Abstract

The use of nanowires as active medium seems very promising for the development of high brightness LEDs. With a lower effective refractive index than bulk, semiconductor nanowire layers may lead to a high light extraction efficiency. We hereafter discuss the anisotropic properties of dense arrays of molecular beam epitaxy (MBE) grown gallium nitride (GaN) nanowires and the consequences on the optical design of nanowire based LEDs. In particular we show numerically that light extraction efficiency as high as 72% can be expected for GaN nanowires layer grown on a low cost Si substrate.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. J. Pauzauskie, and P. Yang, "Nanowire photonics," Mater. Today 9, 36-45 (2006).
    [CrossRef]
  2. T. Kuykendall, P. Ulrich, S. Aloni, and P. Yang, "Complete composition tunability of InGaN nanowires using a combinatorial approach," Nat. Mater. 6, 951-956 (2007).
    [CrossRef] [PubMed]
  3. M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
    [CrossRef]
  4. A. L. Henneghien, B. Gayral, Y. Désieres, and J. M. Gérard, "Simulation of waveguiding and emitting properties of semiconductor nanowires with hexagonal or circular sections," J. Opt. Soc. Am. B 26, 2396-2403 (2009).
    [CrossRef]
  5. S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
    [CrossRef] [PubMed]
  6. O. L. Muskens, M. T. Borgström, E. P. A. M. Bakkers, and J. Gómez Rivas, "Giant optical birefringence in ensembles of semiconductor nanowires," Appl. Phys. Lett. 89, 233117 (2006).
    [CrossRef]
  7. H. Sekiguchi, K. Kishino, and A. Kikuchi, "GaN/AlGaN nanocolumn ultraviolet light-emitting diodes grown on n-(111) Si by RF-plasma-assisted molecular beam epitaxy," Electron. Lett. 44, 151-152 (2008).
    [CrossRef]
  8. A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
    [CrossRef]
  9. R. Songmuang, O. Landré, and B. Daudin, "From nucleation to growth of catalyst-free GaN nanowires on thin AlN buffer layer," Appl. Phys. Lett. 91, 215902 (2007).
    [CrossRef]
  10. O. Landré, C. Bougerol, H. Renevier, and B. Daudin, "Nucleation mechanism of GaN nanowires grown on (111) Si by molecular beam epitaxy," Nanotechnology 20, 415602 (2009).
    [CrossRef] [PubMed]
  11. P. Yeh, Optical waves in layered media, (Wiley Interscience, 1988).
  12. A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
    [CrossRef]
  13. G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
    [CrossRef]
  14. G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
    [CrossRef]
  15. S. Huart, Polarisation de la lumiere, (Masson, 1994).
  16. J. Hao, and L. Zhou, "Electromagnetic wave scatterings by anisotropic metamaterials: Generalized 4 x 4 transfer-matrix method," Phys. Rev. B 77, 094201 (2008).
    [CrossRef]
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover publications, 1981).
  18. H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
    [CrossRef] [PubMed]
  19. K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding," IEEE J. Quantum Electron. 15, 566 (1979).
    [CrossRef]
  20. B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
    [CrossRef]
  21. J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, "Effects of dipole orientation and birefringence on the optical emission from thin films," Opt. Commun. 183, 109-121 (2000).
    [CrossRef]
  22. D. E. Aspnes, "Optical Properties of Thin Films," Thin Solid Films 89, 249-262 (1982).
    [CrossRef]
  23. Z. Liu, K. Wang, X. Luo, and S. Liu, "Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing," Opt. Express 9, 9398-9412 (2010).
    [CrossRef]
  24. H. Benisty, R. Stanley, and M. Mayer, "Method of source terms for dipole emission modification in modes of arbitrary planar structures," J. Opt. Soc. Am. A 5, 1192-1201 (1998).
    [CrossRef]

2010 (2)

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Z. Liu, K. Wang, X. Luo, and S. Liu, "Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing," Opt. Express 9, 9398-9412 (2010).
[CrossRef]

2009 (2)

A. L. Henneghien, B. Gayral, Y. Désieres, and J. M. Gérard, "Simulation of waveguiding and emitting properties of semiconductor nanowires with hexagonal or circular sections," J. Opt. Soc. Am. B 26, 2396-2403 (2009).
[CrossRef]

