Abstract

The finite-difference time-domain method can provide broadband results if the excitation source is a pulse. This demands that the parameters of modeled materials have to be applicable over broad frequency bands. We optimize the modified Debye model parameters for gold, silver, copper, platinum, and aluminum using a large-scale nonlinear optimization algorithm. The complex relative permittivities calculated using the optimized parameters agree well with experimental values over broad frequency bands. The associated root-mean-square deviations are 0.49%, 3.52%, 4.13%, 1.64%, and 0.66%, respectively. We also provide an example of broadband calculations. The obtained broadband results are verified by a series of steady-state calculations.

© 2007 Optical Society of America

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  1. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
    [CrossRef]
  2. G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. EMC-23, 377-382 (1981).
    [CrossRef]
  3. Z. P. Liao, H. L. Wong, G. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. 28, 1063-1076 (1984).
  4. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  5. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).
  6. J. T. K. Li, E. J. Sánchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation," J. Chem. Phys. 116, 10895-10901 (2002).
    [CrossRef]
  7. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  8. E. X. Jin and X. Xu, "Plasmonic effects in near-field optical transmission enhancement through a single bowtie-shaped aperture," Appl. Phys. B 84, 3-9 (2006).
    [CrossRef]
  9. X. Shi, R. L. Thornton, and L. Hesselink, "A nano-aperture with 1000× power throughput enhancement for very small aperture laser system (VSAL)," Proc. SPIE 4342, 320-327 (2002).
    [CrossRef]
  10. X. Shi and L. Hesselink, "Design of a C aperture to achieve λ/10 resolution and resonant transmission," J. Opt. Soc. Am. B 21, 1305-1317 (2004).
    [CrossRef]
  11. J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM J. Sci. Stat. Comput. 3, 553-572 (1983).
  12. T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
    [CrossRef]
  13. T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," Math. Program. 67, 189-224 (1994).
    [CrossRef]
  14. M. A. Branch, T. F. Coleman, and Y. Li, "A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems," SIAM J. Sci. Comput. 21, 1-23 (1999).
    [CrossRef]
  15. R. H. Byrd, R. B. Schnabel, and G. A. Shultz, "Approximate solution of the trust region problem by minimization over two-dimensional subspace," Math. Program. 40, 247-263 (1988).
    [CrossRef]
  16. R.-J. Zhu, J. Wang, and G.-F. Jin, "Mie scattering calculation by FDTD employing a modified Debye model for Gold material," Optik 116, 419-422 (2005).
    [CrossRef]
  17. Computer code xfdtd 6.2, REMCOM, 2000.
  18. X. Xu, E. X. Jin, L. Wang, and S. Uppuluri, "Design, fabrication, and characterization of nanometer-scale ridged aperture optical antennae," Proc. of SPIE 6106, 61061 (2006).
    [CrossRef]
  19. H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
    [CrossRef]
  20. C. J. Bouwkamp, "On Bethe's theory of diffraction by small holes," Philips Res. Rep. 5, 321-332 (1950).

2006

E. X. Jin and X. Xu, "Plasmonic effects in near-field optical transmission enhancement through a single bowtie-shaped aperture," Appl. Phys. B 84, 3-9 (2006).
[CrossRef]

X. Xu, E. X. Jin, L. Wang, and S. Uppuluri, "Design, fabrication, and characterization of nanometer-scale ridged aperture optical antennae," Proc. of SPIE 6106, 61061 (2006).
[CrossRef]

2005

R.-J. Zhu, J. Wang, and G.-F. Jin, "Mie scattering calculation by FDTD employing a modified Debye model for Gold material," Optik 116, 419-422 (2005).
[CrossRef]

2004

2002

J. T. K. Li, E. J. Sánchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation," J. Chem. Phys. 116, 10895-10901 (2002).
[CrossRef]

X. Shi, R. L. Thornton, and L. Hesselink, "A nano-aperture with 1000× power throughput enhancement for very small aperture laser system (VSAL)," Proc. SPIE 4342, 320-327 (2002).
[CrossRef]

1999

M. A. Branch, T. F. Coleman, and Y. Li, "A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems," SIAM J. Sci. Comput. 21, 1-23 (1999).
[CrossRef]

1996

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
[CrossRef]

1994

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," Math. Program. 67, 189-224 (1994).
[CrossRef]

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

1988

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, "Approximate solution of the trust region problem by minimization over two-dimensional subspace," Math. Program. 40, 247-263 (1988).
[CrossRef]

1984

Z. P. Liao, H. L. Wong, G. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. 28, 1063-1076 (1984).

1983

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM J. Sci. Stat. Comput. 3, 553-572 (1983).

1981

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. EMC-23, 377-382 (1981).
[CrossRef]

1966

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

1950

C. J. Bouwkamp, "On Bethe's theory of diffraction by small holes," Philips Res. Rep. 5, 321-332 (1950).

1944

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Berenger, J. P.

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

Bethe, H. A.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, "On Bethe's theory of diffraction by small holes," Philips Res. Rep. 5, 321-332 (1950).

Branch, M. A.

