Abstract

We have developed a rigorous scattering matrix theory of light emission from periodically structured media using a Green’s function approach. We computationally simulate the spectral power inside the structure, incorporating Purcell factor enhancements, and find internal waveguiding modes and plasmonic losses. We simulate the light-outcoupling factor ${\eta _{\text{out}}}$ and describe how corrugations and structured media can enhance ${\eta _{\text{out}}}$. We have extended our framework to describe non-periodic disordered arrays as well as aperiodic quasi-crystalline arrays. Flat organic light-emitting diodes (OLEDs) have low outcoupling with ${\eta _{\text{out}}} \sim{20}\%$ since most of the light is trapped in waveguided modes in higher index layers or lost to plasmonic excitations at the metal cathode. Periodically corrugated OLEDs can achieve highly enhanced ${\eta _{\text{out}}} \sim {65}\% {-} {70}\%$ for pitch values between 1000 and 2000 nm, representing an enhancement factor $ \gt {3}$ over planar structures. Periodic corrugations strongly diffract trapped waveguided and plasmonic modes to the emissive air cone. Disordered templates also can cause significant enhanced outcoupling with ${\eta _{\text{out}}} \sim {50}\% {-} {55}\%$ for smaller nearest-neighbor separations of 300–400 nm. Quasi-crystalline tilings can also lead to enhancements of ${\eta _{\text{out}}} \sim {50}\% {-} {55}\%$. This framework can be utilized to design novel structured media that can generate high light extraction.

© 2021 Optical Society of America

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References

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2021 (1)

Y. Zhang and R. Biswas, “High light outcoupling efficiency from periodically corrugated OLEDs,” ACS Omega 6, 9291–9301 (2021).
[Crossref]

2020 (1)

C. W. Duncan, S. Manna, and A. E. B. Nielsen, “Topological models in rotationally symmetric quasicrystals,” Phys. Rev. B 101, 115413 (2020).
[Crossref]

2018 (2)

Y. Qu, J. Kim, C. Coburn, and S. R. Forrest, “Efficient, nonintrusive outcoupling in organic light emitting devices using embedded microlens arrays,” ACS Photon. 5, 2453–2458 (2018).
[Crossref]

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

2017 (2)

2015 (2)

M. Gather and S. Reineke, “Recent advances in light outcoupling from white organic light-emitting diodes,” J. Photon. Energy 5, 057607 (2015).
[Crossref]

W. Youan, J. Lee, M. Xu, R. Singh, and F. So, “Corrugated sapphire substrates for OLED light extraction,” ACS Appl. Mater. Interfaces 7, 8974–8978 (2015).
[Crossref]

2014 (2)

A. Egel and U. Lemmer, “Dipole emission in stratified media with multiple spherical scatterers: enhanced outcoupling from OLEDs,” J. Quant. Spectrosc. Radiat. Transfer 148, 165–176 (2014).
[Crossref]

A. Peer and R. Biswas, “Nano-photonic organic solar cell architecture for advanced light trapping with dual photonic crystals,” ACS Photon. 1, 840–847 (2014).
[Crossref]

2013 (1)

Z. V. Vardeny, A. Nahata, and A. Agarwal, “Optics of photonic quasicrystals,” Nat. Photonics 7, 177–188 (2013).
[Crossref]

2012 (2)

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

C. H. Chang, K.-Y. Chang, Y.-J. Lo, S.-J. Chang, and H.-H. Chang, “Fourfold power efficiency improvement in organic light-emitting devices using an embedded nanocomposite scattering layer,” Org. Electr. 13, 1073–1080 (2012).
[Crossref]

2011 (1)

2010 (1)

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

2003 (1)

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[Crossref]

2002 (1)

A. Moller and S. R. Forrest, “Improved Light out-coupling in OLEDS employing ordered microlens arrays,” J. Appl. Phys. 91, 3324–3327 (2002).
[Crossref]

1998 (3)

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

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

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80, 956–959 (1998).
[Crossref]

1997 (1)

1996 (1)

D. L. D. Kaspar and E. Fontano, “Five-fold symmetry in crystalline quasicrystal lattices,” Proc. Natl. Acad. Sci. USA 93, 14271–14278 (1996).
[Crossref]

1977 (1)

M. Gardner, “Extraordinary nonperiodic tiling that enriches the theory of tiles,” Sci. Am. 236, 110–119 (1977).
[Crossref]

1946 (1)

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

Agarwal, A.