O. Landré, C. Bougerol, H. Renevier, and B. Daudin, "Nucleation mechanism of GaN nanowires grown on (111) Si by molecular beam epitaxy," Nanotechnology 20, 415602 (2009).
[CrossRef] [PubMed]

2008 (4)

J. Hao, and L. Zhou, "Electromagnetic wave scatterings by anisotropic metamaterials: Generalized 4 x 4 transfer-matrix method," Phys. Rev. B 77, 094201 (2008).
[CrossRef]

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[CrossRef] [PubMed]

H. Sekiguchi, K. Kishino, and A. Kikuchi, "GaN/AlGaN nanocolumn ultraviolet light-emitting diodes grown on n-(111) Si by RF-plasma-assisted molecular beam epitaxy," Electron. Lett. 44, 151-152 (2008).
[CrossRef]

2007 (3)

R. Songmuang, O. Landré, and B. Daudin, "From nucleation to growth of catalyst-free GaN nanowires on thin AlN buffer layer," Appl. Phys. Lett. 91, 215902 (2007).
[CrossRef]

T. Kuykendall, P. Ulrich, S. Aloni, and P. Yang, "Complete composition tunability of InGaN nanowires using a combinatorial approach," Nat. Mater. 6, 951-956 (2007).
[CrossRef] [PubMed]

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

2006 (2)

O. L. Muskens, M. T. Borgström, E. P. A. M. Bakkers, and J. Gómez Rivas, "Giant optical birefringence in ensembles of semiconductor nanowires," Appl. Phys. Lett. 89, 233117 (2006).
[CrossRef]

P. J. Pauzauskie, and P. Yang, "Nanowire photonics," Mater. Today 9, 36-45 (2006).
[CrossRef]

2000 (2)

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, "Effects of dipole orientation and birefringence on the optical emission from thin films," Opt. Commun. 183, 109-121 (2000).
[CrossRef]

1998 (2)

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

H. Benisty, R. Stanley, and M. Mayer, "Method of source terms for dipole emission modification in modes of arbitrary planar structures," J. Opt. Soc. Am. A 5, 1192-1201 (1998).
[CrossRef]

1997 (2)

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

1982 (1)

D. E. Aspnes, "Optical Properties of Thin Films," Thin Solid Films 89, 249-262 (1982).
[CrossRef]

1979 (1)

K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding," IEEE J. Quantum Electron. 15, 566 (1979).
[CrossRef]

Aloni, S.

T. Kuykendall, P. Ulrich, S. Aloni, and P. Yang, "Complete composition tunability of InGaN nanowires using a combinatorial approach," Nat. Mater. 6, 951-956 (2007).
[CrossRef] [PubMed]

André, B.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, "Optical Properties of Thin Films," Thin Solid Films 89, 249-262 (1982).
[CrossRef]

Bakkers, E. P. A. M.

O. L. Muskens, M. T. Borgström, E. P. A. M. Bakkers, and J. Gómez Rivas, "Giant optical birefringence in ensembles of semiconductor nanowires," Appl. Phys. Lett. 89, 233117 (2006).
[CrossRef]

Bandaru, P. R.

S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[CrossRef] [PubMed]

Barnes, W. L.

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, "Effects of dipole orientation and birefringence on the optical emission from thin films," Opt. Commun. 183, 109-121 (2000).
[CrossRef]

Bavencove, A.-L.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Benisty, H.

H. Benisty, R. Stanley, and M. Mayer, "Method of source terms for dipole emission modification in modes of arbitrary planar structures," J. Opt. Soc. Am. A 5, 1192-1201 (1998).
[CrossRef]

Borgström, M. T.

O. L. Muskens, M. T. Borgström, E. P. A. M. Bakkers, and J. Gómez Rivas, "Giant optical birefringence in ensembles of semiconductor nanowires," Appl. Phys. Lett. 89, 233117 (2006).
[CrossRef]

Bougerol, C.

O. Landré, C. Bougerol, H. Renevier, and B. Daudin, "Nucleation mechanism of GaN nanowires grown on (111) Si by molecular beam epitaxy," Nanotechnology 20, 415602 (2009).
[CrossRef] [PubMed]

Brandt, O.

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Busch, K.

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

Chen, H.-Y.

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

Chen, J.-S.

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

Chen, W.-C.

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

Cooper, M. L.

