Abstract

Vertically aligned gallium nitride (GaN) nanorod arrays grown by the catalyst-free, self-organized method based on plasma-assisted molecular-beam epitaxy are shown to behave as subwavelength optical media with low effective refractive indices. In the reflection spectra measured in the entire visible spectral region, strong reflectivity modulations are observed for all nanorod arrays, which are attributed to the effects of Fabry-Pérot microcavities formed within the nanorod arrays by the optically flat air/nanorods and nanorods/substrate interfaces. By analyzing the reflectivity interference fringes, we can quantitatively determine the refractive indices of GaN nanorod arrays as functions of light wavelength. We also propose a model for understanding the optical properties of GaN nanorod arrays in the transparent region. Using this model, good numerical fitting can be achieved for the reflectivity spectra.

© 2008 Optical Society of America

Full Article  |  PDF Article
More Like This
Lasing action in gallium nitride quasicrystal nanorod arrays

Shih-Pang Chang, Kuok-Pan Sou, Chieh-Han Chen, Yuh-Jen Cheng, Ji-Kai Huang, Chung-Hsiang Lin, Hao-Chung Kuo, Chun-Yen Chang, and Wen-Feng Hsieh
Opt. Express 20(11) 12457-12462 (2012)

Polarized photoluminescence from single GaN nanorods: Effects of optical confinement

Hung-Ying Chen, Yu-Chen Yang, Hon-Way Lin, Shih-Cheng Chang, and Shangjr Gwo
Opt. Express 16(17) 13465-13475 (2008)

Light transmission enhancement from hybrid ZnO micro-mesh and nanorod arrays with application to GaN-based light-emitting diodes

Zhengmao Yin, Xiaoyan Liu, Huining Wang, Yongzhong Wu, Xiaopeng Hao, Ziwu Ji, and Xiangang Xu
Opt. Express 21(23) 28531-28542 (2013)

View More...

References

  • View by:
  • Article Order
  • Year
  • Author
  • Publication
  1. P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
    [Crossref]
  2. S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
    [Crossref]
  3. M. Srinivasarao, “Nano-optics in the biological world: Beetles, butterflies, birds, and moths,” Chem. Rev. 99, 1935–1961 (1999).
    [Crossref]
  4. M. E. Motamedi, W. H. Southwell, and W. J. Gunning, “Antireflection surfaces in silicon using binary optics technique,” Appl. Opt. 31, 4371–4376 (1992).
    [Crossref] [PubMed]
  5. P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
    [Crossref]
  6. Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001).
    [Crossref]
  7. D. E. Aspnes, “Optical properties of thin-films,” Thin Solid Films 89, 249–262 (1982).
    [Crossref]
  8. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [Crossref] [PubMed]
  9. D. R. Smith, J. B Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
    [Crossref] [PubMed]
  10. J.-Q. Xi, J. K. Kim, and E. F. Schubert, “Silica nanorod-array films with very low refractive indices,” Nano Lett. 5, 1385–1387 (2005).
    [Crossref] [PubMed]
  11. J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T.-M. Lu, and S.-Y. Lin, “Very low-refractive-index optical thin films consisting of an array of SiO2 nanorods,” Opt. Lett. 31, 601–603 (2006).
    [Crossref] [PubMed]
  12. J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).
  13. C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
    [Crossref]
  14. Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
    [Crossref]
  15. H.-Y. Chen, H.-W. Lin, C.-H. Shen, and S. Gwo, “Structure and photoluminescence properties of epitaxially oriented GaN nanorods grown on Si (111) by plasma-assisted molecular-beam epitaxy,” Appl. Phys. Lett. 89, 243105 (2006), and references therein.
    [Crossref]
  16. E. F. Schubert, Light-Emitting Diodes, 2nd Edition (Cambridge University Press, Cambridge, 2006).
    [Crossref]
  17. T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
    [Crossref]
  18. T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468 (2004).
    [Crossref]
  19. J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
    [Crossref]
  20. A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
    [Crossref]
  21. J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
    [Crossref]
  22. J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
    [Crossref]
  23. H.-M. Kim, T. W. Kang, and K. S. Chung, “Nanoscale ultraviolet-light-emitting diodes using wide-bandgap gallium nitride nanorods,” Adv. Mater. 15, 567–569 (2003).
    [Crossref]
  24. H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
    [Crossref]
  25. A. Kikuchi, M. Kawai, M. Tada, and K. Kishino, “InGaN/GaN multiple quantum disk nanocolumn light-emitting diodes grown on (111) Si substrate,” Jpn. J. Appl. Phys. 43, L1524–L1526 (2004).
    [Crossref]
  26. A. Kikuchi, M. Tada, K. Miwa, and K. Kishino, “Growth and characterization of InGaN/GaN nanocolumn LED,” Proc. SPIE 6129, 612905 (2006).
    [Crossref]
  27. K. Kishino, A. Kikuchi, H. Sekiguchi, and S. Ishizawa, “InGaN/GaN nanocolumn LEDs emitting from blue to red,” Proc. SPIE 6473, 64730T (2007).
    [Crossref]
  28. 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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
    [Crossref]
  29. T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
    [Crossref]
  30. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, New York, 1981).

