Abstract

A finite element-based modal formulation of diffraction of a plane wave by an absorbing photonic crystal slab of arbitrary geometry is developed for photovoltaic applications. The semianalytic approach allows efficient and accurate calculation of the absorption of an array with a complex unit cell. This approach gives direct physical insight into the absorption mechanism in such structures, which can be used to enhance the absorption. The verification and validation of this approach is applied to a silicon nanowire array, and the efficiency and accuracy of the method is demonstrated. The method is ideally suited to studying the manner in which spectral properties (e.g., absorption) vary with the thickness of the array, and we demonstrate this with efficient calculations that can identify an optimal geometry.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  8. B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
    [CrossRef]
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    [CrossRef]
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  12. J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).
  13. R. C. McPhedran, D. H. Dawes, L. C. Botten, and N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” Journal of Electromagnetic Waves and Applications 10, 1085–1111(27) (1996).
    [CrossRef]
  14. L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: Modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882(36) (1998).
    [CrossRef]
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    [CrossRef]
  17. D. J. Kan, A. A. Asatryan, C. G. Poulton, and L. C. Botten, “Multipole method for modeling linear defects in photonic woodpiles,” J. Opt. Soc. Am. B 27, 246–258 (2010).
  18. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta. 28, 1087–1102 (1981).
    [CrossRef]
  19. L. Vardapetyan and L. Demkowicz, “Full-wave analysis of dielectric waveguides at a given frequency,” Math. Comput. 72, 105–129 (electronic) (2003).
  20. K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
    [CrossRef]
  21. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide (Society for Industrial and Applied Mathematics (SIAM), 1998).
  22. A. Bossavit, Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements (Academic, 1998).
  23. D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications (Springer-Verlag, 2008).
  24. G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics : An Introduction (Springer, 2002).
  25. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011).
    [CrossRef]
  26. M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 189–192 (1995).
    [CrossRef]
  27. J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
    [CrossRef]
  28. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
    [CrossRef]
  29. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
  30. R. Asadi, M. Malek-Mohammad, and S. Khorasani, “All optical switch based on Fano resonance in metal nanocomposite photonic crystals,” Opt. Commun. 284, 2230–2235 (2011).

2011 (2)

R. Asadi, M. Malek-Mohammad, and S. Khorasani, “All optical switch based on Fano resonance in metal nanocomposite photonic crystals,” Opt. Commun. 284, 2230–2235 (2011).

B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011).
[CrossRef]

2010 (2)

2009 (3)

2008 (1)

A. Chutinan and S. John, “Light trapping and absorption optimization in certain thin-film photonic crystal architectures,” Phys. Rev. A 78, 023825 (2008).
[CrossRef]

2007 (3)

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
[CrossRef]

N. S. Lewis, “Toward cost-effective solar energy use,” Science 315, 798–801 (2007).
[CrossRef]

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666 (2007).

2006 (1)

2005 (1)

K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
[CrossRef]

2003 (2)

L. Vardapetyan and L. Demkowicz, “Full-wave analysis of dielectric waveguides at a given frequency,” Math. Comput. 72, 105–129 (electronic) (2003).

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

2002 (1)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).

1998 (1)

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: Modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882(36) (1998).
[CrossRef]

1996 (1)

R. C. McPhedran, D. H. Dawes, L. C. Botten, and N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” Journal of Electromagnetic Waves and Applications 10, 1085–1111(27) (1996).
[CrossRef]

1995 (1)

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 189–192 (1995).
[CrossRef]

1982 (1)

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta. 28, 1087–1102 (1981).
[CrossRef]

1972 (1)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[CrossRef]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

Abrams, N.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta. 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta. 28, 1087–1102 (1981).
[CrossRef]

Asadi, R.

R. Asadi, M. Malek-Mohammad, and S. Khorasani, “All optical switch based on Fano resonance in metal nanocomposite photonic crystals,” Opt. Commun. 284, 2230–2235 (2011).

Asano, T.

Asatryan, A. A.

Benkstein, K. D.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

Blad, J.

Boffi, D.

D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications (Springer-Verlag, 2008).

Bossavit, A.

A. Bossavit, Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements (Academic, 1998).

Botten, L. C.

B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011).
[CrossRef]

D. J. Kan, A. A. Asatryan, C. G. Poulton, and L. C. Botten, “Multipole method for modeling linear defects in photonic woodpiles,” J. Opt. Soc. Am. B 27, 246–258 (2010).

