Abstract

We present a novel numerical scheme for the simulation of the field enhancement by metal nano-particles in the time domain. The algorithm is based on a combination of the finite-difference time-domain method and the pseudo-spectral time-domain method for dispersive materials. The hybrid solver leads to an efficient subgridding algorithm that does not suffer from spurious field spikes as do FDTD schemes. Simulation of the field enhancement by gold particles shows the expected exponential field profile. The enhancement factors are computed for single particles and particle arrays. Due to the geometry conforming mesh the algorithm is stable for long integration times and thus suitable for the simulation of resonance phenomena in coupled nano-particle structures.

© 2007 Optical Society of America

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References

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  1. H. Metiu, "Surface enhanced spectroscopy," Prog. Surf. Sci. 17, 153-320 (1984).
    [CrossRef]
  2. M. Moskovits, "Surface enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).
    [CrossRef]
  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]
  4. K. Kneip, Y. Wang, H. Kneip, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, "Single molecule detection using surface-enhanced Raman scattering," Phys. Rev. Lett. 78, 1667-1670 (1997).
    [CrossRef]
  5. S. Nie and S. R. Emory, "Probing single molecules and single nanoparticles by surface-enhanced Raman scattering," Science 275, 1102-1106 (1997).
    [CrossRef] [PubMed]
  6. F. J. Garcia-Vidal and J. B. Pendry, "Collective theory for surface enhanced Raman scattering," Phys. Rev. Lett. 77, 1163-1166 (1996).
    [CrossRef] [PubMed]
  7. F. J. Garcia-Vidal, J. M. Pitarke, and J. B. Pendry, "Silver-filled carbon nanotubes used as spectroscopic enhancers," Phys. Rev. B 58, 6783-6786 (1998).
    [CrossRef]
  8. K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, "Optical antennas: Resonators for local field enhancement," J. Appl. Phys. 94(7), 463242.
  9. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, MA, 1995).
  10. J. T. Krug, E. J. Sanchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain," J. Chem. Phys. 116, 10,895 (2002).
    [CrossRef]
  11. A. C. Cangellaris and D. B. Wright, "Analysis of the numerical error caused by the stair-steppedapproximation of a conducting boundary in FDTD simulations ofelectromagnetic phenomena," IEEE Trans. Antennas Propag. 39(10), 1518-1525 (1991).
    [CrossRef]
  12. R. Holland, "Finite Difference Solutions of Maxwell’s Equations in Generalized Nonorthogonal Coordinates," IEEE Trans. Nucl. Sci. NS-30(6), 4589-4591 (1983).
    [CrossRef]
  13. M. A. Fusco, M. V. Smith, and L. W. Gordon, "A Three-Dimensional FDTD Algorithm in Curvilinear Coordinates," IEEE Trans. Antennas Propag. 39(10), 1463-1471 (1991).
    [CrossRef]
  14. K. M. Krishnaiah and C. J. Railton, "A Stable Subgridding Algorithm and Its Application to Eigenvalue Problems," IEEE Trans. Microwave Theory Techn. 47, 620-628 (1999).
    [CrossRef]
  15. B. Yang, D. Gottlieb, and J. S. Hesthaven, "Spectral Simulations of ElectromagneticWave Scattering," J. Comput. Phys. 134, 216-230 (1997).
    [CrossRef]
  16. G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, "Multidomain Pseudospectral Time-Domain Simulations of Scattering by Objects Buried in Lossy Media," IEEE Trans. Geosci. Remote Sens. 40(6), 1366-1373 (2002).
  17. W. Pernice, F. Payne, and D. Gallgher, "A general framework for the finite-difference time-domain simulation of real metals," IEEE Trans. Antennas Propag. 55, (2007).
    [CrossRef]
  18. J. S. Hesthaven, P. G. Dinensen, and J. P. Lynov, "Spectral collocation time-domain modeling of diffractive optical elements," J. Comput. Phys. 155, 287-306 (1999).
    [CrossRef]
  19. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics ed. (Springer-Verlag, New York, 1987).
  20. J. Boyd, Chebyshev and Fourier Spectral Methods (Dover Publications, Inc, 2001).
  21. A. R. Zakharian,M. Brio, and J. V. Moloney, "FDTD based second-order accurate local mesh refinement method for Maxwell’s equations in two space dimensions," Comm. Math. Sci. 2(3), 497-513 (2004).
  22. OmniSim and FIMMWAVE, Photon Design, 34 Leopold Street, Oxford OX4 1TW, UK.
  23. <jrn>. S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205,402 (2003).

2004 (1)

A. R. Zakharian,M. Brio, and J. V. Moloney, "FDTD based second-order accurate local mesh refinement method for Maxwell’s equations in two space dimensions," Comm. Math. Sci. 2(3), 497-513 (2004).

