Abstract

We present an intuitive, simple theoretical model, coupled leaky mode theory (CLMT), to analyze the light absorption of 2D, 1D, and 0D semiconductor nanostructures. This model correlates the light absorption of nanostructures to the optical coupling between incident light and leaky modes of the nanostructure. Unlike conventional methods such as Mie theory that requests specific physical features of nanostructures to evaluate the absorption, the CLMT model provides an unprecedented capability to analyze the absorption using eigen values of the leaky modes. Because the eigenvalue shows very mild dependence on the physical features of nanostructures, we can generally apply one set of eigenvalues calculated using a real, constant refractive index to calculations for the absorption of various nanostructures with different sizes, different materials, and wavelength-dependent complex refractive index. This CLMT model is general, simple, yet reasonably accurate, and offers new intuitive physical insights that the light absorption of nanostructures is governed by the coupling efficiency between incident light and leaky modes of the structure.

© 2012 OSA

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References

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  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  2. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).
  3. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. 9(3), 205–213 (2010).
    [CrossRef]
  4. L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
    [CrossRef]
  5. L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
    [CrossRef] [PubMed]
  6. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
    [CrossRef]
  7. L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
    [CrossRef] [PubMed]
  8. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  9. L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
    [CrossRef] [PubMed]
  10. L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
    [CrossRef] [PubMed]
  11. A. W. Snyder, Optical Waveguide Theory (Springer, Berlin, 1983).
  12. U. S. Inan and A. S. Inan, Electromagnetic Waves (Prentice Hall, 2000).
  13. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
  14. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
    [CrossRef]
  15. Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
    [CrossRef]
  16. H. A. Haus, Wave and Fields in Optoelectronics (Prentice-Hall, 1984).

2011 (1)

L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[CrossRef]

2010 (6)

L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
[CrossRef] [PubMed]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
[CrossRef] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef]

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[CrossRef]

2009 (1)

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

2007 (1)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[CrossRef]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef]

Barnard, E. S.

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
[CrossRef] [PubMed]

Brongersma, M. L.

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
[CrossRef] [PubMed]

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
[CrossRef] [PubMed]

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

Brown, A. M.

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
[CrossRef] [PubMed]

Cai, W. S.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

Cao, L.

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
[CrossRef] [PubMed]

L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
[CrossRef] [PubMed]

Cao, L. Y.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

Clemens, B.

L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
[CrossRef] [PubMed]

Clemens, B. M.

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

Fan, P.

L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
[CrossRef] [PubMed]

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
[CrossRef] [PubMed]

Fan, P. Y.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

Fan, S. H.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[CrossRef]

Gardes, F. Y.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Hamam, R. E.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[CrossRef]

Hulst, N.

L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[CrossRef]

Joannopoulos, J. D.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[CrossRef]

Karalis, A.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[CrossRef]

Mashanovich, G.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Novotny, L.

L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[CrossRef]

Park, J. S.

L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
[CrossRef] [PubMed]

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef]

Reed, G. T.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Ruan, Z. C.

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[CrossRef]

Schuller, J. A.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

Soljacic, M.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[CrossRef]

Thomson, D. J.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Vasudev, A. P.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

White, J. S.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

Yu, Z. F.

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

J. Phys. Chem. C (1)

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[CrossRef]

Nano Lett. (3)

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano Lett. 10(7), 2649–2654 (2010).
[CrossRef] [PubMed]

L. Cao, J. S. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant germanium nanoantenna photodetectors,” Nano Lett. 10(4), 1229–1233 (2010).
[CrossRef] [PubMed]

L. Y. Cao, P. Y. Fan, A. P. Vasudev, J. S. White, Z. F. Yu, W. S. Cai, J. A. Schuller, S. H. Fan, and M. L. Brongersma, “Semiconductor nanowire optical antenna solar absorbers,” Nano Lett. 10(2), 439–445 (2010).
[CrossRef] [PubMed]

Nat. Mater. (2)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltais devices,” Nat. Mater. 9(3), 205–213 (2010).
[CrossRef]

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[CrossRef] [PubMed]

Nat. Photonics (2)

L. Novotny and N. Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[CrossRef]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[CrossRef]

Phys. Rev. A (1)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[CrossRef]

Other (7)

H. A. Haus, Wave and Fields in Optoelectronics (Prentice-Hall, 1984).

A. W. Snyder, Optical Waveguide Theory (Springer, Berlin, 1983).

U. S. Inan and A. S. Inan, Electromagnetic Waves (Prentice Hall, 2000).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).

