Abstract

We investigate the accuracy of the two-dimensional Finite-Difference Time-Domain (FDTD) method in modelling Surface Plasmon Polaritons (SPPs) in the case of a single metal-dielectric interface and of a thin metal film showing that FDTD has difficulties in the low-group-velocity region of the SPP. We combine a contour-path approach with Z transform to handle both the electromagnetic boundary conditions at the interface and the negative dispersive dielectric function of the metal. The relative error is thus significantly reduced.

© 2006 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988)
  2. W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [CrossRef] [PubMed]
  3. A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005).
  4. D. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, 2000).
    [CrossRef]
  5. T.G. Jurgens, A. Taflove, K. Umaschankar, and T.G. Moore, “Finite-Difference Time-Domain Modeling of Curved Surfaces,” IEEE Trans. Antennas Propag. 40, 357–365 (1992).
    [CrossRef]
  6. C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B 108, 17740–17747 (2004).
    [CrossRef]
  7. H. Shin and S. Fan, “All-Angle Negative Refraction for Surface Plasmon Waves Using a Metal-Dielectric-Metal Structure,” Phys. Rev. Lett. 96, 073907(4) (2006).
  8. K.-P. Hwang and A.C. Cangellaris, “Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces,” IEEE Microwave Wirel. Compon. Lett. 11, 158–160 (2001).
    [CrossRef]
  9. A. Mohammadi and M. Agio, “Contour-path effective permittivities for the twodimensional finite-difference time-domain method,” Opt. Express 13, 10367–10381 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10367, and references therein.
    [CrossRef] [PubMed]
  10. M.K. Kärkkäinen, Subcell FDTD Modeling of Electrically Thin Dispersive Layers,” IEEE Trans. Microwave Theory Tech. 51, 1774–1780 (2003).
    [CrossRef]
  11. D. Popovic and M. Okoniewski, “Effective Permittivity at the Interface of Dispersive Dielectrics in FDTD,” IEEE Microwave Wirel. Compon. Lett. 13, 265–267 (2003).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), 7th ed.
  13. C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
    [CrossRef]
  14. O. Ramadan and A.Y. Oztoprak, “Z-transform implementation of the perfectly matched layer for truncating FDTD domains,” IEEE Microwave Wirel. Compon. Lett. 13, 402–404 (2003).
    [CrossRef]
  15. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996), 7th ed.
  16. In the error calculation we choose wavevectors k≥1.5ks (Fig. 3 and Fig. 4) and k≥2.5ks (Fig. 6) to focus on the low-group-velocity region.
  17. The fit is performed using the logarithm of the error, so that the exponent α is obtained from logy=α log x+loga. The errors for Δ=2–4nm are excluded from the fit since the Fourier error is of the same order of magnitude and thus it may have an effect on the actual FDTD accuracy.
  18. C. Hulse and A. Knoesen, “Dispersive models for the finite-difference time-domain method: design, analysis, and implementation,” J. Opt. Soc. Am. A 11, 1802–1811 (1994).
    [CrossRef]

2006 (1)

H. Shin and S. Fan, “All-Angle Negative Refraction for Surface Plasmon Waves Using a Metal-Dielectric-Metal Structure,” Phys. Rev. Lett. 96, 073907(4) (2006).

2005 (1)

2004 (1)

C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[CrossRef]

2003 (4)

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

M.K. Kärkkäinen, Subcell FDTD Modeling of Electrically Thin Dispersive Layers,” IEEE Trans. Microwave Theory Tech. 51, 1774–1780 (2003).
[CrossRef]

D. Popovic and M. Okoniewski, “Effective Permittivity at the Interface of Dispersive Dielectrics in FDTD,” IEEE Microwave Wirel. Compon. Lett. 13, 265–267 (2003).
[CrossRef]

O. Ramadan and A.Y. Oztoprak, “Z-transform implementation of the perfectly matched layer for truncating FDTD domains,” IEEE Microwave Wirel. Compon. Lett. 13, 402–404 (2003).
[CrossRef]

2001 (1)

K.-P. Hwang and A.C. Cangellaris, “Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces,” IEEE Microwave Wirel. Compon. Lett. 11, 158–160 (2001).
[CrossRef]

1995 (1)

C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[CrossRef]

1994 (1)

1992 (1)

T.G. Jurgens, A. Taflove, K. Umaschankar, and T.G. Moore, “Finite-Difference Time-Domain Modeling of Curved Surfaces,” IEEE Trans. Antennas Propag. 40, 357–365 (1992).
[CrossRef]

Agio, M.

Barnes, W.L.

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), 7th ed.

Cangellaris, A.C.

K.-P. Hwang and A.C. Cangellaris, “Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces,” IEEE Microwave Wirel. Compon. Lett. 11, 158–160 (2001).
[CrossRef]

Chan, C.T.

C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[CrossRef]

Dereux, A.

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Ebbesen, T.W.

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Fan, S.