2002 (2)

G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, "Multidomain Pseudospectral Time-Domain Simulations of Scattering by Objects Buried in Lossy Media," IEEE Trans. Geosci. Remote Sens. 40(6), 1366-1373 (2002).

J. T. Krug, E. J. Sanchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain," J. Chem. Phys. 116, 10,895 (2002).
[CrossRef]

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]

1999 (2)

J. S. Hesthaven, P. G. Dinensen, and J. P. Lynov, "Spectral collocation time-domain modeling of diffractive optical elements," J. Comput. Phys. 155, 287-306 (1999).
[CrossRef]

K. M. Krishnaiah and C. J. Railton, "A Stable Subgridding Algorithm and Its Application to Eigenvalue Problems," IEEE Trans. Microwave Theory Techn. 47, 620-628 (1999).
[CrossRef]

1998 (1)

F. J. Garcia-Vidal, J. M. Pitarke, and J. B. Pendry, "Silver-filled carbon nanotubes used as spectroscopic enhancers," Phys. Rev. B 58, 6783-6786 (1998).
[CrossRef]

1997 (3)

B. Yang, D. Gottlieb, and J. S. Hesthaven, "Spectral Simulations of ElectromagneticWave Scattering," J. Comput. Phys. 134, 216-230 (1997).
[CrossRef]

K. Kneip, Y. Wang, H. Kneip, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, "Single molecule detection using surface-enhanced Raman scattering," Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

S. Nie and S. R. Emory, "Probing single molecules and single nanoparticles by surface-enhanced Raman scattering," Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

1996 (1)

F. J. Garcia-Vidal and J. B. Pendry, "Collective theory for surface enhanced Raman scattering," Phys. Rev. Lett. 77, 1163-1166 (1996).
[CrossRef] [PubMed]

1991 (2)

A. C. Cangellaris and D. B. Wright, "Analysis of the numerical error caused by the stair-steppedapproximation of a conducting boundary in FDTD simulations ofelectromagnetic phenomena," IEEE Trans. Antennas Propag. 39(10), 1518-1525 (1991).
[CrossRef]

M. A. Fusco, M. V. Smith, and L. W. Gordon, "A Three-Dimensional FDTD Algorithm in Curvilinear Coordinates," IEEE Trans. Antennas Propag. 39(10), 1463-1471 (1991).
[CrossRef]

1985 (1)

M. Moskovits, "Surface enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).
[CrossRef]

1984 (1)

H. Metiu, "Surface enhanced spectroscopy," Prog. Surf. Sci. 17, 153-320 (1984).
[CrossRef]

1983 (1)

R. Holland, "Finite Difference Solutions of Maxwell’s Equations in Generalized Nonorthogonal Coordinates," IEEE Trans. Nucl. Sci. NS-30(6), 4589-4591 (1983).
[CrossRef]

Comm. Math. Sci. (1)

A. R. Zakharian,M. Brio, and J. V. Moloney, "FDTD based second-order accurate local mesh refinement method for Maxwell’s equations in two space dimensions," Comm. Math. Sci. 2(3), 497-513 (2004).

IEEE Trans. Antennas Propag. (2)

A. C. Cangellaris and D. B. Wright, "Analysis of the numerical error caused by the stair-steppedapproximation of a conducting boundary in FDTD simulations ofelectromagnetic phenomena," IEEE Trans. Antennas Propag. 39(10), 1518-1525 (1991).
[CrossRef]

M. A. Fusco, M. V. Smith, and L. W. Gordon, "A Three-Dimensional FDTD Algorithm in Curvilinear Coordinates," IEEE Trans. Antennas Propag. 39(10), 1463-1471 (1991).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

G.-X. Fan, Q. H. Liu, and J. S. Hesthaven, "Multidomain Pseudospectral Time-Domain Simulations of Scattering by Objects Buried in Lossy Media," IEEE Trans. Geosci. Remote Sens. 40(6), 1366-1373 (2002).

IEEE Trans. Microwave Theory Techn. (1)

K. M. Krishnaiah and C. J. Railton, "A Stable Subgridding Algorithm and Its Application to Eigenvalue Problems," IEEE Trans. Microwave Theory Techn. 47, 620-628 (1999).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

R. Holland, "Finite Difference Solutions of Maxwell’s Equations in Generalized Nonorthogonal Coordinates," IEEE Trans. Nucl. Sci. NS-30(6), 4589-4591 (1983).
[CrossRef]

J. Appl. Phys. (1)

K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, "Optical antennas: Resonators for local field enhancement," J. Appl. Phys. 94(7), 463242.