2007 (4)

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

K. Kishino, A. Kikuchi, H. Sekiguchi, and S. Ishizawa, “InGaN/GaN nanocolumn LEDs emitting from blue to red,” Proc. SPIE 6473, 64730T (2007).
[Crossref]

2006 (5)

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T.-M. Lu, and S.-Y. Lin, “Very low-refractive-index optical thin films consisting of an array of SiO2 nanorods,” Opt. Lett. 31, 601–603 (2006).
[Crossref] [PubMed]

A. Kikuchi, M. Tada, K. Miwa, and K. Kishino, “Growth and characterization of InGaN/GaN nanocolumn LED,” Proc. SPIE 6129, 612905 (2006).
[Crossref]

H.-Y. Chen, H.-W. Lin, C.-H. Shen, and S. Gwo, “Structure and photoluminescence properties of epitaxially oriented GaN nanorods grown on Si (111) by plasma-assisted molecular-beam epitaxy,” Appl. Phys. Lett. 89, 243105 (2006), and references therein.
[Crossref]

2005 (1)

J.-Q. Xi, J. K. Kim, and E. F. Schubert, “Silica nanorod-array films with very low refractive indices,” Nano Lett. 5, 1385–1387 (2005).
[Crossref] [PubMed]

2004 (7)

D. R. Smith, J. B Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468 (2004).
[Crossref]

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

A. Kikuchi, M. Kawai, M. Tada, and K. Kishino, “InGaN/GaN multiple quantum disk nanocolumn light-emitting diodes grown on (111) Si substrate,” Jpn. J. Appl. Phys. 43, L1524–L1526 (2004).
[Crossref]

2003 (1)

H.-M. Kim, T. W. Kang, and K. S. Chung, “Nanoscale ultraviolet-light-emitting diodes using wide-bandgap gallium nitride nanorods,” Adv. Mater. 15, 567–569 (2003).
[Crossref]

2001 (1)

Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001).
[Crossref]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref] [PubMed]

1999 (1)

M. Srinivasarao, “Nano-optics in the biological world: Beetles, butterflies, birds, and moths,” Chem. Rev. 99, 1935–1961 (1999).
[Crossref]

1997 (2)

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
[Crossref]

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
[Crossref]

1992 (1)

1982 (2)

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[Crossref]

D. E. Aspnes, “Optical properties of thin-films,” Thin Solid Films 89, 249–262 (1982).
[Crossref]

1973 (1)

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[Crossref]

Adachi, S.

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, “Optical properties of thin-films,” Thin Solid Films 89, 249–262 (1982).
[Crossref]

Benisty, H.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

Chang, Y.-H.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Chattopadhyay, S.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Chen, C.-F.

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Chen, H.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Chen, H.-Y.

H.-Y. Chen, H.-W. Lin, C.-H. Shen, and S. Gwo, “Structure and photoluminescence properties of epitaxially oriented GaN nanorods grown on Si (111) by plasma-assisted molecular-beam epitaxy,” Appl. Phys. Lett. 89, 243105 (2006), and references therein.
[Crossref]

Chen, K.-H.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Chen, L.-C.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Chen, M.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

Cho, J.