Z. V. Vardeny, A. Nahata, and A. Agarwal, “Optics of photonic quasicrystals,” Nat. Photonics 7, 177–188 (2013).
[Crossref]

Araoka, F.

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

Biswas, R.

Y. Zhang and R. Biswas, “High light outcoupling efficiency from periodically corrugated OLEDs,” ACS Omega 6, 9291–9301 (2021).
[Crossref]

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

A. Peer, R. Biswas, J.-M. Park, R. Shinar, and J. Shinar, “Light management in perovskite solar cells and organic LEDs with microlens arrays,” Opt. Express 25, 10704–10709 (2017).
[Crossref]

A. Peer and R. Biswas, “Nano-photonic organic solar cell architecture for advanced light trapping with dual photonic crystals,” ACS Photon. 1, 840–847 (2014).
[Crossref]

Chan, C. T.

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80, 956–959 (1998).
[Crossref]

Chan, Y. S.

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80, 956–959 (1998).
[Crossref]

Chang, C. H.

C. H. Chang, K.-Y. Chang, Y.-J. Lo, S.-J. Chang, and H.-H. Chang, “Fourfold power efficiency improvement in organic light-emitting devices using an embedded nanocomposite scattering layer,” Org. Electr. 13, 1073–1080 (2012).
[Crossref]

Chang, H.-H.

C. H. Chang, K.-Y. Chang, Y.-J. Lo, S.-J. Chang, and H.-H. Chang, “Fourfold power efficiency improvement in organic light-emitting devices using an embedded nanocomposite scattering layer,” Org. Electr. 13, 1073–1080 (2012).
[Crossref]

Chang, K.-Y.

C. H. Chang, K.-Y. Chang, Y.-J. Lo, S.-J. Chang, and H.-H. Chang, “Fourfold power efficiency improvement in organic light-emitting devices using an embedded nanocomposite scattering layer,” Org. Electr. 13, 1073–1080 (2012).
[Crossref]

Chang, S.-J.

C. H. Chang, K.-Y. Chang, Y.-J. Lo, S.-J. Chang, and H.-H. Chang, “Fourfold power efficiency improvement in organic light-emitting devices using an embedded nanocomposite scattering layer,” Org. Electr. 13, 1073–1080 (2012).
[Crossref]

Coburn, C.

Y. Qu, J. Kim, C. Coburn, and S. R. Forrest, “Efficient, nonintrusive outcoupling in organic light emitting devices using embedded microlens arrays,” ACS Photon. 5, 2453–2458 (2018).
[Crossref]

Constant, K.

Duncan, C. W.

C. W. Duncan, S. Manna, and A. E. B. Nielsen, “Topological models in rotationally symmetric quasicrystals,” Phys. Rev. B 101, 115413 (2020).
[Crossref]

Egel, A.

A. Egel and U. Lemmer, “Dipole emission in stratified media with multiple spherical scatterers: enhanced outcoupling from OLEDs,” J. Quant. Spectrosc. Radiat. Transfer 148, 165–176 (2014).
[Crossref]

Fontano, E.

D. L. D. Kaspar and E. Fontano, “Five-fold symmetry in crystalline quasicrystal lattices,” Proc. Natl. Acad. Sci. USA 93, 14271–14278 (1996).
[Crossref]

Forrest, S. R.