J. Chem. Phys. (1)

J. T. Krug, E. J. Sanchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain," J. Chem. Phys. 116, 10,895 (2002).
[CrossRef]

J. Comput. Phys. (2)

B. Yang, D. Gottlieb, and J. S. Hesthaven, "Spectral Simulations of ElectromagneticWave Scattering," J. Comput. Phys. 134, 216-230 (1997).
[CrossRef]

J. S. Hesthaven, P. G. Dinensen, and J. P. Lynov, "Spectral collocation time-domain modeling of diffractive optical elements," J. Comput. Phys. 155, 287-306 (1999).
[CrossRef]

Phys. Rev. B (2)

F. J. Garcia-Vidal, J. M. Pitarke, and J. B. Pendry, "Silver-filled carbon nanotubes used as spectroscopic enhancers," Phys. Rev. B 58, 6783-6786 (1998).
[CrossRef]

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]

Phys. Rev. Lett. (2)

K. Kneip, Y. Wang, H. Kneip, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, "Single molecule detection using surface-enhanced Raman scattering," Phys. Rev. Lett. 78, 1667-1670 (1997).
[CrossRef]

F. J. Garcia-Vidal and J. B. Pendry, "Collective theory for surface enhanced Raman scattering," Phys. Rev. Lett. 77, 1163-1166 (1996).
[CrossRef] [PubMed]

Prog. Surf. Sci. (1)

H. Metiu, "Surface enhanced spectroscopy," Prog. Surf. Sci. 17, 153-320 (1984).
[CrossRef]

Rev. Mod. Phys. (1)

M. Moskovits, "Surface enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985).
[CrossRef]

Science (1)

S. Nie and S. R. Emory, "Probing single molecules and single nanoparticles by surface-enhanced Raman scattering," Science 275, 1102-1106 (1997).
[CrossRef] [PubMed]

Other (6)

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, MA, 1995).

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics ed. (Springer-Verlag, New York, 1987).

J. Boyd, Chebyshev and Fourier Spectral Methods (Dover Publications, Inc, 2001).

W. Pernice, F. Payne, and D. Gallgher, "A general framework for the finite-difference time-domain simulation of real metals," IEEE Trans. Antennas Propag. 55, (2007).
[CrossRef]

OmniSim and FIMMWAVE, Photon Design, 34 Leopold Street, Oxford OX4 1TW, UK.

<jrn>. S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205,402 (2003).

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Figures (7)

Fig. 1.
Fig. 1.

Illustration of the positions of the primary FDTD and secondary PSTD grids for the TE case. The arrows indicate the fields that are used to calculate electric replacement fields for the primary grid or interpolate magnetic boundary fields for the secondary grid.

Fig. 2.
Fig. 2.

Transmission through an infinite chain of cylindrical nano-particles of 100 nm diameter. Shown are numerical calculations for both Nickel and Aluminium. The simulation was compared to the commercial products FIMMWAVE and OmniSim, using EME and FDTD, respectively.

Fig. 3.
Fig. 3.

The fitted dielectric function of Gold. Shown are the Drude-Lorentz model fit from OMNISIM [22] and the Drude-model fit from Maier [23].

Fig. 4.
Fig. 4.

The geometrical details of the computational domain used for the simulation of cylindrical nano-particles. The spectral element mesh of the PSTD subgrid is shown in the right figure.

Fig. 5.
Fig. 5.

Results from the simulation of a single gold particle embedded in air. Simulation results are shown for both the FDTD method and the new hybrid algorithm. The hybrid approach is virtually spike-free.

Fig. 6.
Fig. 6.

Numerical investigation of the dependency of the field enhancement factor for the Ex component on the cylinder diameter. The factor shows an almost linear behavior.

Fig. 7.
Fig. 7.

The magnitude of the electric Ex fields after an integration time of 4 fs. Shown is the enhanced field for an array of six gold cylinders with an inter-particle spacing of 100 nm computed using the FDTD method (a) and the hybrid method (b).

Tables (2)

Tables Icon

Table 1. Fitted material parameters for the the Drude-Lorentz model for Gold, the Drude model of Gold ([23]) and Silver.

Tables Icon

Table 2. Comparison of the performance of the hybrid algorithm with the FDTD method and a refined FDTD method.

Equations (5)

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

ε ( ω ) = ε ω P 2 ν ( ν + i ω ) + ω P 2 ν ( i ω ) + k = 1 N A k ω k 2 + 2 i Γ k ω ω 2
t μ H = × E
ε t E + t P D + k = 1 N t P k = × H σ E
t P D + ν P D = ω P 2 ν E
t 2 P k + 2 Γ k t P k + ω k 2 P k = A k E

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