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: Modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882(36) (1998).
[CrossRef]

R. C. McPhedran, D. H. Dawes, L. C. Botten, and N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” Journal of Electromagnetic Waves and Applications 10, 1085–1111(27) (1996).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta. 28, 1087–1102 (1981).
[CrossRef]

Brezzi, F.

D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications (Springer-Verlag, 2008).

Chutinan, A.

A. Chutinan and S. John, “Light trapping and absorption optimization in certain thin-film photonic crystal architectures,” Phys. Rev. A 78, 023825 (2008).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta. 28, 1087–1102 (1981).
[CrossRef]

Dawes, D. H.

R. C. McPhedran, D. H. Dawes, L. C. Botten, and N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” Journal of Electromagnetic Waves and Applications 10, 1085–1111(27) (1996).
[CrossRef]

de Sterke, C. M.

Demkowicz, L.

L. Vardapetyan and L. Demkowicz, “Full-wave analysis of dielectric waveguides at a given frequency,” Math. Comput. 72, 105–129 (electronic) (2003).

Demkowicz, L. F.

D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications (Springer-Verlag, 2008).

Dossou, K.

K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
[CrossRef]

Dossou, K. B.

Durán, R. G.

D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications (Springer-Verlag, 2008).

Falk, R. S.

D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications (Springer-Verlag, 2008).

Fan, S.

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18, A366–A380 (2010).
[CrossRef]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).

Fang, Y.

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
[CrossRef]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, 1953).

Fontaine, M.

K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
[CrossRef]

Fortin, M.

D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications (Springer-Verlag, 2008).

Frank, A. J.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

Green, M. A.

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 189–192 (1995).
[CrossRef]

Halaoui, L. I.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

Hanson, G. W.

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics : An Introduction (Springer, 2002).

Huang, J.

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
[CrossRef]

Joannopoulos, J. D.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

John, S.

A. Chutinan and S. John, “Light trapping and absorption optimization in certain thin-film photonic crystal architectures,” Phys. Rev. A 78, 023825 (2008).
[CrossRef]

Kan, D. J.

Keevers, M. J.

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 189–192 (1995).
[CrossRef]

Kempa, T. J.

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
[CrossRef]

Khorasani, S.

R. Asadi, M. Malek-Mohammad, and S. Khorasani, “All optical switch based on Fano resonance in metal nanocomposite photonic crystals,” Opt. Commun. 284, 2230–2235 (2011).

Krauss, T. F.

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666 (2007).

Kwong, D.-L.

J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).

Lehoucq, R. B.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide (Society for Industrial and Applied Mathematics (SIAM), 1998).

Lewis, B. A.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

Lewis, N. S.

N. S. Lewis, “Toward cost-effective solar energy use,” Science 315, 798–801 (2007).
[CrossRef]

Li, J.

J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).

Li, X.

J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).

Lieber, C. M.

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
[CrossRef]

Lin, C.

Lo, P. G.-Q.

J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).

Malek-Mohammad, M.

R. Asadi, M. Malek-Mohammad, and S. Khorasani, “All optical switch based on Fano resonance in metal nanocomposite photonic crystals,” Opt. Commun. 284, 2230–2235 (2011).

Mallouk, T. E.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

McPhedran, R. C.

B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: Modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882(36) (1998).
[CrossRef]

R. C. McPhedran, D. H. Dawes, L. C. Botten, and N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” Journal of Electromagnetic Waves and Applications 10, 1085–1111(27) (1996).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta. 28, 1087–1102 (1981).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Meixner, J.

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, 1953).

Movchan, A. B.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: Modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882(36) (1998).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and A. B. Movchan, “Off-axis diffraction by perfectly conducting capacitive grids: Modal formulation and verification,” J. Electromagn. Waves Appl. 12, 847–882(36) (1998).
[CrossRef]

R. C. McPhedran, D. H. Dawes, L. C. Botten, and N. A. Nicorovici, “On-axis diffraction by perfectly conducting capacitive grids,” Journal of Electromagnetic Waves and Applications 10, 1085–1111(27) (1996).
[CrossRef]

Nishimura, S.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

Noda, S.

Poulton, C. G.

Povinelli, M. L.

Raman, A.

Song, B.-S.

Sorensen, D. C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide (Society for Industrial and Applied Mathematics (SIAM), 1998).

Sturmberg, B. C. P.

Sudbø, A. S.

Tian, B.