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

Cho, Y.-H.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

Choi, C. K.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Chung, K. S.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

H.-M. Kim, T. W. Kang, and K. S. Chung, “Nanoscale ultraviolet-light-emitting diodes using wide-bandgap gallium nitride nanorods,” Adv. Mater. 15, 567–569 (2003).
[Crossref]

Clapham, P. B.

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[Crossref]

Craford, M. G.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Das, D.

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

David, A.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

DenBaars, S. P.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

Egawa, 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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

Epler, J. E.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Fujii, T.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

Fuke, S.

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
[Crossref]

Gao, Y.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

Gardner, N. F.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Gessmann, T.

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

Gunning, W. J.

Gwo, S.

H.-Y. Chen, H.-W. Lin, C.-H. Shen, and S. Gwo, “Structure and photoluminescence properties of epitaxially oriented GaN nanorods grown on Si (111) by plasma-assisted molecular-beam epitaxy,” Appl. Phys. Lett. 89, 243105 (2006), and references therein.
[Crossref]

Hane, K.

Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001).
[Crossref]

Hsu, C.-H.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Hsu, Y.-K.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Hu, E. L.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

Huang, Y.-F.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Hutley, M. C.

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[Crossref]

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[Crossref]

Hwang, J.-S.

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Ishikawa, H.

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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

Ishizawa, S.

K. Kishino, A. Kikuchi, H. Sekiguchi, and S. Ishizawa, “InGaN/GaN nanocolumn LEDs emitting from blue to red,” Proc. SPIE 6473, 64730T (2007).
[Crossref]

Jen, Y.-J.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Jiang, H. X.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468 (2004).
[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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

Kanamori, Y.

Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001).
[Crossref]

Kang, T. W.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

H.-M. Kim, T. W. Kang, and K. S. Chung, “Nanoscale ultraviolet-light-emitting diodes using wide-bandgap gallium nitride nanorods,” Adv. Mater. 15, 567–569 (2003).
[Crossref]

Kawai, M.

A. Kikuchi, M. Kawai, M. Tada, and K. Kishino, “InGaN/GaN multiple quantum disk nanocolumn light-emitting diodes grown on (111) Si substrate,” Jpn. J. Appl. Phys. 43, L1524–L1526 (2004).
[Crossref]

Kawashima, T.

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
[Crossref]

Kikuchi, A.

K. Kishino, A. Kikuchi, H. Sekiguchi, and S. Ishizawa, “InGaN/GaN nanocolumn LEDs emitting from blue to red,” Proc. SPIE 6473, 64730T (2007).
[Crossref]

A. Kikuchi, M. Tada, K. Miwa, and K. Kishino, “Growth and characterization of InGaN/GaN nanocolumn LED,” Proc. SPIE 6129, 612905 (2006).
[Crossref]

A. Kikuchi, M. Kawai, M. Tada, and K. Kishino, “InGaN/GaN multiple quantum disk nanocolumn light-emitting diodes grown on (111) Si substrate,” Jpn. J. Appl. Phys. 43, L1524–L1526 (2004).
[Crossref]

Kim, D. Y.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

Kim, H.-M.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

H.-M. Kim, T. W. Kang, and K. S. Chung, “Nanoscale ultraviolet-light-emitting diodes using wide-bandgap gallium nitride nanorods,” Adv. Mater. 15, 567–569 (2003).
[Crossref]

Kim, J. K.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T.-M. Lu, and S.-Y. Lin, “Very low-refractive-index optical thin films consisting of an array of SiO2 nanorods,” Opt. Lett. 31, 601–603 (2006).
[Crossref] [PubMed]

J.-Q. Xi, J. K. Kim, and E. F. Schubert, “Silica nanorod-array films with very low refractive indices,” Nano Lett. 5, 1385–1387 (2005).
[Crossref] [PubMed]

Kim, J. Y.

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

Kim, K. H.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468 (2004).
[Crossref]

Kim, S. I.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

Kishino, K.

