Abstract

The interactions between electromagnetic field and arbitrarily shaped metallic nanoparticles are numerically investigated. The scattering and near field intensity of nanoparticles are characterized by using volume integral equation which is formulated by considering the total electric field, i.e. the sum of incident fields and radiated fields by equivalent electric volume currents, within the scatterers. The resultant volume integral equation is then discretized using divergence-conforming vector basis functions and is subsequently solved numerically. Numerical examples are presented to demonstrate the application of volume integral equation to capture and analyze the surface plasmon resonance of arbitrarily shaped metallic nanoparticles. The effects of illumination angles and background media to the surface plasmon resonance are also investigated. The results show that our proposed method is particularly useful and accurate in characterizing the surface plasmon properties of metallic nanoparticles.

© 2007 Optical Society of America

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References

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  1. V. D. Hulst, Light Scattering by Small Particles (John Wiley, New York, 1957).
  2. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  3. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2000), 2nd ed.
  4. S. K. Gray and T. Kupka, "Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders," Phys. Rev. B 68, 045,415 (2003).
    [CrossRef]
  5. E. Moreno, D. E. Erni, C. Hafner, and R. Vahldieck, "Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures," J. Opt. Soc. Am. A 19, 101-111 (2002).
    [CrossRef]
  6. W. H. Yang, G. C. Schatz, and R. P. V. Duyne, "Discrete dipole approximation for calculating extinction and raman intensities for small particles with arbitrary shape," J. Chem. Phys. 103, 869-875 (1995).
    [CrossRef]
  7. C. Rockstuhl, M. G. Salt, and H. P. Herzig, "Application of the boundary-element method to the interaction of light with single and coupled metallic nanoparticles," J. Opt. Soc. Am. A 20, 1969-1973 (2003).
    [CrossRef]
  8. J.-W. Liaw, "Simulation of surface plasmon resonance of metallic nanoparticles by the boundary-element method," J. Opt. Soc. Am. A 23, 108-116 (2006).
    [CrossRef]
  9. H. S. Chu, W. B. Ewe, E. P. Li, and R. Vahldieck, "Analysis of sub-wavelength light propagation through long double-chain nanowires with funnel feeding," Opt. Express 15, 4216-4223 (2007).
    [CrossRef] [PubMed]
  10. J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
    [CrossRef]
  11. J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticleswith a non-regular shape," Opt. Express 6, 213-219 (2000).
    [CrossRef] [PubMed]
  12. J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
    [CrossRef]
  13. C. A. Balanis, Advanced Engineering Electromagnetics (John Wiley, New York, 1989).
  14. A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE-Oxford University Press, 1998).
  15. R. F. Harrington, Field Computation by Moment Methods (MacMillan, New York, 1968).
  16. A. W. Glisson and D. R. Wilton, "Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces," IEEE Trans. Antennas Propag. 28, 593-603 (1980).
    [CrossRef]
  17. S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propagat. 30, 409-418 (1982).
    [CrossRef]
  18. V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 86, 414-439 (1990).
    [CrossRef]
  19. F. Ling, C. F. Wang, and J. M. Jin, "Application of adaptive integral method to scattering and radiation analysis of arbitrarily shaped planar structures," J. Electromag. Waves Applicat. 12, 1021-1037 (1998).
    [CrossRef]
  20. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  21. C. L. Nehl, H. Liao, and J. H. Hafner, "Optical properties of star-shaped gold nanoparticles," Nano Lett. 6, 683-688 (2006).
    [CrossRef] [PubMed]

2007 (1)

2006 (2)

J.-W. Liaw, "Simulation of surface plasmon resonance of metallic nanoparticles by the boundary-element method," J. Opt. Soc. Am. A 23, 108-116 (2006).
[CrossRef]

C. L. Nehl, H. Liao, and J. H. Hafner, "Optical properties of star-shaped gold nanoparticles," Nano Lett. 6, 683-688 (2006).
[CrossRef] [PubMed]

2003 (2)

C. Rockstuhl, M. G. Salt, and H. P. Herzig, "Application of the boundary-element method to the interaction of light with single and coupled metallic nanoparticles," J. Opt. Soc. Am. A 20, 1969-1973 (2003).
[CrossRef]

S. K. Gray and T. Kupka, "Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders," Phys. Rev. B 68, 045,415 (2003).
[CrossRef]