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
[CrossRef]

van de Lagemaat, J.

S. Nishimura, N. Abrams, B. A. Lewis, L. I. Halaoui, T. E. Mallouk, K. D. Benkstein, J. van de Lagemaat, and A. J. Frank, “Standing wave enhancement of red absorbance and photocurrent in dye-sensitized titanium dioxide photoelectrodes coupled to photonic crystals,” J. Am. Chem. Soc. 125, 6306–6310 (2003).
[CrossRef]

Vardapetyan, L.

L. Vardapetyan and L. Demkowicz, “Full-wave analysis of dielectric waveguides at a given frequency,” Math. Comput. 72, 105–129 (electronic) (2003).

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Wong, S. M.

J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).

Würfel, P.

P. Würfel, Physics of Solar Cells: From Basic Principles to Advanced Concepts (Wiley-VCH, 2009).

Yablonovitch, E.

Yakovlev, A. B.

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics : An Introduction (Springer, 2002).

Yang, C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide (Society for Industrial and Applied Mathematics (SIAM), 1998).

Yu, G.

B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber, “Coaxial silicon nanowires as solar cells and nanoelectronic power sources,” Nature 449, 885–889 (2007).
[CrossRef]

Yu, H.

J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).

Yu, N.

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Figures (10)

Fig. 1.
Fig. 1.

Geometry of the problem. The dielectric permittivity and the magnetic permeability of the slab can be dispersive and can have absorption. The unit cell of the array (right panel) can have arbitrary number of inclusions with arbitrary shapes. The direction of the plane wave incident from above the structure can be arbitrarily chosen.

Fig. 2.
Fig. 2.

Schematic of the field representation near the top and bottom interfaces.

Fig. 3.
Fig. 3.

Dispersion curves (lossless cylinders). The solid red curves represent solutions such that ζ 2 is real. The dashed blue curves represent solutions such that Im ( ζ 2 ) is not zero. The complex solutions ζ 2 occur as conjugate pairs and, in order to differentiate the pairs, we use the term Re ( ζ 2 ) + Im ( ζ 2 ) for the x -axis instead of Re ( ζ 2 ) .

Fig. 4.
Fig. 4.

Dotted black curve is the absorption spectrum at normal incidence for a dilute SiNW array (silicon fill fraction is approximately 3.1%). For comparison, the absorptance curves of a homogeneous slab of equal thickness (thin blue curve) and of a homogeneous slab comprising equal volume of silicon (thick red curve) are shown.

Fig. 5.
Fig. 5.

The black dotted curve and the red solid curve are the absorption spectrum, at normal incidence, of dilute SiNW arrays consisting, respectively, of circular cylinders with radius a = 60 nm and square cylinders with side length a π . The two types of cylinders have the same cross-section area and the same height.

Fig. 6.
Fig. 6.

Dilute SiNW array: Contour plot of the absorptance as a function of the wavelength and the cylinder height h . The cylinder radius and the lattice constant are, respectively, a = 60 nm and d = 600 nm .

Fig. 7.
Fig. 7.

Dilute SiNW array: Convergence as the plane wave truncation number N PM increases for a fixed wavelength λ = 700 nm and waveguide truncation number N array = 150 . The computed absorptance for N PM = 10 and N array = 160 is A = 0.13940 and this value is considered as “exact” and used to compute the error curve.

Fig. 8.
Fig. 8.

Dilute SiNW array: Convergence as the array mode truncation number N array increases. The details given in the caption of Fig. 7 also apply here except that the plane wave truncation number is fixed at N PM = 10 . Note that ζ 120 = ζ 121 is a degenerate eigenvalue. The isolated blue dot corresponds to the truncation value N array = 120 where the error is unexpectedly high; thus the truncation N array = 121 is instead used for the error curve.

Fig. 9.
Fig. 9.

Off-normal incidence on a dilute SiNW: Absorption spectrum for 45° off-normal orientated along x -axis (azimuthal  angle = 0 ). The solid red and dashed blue curves represent an incidence by TE-polarized and TM-polarized plane wave, respectively. The absorption spectrum for normal incidence (Fig. 4) is also shown (dotted black).

Fig. 10.
Fig. 10.

Fano resonances: Transmittance versus normalized frequency for the normal incidence. The radius of the cylinders is a / d = 0.05 . The resonances in transmission are well resolved.

