Abstract

We use the finite-difference time-domain (FDTD) method to model the spectral properties of frequency-selective surfaces (FSSs) at normal incidence in the 110  μm wavelength. At these wavelengths the usual assumption that the metallic portions of a FSS are infinitesimally thin perfect conductors are no longer valid. We include the effects of dispersive complex conductivity for real metals and dispersive permittivity for dielectric materials by developing a unified approach that is especially suited for use in FDTD simulations. We concentrate on the finite nature of the metallic conductivity and its variation with wavelength in FSS structures. Our simulation results indicate that the resonant spectrum of a FSS in this wavelength range depends not only on the geometry of the structure and the dielectric substrate present, but also critically on the dispersive properties of the metal species used for the conductors.

© 2006 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 Propag. AP-14, 302-307 (1966).
  2. A. Taflove and M. Brodwin, "Scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. MTT-23, 623-630 (1975).
    [CrossRef]
  3. W. Tsay and D. Pozar, "Application of the FDTD technique to periodic problems in scattering and radiation," IEEE Microwave Guid. Wave Lett. 3, 250-252 (1993).
  4. K. Chamberlin and L. Gordon, "Modeling good conductors using the finite difference time-domain technique," IEEE Trans. Electromagn. Compat. 37, 210-216 (1995).
    [CrossRef]
  5. S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
    [CrossRef]
  6. J. Judkins and R. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995).
  7. R. Luebbers, F. Hunsberger, and K. Kunz, "A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma," IEEE Trans. Antennas Propag. 39, 29-34 (1991).
    [CrossRef]
  8. A. Taflove and S. Hagness, Computational Electrodynamics (Artech, 2000).
  9. M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997).
    [CrossRef]
  10. G. Fowles, Introduction to Modern Optics (Dover, 1989).
  11. A. Miller, "Fundamental optical properties of solids," in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, p. 9.15.
  12. D. Smith, E. Shiles, and M. Inokuti, "The optical properties of metallic aluminum," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 398-401.
  13. D. Lynch and W. Hunter, "Comments on the optical constants of metals and an introduction to the data for several metals," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 294-295.
  14. A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).
  15. C. Rhoads, E. Damon, and B. Munk, "Mid-infrared filters using conductive elements," Appl. Opt. 21, 2814-2816 (1982).
    [CrossRef] [PubMed]
  16. R. Ulrich, "Far-infrared properties of metallic mesh and its complementary structure," Infrared Phys. 7, 37-55 (1967).
    [CrossRef]

2001

S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
[CrossRef]

1997

M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997).
[CrossRef]

1995

K. Chamberlin and L. Gordon, "Modeling good conductors using the finite difference time-domain technique," IEEE Trans. Electromagn. Compat. 37, 210-216 (1995).
[CrossRef]

1993

W. Tsay and D. Pozar, "Application of the FDTD technique to periodic problems in scattering and radiation," IEEE Microwave Guid. Wave Lett. 3, 250-252 (1993).

1991

R. Luebbers, F. Hunsberger, and K. Kunz, "A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma," IEEE Trans. Antennas Propag. 39, 29-34 (1991).
[CrossRef]

1982

1975

A. Taflove and M. Brodwin, "Scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. MTT-23, 623-630 (1975).
[CrossRef]

1967

R. Ulrich, "Far-infrared properties of metallic mesh and its complementary structure," Infrared Phys. 7, 37-55 (1967).
[CrossRef]

1966

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

Brodwin, M.

A. Taflove and M. Brodwin, "Scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. MTT-23, 623-630 (1975).
[CrossRef]

Chamberlin, K.

K. Chamberlin and L. Gordon, "Modeling good conductors using the finite difference time-domain technique," IEEE Trans. Electromagn. Compat. 37, 210-216 (1995).
[CrossRef]

Damon, E.

Fowles, G.

G. Fowles, Introduction to Modern Optics (Dover, 1989).

Fu, J.

S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
[CrossRef]

Ghatak, A.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

Gordon, L.

K. Chamberlin and L. Gordon, "Modeling good conductors using the finite difference time-domain technique," IEEE Trans. Electromagn. Compat. 37, 210-216 (1995).
[CrossRef]

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics (Artech, 2000).