2002 (1)

2001 (1)

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

2000 (2)

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticleswith a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

1998 (1)

F. Ling, C. F. Wang, and J. M. Jin, "Application of adaptive integral method to scattering and radiation analysis of arbitrarily shaped planar structures," J. Electromag. Waves Applicat. 12, 1021-1037 (1998).
[CrossRef]

1995 (1)

W. H. Yang, G. C. Schatz, and R. P. V. Duyne, "Discrete dipole approximation for calculating extinction and raman intensities for small particles with arbitrary shape," J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

1990 (1)

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 86, 414-439 (1990).
[CrossRef]

1982 (1)

S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propagat. 30, 409-418 (1982).
[CrossRef]

1980 (1)

A. W. Glisson and D. R. Wilton, "Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces," IEEE Trans. Antennas Propag. 28, 593-603 (1980).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Chu, H. S.

Duyne, R. P. V.

W. H. Yang, G. C. Schatz, and R. P. V. Duyne, "Discrete dipole approximation for calculating extinction and raman intensities for small particles with arbitrary shape," J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

Erni, D. E.

Ewe, W. B.

Glisson, A. W.

S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propagat. 30, 409-418 (1982).
[CrossRef]

A. W. Glisson and D. R. Wilton, "Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces," IEEE Trans. Antennas Propag. 28, 593-603 (1980).
[CrossRef]

Gray, S. K.

S. K. Gray and T. Kupka, "Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders," Phys. Rev. B 68, 045,415 (2003).
[CrossRef]

Hafner, C.

Hafner, J. H.

C. L. Nehl, H. Liao, and J. H. Hafner, "Optical properties of star-shaped gold nanoparticles," Nano Lett. 6, 683-688 (2006).
[CrossRef] [PubMed]

Herzig, H. P.

Jin, J. M.

F. Ling, C. F. Wang, and J. M. Jin, "Application of adaptive integral method to scattering and radiation analysis of arbitrarily shaped planar structures," J. Electromag. Waves Applicat. 12, 1021-1037 (1998).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kottmann, J. P.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticleswith a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

Kupka, T.

S. K. Gray and T. Kupka, "Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders," Phys. Rev. B 68, 045,415 (2003).
[CrossRef]

Li, E. P.

Liao, H.

C. L. Nehl, H. Liao, and J. H. Hafner, "Optical properties of star-shaped gold nanoparticles," Nano Lett. 6, 683-688 (2006).
[CrossRef] [PubMed]

Liaw, J.-W.

Ling, F.

F. Ling, C. F. Wang, and J. M. Jin, "Application of adaptive integral method to scattering and radiation analysis of arbitrarily shaped planar structures," J. Electromag. Waves Applicat. 12, 1021-1037 (1998).
[CrossRef]

Martin, O. J. F.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticleswith a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

Moreno, E.

Nehl, C. L.

C. L. Nehl, H. Liao, and J. H. Hafner, "Optical properties of star-shaped gold nanoparticles," Nano Lett. 6, 683-688 (2006).
[CrossRef] [PubMed]

Rao, S. M.

S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propagat. 30, 409-418 (1982).
[CrossRef]

Rockstuhl, C.

Rokhlin, V.

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 86, 414-439 (1990).
[CrossRef]

Salt, M. G.

Schatz, G. C.

W. H. Yang, G. C. Schatz, and R. P. V. Duyne, "Discrete dipole approximation for calculating extinction and raman intensities for small particles with arbitrary shape," J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

Schultz, S.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticleswith a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

Smith, D. R.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticleswith a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

Vahldieck, R.

Wang, C. F.

F. Ling, C. F. Wang, and J. M. Jin, "Application of adaptive integral method to scattering and radiation analysis of arbitrarily shaped planar structures," J. Electromag. Waves Applicat. 12, 1021-1037 (1998).
[CrossRef]

Wilton, D. R.

S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propagat. 30, 409-418 (1982).
[CrossRef]

A. W. Glisson and D. R. Wilton, "Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces," IEEE Trans. Antennas Propag. 28, 593-603 (1980).
[CrossRef]

Yang, W. H.