S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[CrossRef] [PubMed]

Craford, M. G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Dang, L. S.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Daudin, B.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

O. Landré, C. Bougerol, H. Renevier, and B. Daudin, "Nucleation mechanism of GaN nanowires grown on (111) Si by molecular beam epitaxy," Nanotechnology 20, 415602 (2009).
[CrossRef] [PubMed]

R. Songmuang, O. Landré, and B. Daudin, "From nucleation to growth of catalyst-free GaN nanowires on thin AlN buffer layer," Appl. Phys. Lett. 91, 215902 (2007).
[CrossRef]

Désieres, Y.

A. L. Henneghien, B. Gayral, Y. Désieres, and J. M. Gérard, "Simulation of waveguiding and emitting properties of semiconductor nanowires with hexagonal or circular sections," J. Opt. Soc. Am. B 26, 2396-2403 (2009).
[CrossRef]

Egawa, T.

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

Feuillet, G.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Garcia, J.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Gayral, B.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

A. L. Henneghien, B. Gayral, Y. Désieres, and J. M. Gérard, "Simulation of waveguiding and emitting properties of semiconductor nanowires with hexagonal or circular sections," J. Opt. Soc. Am. B 26, 2396-2403 (2009).
[CrossRef]

Gérard, J. M.

A. L. Henneghien, B. Gayral, Y. Désieres, and J. M. Gérard, "Simulation of waveguiding and emitting properties of semiconductor nanowires with hexagonal or circular sections," J. Opt. Soc. Am. B 26, 2396-2403 (2009).
[CrossRef]

Gilet, P.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Gómez Rivas, J.

O. L. Muskens, M. T. Borgström, E. P. A. M. Bakkers, and J. Gómez Rivas, "Giant optical birefringence in ensembles of semiconductor nanowires," Appl. Phys. Lett. 89, 233117 (2006).
[CrossRef]

Gwo, S.

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

Hao, J.

J. Hao, and L. Zhou, "Electromagnetic wave scatterings by anisotropic metamaterials: Generalized 4 x 4 transfer-matrix method," Phys. Rev. B 77, 094201 (2008).
[CrossRef]

Harbers, G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Henneberger, F.

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Henneghien, A. L.

A. L. Henneghien, B. Gayral, Y. Désieres, and J. M. Gérard, "Simulation of waveguiding and emitting properties of semiconductor nanowires with hexagonal or circular sections," J. Opt. Soc. Am. B 26, 2396-2403 (2009).
[CrossRef]

Ishikawa, H.

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

Jimbo, T.

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

Kikuchi, A.

H. Sekiguchi, K. Kishino, and A. Kikuchi, "GaN/AlGaN nanocolumn ultraviolet light-emitting diodes grown on n-(111) Si by RF-plasma-assisted molecular beam epitaxy," Electron. Lett. 44, 151-152 (2008).
[CrossRef]

Kirchner, A.

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

Kishino, K.

H. Sekiguchi, K. Kishino, and A. Kikuchi, "GaN/AlGaN nanocolumn ultraviolet light-emitting diodes grown on n-(111) Si by RF-plasma-assisted molecular beam epitaxy," Electron. Lett. 44, 151-152 (2008).
[CrossRef]

Krames, M. R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Kuykendall, T.

T. Kuykendall, P. Ulrich, S. Aloni, and P. Yang, "Complete composition tunability of InGaN nanowires using a combinatorial approach," Nat. Mater. 6, 951-956 (2007).
[CrossRef] [PubMed]

Landré, O.

O. Landré, C. Bougerol, H. Renevier, and B. Daudin, "Nucleation mechanism of GaN nanowires grown on (111) Si by molecular beam epitaxy," Nanotechnology 20, 415602 (2009).
[CrossRef] [PubMed]

R. Songmuang, O. Landré, and B. Daudin, "From nucleation to growth of catalyst-free GaN nanowires on thin AlN buffer layer," Appl. Phys. Lett. 91, 215902 (2007).
[CrossRef]

Levy, F.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Lin, H.-W.

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

Liu, S.

Z. Liu, K. Wang, X. Luo, and S. Liu, "Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing," Opt. Express 9, 9398-9412 (2010).
[CrossRef]

Liu, Z.

Z. Liu, K. Wang, X. Luo, and S. Liu, "Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing," Opt. Express 9, 9398-9412 (2010).
[CrossRef]

Luo, X.

Z. Liu, K. Wang, X. Luo, and S. Liu, "Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing," Opt. Express 9, 9398-9412 (2010).
[CrossRef]

Mayer, M.