Equations (146)

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2 E + k 2 E = 0 ,
E ( r + R ) = E ( r ) e i k 0 · R ,
α p = α 0 + p 2 π d ,
β q = β 0 + q 2 π d
R s E ( x , y ) = i Q s e z × V s = e z × Q s Q s V s ( x , y ) ,
R s M ( x , y ) = i Q s V s = Q s Q s V s ( x , y ) ,
Q s = ( α p , β q ) ,
R p a · R ¯ q b d S = δ a b δ p q ,
E ( r ) = s χ s 1 / 2 [ f s E e i γ s ( z z 0 ) + f s E + e i γ s ( z z 0 ) ] R s E ( r ) + χ s 1 / 2 [ f s M e i γ s ( z z 0 ) + f s M + e i γ s ( z z 0 ) ] R s M ( r ) ,
e z × H ( r ) = s χ s 1 / 2 [ f s E e i γ s ( z z 0 ) f s E + e i γ s ( z z 0 ) ] R s E ( r ) + χ s 1 / 2 [ f s M e i γ s ( z z 0 ) f s M + e i γ s ( z z 0 ) ] R s M ( r ) ,
γ s = k 2 α p 2 β q 2 ,
χ s = γ s k
× H = i k ε E , · ( ε E ) = 0 ,
× E = i k μ H , · ( μ H ) = 0 ,
× H c = i k ε ¯ E c , · ( ε ¯ E c ) = 0 ,
× E c = i k μ ¯ H c , · ( μ ¯ H c ) = 0 .
× H = i k ε E , · ( ε E ) = 0 ,
× E = i k μ H , · ( μ H ) = 0 .
× ( μ 1 × E ) k 2 ε E = 0 , on the domain Ω ,
E ( x , y , z ) = E ( x , y ) e i ζ z .
E · τ and E z are k -quasi-periodic over Ω .
{ × ( μ 1 ( × E ) ) i ζ μ 1 E z + ( ζ 2 μ 1 k 2 ε ) E = 0 , i ζ · ( μ 1 E ) · ( μ 1 E z ) k 2 ε E z = 0
× e z F z = [ F z y F z x ] ,
× F = ( F y x F x y ) e z .
E z = i ζ E ^ z
{ × ( μ 1 ( × E ) ) k 2 ε E = ζ 2 ( μ 1 E ^ z μ 1 E ) , · ( μ 1 E ) + · ( μ 1 E ^ z ) + k 2 ε E ^ z = 0 .
{ × ( μ 1 ( × E ) ) k 2 ε E = ζ 2 ( μ 1 E ^ z μ 1 E ) , 0 = ζ 2 ( · ( μ 1 E ) + · ( μ 1 E ^ z ) + k 2 ε E ^ z ) .
V = { F z L 2 ( Ω ) | F z L 2 ( Ω ) ; F z is k -quasi-periodic over Ω } ,
W = { F ( L 2 ( Ω ) ) 2 | × F L 2 ( Ω ) ; F · τ is k -quasi-periodic over Ω } .
{ ( ( × F ) , μ 1 ( × E ) ) k 2 ( F , ε E ) = ζ 2 ( F , μ 1 ( E ^ z E ) ) , ( F z , μ 1 E ) ( F z , μ 1 E ^ z ) + k 2 ( F z , ε E ^ z ) = 0 ,
( F , E ) = Ω F ¯ · E d A .
[ K t t 0 K z t K z z ] [ E , n E ^ z , n ] = ζ n 2 [ M t t K z t H 0 0 ] [ E , n E ^ z , n ] ,
( K t t ) i j = ( ( × G , i ) , μ 1 ( × G , j ) ) k 2 ( G , i , ε G , j ) ,
( M t t ) i j = ( G , i , μ 1 ( G , j ) ) ,
( K z t ) i j = ( G ^ z , i , μ 1 G , j ) ,
( K z z ) i j = ( G ^ z , i , μ 1 G ^ z , j ) + k 2 ( G ^ z , i , ε G ^ z , j ) .
E ( x , y , z ) = n c n E n ( x , y ) exp ( i ζ n z ) ,
H ( Grad , Ω ) H ( Curl , Ω ) × H ( Div , Ω ) · L ( Ω ) ,
[ K t t 0 0 0 ] [ E , n E ^ z , n ] = ζ n 2 [ M t t K z t H K z t K z z ] [ E , n E ^ z , n ] ,
L E n = ζ n 2 M E n ,
L E = [ × ( μ 1 ( × E ) ) k 2 ε E 0 0 0 ] ,