W. H. Yang, G. C. Schatz, and R. P. V. Duyne, "Discrete dipole approximation for calculating extinction and raman intensities for small particles with arbitrary shape," J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

J. P. Kottmann and O. J. F. Martin, "Accurate solution of the volume integral equation for high-permittivity scatterers," IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

A. W. Glisson and D. R. Wilton, "Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces," IEEE Trans. Antennas Propag. 28, 593-603 (1980).
[CrossRef]

IEEE Trans. Antennas Propagat. (1)

S. M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propagat. 30, 409-418 (1982).
[CrossRef]

J. Chem. Phys. (1)

W. H. Yang, G. C. Schatz, and R. P. V. Duyne, "Discrete dipole approximation for calculating extinction and raman intensities for small particles with arbitrary shape," J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

J. Comput. Phys. (1)

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 86, 414-439 (1990).
[CrossRef]

J. Electromag. Waves Applicat. (1)

F. Ling, C. F. Wang, and J. M. Jin, "Application of adaptive integral method to scattering and radiation analysis of arbitrarily shaped planar structures," J. Electromag. Waves Applicat. 12, 1021-1037 (1998).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nano Lett. (1)

C. L. Nehl, H. Liao, and J. H. Hafner, "Optical properties of star-shaped gold nanoparticles," Nano Lett. 6, 683-688 (2006).
[CrossRef] [PubMed]

Opt. Express (2)

Phys. Rev. B (3)

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

S. K. Gray and T. Kupka, "Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders," Phys. Rev. B 68, 045,415 (2003).
[CrossRef]

Other (6)

V. D. Hulst, Light Scattering by Small Particles (John Wiley, New York, 1957).

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2000), 2nd ed.

C. A. Balanis, Advanced Engineering Electromagnetics (John Wiley, New York, 1989).

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE-Oxford University Press, 1998).

R. F. Harrington, Field Computation by Moment Methods (MacMillan, New York, 1968).

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Figures (10)

Fig. 1.
Fig. 1.

A triangular rooftop basis function.

Fig. 2.
Fig. 2.

SCS as a function of wavelength for a silver circular cylinder with radius =50 nm. The plasmon resonance at λ=347 nm is well reproduced by our method.

Fig. 3.
Fig. 3.

SCS as a function of wavelength for different shaped cylinders computed by VIE (solid lines) and boundary element method (square symbols). For both elliptical and rectangular cylinders, their surface plasmon resonances are redshifted and their FWHM become wider when the light incident angle varies from 180° to 90°.

Fig. 4.
Fig. 4.

The normalized near-field distribution of nanoparticles at their corresponding resonant wavelengths with light incident from incident angle 180° as shown in inset of Fig. 3(a).

Fig. 5.
Fig. 5.

The normalized near-field distribution of nanoparticles at their corresponding resonant wavelengths with light incident from incident angle 90° as shown in the inset of Fig. 3(b).

Fig. 6.
Fig. 6.

Geometry and scattering cross section as a function of wavelength for a hollow (solid) silver cylinder. The cylinder is considered solid when n 1=n silver. The solid and hollow cylinders are immersed in different background materials.

Fig. 7.
Fig. 7.

The normalized near-field distribution of a solid silver cylinder (n 1=n silver) immersed in different background materials.

Fig. 8.
Fig. 8.

The normalized near-field distribution of a hollow silver cylinder (n 1n silver) immersed in different background materials.

Fig. 9.
Fig. 9.

Geometry and scattering cross section as a function of wavelength for silver five-pointed and six-pointed stars. The corners of the stars are filleted with different radii r.

Fig. 10.
Fig. 10.

The normalized near-field distribution of five- and six-pointed stars with different fillet radii r.

Equations (10)

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E inc = D ε + j k b η b V J V G + 1 k b 2 · ( J V G ) d V
J V ( r ) = j ω κ e ( r ) D ( r )
κ e ( r ) = ε ( r ) ε b ε ( r ) .
f n ( r ) = { l n 2 A n ± ρ n ± , r in T n ± 0 , otherwise ,
· f n ( r ) = { ± l n A n ± , r in T n ± 0 , otherwise ,
D ( r ) = n = 1 N D n f n ( r ) ,
J V ( r ) = j ω n = 1 N κ e ( r ) D n f n ( r ) .
Z ¯ I = V ,
Z mn = T m f m · f n ε dV ω k b η b T m f m · T n κ e f n G + 1 k b 2 · ( κ e f n G ) dV dV ,
V m = T m f m · E inc dV .

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