H. Benisty, R. Stanley, and M. Mayer, "Method of source terms for dipole emission modification in modes of arbitrary planar structures," J. Opt. Soc. Am. A 5, 1192-1201 (1998).
[CrossRef]

Mookherjea, S.

S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[CrossRef] [PubMed]

Mueller, G. O.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Mueller-Mach, R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Muskens, O. L.

O. L. Muskens, M. T. Borgström, E. P. A. M. Bakkers, and J. Gómez Rivas, "Giant optical birefringence in ensembles of semiconductor nanowires," Appl. Phys. Lett. 89, 233117 (2006).
[CrossRef]

Pauzauskie, P. J.

P. J. Pauzauskie, and P. Yang, "Nanowire photonics," Mater. Today 9, 36-45 (2006).
[CrossRef]

Petermann, K.

K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding," IEEE J. Quantum Electron. 15, 566 (1979).
[CrossRef]

Ploog, K. H.

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Pougeoise, E.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Puls, J.

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Ramsteiner, M.

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Rau, B.

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Renevier, H.

O. Landré, C. Bougerol, H. Renevier, and B. Daudin, "Nucleation mechanism of GaN nanowires grown on (111) Si by molecular beam epitaxy," Nanotechnology 20, 415602 (2009).
[CrossRef] [PubMed]

Safonov, A.

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, "Effects of dipole orientation and birefringence on the optical emission from thin films," Opt. Commun. 183, 109-121 (2000).
[CrossRef]

Samuel, I. D. W.

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, "Effects of dipole orientation and birefringence on the optical emission from thin films," Opt. Commun. 183, 109-121 (2000).
[CrossRef]

Sekiguchi, H.

H. Sekiguchi, K. Kishino, and A. Kikuchi, "GaN/AlGaN nanocolumn ultraviolet light-emitting diodes grown on n-(111) Si by RF-plasma-assisted molecular beam epitaxy," Electron. Lett. 44, 151-152 (2008).
[CrossRef]

Shchekin, O. B.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Soga, T.

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

Songmuang, R.

R. Songmuang, O. Landré, and B. Daudin, "From nucleation to growth of catalyst-free GaN nanowires on thin AlN buffer layer," Appl. Phys. Lett. 91, 215902 (2007).
[CrossRef]

Soukoulis, C. M.

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

Stanley, R.

H. Benisty, R. Stanley, and M. Mayer, "Method of source terms for dipole emission modification in modes of arbitrary planar structures," J. Opt. Soc. Am. A 5, 1192-1201 (1998).
[CrossRef]

Tourbot, G.

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Ulrich, P.

T. Kuykendall, P. Ulrich, S. Aloni, and P. Yang, "Complete composition tunability of InGaN nanowires using a combinatorial approach," Nat. Mater. 6, 951-956 (2007).
[CrossRef] [PubMed]

Umeno, M.

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

Waltereit, P.

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Wang, G.

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

Wang, K.

Z. Liu, K. Wang, X. Luo, and S. Liu, "Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing," Opt. Express 9, 9398-9412 (2010).
[CrossRef]

Wasey, J. A. E.

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, "Effects of dipole orientation and birefringence on the optical emission from thin films," Opt. Commun. 183, 109-121 (2000).
[CrossRef]

Watanabe, J.

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

Wu, C.-Y.

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

Yang, P.

T. Kuykendall, P. Ulrich, S. Aloni, and P. Yang, "Complete composition tunability of InGaN nanowires using a combinatorial approach," Nat. Mater. 6, 951-956 (2007).
[CrossRef] [PubMed]

P. J. Pauzauskie, and P. Yang, "Nanowire photonics," Mater. Today 9, 36-45 (2006).
[CrossRef]

Yang, S.-H.

S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[CrossRef] [PubMed]

Yu, G.

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

Zhou, L.