He, J.

S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
[CrossRef]

He, S.

S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
[CrossRef]

Hunsberger, F.

R. Luebbers, F. Hunsberger, and K. Kunz, "A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma," IEEE Trans. Antennas Propag. 39, 29-34 (1991).
[CrossRef]

Hunter, W.

D. Lynch and W. Hunter, "Comments on the optical constants of metals and an introduction to the data for several metals," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 294-295.

Inokuti, M.

D. Smith, E. Shiles, and M. Inokuti, "The optical properties of metallic aluminum," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 398-401.

Judkins, J.

J. Judkins and R. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995).

Kunz, K.

R. Luebbers, F. Hunsberger, and K. Kunz, "A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma," IEEE Trans. Antennas Propag. 39, 29-34 (1991).
[CrossRef]

Luebbers, R.

R. Luebbers, F. Hunsberger, and K. Kunz, "A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma," IEEE Trans. Antennas Propag. 39, 29-34 (1991).
[CrossRef]

Lynch, D.

D. Lynch and W. Hunter, "Comments on the optical constants of metals and an introduction to the data for several metals," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 294-295.

Miller, A.

A. Miller, "Fundamental optical properties of solids," in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, p. 9.15.

Mrozowski, M.

M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997).
[CrossRef]

Munk, B.

Okoniewski, M.

M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997).
[CrossRef]

Pozar, D.

W. Tsay and D. Pozar, "Application of the FDTD technique to periodic problems in scattering and radiation," IEEE Microwave Guid. Wave Lett. 3, 250-252 (1993).

Rhoads, C.

Shen, L.

S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
[CrossRef]

Shiles, E.

D. Smith, E. Shiles, and M. Inokuti, "The optical properties of metallic aluminum," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 398-401.

Smith, D.

D. Smith, E. Shiles, and M. Inokuti, "The optical properties of metallic aluminum," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 398-401.

Stuchly, M.

M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997).
[CrossRef]

Taflove, A.

A. Taflove and M. Brodwin, "Scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. MTT-23, 623-630 (1975).
[CrossRef]

A. Taflove and S. Hagness, Computational Electrodynamics (Artech, 2000).

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

Tsay, W.

W. Tsay and D. Pozar, "Application of the FDTD technique to periodic problems in scattering and radiation," IEEE Microwave Guid. Wave Lett. 3, 250-252 (1993).

Ulrich, R.

R. Ulrich, "Far-infrared properties of metallic mesh and its complementary structure," Infrared Phys. 7, 37-55 (1967).
[CrossRef]

Xiao, S.

S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
[CrossRef]

Yee, K. S.

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

Ziolkowski, R.

J. Judkins and R. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995).

Antennas Propag.

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

R. Luebbers, F. Hunsberger, and K. Kunz, "A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma," IEEE Trans. Antennas Propag. 39, 29-34 (1991).
[CrossRef]

Appl. Opt.

Electromagn. Compat.

K. Chamberlin and L. Gordon, "Modeling good conductors using the finite difference time-domain technique," IEEE Trans. Electromagn. Compat. 37, 210-216 (1995).
[CrossRef]

Infrared Phys.

R. Ulrich, "Far-infrared properties of metallic mesh and its complementary structure," Infrared Phys. 7, 37-55 (1967).
[CrossRef]

J. Opt. Soc. Am. A

J. Judkins and R. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995).

J. Phys. A

S. He, S. Xiao, L. Shen, J. He, and J. Fu, "A new finite-difference time-domain method for photonic crystals consisting of nearly-free-electron-metals," J. Phys. A 34, 9713-9721 (2001).
[CrossRef]

Microwave Theory Tech.

A. Taflove and M. Brodwin, "Scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. MTT-23, 623-630 (1975).
[CrossRef]

Wave Lett.

M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guid. Wave Lett. 7, 121-123 (1997).
[CrossRef]

Other

G. Fowles, Introduction to Modern Optics (Dover, 1989).

A. Miller, "Fundamental optical properties of solids," in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, p. 9.15.

D. Smith, E. Shiles, and M. Inokuti, "The optical properties of metallic aluminum," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 398-401.