M. A. Branch, T. F. Coleman, and Y. Li, "A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems," SIAM J. Sci. Comput. 21, 1-23 (1999).
[CrossRef]

Byrd, R. H.

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, "Approximate solution of the trust region problem by minimization over two-dimensional subspace," Math. Program. 40, 247-263 (1988).
[CrossRef]

Coleman, T. F.

M. A. Branch, T. F. Coleman, and Y. Li, "A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems," SIAM J. Sci. Comput. 21, 1-23 (1999).
[CrossRef]

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
[CrossRef]

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," Math. Program. 67, 189-224 (1994).
[CrossRef]

Hesselink, L.

X. Shi and L. Hesselink, "Design of a C aperture to achieve λ/10 resolution and resonant transmission," J. Opt. Soc. Am. B 21, 1305-1317 (2004).
[CrossRef]

X. Shi, R. L. Thornton, and L. Hesselink, "A nano-aperture with 1000× power throughput enhancement for very small aperture laser system (VSAL)," Proc. SPIE 4342, 320-327 (2002).
[CrossRef]

Jin, E. X.

X. Xu, E. X. Jin, L. Wang, and S. Uppuluri, "Design, fabrication, and characterization of nanometer-scale ridged aperture optical antennae," Proc. of SPIE 6106, 61061 (2006).
[CrossRef]

E. X. Jin and X. Xu, "Plasmonic effects in near-field optical transmission enhancement through a single bowtie-shaped aperture," Appl. Phys. B 84, 3-9 (2006).
[CrossRef]

Jin, G.-F.

R.-J. Zhu, J. Wang, and G.-F. Jin, "Mie scattering calculation by FDTD employing a modified Debye model for Gold material," Optik 116, 419-422 (2005).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Li, J. T. K.

J. T. K. Li, E. J. Sánchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation," J. Chem. Phys. 116, 10895-10901 (2002).
[CrossRef]

Li, Y.

M. A. Branch, T. F. Coleman, and Y. Li, "A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems," SIAM J. Sci. Comput. 21, 1-23 (1999).
[CrossRef]

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
[CrossRef]

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," Math. Program. 67, 189-224 (1994).
[CrossRef]

Liao, Z. P.

Z. P. Liao, H. L. Wong, G. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. 28, 1063-1076 (1984).

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Moré, J. J.

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM J. Sci. Stat. Comput. 3, 553-572 (1983).

Mur, G.

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. EMC-23, 377-382 (1981).
[CrossRef]

Palik, E. D.

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

Sánchez, E. J.

J. T. K. Li, E. J. Sánchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation," J. Chem. Phys. 116, 10895-10901 (2002).
[CrossRef]

Schnabel, R. B.

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, "Approximate solution of the trust region problem by minimization over two-dimensional subspace," Math. Program. 40, 247-263 (1988).
[CrossRef]

Shi, X.

X. Shi and L. Hesselink, "Design of a C aperture to achieve λ/10 resolution and resonant transmission," J. Opt. Soc. Am. B 21, 1305-1317 (2004).
[CrossRef]

X. Shi, R. L. Thornton, and L. Hesselink, "A nano-aperture with 1000× power throughput enhancement for very small aperture laser system (VSAL)," Proc. SPIE 4342, 320-327 (2002).
[CrossRef]

Shultz, G. A.

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, "Approximate solution of the trust region problem by minimization over two-dimensional subspace," Math. Program. 40, 247-263 (1988).
[CrossRef]

Sorensen, D. C.

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM J. Sci. Stat. Comput. 3, 553-572 (1983).

Thornton, R. L.

X. Shi, R. L. Thornton, and L. Hesselink, "A nano-aperture with 1000× power throughput enhancement for very small aperture laser system (VSAL)," Proc. SPIE 4342, 320-327 (2002).
[CrossRef]

Uppuluri, S.

X. Xu, E. X. Jin, L. Wang, and S. Uppuluri, "Design, fabrication, and characterization of nanometer-scale ridged aperture optical antennae," Proc. of SPIE 6106, 61061 (2006).
[CrossRef]

Wang, J.

R.-J. Zhu, J. Wang, and G.-F. Jin, "Mie scattering calculation by FDTD employing a modified Debye model for Gold material," Optik 116, 419-422 (2005).
[CrossRef]

Wang, L.

X. Xu, E. X. Jin, L. Wang, and S. Uppuluri, "Design, fabrication, and characterization of nanometer-scale ridged aperture optical antennae," Proc. of SPIE 6106, 61061 (2006).
[CrossRef]

Wong, H. L.

Z. P. Liao, H. L. Wong, G. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. 28, 1063-1076 (1984).

Xie, X. S.

J. T. K. Li, E. J. Sánchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation," J. Chem. Phys. 116, 10895-10901 (2002).
[CrossRef]

Xu, X.

E. X. Jin and X. Xu, "Plasmonic effects in near-field optical transmission enhancement through a single bowtie-shaped aperture," Appl. Phys. B 84, 3-9 (2006).
[CrossRef]

X. Xu, E. X. Jin, L. Wang, and S. Uppuluri, "Design, fabrication, and characterization of nanometer-scale ridged aperture optical antennae," Proc. of SPIE 6106, 61061 (2006).
[CrossRef]

Yang, G. P.