J. Hao, and L. Zhou, "Electromagnetic wave scatterings by anisotropic metamaterials: Generalized 4 x 4 transfer-matrix method," Phys. Rev. B 77, 094201 (2008).
[CrossRef]

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

Appl. Phys. Lett. (4)

O. L. Muskens, M. T. Borgström, E. P. A. M. Bakkers, and J. Gómez Rivas, "Giant optical birefringence in ensembles of semiconductor nanowires," Appl. Phys. Lett. 89, 233117 (2006).
[CrossRef]

R. Songmuang, O. Landré, and B. Daudin, "From nucleation to growth of catalyst-free GaN nanowires on thin AlN buffer layer," Appl. Phys. Lett. 91, 215902 (2007).
[CrossRef]

G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, "Optical properties of wurtzite structure GaN on sapphire around fundamental absorption edge (0.78-4.77 eV) by spectroscopic ellipsometry and the optical transmission method," Appl. Phys. Lett. 70, 3209-3211 (1997).
[CrossRef]

B. Rau, P. Waltereit, O. Brandt, M. Ramsteiner, K. H. Ploog, J. Puls, and F. Henneberger, "In-plane polarization anisotropy of the spontaneous emission of M-plane GaN/(Al,Ga)N quantum wells," Appl. Phys. Lett. 77, 3343-3345 (2000).
[CrossRef]

Electron. Lett. (1)

H. Sekiguchi, K. Kishino, and A. Kikuchi, "GaN/AlGaN nanocolumn ultraviolet light-emitting diodes grown on n-(111) Si by RF-plasma-assisted molecular beam epitaxy," Electron. Lett. 44, 151-152 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding," IEEE J. Quantum Electron. 15, 566 (1979).
[CrossRef]

J. Disp. Technol. (1)

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol. 3, 160-175 (2007).
[CrossRef]

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

H. Benisty, R. Stanley, and M. Mayer, "Method of source terms for dipole emission modification in modes of arbitrary planar structures," J. Opt. Soc. Am. A 5, 1192-1201 (1998).
[CrossRef]

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

A. L. Henneghien, B. Gayral, Y. Désieres, and J. M. Gérard, "Simulation of waveguiding and emitting properties of semiconductor nanowires with hexagonal or circular sections," J. Opt. Soc. Am. B 26, 2396-2403 (2009).
[CrossRef]

Jpn. J. Appl. Phys. (1)

G. Yu, H. Ishikawa, T. Egawa, T. Soga, J. Watanabe, T. Jimbo, and M. Umeno, "Polarized Reflectance Spectroscopy and Spectroscopic Ellipsometry Determination of the Optical Anisotropy of Gallium nitride on Sapphire," Jpn. J. Appl. Phys. 36, L1029-L1031 (1997).
[CrossRef]

Mater. Today (1)

P. J. Pauzauskie, and P. Yang, "Nanowire photonics," Mater. Today 9, 36-45 (2006).
[CrossRef]

Nanotechnology (1)

O. Landré, C. Bougerol, H. Renevier, and B. Daudin, "Nucleation mechanism of GaN nanowires grown on (111) Si by molecular beam epitaxy," Nanotechnology 20, 415602 (2009).
[CrossRef] [PubMed]

Nat. Mater. (1)

T. Kuykendall, P. Ulrich, S. Aloni, and P. Yang, "Complete composition tunability of InGaN nanowires using a combinatorial approach," Nat. Mater. 6, 951-956 (2007).
[CrossRef] [PubMed]

Opt. Commun. (1)

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, "Effects of dipole orientation and birefringence on the optical emission from thin films," Opt. Commun. 183, 109-121 (2000).
[CrossRef]

Opt. Express (3)

H.-Y. Chen, H.-W. Lin, C.-Y. Wu, W.-C. Chen, J.-S. Chen, and S. Gwo, "Gallium nitride nanorod arrays as low-refractive-index-transparent media in the entire visible spectral region," Opt. Express 16, 8106-8116 (2008).
[CrossRef] [PubMed]

S.-H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[CrossRef] [PubMed]

Z. Liu, K. Wang, X. Luo, and S. Liu, "Precise optical modeling of blue light-emitting diodes by Monte Carlo ray-tracing," Opt. Express 9, 9398-9412 (2010).
[CrossRef]

Phys. Rev. B (2)

A. Kirchner, K. Busch, and C. M. Soukoulis, "Transport properties of random arrays of dielectric cylinders," Phys. Rev. B 57, 277-288 (1998).
[CrossRef]

J. Hao, and L. Zhou, "Electromagnetic wave scatterings by anisotropic metamaterials: Generalized 4 x 4 transfer-matrix method," Phys. Rev. B 77, 094201 (2008).
[CrossRef]

Phys. Stat. Solidi A (1)