H. Shin and S. Fan, “All-Angle Negative Refraction for Surface Plasmon Waves Using a Metal-Dielectric-Metal Structure,” Phys. Rev. Lett. 96, 073907(4) (2006).

Hagness, S.C.

A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005).

Ho, K.M.

C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[CrossRef]

Hulse, C.

Hwang, K.-P.

K.-P. Hwang and A.C. Cangellaris, “Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces,” IEEE Microwave Wirel. Compon. Lett. 11, 158–160 (2001).
[CrossRef]

Jurgens, T.G.

T.G. Jurgens, A. Taflove, K. Umaschankar, and T.G. Moore, “Finite-Difference Time-Domain Modeling of Curved Surfaces,” IEEE Trans. Antennas Propag. 40, 357–365 (1992).
[CrossRef]

Kärkkäinen, M.K.

M.K. Kärkkäinen, Subcell FDTD Modeling of Electrically Thin Dispersive Layers,” IEEE Trans. Microwave Theory Tech. 51, 1774–1780 (2003).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996), 7th ed.

Knoesen, A.

Mohammadi, A.

Moore, T.G.

T.G. Jurgens, A. Taflove, K. Umaschankar, and T.G. Moore, “Finite-Difference Time-Domain Modeling of Curved Surfaces,” IEEE Trans. Antennas Propag. 40, 357–365 (1992).
[CrossRef]

Nordlander, P.

C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[CrossRef]

Okoniewski, M.

D. Popovic and M. Okoniewski, “Effective Permittivity at the Interface of Dispersive Dielectrics in FDTD,” IEEE Microwave Wirel. Compon. Lett. 13, 265–267 (2003).
[CrossRef]

Oubre, C.

C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[CrossRef]

Oztoprak, A.Y.

O. Ramadan and A.Y. Oztoprak, “Z-transform implementation of the perfectly matched layer for truncating FDTD domains,” IEEE Microwave Wirel. Compon. Lett. 13, 402–404 (2003).
[CrossRef]

Popovic, D.

D. Popovic and M. Okoniewski, “Effective Permittivity at the Interface of Dispersive Dielectrics in FDTD,” IEEE Microwave Wirel. Compon. Lett. 13, 265–267 (2003).
[CrossRef]

Raether, H.

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

Ramadan, O.

O. Ramadan and A.Y. Oztoprak, “Z-transform implementation of the perfectly matched layer for truncating FDTD domains,” IEEE Microwave Wirel. Compon. Lett. 13, 402–404 (2003).
[CrossRef]

Shin, H.

H. Shin and S. Fan, “All-Angle Negative Refraction for Surface Plasmon Waves Using a Metal-Dielectric-Metal Structure,” Phys. Rev. Lett. 96, 073907(4) (2006).

Sullivan, D.

D. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, 2000).
[CrossRef]

Taflove, A.

T.G. Jurgens, A. Taflove, K. Umaschankar, and T.G. Moore, “Finite-Difference Time-Domain Modeling of Curved Surfaces,” IEEE Trans. Antennas Propag. 40, 357–365 (1992).
[CrossRef]

A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005).

Umaschankar, K.

T.G. Jurgens, A. Taflove, K. Umaschankar, and T.G. Moore, “Finite-Difference Time-Domain Modeling of Curved Surfaces,” IEEE Trans. Antennas Propag. 40, 357–365 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), 7th ed.

Yu, Q.L.

C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[CrossRef]

IEEE Microwave Wirel. Compon. Lett. (3)

K.-P. Hwang and A.C. Cangellaris, “Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces,” IEEE Microwave Wirel. Compon. Lett. 11, 158–160 (2001).
[CrossRef]

D. Popovic and M. Okoniewski, “Effective Permittivity at the Interface of Dispersive Dielectrics in FDTD,” IEEE Microwave Wirel. Compon. Lett. 13, 265–267 (2003).
[CrossRef]

O. Ramadan and A.Y. Oztoprak, “Z-transform implementation of the perfectly matched layer for truncating FDTD domains,” IEEE Microwave Wirel. Compon. Lett. 13, 402–404 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T.G. Jurgens, A. Taflove, K. Umaschankar, and T.G. Moore, “Finite-Difference Time-Domain Modeling of Curved Surfaces,” IEEE Trans. Antennas Propag. 40, 357–365 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M.K. Kärkkäinen, Subcell FDTD Modeling of Electrically Thin Dispersive Layers,” IEEE Trans. Microwave Theory Tech. 51, 1774–1780 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. Chem. B (1)

C. Oubre and P. Nordlander, “Optical Properties of Metallodielectric Nanostructures Calculated Using the Finite Difference Time Domain Method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[CrossRef]

Nature (1)

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. B (1)

C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

H. Shin and S. Fan, “All-Angle Negative Refraction for Surface Plasmon Waves Using a Metal-Dielectric-Metal Structure,” Phys. Rev. Lett. 96, 073907(4) (2006).

Other (7)

A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005).

D. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, 2000).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), 7th ed.

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

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996), 7th ed.

In the error calculation we choose wavevectors k≥1.5ks (Fig. 3 and Fig. 4) and k≥2.5ks (Fig. 6) to focus on the low-group-velocity region.

The fit is performed using the logarithm of the error, so that the exponent α is obtained from logy=α log x+loga. The errors for Δ=2–4nm are excluded from the fit since the Fourier error is of the same order of magnitude and thus it may have an effect on the actual FDTD accuracy.

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Figures (6)

Fig. 1.
Fig. 1.

Layout of the FDTD mesh on the metal-dielectric interface (metal at the left). The integration paths for Ey and Ex are marked by thick lines.

Fig. 2.
Fig. 2.

Dispersion relation for a SPP at a glass/copper interface. Frequencies and wavevectors are expressed in units of ω s=ω p/(ε d+ε )1/2 and k s = ω s ε d c s=2π/k s ≃ 450nm). The discretization corresponds to Δ=5nm. Inset: simulation scheme, see text for details.

Fig. 3.
Fig. 3.

Relative error as a function of discretization Δ. Only discretizations that divide the mesh along k are considered in order to keep its length constant. The frequency relative error in the Fourier transform is kept below 0.2%. Inset: zoomed error for the CP method. We can assign a positive or negative sign to the error because the FDTD result is always above or below the exact one, as shown in Fig. 2.

Fig. 4.
Fig. 4.

Relative error for the CP method as a function of the cell cut f for various discretizations Δ. The label to each curve refers to Δ expressed in nm. f/Δ=0.0 is equivalent to f/Δ=1.0. We can assign a positive or negative sign to the error because the FDTD result is always above or below the exact one, as shown in Fig. 2.

Fig. 5.
Fig. 5.

Dispersion relation for symmetric (ω -) and antisymmetric (ω +) SPP at a glass/copper/glass thin film (thickness d=50nm). Frequencies and wavevectors are expressed in the same units of Fig. 2. The discretization corresponds to Δ=5nm. Inset: simulation scheme, see text for details.

Fig. 6.
Fig. 6.

Relative error as a function of discretization Δ. The frequency relative error in the Fourier transform is kept below 0.2%. Lower inset: zoomed error for the CP method.

Equations (22)

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D ( z ) = ε m ( z ) E ( z ) Δ t ,
ε m ( z ) = ε Δ t + ω p 2 γ ( 1 1 z 1 1 1 z 1 e γ Δ t ) .
E ( z ) = D a D ( z ) z 1 S ( z ) ,
S ( z ) = C a z 1 S ( z ) C b z 2 S ( z ) + C c E ( z ) ,
C a = 1 + e γ Δ t , C b = e γ Δ t , C c = ω p 2 Δ t γ ε ( 1 e γ Δ t ) , D a = 1 ε .
E n + 1 = D a D n + 1 S n ,
S n + 1 = C a S n C b S n 1 + C c E n + 1 ,
D = d Δ D m + ( 1 d Δ ) D d ,
D ( z ) = ε E ( z ) + d Δ ε z 1 S m ( z ) ,
C c = d Δ ω p 2 Δ t γ ε ( 1 e γ Δ t ) , D a = 1 ε .
E = f Δ E m + ( 1 f Δ ) E d ,
E ( z ) = D ( z ) ε f Δ z 1 S m ( z ) ,
C a = 1 + e γ Δ t ( 1 f Δ ε ε ) ω p 2 Δ t γ ε ( 1 e γ Δ t ) ,
C c = ε ε f Δ ω p 2 Δ t γ ε ( 1 e γ Δ t ) , D a = 1 ε .
k = ω c [ ε d ε m ( ω ) ε d + ε m ( ω ) ] 1 2 .
E x i , j + 1 2 n + 1 = E x , 0 exp { ι [ k i Δ ω ~ ( n + 1 ) Δ t ] k ~ ( j + 1 2 ) Δ } ,
E y i + 1 2 , j n + 1 = E y , 0 exp { ι [ k ( i + 1 2 ) Δ ω ~ ( n + 1 ) Δ t ] k ~ j Δ } ,
H z i , j n + 1 2 = H z , 0 exp { ι [ k i Δ ω ~ ( n + 1 2 ) Δ t ] k ~ j Δ } ,
sin ( k Δ 2 ) = 1 S c [ ε d ε m ( ω ~ ) ε d + ε m ( ω ~ ) ] 1 2 sin ( ω ~ Δ t 2 ) ,
sin ( k Δ 2 ) = 1 S c [ ε d ( ε m ( ω ~ ) + ω p 2 Δ t 2 4 ) ε d + ε m ( ω ~ ) + ω p 2 Δ t 2 4 ] 1 2 sin ( ω ~ Δ t 2 ) .
ω + : ε m k d + ε d k m tanh k m d 2 ι = 0 ,
ω : ε m k d + ε d k m coth k m d 2 ι = 0 ,

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