Z. P. Liao, H. L. Wong, G. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. 28, 1063-1076 (1984).

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

Yuan, Y. F.

Z. P. Liao, H. L. Wong, G. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. 28, 1063-1076 (1984).

Zhu, R.-J.

R.-J. Zhu, J. Wang, and G.-F. Jin, "Mie scattering calculation by FDTD employing a modified Debye model for Gold material," Optik 116, 419-422 (2005).
[CrossRef]

Appl. Phys. B

E. X. Jin and X. Xu, "Plasmonic effects in near-field optical transmission enhancement through a single bowtie-shaped aperture," Appl. Phys. B 84, 3-9 (2006).
[CrossRef]

IEEE Trans. Antennas Propagat.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

IEEE Trans. Electromagn. Compat.

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat. EMC-23, 377-382 (1981).
[CrossRef]

J. Chem. Phys.

J. T. K. Li, E. J. Sánchez, and X. S. Xie, "Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation," J. Chem. Phys. 116, 10895-10901 (2002).
[CrossRef]

J. Comput. Phys.

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Math. Program.

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," Math. Program. 67, 189-224 (1994).
[CrossRef]

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, "Approximate solution of the trust region problem by minimization over two-dimensional subspace," Math. Program. 40, 247-263 (1988).
[CrossRef]

Optik

R.-J. Zhu, J. Wang, and G.-F. Jin, "Mie scattering calculation by FDTD employing a modified Debye model for Gold material," Optik 116, 419-422 (2005).
[CrossRef]

Philips Res. Rep.

C. J. Bouwkamp, "On Bethe's theory of diffraction by small holes," Philips Res. Rep. 5, 321-332 (1950).

Phys. Rev.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Proc. of SPIE

X. Xu, E. X. Jin, L. Wang, and S. Uppuluri, "Design, fabrication, and characterization of nanometer-scale ridged aperture optical antennae," Proc. of SPIE 6106, 61061 (2006).
[CrossRef]

Proc. SPIE

X. Shi, R. L. Thornton, and L. Hesselink, "A nano-aperture with 1000× power throughput enhancement for very small aperture laser system (VSAL)," Proc. SPIE 4342, 320-327 (2002).
[CrossRef]

Sci. Sin.

Z. P. Liao, H. L. Wong, G. P. Yang, and Y. F. Yuan, "A transmitting boundary for transient wave analysis," Sci. Sin. 28, 1063-1076 (1984).

SIAM J. Optim.

T. F. Coleman and Y. Li, "On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
[CrossRef]

SIAM J. Sci. Comput.

M. A. Branch, T. F. Coleman, and Y. Li, "A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems," SIAM J. Sci. Comput. 21, 1-23 (1999).
[CrossRef]

SIAM J. Sci. Stat. Comput.

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM J. Sci. Stat. Comput. 3, 553-572 (1983).

Other

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

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Computer code xfdtd 6.2, REMCOM, 2000.

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Figures (3)

Fig. 1
Fig. 1

(Color online) Comparison between our results and the experimental determined complex relative permittivities[7] of (a) gold, (b) silver, (c) copper, (d) platinum, and (e) aluminum. The squares are the real parts of the experimental values; the diamonds are the imaginary parts. The solid curves are our results.

Fig. 2
Fig. 2

(Color online) FDTD simulation model: a circular aperture in a silver film.

Fig. 3
Fig. 3

(Color online) Comparison between the power throughputs calculated using two types of excitation sources:a pulse and a time harmonic wave. The solid curve is the broadband results obtained with the pulse excitation. The squares are the discrete results obtained with the time harmonic wave excitation. The inset shows the incident pulse field.

Tables (1)

Tables Icon

Table 1 Modified Debye Model Parameters of Metals Applicable over Broad Frequency Bands

Equations (41)

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

550 950   nm
ε ^ ( ω ) = ε + ε s ε 1 + i ω τ + σ i ω ε 0 = ε i ε ,
ε ^
ε
ε s
ε 0
ε
ε
ε
ε s
ε ^ ( ω ) = 1 + ω p 2 ω ( i ν c ω ) = 1 ( ω p / ν c ) 2 1 + i ω ν c + ( ω p 2 / ν c ) i ω ,
ω p
ν c
σ τ = ε 0 ( ε ε s ) .
ε
ε s
f ( ε , ε s , τ )
min f ( ε , ε s , τ ) = 1 2 j ε ^ j ( ε , ε s , τ ) ( ε j i ε j ) 2 2 .
ε > 1
ε s < 0
τ > 0
ε = 11.575
ε s = 15789
τ = 8.71 × 10 15
σ = 1.6062 × 10 7 S / m
700 1200   nm
700 1000   nm
700 1000   nm
550 9 5 0   nm
100   nm
50   nm
1000   nm × 1000   nm × 1250   nm
5   nm × 5   nm × 5   nm
450 1200   nm
8.66625 × 10 18   s
10   nm
450   nm
1200   nm
50   nm
ε s
ε

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