Y. Qu, J. Kim, C. Coburn, and S. R. Forrest, “Efficient, nonintrusive outcoupling in organic light emitting devices using embedded microlens arrays,” ACS Photon. 5, 2453–2458 (2018).
[Crossref]

A. Moller and S. R. Forrest, “Improved Light out-coupling in OLEDS employing ordered microlens arrays,” J. Appl. Phys. 91, 3324–3327 (2002).
[Crossref]

Furno, M.

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

Gan, Z.

Gardner, M.

M. Gardner, “Extraordinary nonperiodic tiling that enriches the theory of tiles,” Sci. Am. 236, 110–119 (1977).
[Crossref]

Gather, M.

M. Gather and S. Reineke, “Recent advances in light outcoupling from white organic light-emitting diodes,” J. Photon. Energy 5, 057607 (2015).
[Crossref]

Hall, D. G.

Hippola, C.

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

Ho, K. M.

Hofmann, S.

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

Ishikawa, K.

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

Jeong, S. M.

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

Kaspar, D. L. D.

D. L. D. Kaspar and E. Fontano, “Five-fold symmetry in crystalline quasicrystal lattices,” Proc. Natl. Acad. Sci. USA 93, 14271–14278 (1996).
[Crossref]

Kaudal, R.

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

Kim, J.

Y. Qu, J. Kim, C. Coburn, and S. R. Forrest, “Efficient, nonintrusive outcoupling in organic light emitting devices using embedded microlens arrays,” ACS Photon. 5, 2453–2458 (2018).
[Crossref]

Koo, W. H.

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

Lee, J.

W. Youan, J. Lee, M. Xu, R. Singh, and F. So, “Corrugated sapphire substrates for OLED light extraction,” ACS Appl. Mater. Interfaces 7, 8974–8978 (2015).
[Crossref]

Lemmer, U.

A. Egel and U. Lemmer, “Dipole emission in stratified media with multiple spherical scatterers: enhanced outcoupling from OLEDs,” J. Quant. Spectrosc. Radiat. Transfer 148, 165–176 (2014).
[Crossref]

Leo, K.

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

Leung, W. Y.

Li, Z.-Y.

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[Crossref]

Lin, L.-L.

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[Crossref]

Liu, R.

Liu, Z. Y.

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80, 956–959 (1998).
[Crossref]

Lo, Y.-J.

C. H. Chang, K.-Y. Chang, Y.-J. Lo, S.-J. Chang, and H.-H. Chang, “Fourfold power efficiency improvement in organic light-emitting devices using an embedded nanocomposite scattering layer,” Org. Electr. 13, 1073–1080 (2012).
[Crossref]

Lüssem, S. B.

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

Manna, E.

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

Manna, S.

C. W. Duncan, S. Manna, and A. E. B. Nielsen, “Topological models in rotationally symmetric quasicrystals,” Phys. Rev. B 101, 115413 (2020).
[Crossref]

Meerheim, R.

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

Moller, A.

A. Moller and S. R. Forrest, “Improved Light out-coupling in OLEDS employing ordered microlens arrays,” J. Appl. Phys. 91, 3324–3327 (2002).
[Crossref]

Nahata, A.

Z. V. Vardeny, A. Nahata, and A. Agarwal, “Optics of photonic quasicrystals,” Nat. Photonics 7, 177–188 (2013).
[Crossref]

Nielsen, A. E. B.

C. W. Duncan, S. Manna, and A. E. B. Nielsen, “Topological models in rotationally symmetric quasicrystals,” Phys. Rev. B 101, 115413 (2020).
[Crossref]

Nishimura, S.

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

Nyets, K. A.

Park, J. M.

Park, J.-M.

Peer, A.

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

A. Peer, R. Biswas, J.-M. Park, R. Shinar, and J. Shinar, “Light management in perovskite solar cells and organic LEDs with microlens arrays,” Opt. Express 25, 10704–10709 (2017).
[Crossref]

A. Peer and R. Biswas, “Nano-photonic organic solar cell architecture for advanced light trapping with dual photonic crystals,” ACS Photon. 1, 840–847 (2014).
[Crossref]

Pendry, J. B.