D. Lynch and W. Hunter, "Comments on the optical constants of metals and an introduction to the data for several metals," in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), pp. 294-295.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

W. Tsay and D. Pozar, "Application of the FDTD technique to periodic problems in scattering and radiation," IEEE Microwave Guid. Wave Lett. 3, 250-252 (1993).

A. Taflove and S. Hagness, Computational Electrodynamics (Artech, 2000).

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Figures (7)

Fig. 1
Fig. 1

χ2 versus number of fit terms (P) for aluminum and gold.

Fig. 2
Fig. 2

Fractional error of complex refractive indices for aluminum and gold for one and two term fits.

Fig. 3
Fig. 3

Comparison of reflectivity and reflection phase for aluminum and gold.

Fig. 4
Fig. 4

Geometry of a simulated FSS.

Fig. 5
Fig. 5

FDTD-calculated FSS reflectivity for PEC, R-98, aluminum, and gold.

Fig. 6
Fig. 6

Transmission line models for PEC, real conductivity, and complex conductivity.

Fig. 7
Fig. 7

Wavelength-dependent complex conductivity for aluminum and gold.

Equations (22)

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

× H ( t , r ) = D ( t , r ) t + J ( t , r ) ,
× H ( t ) = ε 0 E ( t ) t + P ( t ) t + J ( t ) .
× H ( ω ) = j ω ε 0 [ E ( ω ) + 1 ε 0 P ( ω ) ] + J ( ω ) .
P ( ω ) = N e 2 / m ω 0     2 ω 2 + j ω γ E ( ω ) ,
× H ( ω ) = j ω ε 0 [ 1 + ( 1 ε 0 ) N e 2 / m ω 0     2 ω 2 + j ω γ ] E ( ω ) = j ω ε 0 ε ˜ r ( ω ) E ( ω ) ,
J ( ω ) = σ s 1 + j ω τ E ( ω ) = σ ˜ ( ω ) E ( ω ) ,
× H ( ω ) = [ j ω ε 0 + σ ˜ ( ω ) ] E ( ω ) = j ω ε 0 [ 1 + ( 1 ε 0 ) σ s / τ ω 2 + j ω / τ ] E ( ω ) .
ξ ˜ ( ω ) = 1 + b d ω 2 + j ω C ,
× H ( ω ) = j ω ε 0 ξ ˜ ( ω ) E ( ω ) .
ξ ˜ ( ω ) = n ˜ 2 ( ω ) = [ n r e ( ω ) j n i m ( ω ) ] 2 ,
ε r ( ω ) = ξ ˜ ( ω ) = 1 + b d ω 2 + j ω C ,
ξ ˜ ( ω ) = 1 + b ω 2 + j ω C .
σ ˜ ( ω ) = j ω ε 0 b ω 2 + j ω C = j ω ε 0 [ ξ ˜ ( ω ) 1 ] .
σ ˜ ( ω ) j ω ε 0 ξ ˜ ( ω ) .
ξ ˜ ( ω ) = 1 + p = 1 p b p d p ω 2 + j ω C p ,
n 2 ( λ 0 ) = ε r ( λ 0 ) = 1 + p = 1 p S p λ 0     2 λ 0     2 λ p     2 ,
n 2 ( ω ) = ε r ( ω ) = 1 + p = 1 p S p ω p     2 ω p     2 ω 2 ,
ε r ( ω ) = ε + p = 1 p Δ ε p ω p     2 ω p     2 ω 2 + 2 j ω δ p ,
n re ( ω ) = Re { ξ ˜ ( ω ) } ,
n im ( ω ) = Im { ξ ˜ ( ω ) } ,
χ 2 = 1 N i = 1 N { [ ( n ^ re ( ω i ) n re ( ω i ) ) / n re ( ω i ) ] 2 + [ ( n ^ im ( ω i ) n im ( ω i ) ) / n im ( ω i ) ] 2 } ,
σ ˜ ( ω ) j ω ε 0 n ˜ 2 ( ω ) .

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