A.-L. Bavencove, G. Tourbot, E. Pougeoise, J. Garcia, P. Gilet, F. Levy, B. André, G. Feuillet, B. Gayral, B. Daudin, and L. S. Dang, "GaN-based nanowires: from nanometric scale characterization to light emitting devices," Phys. Stat. Solidi A 207, 1425-1427 (2010).
[CrossRef]

Thin Solid Films (1)

D. E. Aspnes, "Optical Properties of Thin Films," Thin Solid Films 89, 249-262 (1982).
[CrossRef]

Other (3)

H. C. van de Hulst, Light Scattering by Small Particles (Dover publications, 1981).

S. Huart, Polarisation de la lumiere, (Masson, 1994).

P. Yeh, Optical waves in layered media, (Wiley Interscience, 1988).

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 (10)

Fig. 1
Fig. 1

SEM images of the two samples, scale bar=1μm.

Fig. 2
Fig. 2

Huygens’ construction.

Fig. 3
Fig. 3

Experimental setup: polarized-goniometry. Grey broken lines for excitation beam; black solid lines for reflective beam. ES, EP directions of the electric field vector for S and P polarization respectively. Detection apparatus: integrating sphere + photodiode + PC connection.

Fig. 4
Fig. 4

Reflectivity of the sample A at the wavelength of 488nm. The experimental values (resp. simulated values) are plotted in full line (resp. dashed line). The half dotted line represents the reflectivity of an isotropic medium with a refractive index n = n.

Fig. 5
Fig. 5

Reflectivity of the sample B at the wavelength of 488nm. The experimental values (resp. simulated values) are plotted in full line (resp. dashed line). The half dotted line represents the reflectivity of an isotropic medium with a refractive index n = n.

Fig. 6
Fig. 6

Reflectivity of the sample C at the wavelength of 488nm. The experimental values (resp. simulated values) are plotted in full line (resp. dashed line). The half dotted line represents the reflectivity of an isotropic medium with a refractive index n = n.

Fig. 7
Fig. 7

Description of the three nanowire layer cases. The nanowires are always surrounded by air. The superstrate can be made of air, SiO2 or a bilayer composed of 100 nm of ITO and of a semi-infinite epoxy medium. The source represents the QW.

Fig. 8
Fig. 8

Light extraction efficiency (ηext) toward the superstrate, for the array of semi-infinite nanowires sketched in Fig. 7(a). ηext is plotted as a function of the filling factor of the nanowire array, for an air (solid line), a SiO2 superstrate (dashed line) and a bilayer composed of 100 nm of ITO and of a semi-infinite epoxy medium (dashdotted line). Note that for this semi-infinite nanowire geometry, ηext is at most equal to 0.5.

Fig. 9
Fig. 9

Extraction efficiency (ηext) depending on the filling factor for an air (solid line), a SiO2 superstrate (dashed line) and a bilayer composed of 100 nm of ITO and of a semi-infinite epoxy medium (dashdotted line). The nanowires are 1.3 μm length on silicon substrate which corresponds to the geometry sketched in Fig. 7(b).

Fig. 10
Fig. 10

Extraction efficiency (ηext) depending on the filling factor for an air (solid line), a SiO2 superstrate (dashed line) and a bilayer composed of 100 nm of ITO and of a semi-infinite epoxy medium (dashdotted line). The nanowires are 1.3 μm length on silicon substrate and correspond to fig. 7(c). The p-doped section has a 6° aperture.

Tables (1)

Tables Icon

Table 1 Experimental and reference values of the ordinary and extraordinary permittivities. The experimental values have been determined by fitting the experimental reflectivity with an anisotropic model. The reference values are obtained from Eqs. (4) and (3) applied to a filling factor which is determined by the SEM images

Equations (8)

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

D = ɛ 0 ɛ E
ɛ = ( ɛ x 0 0 0 ɛ y 0 0 0 ɛ z )
ɛ = ɛ srg ( 1 + 2 f α 1 f α )
ɛ | | = f ɛ nw + ( 1 f ) ɛ srg
s x 2 n 2 ɛ x + s y 2 n 2 ɛ y + s z 2 n 2 ɛ z = 1 n 2
1 n e 2 = sin 2 θ ɛ || + cos 2 θ ɛ
n sup 2 = n o 2 = ɛ = ɛ srg ( 1 + 2 f α 1 f α )
n sup 2 = ɛ || = f ɛ nw + ( 1 f ) ɛ srg

Metrics