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

Purcell, E. M.

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

Qu, Y.

Y. Qu, J. Kim, C. Coburn, and S. R. Forrest, “Efficient, nonintrusive outcoupling in organic light emitting devices using embedded microlens arrays,” ACS Photon. 5, 2453–2458 (2018).
[Crossref]

Reineke, S.

M. Gather and S. Reineke, “Recent advances in light outcoupling from white organic light-emitting diodes,” J. Photon. Energy 5, 057607 (2015).
[Crossref]

Shinar, J.

Shinar, R.

Singh, R.

W. Youan, J. Lee, M. Xu, R. Singh, and F. So, “Corrugated sapphire substrates for OLED light extraction,” ACS Appl. Mater. Interfaces 7, 8974–8978 (2015).
[Crossref]

Slafer, W. D.

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

So, F.

W. Youan, J. Lee, M. Xu, R. Singh, and F. So, “Corrugated sapphire substrates for OLED light extraction,” ACS Appl. Mater. Interfaces 7, 8974–8978 (2015).
[Crossref]

Sullivan, K. G.

Takezoe, H.

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

Tan, S.

Toyooka, T.

W. H. Koo, S. M. Jeong, F. Araoka, K. Ishikawa, S. Nishimura, T. Toyooka, and H. Takezoe, “Light extraction from organic light-emitting diodes enhanced by spontaneously formed buckles,” Nat. Photonics 4, 222–226 (2010).
[Crossref]

Trovato, T.

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

Tsang, L.

Vardeny, Z. V.

Z. V. Vardeny, A. Nahata, and A. Agarwal, “Optics of photonic quasicrystals,” Nat. Photonics 7, 177–188 (2013).
[Crossref]

Ward, A. J.

A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[Crossref]

Xiao, T.

C. Hippola, R. Kaudal, E. Manna, T. Xiao, A. Peer, R. Biswas, W. D. Slafer, T. Trovato, J. Shinar, and R. Shinar, “Enhanced light extraction from OLEDs fabricated on plastic substrates,” Adv. Opt. Mater. 6, 1701244 (2018).
[Crossref]

Xu, M.

W. Youan, J. Lee, M. Xu, R. Singh, and F. So, “Corrugated sapphire substrates for OLED light extraction,” ACS Appl. Mater. Interfaces 7, 8974–8978 (2015).
[Crossref]

Ye, Z.

Youan, W.

W. Youan, J. Lee, M. Xu, R. Singh, and F. So, “Corrugated sapphire substrates for OLED light extraction,” ACS Appl. Mater. Interfaces 7, 8974–8978 (2015).
[Crossref]

Zhang, Y.

Y. Zhang and R. Biswas, “High light outcoupling efficiency from periodically corrugated OLEDs,” ACS Omega 6, 9291–9301 (2021).
[Crossref]

ACS Appl. Mater. Interfaces (1)

W. Youan, J. Lee, M. Xu, R. Singh, and F. So, “Corrugated sapphire substrates for OLED light extraction,” ACS Appl. Mater. Interfaces 7, 8974–8978 (2015).
[Crossref]

ACS Omega (1)

Y. Zhang and R. Biswas, “High light outcoupling efficiency from periodically corrugated OLEDs,” ACS Omega 6, 9291–9301 (2021).
[Crossref]

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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)