K. Kishino, A. Kikuchi, H. Sekiguchi, and S. Ishizawa, “InGaN/GaN nanocolumn LEDs emitting from blue to red,” Proc. SPIE 6473, 64730T (2007).
[Crossref]

A. Kikuchi, M. Tada, K. Miwa, and K. Kishino, “Growth and characterization of InGaN/GaN nanocolumn LED,” Proc. SPIE 6129, 612905 (2006).
[Crossref]

A. Kikuchi, M. Kawai, M. Tada, and K. Kishino, “InGaN/GaN multiple quantum disk nanocolumn light-emitting diodes grown on (111) Si substrate,” Jpn. J. Appl. Phys. 43, L1524–L1526 (2004).
[Crossref]

Krames, M. R.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Lalanne, P.

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
[Crossref]

Lee, C.-S.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Lee, H.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

Lin, H.-W.

H.-Y. Chen, H.-W. Lin, C.-H. Shen, and S. Gwo, “Structure and photoluminescence properties of epitaxially oriented GaN nanorods grown on Si (111) by plasma-assisted molecular-beam epitaxy,” Appl. Phys. Lett. 89, 243105 (2006), and references therein.
[Crossref]

Lin, J. Y.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468 (2004).
[Crossref]

Lin, S.-Y.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T.-M. Lu, and S.-Y. Lin, “Very low-refractive-index optical thin films consisting of an array of SiO2 nanorods,” Opt. Lett. 31, 601–603 (2006).
[Crossref] [PubMed]

Liu, T.-A.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Liu, W.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

Lo, H.-C.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Lu, T.-M.

Lu, Y.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Luo, H.

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

Mackie, D. M.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

McGroddy, K.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

Miwa, K.

A. Kikuchi, M. Tada, K. Miwa, and K. Kishino, “Growth and characterization of InGaN/GaN nanocolumn LED,” Proc. SPIE 6129, 612905 (2006).
[Crossref]

Morris, G. M.

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
[Crossref]

Motamedi, M. E.

Nakamura, S.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref] [PubMed]

Oder, T. N.

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468 (2004).
[Crossref]

Ohtsuka, K.

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref] [PubMed]

Pan, C.-L.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Park, Y.

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

Pendry, J. B

D. R. Smith, J. B Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Peng, C.-Y.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Ryu, S. R.

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

Sai, H.

Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001).
[Crossref]

Saraf, G.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Schubert, E. F.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T.-M. Lu, and S.-Y. Lin, “Very low-refractive-index optical thin films consisting of an array of SiO2 nanorods,” Opt. Lett. 31, 601–603 (2006).
[Crossref] [PubMed]

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

J.-Q. Xi, J. K. Kim, and E. F. Schubert, “Silica nanorod-array films with very low refractive indices,” Nano Lett. 5, 1385–1387 (2005).
[Crossref] [PubMed]

E. F. Schubert, Light-Emitting Diodes, 2nd Edition (Cambridge University Press, Cambridge, 2006).
[Crossref]

Schubert, M. F.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref] [PubMed]

Sekiguchi, H.

K. Kishino, A. Kikuchi, H. Sekiguchi, and S. Ishizawa, “InGaN/GaN nanocolumn LEDs emitting from blue to red,” Proc. SPIE 6473, 64730T (2007).
[Crossref]

Sharma, R.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

Shen, C.-H.

H.-Y. Chen, H.-W. Lin, C.-H. Shen, and S. Gwo, “Structure and photoluminescence properties of epitaxially oriented GaN nanorods grown on Si (111) by plasma-assisted molecular-beam epitaxy,” Appl. Phys. Lett. 89, 243105 (2006), and references therein.
[Crossref]

Shen, H.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Sigalas, M. M.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Simmons, J. A.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Smart, J. A.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

Smith, D. R.

D. R. Smith, J. B Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref] [PubMed]

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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

Sone, C.

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

Song, J. J.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Southwell, W. H.

Srinivasarao, M.

M. Srinivasarao, “Nano-optics in the biological world: Beetles, butterflies, birds, and moths,” Chem. Rev. 99, 1935–1961 (1999).
[Crossref]

Tada, M.

A. Kikuchi, M. Tada, K. Miwa, and K. Kishino, “Growth and characterization of InGaN/GaN nanocolumn LED,” Proc. SPIE 6129, 612905 (2006).
[Crossref]

A. Kikuchi, M. Kawai, M. Tada, and K. Kishino, “InGaN/GaN multiple quantum disk nanocolumn light-emitting diodes grown on (111) Si substrate,” Jpn. J. Appl. Phys. 43, L1524–L1526 (2004).
[Crossref]

Tsai, J.

