spectral power density

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Table 5 Calculated relative excitation efficiency based on the electromagnetic point dipole model

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Table 6 The ratio of mode coupling efficiency calculated by the mode expansion method based on the quantum approach.

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Table 7 The ratio of mode coupling efficiency calculated by the point dipole method based on the classical electromagnetic approach.

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Table 8 Calculated effective mode index, phase index, group index, and photonic density of states of three confined modes in the thin bottom-emitting OLED shown in Fig. 6(a).

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Table 9 Calculated electric field intensity of three confined modes at three dipole positions in the thin bottom-emitting OLED

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Table 10 Calculated relative excitation efficiencies based on the quantum-mechanical mode expansion method in the thin bottom-emitting OLED

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Table 11 The values of the phase index and extinction coefficient of three confined modes obtained by the fitting to the spectral power density in the thin bottom-emitting OLED shown in Fig. 6(a).

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Table 12 Calculated relative excitation efficiency based on the electromagnetic point dipole model in the thin bottom-emitting OLED.

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Table 13 The ratio of mode coupling efficiency calculated by the mode expansion method based on the quantum approach in the thin bottom-emitting OLED.

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Table 14 The ratio of mode coupling efficiency calculated by the point dipole method based on the classical electromagnetic approach in the thin bottom-emitting OLED.

Equations (11)

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Γ= 2π ρ( ν ) | j| μ E ( r e )|i | 2 ,
Γ= Γ 0 λ 2 { l n phase TE (l) n group TE (l) | E ( r e ) | 2 + m n phase TM (m) n group TM (m) | E ( r e ) | 2 },
K v TM ( λ,u )= 3 4 Re[ u 2 1 u 2 ( 1+ a + TM )( 1+ a TM ) 1 a TM ],
K h TM ( λ,u )= 3 8 Re[ 1 u 2 ( 1 a + TM )( 1 a TM ) 1 a TM ],
K h TE ( λ,u )= 3 8 Re[ 1 1 u 2 ( 1+ a + TE )( 1+ a TE ) 1 a TE ],
a +() TE(TM) = r +() TE(TM) exp(2j k z,EML z +() ),
a TE(TM) = r + TE(TM) r TE(TM) exp(2j k z,EML d EML ),
K( λ,u )= 2 3 K h TE ( λ,u )+ 1 3 [ K v TM ( λ,u )+2 K h TM ( λ,u ) ].
F( λ )= 0 K( λ,u ) d u 2 =2 0 uK( λ,u ) du.
k z,EML = | k EML | 2 | k t,EML | 2 = k 0 ( n EML ) 2 ( n sp ) 2 ,
K( u ) l=1 2 σ l TE ( n EML u n phase,l TE ) 2 + ( κ l TE ) 2 + m=1 3 σ m TM ( n EML u n phase,m TM ) 2 + ( κ m TM ) 2 ,

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