Fig. 1.
Fig. 1. Schematic corrugated OLED structure depicting the scattering matrices F, B and the incident fields ${a}_{\text{in}}^ +$, ${a}_{\text{in}}^ -$ and total fields ${{b}^ +}$${{b}^ -}$ in the emissive layer. The profiles of the dipole emitter within the organic layer are shown.
Fig. 2.
Fig. 2. Spectral power inside a flat OLED as a function of the scaled in-plane wave vector $u$ for ETL thicknesses of (a) 20 nm, (b) 70 nm, (d) 160 nm, and (e) 250 nm. The schematic flat OLED is shown in (c).
Fig. 3.
Fig. 3. (a) Outcoupling as a function of the corrugation pitch $a$ for three different directions of the parallel wave vector $k_{||}$ in a conformally corrugated OLED. (b) Division of emitted power into outcoupled power, waveguided power, and plasmonic losses as a function of the pitch.
Fig. 4.
Fig. 4. (a) Nanostructured sapphire substrate, (b) corrugated sapphire substrate fabricated with 275 nm diameter silica sphere with FFTs shown in the inset, and (c) power spectra of the FFT as a function of the periodicity, indicating the nominal period. Reprinted with permission from Youan et al., ACS Appl. Mater. Interfaces 7, 8974–8978 (2015) [17]. Copyright 2015 American Chemical Society.
Fig. 5.
Fig. 5. (a) Uniform distribution of wave vectors (G) used to describe the diffraction from nearest-neighbor shell, illustrated with ${{\rm N}_G} = {38}$ G vectors. (b) Outcoupling and (c) losses as a function of the nearest-neighbor separation $l$ for green OLED emission at a wavelength of 530 nm. (d) Outcoupling and losses for blue emission with $\lambda = {470}\;{\rm nm}$. (e) Outcoupling and losses for red emission with $\lambda = {610}\;{\rm nm}$. ${{\rm N}_G} = {110}\;\textbf{G}$ vectors were used in calculations (b)–(e).
Fig. 6.
Fig. 6. Diffraction of waveguided modes from the high-index region of phase space to the center of the emission zone.
Fig. 7.
Fig. 7. (a) Penrose tile [16] composed of tiling a pentagon and rhombus; (b) schematic depiction of the diffraction spots of a quasi-crystalline structure. (c) Outcoupled power and losses for green OLED emission at 530 nm as a function of the nearest-neighbor separation $l$ for the second shell of diffraction G vectors in a quasi-crystalline structure. (d) Outcoupled power and losses for blue OLED emission at 470 nm. (e) Outcoupled power and losses for red OLED emission at 610 nm.

Equations (22)

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η E Q E = γ η s / t η r η out .
( c + b ) = ( F 11 F 12 F 21 F 22 ) ( a in + + b + 0 ) ,
( b + c ) = ( B 11 B 12 B 21 B 22 ) ( 0 a in + b ) .
c + = F 11 a in + + F 11 b + .
b = F 21 a in + + F 21 b + ,
b + = B 12 a in + B 12 b .
c = B 22 a in + B 22 b .
b + = ( I B 12 F 21 ) 1 ( B 21 a in + B 12 F 21 a in + ) ,
b = ( 1 F 21 B 12 ) 1 ( F 21 a in + + F 21 B 12 a in ) .
c + = F 11 a in + + F 11 ( 1 F 21 B 12 ) 1 ( F 21 a in + + F 21 B 12 a in ) .
u = k n ( o r g ) k 0 .
P ( j ) = 3 2 0 d u g j ( u ) { 1 + b TM + + b TM } ,
g TMv ( u ) = u 3 1 u 2 ; g TMh ( u ) = 1 2 u ( 1 u 2 ) 1 u 2 ; g TEh = u 1 u 2 .
h ( G ) = d x exp ( i G x ) h ( x ) .
Γ tot = d x d u g j ( u ) f ( u , x ) h ( x ) .
Γ tot = d u g ( u ) G f ( u , G ) h ( G ) .
P ( j ) = 0 d u g j ( u ) { 1 + G [ b i + ( u , G ) + b i ( u , G ) ] H ( G ) } .
P air ( j ) = 0 d u g j ( u ) { G k z 2 > 0 c j + ( u , G ) } .
k z 2 = ( ω c ) 2 ( u + G x ) 2 G y 2 .
η out = j = 1 3 P air ( j ) j = 1 3 P ( j ) .
| G | k | | = n org k 0 .
l = λ n org .

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