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

van de Hulst, H. C.

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

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref] [PubMed]

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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

Weisbuch, C.

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

Wendt, J. R.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Wierer, J. J.

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

Wilson, S. J.

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[Crossref]

Wiltshire, M. C. K.

D. R. Smith, J. B Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Wu, C. T.

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Xi, J.-Q.

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T.-M. Lu, and S.-Y. Lin, “Very low-refractive-index optical thin films consisting of an array of SiO2 nanorods,” Opt. Lett. 31, 601–603 (2006).
[Crossref] [PubMed]

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

J.-Q. Xi, J. K. Kim, and E. F. Schubert, “Silica nanorod-array films with very low refractive indices,” Nano Lett. 5, 1385–1387 (2005).
[Crossref] [PubMed]

Ye, D.

Yoshikawa, H.

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
[Crossref]

Yu, 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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

Yugami, H.

Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001).
[Crossref]

Zhong, J.

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Adv. Mater. (1)

H.-M. Kim, T. W. Kang, and K. S. Chung, “Nanoscale ultraviolet-light-emitting diodes using wide-bandgap gallium nitride nanorods,” Adv. Mater. 15, 567–569 (2003).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (8)

Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001).
[Crossref]

H.-Y. Chen, H.-W. Lin, C.-H. Shen, and S. Gwo, “Structure and photoluminescence properties of epitaxially oriented GaN nanorods grown on Si (111) by plasma-assisted molecular-beam epitaxy,” Appl. Phys. Lett. 89, 243105 (2006), and references therein.
[Crossref]

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004).
[Crossref]

T. N. Oder, K. H. Kim, J. Y. Lin, and H. X. Jiang, “III-nitride blue and ultraviolet photonic crystal light emitting diodes,” Appl. Phys. Lett. 84, 466–468 (2004).
[Crossref]

J. J. Wierer, M. R. Krames, J. E. Epler, N. F. Gardner, M. G. Craford, J. R. Wendt, J. A. Simmons, and M. M. Sigalas, “InGaN/GaN quantum-well heterostructure light-emitting diodes employing photonic crystal structures,” Appl. Phys. Lett. 84, 3885–3887 (2004).
[Crossref]

A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S. P. DenBaars, E. L. Hu, C. Weisbuch, and H. Benisty, “Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution,” Appl. Phys. Lett. 88, 061124 (2006).
[Crossref]

J. Y. Kim, T. Gessmann, E. F. Schubert, J.-Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88, 013501 (2006).
[Crossref]

J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90, 203515 (2007).
[Crossref]

Chem. Rev. (1)

M. Srinivasarao, “Nano-optics in the biological world: Beetles, butterflies, birds, and moths,” Chem. Rev. 99, 1935–1961 (1999).
[Crossref]

J. Appl. Phys. (1)

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82, 3528–3535 (1997).
[Crossref]

Jpn. J. Appl. Phys. (1)

A. Kikuchi, M. Kawai, M. Tada, and K. Kishino, “InGaN/GaN multiple quantum disk nanocolumn light-emitting diodes grown on (111) Si substrate,” Jpn. J. Appl. Phys. 43, L1524–L1526 (2004).
[Crossref]

Nano Lett. (3)

H.-M. Kim, Y.-H. Cho, H. Lee, S. I. Kim, S. R. Ryu, D. Y. Kim, T. W. Kang, and K. S. Chung, “High-brightness light emitting diodes using dislocation-free indium gallium nitride/gallium nitride multiquantum-well nanorod arrays,” Nano Lett. 4, 1059–1062 (2004).
[Crossref]

J.-Q. Xi, J. K. Kim, and E. F. Schubert, “Silica nanorod-array films with very low refractive indices,” Nano Lett. 5, 1385–1387 (2005).
[Crossref] [PubMed]

C.-H. Hsu, H.-C. Lo, C.-F. Chen, C. T. Wu, J.-S. Hwang, D. Das, J. Tsai, L.-C. Chen, and K.-H. Chen, “Generally applicable self-masked dry etching technique for nanotip array fabrication,” Nano Lett. 4, 471–475 (2004).
[Crossref]

Nanotechnology (1)

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8, 53–56 (1997).
[Crossref]

Nat. Nanotechnol. (1)

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Nat. Photonics (1)

J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176–179 (2007).

