Abstract

Simulating the interaction of electromagnetic terahertz radiation with metals poses difficulties not encountered in the optical regime. Owing to a penetration depth small compared to the wavelength, such simulations in the terahertz frequency range require large discretization volumes with very small grid spacings. We present a unique subgridding scheme that accurately describes this interaction while keeping computational costs minimal. Bidirectional coupling between grids allows for the complete integration of subdomains into the simulation volume. Implementation in one and two dimensions is demonstrated, and a comparison with theoretical and experimental results [Opt. Express 15, 6552 (2007) ] [Phys. Rev. B 69, 155427 (2004) ] is given. Using our technique, we are able to accurately simulate plasmonic effects in terahertz experiments for the first time.

© 2008 Optical Society of America

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References

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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).
    [CrossRef]
  2. A. Taflove, "Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems," IEEE Trans. Electromagn. Compat. EMC-22, 191-202 (1980).
    [CrossRef]
  3. D. M. Sullivan, "Frequency-dependent FDTD methods using Z transforms," IEEE Trans. Antennas Propag. AP-40, 1223-1230 (1992).
    [CrossRef]
  4. W. Tong, Z. Wenjun, and L. Weiliang, "The frequency-dependent FDTD and the digital signal processing techniques," in Proceedings of International Conference on Computational Electromagnetics and Its Applications (ICCEA, 1999), pp. 48-51.
  5. P. Thoma and T. Weiland, "Numerical stability of finite difference time domain methods," IEEE Trans. Magn. 34, 2740-2743 (1998).
    [CrossRef]
  6. K. S. Kunz and L. Simpson, "A technique for increasing the resolution of finite-difference solutions of the Maxwell equation," IEEE Trans. Electromagn. Compat. EMC-23, 419-422 (1981).
    [CrossRef]
  7. Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, "Transmission of light through slit apertures in metallic films," Opt. Express 12, 6106-6121 (2004).
    [CrossRef] [PubMed]
  8. J. Kröll, J. Darmo, and K. Unterrainer, "Metallic wave-impedance matching layers for broadband terahertz optical systems," Opt. Express 15, 6552-6560 (2007).
    [CrossRef] [PubMed]
  9. J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  10. D.R.Lide, ed., CRC Handbook of Chemistry and Physics, 84th ed. (CRC Press, 2003).
  11. "Freiburg THz-TDS group homepage," http://frhethz.physik.uni-freiburg.de.
  12. J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolívar, and H. Kurz, "Time-domain measurements of surface plasmon polaritons in the terahertz frequency range," Phys. Rev. B 69, 155427 (2004).
    [CrossRef]
  13. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr., and C. A. Ward, "Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared," Appl. Opt. 22, 1099-1120 (1983).
    [CrossRef] [PubMed]
  14. J. D. Jackson, Classical Electrodynamics, 3rd ed. (de Gruyter, 2002).
  15. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  16. E. S. Koteles and W. H. McNeill, "Far infrared surface plasmon propagation," Int. J. Infrared Millim. Waves 2, 361-371 (1981).
    [CrossRef]
  17. K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength size metallic particles," Phys. Rev. B 72, 075110 (2005).
    [CrossRef]

2007

2005

K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength size metallic particles," Phys. Rev. B 72, 075110 (2005).
[CrossRef]

2004

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolívar, and H. Kurz, "Time-domain measurements of surface plasmon polaritons in the terahertz frequency range," Phys. Rev. B 69, 155427 (2004).
[CrossRef]

Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, "Transmission of light through slit apertures in metallic films," Opt. Express 12, 6106-6121 (2004).
[CrossRef] [PubMed]

1998

P. Thoma and T. Weiland, "Numerical stability of finite difference time domain methods," IEEE Trans. Magn. 34, 2740-2743 (1998).
[CrossRef]

1994

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

1992

D. M. Sullivan, "Frequency-dependent FDTD methods using Z transforms," IEEE Trans. Antennas Propag. AP-40, 1223-1230 (1992).
[CrossRef]

1983

1981

E. S. Koteles and W. H. McNeill, "Far infrared surface plasmon propagation," Int. J. Infrared Millim. Waves 2, 361-371 (1981).
[CrossRef]

K. S. Kunz and L. Simpson, "A technique for increasing the resolution of finite-difference solutions of the Maxwell equation," IEEE Trans. Electromagn. Compat. EMC-23, 419-422 (1981).
[CrossRef]

1980

A. Taflove, "Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems," IEEE Trans. Electromagn. Compat. EMC-22, 191-202 (1980).
[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).
[CrossRef]

Appl. Opt.