M E = [ μ 1 E μ 1 E ^ z · ( μ 1 E ) · ( μ 1 E ^ z ) + k 2 ε E ^ z ] .
L F = [ × ( μ ¯ 1 ( × F ) ) k 2 ε ¯ F 0 0 0 ] ,
M F = [ μ ¯ 1 F μ ¯ 1 F ^ z · ( μ ¯ 1 F ) · ( μ ¯ 1 F ^ z ) + k 2 ε ¯ F ^ z ] ,
( L F , E ) = ( F , L E ) , E , F ,
( M F , E ) = ( F , M E ) , E , F .
L E m c = ζ m c 2 M E m c
L E m = ζ m 2 M E m ,
Ω e z · ( E m × H n ) d A = δ m n ,
H = × E i k μ ,
( L E m ¯ , E n ) = ( E m ¯ , L E n ) ,
( ζ m ¯ 2 M E m ¯ , E n ) = ( E m ¯ , ζ n 2 M E n ) .
( ζ n 2 ζ m 2 ) ( E m ¯ , M E n ) = 0 ,
( E m ¯ , M E n ) = 0 , if ζ n 2 ζ m 2 .
e z · ( E m × H n ) = E m , · ( i ζ n E n , E n , z ) i k μ ,
= ζ E m , · ( M E n ) k ,
= ζ E m , · ( M E n ) k ,
s χ s 1 / 2 ( f s E + f s E + ) R s E + χ s 1 / 2 ( f s M + f s M + ) R s M = n c n E n ,
s χ s 1 / 2 ( f s E f s E + ) R s E + χ s 1 / 2 ( f s M f s M + ) R s M = n c n ( e z × H n ) .
Ω e z ( E m × H n ) d A = δ m n .
X 1 / 2 ( f + f + ) = J c ,
J X 1 / 2 ( f f + ) = c ,
f ± = ( f E ± f M ± ) , X = ( χ 0 0 χ 1 ) ,
J = ( J E J M ) , J E / M = [ J s m E / M ] , J s m E / M = R ¯ s E / M E m d S ,
J = ( J E J M ) , J E / M = [ J m s E / M ] , J m s E / M = R s E / M · E m d S ,
χ = diag { χ s } .
f + = R 12 f , c = T 12 f ,
R 12 = I + 2 A ( I + B A ) 1 B = ( A B + I ) 1 ( A B I ) ,
T 12 = 2 ( I + B A ) 1 B ,
where A = X 1 / 2 J , B = J X 1 / 2 .
s χ s 1 / 2 f s E R s E + χ s 1 / 2 f s M R s M = n ( c n + c n + ) E n ,
s χ s 1 / 2 f s E R s E + χ s 1 / 2 f s M R s M = n ( c n c n + ) ( e z × H n ) ,
X 1 / 2 f = J ( c + c + ) ,
J X 1 / 2 f = c c + .
R 21 = ( I B A ) ( I + B A ) 1 ,
T 21 = 2 A ( I + B A ) 1 .
f 1 + = R 12 f 1 + T 21 P c + ,
c = T 12 f 1 + R 21 P c + ,
c + = R 21 P c ,
f 2 = T 21 P c ,
T = T 21 P ( I R 21 P R 21 P ) 1 T 12 ,
R = R 12 + T 21 P ( I R 21 P R 21 P ) 1 R 21 P T 12 .
A = 1 s P [ | r s | 2 + | t s | 2 ] ,
R H I 1 R + T H I 1 T = I 1 + i R H I 1 ¯ i I 1 ¯ R ,
R H I 1 T + T H I 1 R = i T H I 1 ¯ i I 1 ¯ T ,
( I R 21 P R 21 P ) 1
× H n = i k ε E n ,
× E n = i k μ H n
× H m = i k ε E m ,
× E m = i k μ H m .
· ( E n × H m + E m × H n ) = 0 .
E n = E n + E n , H n = H n + H n , = + e z z .
e z · z [ E n × H m + E m × H n ] = · [ E n × H m + E n × H m + E m × H n + E m × H n ] .
i ( ζ n ζ m ) Ω e z · ( E n × H m + E m × H n ) d A = 0 ,
Ω e z · ( E n × H m + E m × H n ) d A = 0 ,
i ( ζ n + ζ m ) Ω e z · ( E n × H n + E m × H n ) d A = 0 .
Ω e z · ( E n × H m E m × H n ) d A = 0 .
Ω e z · ( E m × H n ) d A = 0 ,
Ω ( e z × H n ) · E m d A = δ m n .