Nature (1)

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[Crossref]

Opt. Acta (1)

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity.” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref] [PubMed]

Proc. SPIE (2)

A. Kikuchi, M. Tada, K. Miwa, and K. Kishino, “Growth and characterization of InGaN/GaN nanocolumn LED,” Proc. SPIE 6129, 612905 (2006).
[Crossref]

K. Kishino, A. Kikuchi, H. Sekiguchi, and S. Ishizawa, “InGaN/GaN nanocolumn LEDs emitting from blue to red,” Proc. SPIE 6473, 64730T (2007).
[Crossref]

Science (1)

D. R. Smith, J. B Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Thin Solid Films (1)

D. E. Aspnes, “Optical properties of thin-films,” Thin Solid Films 89, 249–262 (1982).
[Crossref]

Other (3)

E. F. Schubert, Light-Emitting Diodes, 2nd Edition (Cambridge University Press, Cambridge, 2006).
[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.703209–3211 (1997); the refractive index n was found to follow the Sellmeir-type dispersion relationship n2(λ)=2.272+304.72/(λ2-294.02) with the incident photon energy below the fundamental band edge of wurtzite GaN.
[Crossref]

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

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)
Fig. 1.
Fig. 1. FE-SEM images of the vertically aligned GaN nanorod arrays grown on Si(111) by PA-MBE. Plan view (upper panels) and cross-sectional (lower panels) view of (a) 0.4 µm, (b) 1.2 µm, and (c) 2 µm GaN nanorod arrays.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. Normal-incidence reflectivity spectra of vertically-aligned GaN nanorod arrays. The dashed line labels the incident photon energy that corresponds to the GaN band gap energy. To the left of the dashed line is the opaque region, and to the right of the dashed line is the transparent region. The dotted curves in the opaque region are the reflectivities of GaN epilayers measured with the same set-up.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Dispersion relationship of GaN-bulk-film refractive index determined by reflectivity spectroscopy in the transparent region. The result obtained by our method agrees well with that measured by spectroscopic ellipsometry and optical transmission in Ref. [28].

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. Determination of effective reflective index dispersion curves of GaN nanorod array. (a) The relations of 2d/λl and l/2 for the nanorod-array samples. The dashed lines are linear fits (in the long wavelength region) with integer intercepts on the vertical axis. (b) The effective reflective index dispersion curves obtained from the interference fringes of reflectivity spectra for all GaN nanorod-array samples. The solid lines are the dispersion curves obtained by multiplying the dispersion curve of GaN-bulk-film reflective index [n GaN, as shown in Fig. 3(c)] with constant numbers.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5. FE-SEM image of 2-µm-long GaN nanorod array near the sample cleavage edge. The arrow directs toward the top region of vertically aligned nanorod array.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6. Comparison of simulated and experimental reflectivity spectra. The simulation results correspond to (a) 0.4 µm, (b) 1.2 µm, and (c) 2 µm GaN nanorods, respectively. Color dots represent the data point and the solid lines are the simulation results using the model described in the text.

Download Full Size | PPT Slide | PDF

Equations (5)

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

f GaN n GaN 2 − n eff 2 n GaN 2 + 2 n eff 2 + ( 1 − f GaN ) n air 2 − n eff 2 n air 2 + 2 n eff 2 = 0 ,
R = ∣ r ∣ 2 = ∣ r 1 + r 2 e i φ 1 + r 1 r 2 e i φ ∣ 2 ,
l 2 = 2 n eff d λ l − m , l = 0 , 1 , 2 , · · · .
n eff = ( 2 m + l ) λ l 4 d .
r = r 1 + ⌊ ( 1 − r 1 2 ) r 2 e i φ − ( 1 − r 1 2 ) r 1 r 2 2 e i 2 φ + ( 1 − r 1 2 ) r 1 2 r 2 3 e i 3 φ − . . . ⌋ ,

Metrics