IEEE Trans. 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).
[CrossRef]

D. M. Sullivan, "Frequency-dependent FDTD methods using Z transforms," IEEE Trans. Antennas Propag. AP-40, 1223-1230 (1992).
[CrossRef]

IEEE Trans. Electromagn. Compat.

A. Taflove, "Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems," IEEE Trans. Electromagn. Compat. EMC-22, 191-202 (1980).
[CrossRef]

K. S. Kunz and L. Simpson, "A technique for increasing the resolution of finite-difference solutions of the Maxwell equation," IEEE Trans. Electromagn. Compat. EMC-23, 419-422 (1981).
[CrossRef]

IEEE Trans. Magn.

P. Thoma and T. Weiland, "Numerical stability of finite difference time domain methods," IEEE Trans. Magn. 34, 2740-2743 (1998).
[CrossRef]

Int. J. Infrared Millim. Waves

E. S. Koteles and W. H. McNeill, "Far infrared surface plasmon propagation," Int. J. Infrared Millim. Waves 2, 361-371 (1981).
[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]

Opt. Express

Phys. Rev. B

K. J. Chau and A. Y. Elezzabi, "Terahertz transmission through ensembles of subwavelength size metallic particles," Phys. Rev. B 72, 075110 (2005).
[CrossRef]

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. Haring Bolívar, and H. Kurz, "Time-domain measurements of surface plasmon polaritons in the terahertz frequency range," Phys. Rev. B 69, 155427 (2004).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics, 3rd ed. (de Gruyter, 2002).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

D.R.Lide, ed., CRC Handbook of Chemistry and Physics, 84th ed. (CRC Press, 2003).

"Freiburg THz-TDS group homepage," http://frhethz.physik.uni-freiburg.de.

W. Tong, Z. Wenjun, and L. Weiliang, "The frequency-dependent FDTD and the digital signal processing techniques," in Proceedings of International Conference on Computational Electromagnetics and Its Applications (ICCEA, 1999), pp. 48-51.

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Figures (5)

Fig. 1
Fig. 1

Concept of the proposed subgridding scheme with bidirectional coupling in one dimension. Circles represent grid points containing electric field components, vertical dashes are magnetic field components. Ω 1 is the coarse grid, and Ω 2 is the subgrid. The large gray rectangle represents a spatial structure that demands subgridding.

Fig. 2
Fig. 2

Simulation of the system described in [8]. (a) shows the field amplitude of a THz pulse after passing through the substrate without (upper line) and with (lower line) 8 nm chromium coating. (b) shows the effect of chromium films thinner ( 4 nm ) and thicker ( 12 nm ) than the optimal thickness. Here, the field amplitudes are magnified 4 × to the right of the dashed line. A time-resolved animation of the simulation can be viewed on our homepage [11].

Fig. 3
Fig. 3

Simulation of THz pulse passing through a 100 μ m aperture above a gold sheet. The figure shows superimposed snapshots of the simulated field in 4 ps steps, H z component shown. (a), the coarse grid; (b), the refined subgrid. Horizontal lines represent gold surfaces; the vertical line is a perfect metallic conductor. Step sizes in (a) are Δ x = Δ y = 10 μ m ; in (b), Δ x = 10 μ m , Δ y = 40 nm . An animation of the simulation can be viewed on our homepage [11].

Fig. 4
Fig. 4

Simulated field amplitude above Drude metal calculated with subgridding relative to a perfect metallic conductor (circles). The line is a fit according to Eq. (11). Fit parameters R and A correspond to the reflectivity of the gold and the SPP coupling strength.

Fig. 5
Fig. 5

2-D simulated system without subgridding. H z field component shown. The horizontal line represents Drude modeled gold surface; the vertical line represents a perfect metallic conductor.

Equations (13)

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D t = × H ,
B t = × E ,
H = μ 1 B ,
E = ϵ 1 D .
s = 2 Δ t 1 z 1 1 + z 1 ,
ϵ ( z ) = b 0 + b 1 z 1 + b 2 z 2 + + b N z N a 0 + a 1 z 1 + a 2 z 2 + + a M z M ,
( b 0 + b 1 z 1 + b 2 z 2 + + b N z N ) E ( z ) = ( a 0 + a 1 z 1 + a 2 z 2 + + a M z M ) D ( z ) ,
Z { f n k } = z k Z { f n } ,
E n = [ ( a 0 D n + a 1 D n 1 + + a M D n M ) ( b 1 E n 1 + b 2 E n 2 + + b N E n N ) ] b 0 1 .
[ ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ] 1 2 > c max Δ t ,
H z , PMC ( r ) = H 0 * r 1 2 .
H z , SGr ( r ) = H 0 * ( R r 1 2 + A ) ,
H z , SGr ( r ) H z , PMC ( r ) = R + A r 1 2 .

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