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Figures (4)

Fig. 1
Fig. 1

Eigenvalues of leaky modes in 2D (left), 1D (middle), and 0D (right) structures. (a-c) the real part of the eigenvalue, Nreal, is plotted as a function of the mode number m with different refractive indexes n = 2 (red curve), 3(black curve), and 4(blue curve). In the result for the 2D film, the calculations for different refractive index perfectly overlap each other. (d-f) the imaginary part of the eigenvalue, Nimag, is plotted as a function of the mode number m with different refractive index n = 2 (red curve), 3(black curve), and 4(blue curve). For 1D and 0D structures, Nimag is plotted in log scale for visual convenience, and results for only one polarization (TM for 1D, and TE for 0D) are given.

Fig. 2
Fig. 2

Comparison between spectral absorption and leaky modes of 2D (left), 1D (middle), and 0D (right) silicon nanostructures. The thickness of the 2D film is 100 nm, and the radii of the 1D wire and the 0D particle both are 100 nm. The absorption spectra are calculated using transfer matrix for the 2Dstructure, and Mie theory for the 1D and the 0D structures. For the 1D and 0D structures, calculations for only one polarization (TM for 1D, and TE for 0D) are given. Related leaky modes are plotted as ticks above the absorption spectrain terms of the real part of the eigenvalue.

Fig. 3
Fig. 3

Calculated absorption spectra of 2D (left), 1D (middle), and 0D (right) silicon nanostructures under normal illumination of a plane wave using conventional analytical methods and the CLMT model. The analytical method for the 2D structure is transfer matrix, and Mie theory for the 1D and the 0D structures. For the 1D and 0D structures, calculations for only one polarization (TM for 1D, and TE for 0D) are given. The thickness of the 2D film is 100 nm, and the radii of the 1D wire and the 0D particle both are 100 nm.

Fig. 4
Fig. 4

Calculated absorptions for 100-nm-radius nanowires of different materials under normal illumination of a TM-polarized plane wave with Mie theory (blue) and the CLMT model we propose (red). The materials of the nanowire are indicated in the corresponding panel. Most of the CLMT calculations use the eigenvalue of leaky modes for a constant refractive index of 4. For a-Si and CuInGaSe, the CLMT calculations with n = 3 are also given as black dashed curves.

Tables (1)

Tables Icon

Table 1 Eigenvalue of leaky modes in nanostructures

Equations (18)

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2D planar film, even TEM modes: Cot(nkr)=ni
odd TEM modes: Tan(nkr)=ni
1D circular wire, TM modes: J m (nkr) J m ' (nkr) =n H m (nkr) H m ' (nkr)
TE modes: n J m (nkr) J m ' (nkr) = H m (nkr) H m ' (nkr)
0D spherical particle, TM modes: n ψ m (nkr) ψ m ' (nkr) = ξ m (kr) ξ m ' (kr)
TE modes: ψ m (nkr) ψ m ' (nkr) =n ξ m (kr) ξ m ' (kr)
Ρ abs =(2 γ abs ) | a | 2
da dt =(i ω 0 γ rad γ abs )a + K w i
| a | 2 = C γ rad | W i | 2 (ω- ω 0 ) 2 + ( γ rad + γ abs ) 2
P abs = 2C/( q rad q abs ) 4 (α1) 2 + (1/ q rad +1/ q abs ) 2 | W i | 2
P abs = 2C N imag / N real . n imag / n real ( n real kr/ N real 1) 2 + ( N imag / N real + n imag / n real ) 2 | W i | 2
E i = E 0 m= ( i ) m [ H m (1) (kr)+ H m (2) (kr) 2 ] z ^ e imϕ
H i = E 0 ik ωμ m= [ im H m (1) (kr)+ H m (2) (kr) 2kr r ^ H m '(1) (kr)+ H m '(2) (kr) 2kr ϕ ^ ] e imϕ
| W i | 2 = 1 2 Re( E×H* ) r ^ dS= E 0 2 8 ( ( i ) m H m (2) (kr) e imϕ z ^ × ik ωμ H m '(2)* (kr) e imϕ ϕ ^ )dS
2D film Q abs = 2 N imag / N real . n imag / n real ( n real kr/ N real 1) 2 + ( N imag / N real + n imag / n real ) 2
1D wire Q abs = 2 kr N imag / N real . n imag / n real ( n real kr/ N real 1) 2 + ( N imag / N real + n imag / n real ) 2
0D particle Q abs =(2m+1) 2 ( kr ) 2 N imag / N real n imag / n real ( n real kr/ N real 1) 2 + ( N imag / N real + n imag / n real ) 2
Q abs T = m l Q abs,ml

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