R ¯ s E / M = n c n E / M ( e z × H n ) ,
R s E / M = n d n E / M E n .
c m E / M = Ω E m · R ¯ s E / M d A = J s m E / M .
d m E / M = Ω ( e z × H m ) · R s E / M d A = K m s E / M .
R ¯ s E / M = n K n s E / M ( e z × H n ) ,
R s E / M = n J n s E / M E n .
Ω R s E / M · R ¯ s E / M d A = n K n s E / M Ω E n · R ¯ s E / M d A = n K n s E / M J s n E / M = δ s s .
J = [ J E J M ] and K = [ K E K M ] ,
J E / M = [ J s m E / M ] and K E / M = [ K s n E / M ] ,
J K = I ,
E m = s ( J s m E R s E + J s m M R s M ) ,
e z × H n = s ( K n s E R ¯ s E + K n s M R ¯ s M ) .
Ω E m · ( e z × H n ) d A = s ( J s m E K s n E + J s m M K s n M ) = δ n m .
K J = I .
E = n ( c n + c n + ) E n
e z × H = n ( c n c n + ) e z × H n .
S z = Re [ E · ( e z × H ¯ ) ] = 1 2 [ ( c c + ) H U ( c + c + ) + ( c + c + ) H U H ( c c + ) ] ,
U m n = Ω E m · ( e z × H ¯ n ) d A .
S z = Re { [ c H c + H ] V [ c c + ] } ,
V = [ 1 2 ( U + U H ) 1 2 ( U U H ) 1 2 ( U U H ) 1 2 ( U + U H ) ]
U = [ 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ± 1 0 0 0 0 0 0 0 0 0 ± 1 0 0 0 0 0 0 0 0 0 ± 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 ]
S z = [ c c + ] H [ I m i I m ¯ i I m ¯ I m ] [ c c + ] ,
c = T 12 f + R 21 c + ,
f + = R 12 f + T 21 c + ,
[ c c + ] = [ T 12 R 12 0 I ] [ f c + ] .
[ f f + ] = [ I 0 R 12 T 12 ] [ f c + ] ,
S z = [ f f + ] H [ I 1 i I 1 ¯ i I 1 ¯ I 1 ] [ f f + ] .
[ I R 12 H 0 T 21 H ] [ I 1 i I 1 ¯ i I 1 ¯ I 1 ] [ I 0 R 12 T 21 ] = [ T 12 H 0 R 21 H I ] [ I 2 i I 2 ¯ i I 2 ¯ I 2 ] [ T 12 R 21 0 I ] ,
R 12 H I 1 R 12 + T 12 H I 2 T 12 = I 1 + i R 12 H I 1 ¯ i I 1 ¯ R 12 ,
R 12 H I 1 T 21 + T 12 H I 2 R 21 = i T 12 H I 2 ¯ i I 1 ¯ T 21 ,
R 21 H I 2 T 12 + T 21 H I 1 R 12 = i T 21 H I 1 ¯ i I 2 ¯ T 12 ,
R 21 H I 2 R 21 + T 21 H I 1 T 21 = I 2 + i R 21 H I 2 ¯ i I 2 ¯ R 21 .
L E n = ζ n 2 M E n ,
L E n = ζ n 2 M E n
L E n ¯ = ζ n 2 ¯ M E n ¯ ,
E n ζ 2 = E ¯ n ζ ¯ 2
E n ζ = E ¯ n ζ ¯
E n ζ = E ¯ n , ζ ¯
H n ζ = H ¯ n , ζ ¯ .
( ζ m ζ ¯ n ) Ω e z · ( E m × H ¯ n + E ¯ n × H m ) d A = 0 ,
( ζ m + ζ ¯ n ) Ω e z · ( E m × H ¯ n E ¯ n × H m ) d A = 0.
Ω e z · ( E m × H ¯ n ) d A = Ω e z · ( E ¯ n × H m ) d A = 0 ,
Ω e z · ( E n × H ¯ n ) d A
Ω e z · ( E n × H ¯ n ) d A
U m n = Ω E ζ · ( e z × H ¯ ζ ¯ ) d A = Ω E ζ · ( e z × H ζ ) d A .
Ω E ζ · ( e z × H ζ ) d A = 1
U m n = Ω E ζ ¯ · ( e z × H ¯ ζ ) d A = Ω E ζ · ( e z × H ζ ) ¯ d A = 1 .

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