Abstract

Finite-difference time-domain methods suffer from reduced accuracy when discretizing discontinuous materials. We previously showed that accuracy can be significantly improved by using subpixel smoothing of the isotropic dielectric function, but only if the smoothing scheme is properly designed. Using recent developments in perturbation theory that were applied to spectral methods, we extend this idea to anisotropic media and demonstrate that the generalized smoothing consistently reduces the errors and even attains second-order convergence with resolution.

© 2009 Optical Society of America

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References

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  1. A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech, 2005),
  2. C. Kottke, A. Farjadpour, and S. Johnson, Phys. Rev. E 77, 036611 (2008).
    [CrossRef]
  3. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Joannopoulos, S. Johnson, and G. Burr, Opt. Lett. 31, 2972 (2006).
    [CrossRef] [PubMed]
  4. G. Werner and J. Cary, J. Comput. Phys. 226, 1085 (2007).
    [CrossRef]
  5. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 1732001.
    [CrossRef] [PubMed]
  6. A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comp. Physiol. 170, 39 (2001).
  7. A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 773 (1996).
    [CrossRef]
  8. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  9. Meep FDTD, http://ab-initio.mit.edu/meep.
  10. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
    [CrossRef]
  11. S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).
  12. J.-Y. Lee and N.-H. Myung, Microwave Opt. Technol. Lett. 23, 245 (1999).
    [CrossRef]
  13. J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).
  14. V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys. 107, 6756 (1997).
    [CrossRef]
  15. S. Dey and R. Mittra, IEEE Trans. Microwave Theory Tech. 47, 1737 (1999).
    [CrossRef]

2008 (2)

C. Kottke, A. Farjadpour, and S. Johnson, Phys. Rev. E 77, 036611 (2008).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

2007 (1)

G. Werner and J. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

2006 (2)

2005 (1)

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech, 2005),

2001 (2)

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 1732001.
[CrossRef] [PubMed]

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comp. Physiol. 170, 39 (2001).

1999 (2)

J.-Y. Lee and N.-H. Myung, Microwave Opt. Technol. Lett. 23, 245 (1999).
[CrossRef]

S. Dey and R. Mittra, IEEE Trans. Microwave Theory Tech. 47, 1737 (1999).
[CrossRef]

1997 (2)

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys. 107, 6756 (1997).
[CrossRef]

1996 (1)

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 773 (1996).
[CrossRef]

1993 (1)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Alerhand, O. L.

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Bermel, P.

Brommer, K. D.

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Burr, G.

Cary, J.

G. Werner and J. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

Dey, S.

S. Dey and R. Mittra, IEEE Trans. Microwave Theory Tech. 47, 1737 (1999).
[CrossRef]

Ditkowski, A.

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comp. Physiol. 170, 39 (2001).

Dridi, K.

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comp. Physiol. 170, 39 (2001).

Farjadpour, A.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech, 2005),

Hesthaven, J. S.

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comp. Physiol. 170, 39 (2001).

Ibanescu, M.

Joannopoulos, J.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 1732001.
[CrossRef] [PubMed]

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Johnson, S.

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 1732001.
[CrossRef] [PubMed]

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

Kottke, C.

C. Kottke, A. Farjadpour, and S. Johnson, Phys. Rev. E 77, 036611 (2008).
[CrossRef]

Lee, J.-Y.

J.-Y. Lee and N.-H. Myung, Microwave Opt. Technol. Lett. 23, 245 (1999).
[CrossRef]

Mandelshtam, V. A.

V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys. 107, 6756 (1997).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Mittra, R.

S. Dey and R. Mittra, IEEE Trans. Microwave Theory Tech. 47, 1737 (1999).
[CrossRef]

Myung, N.-H.

J.-Y. Lee and N.-H. Myung, Microwave Opt. Technol. Lett. 23, 245 (1999).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 773 (1996).
[CrossRef]

Rappe, A. M.

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

Rodriguez, A.

Roundy, D.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech, 2005),

Taylor, H. S.

V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys. 107, 6756 (1997).
[CrossRef]

Ward, A. J.

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 773 (1996).
[CrossRef]

Werner, G.

G. Werner and J. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

IEEE Trans. Microwave Theory Tech. (1)

S. Dey and R. Mittra, IEEE Trans. Microwave Theory Tech. 47, 1737 (1999).
[CrossRef]

J. Chem. Phys. (1)

V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys. 107, 6756 (1997).
[CrossRef]

J. Comp. Physiol. (1)

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comp. Physiol. 170, 39 (2001).

J. Comput. Phys. (1)

G. Werner and J. Cary, J. Comput. Phys. 226, 1085 (2007).
[CrossRef]

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, J. Mod. Opt. 43, 773 (1996).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

J.-Y. Lee and N.-H. Myung, Microwave Opt. Technol. Lett. 23, 245 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (2)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
[CrossRef]

S. G. Johnson, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997).

Phys. Rev. E (1)

C. Kottke, A. Farjadpour, and S. Johnson, Phys. Rev. E 77, 036611 (2008).
[CrossRef]

Science (1)

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

Other (3)

Meep FDTD, http://ab-initio.mit.edu/meep.

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech, 2005),

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Figures (3)

Fig. 1
Fig. 1

Schematic 2D Yee FDTD discretization near a dielectric interface, showing the method [4] used to compute the part of E x that comes from D y and the locations where various ε 1 components are required.

Fig. 2
Fig. 2

Relative error Δω/ω for an eigenmode calculation with a square lattice (period a) of 2D anisotropic ellipsoids (right inset) versus spatial resolution (units of pixels per vacuum wavelength λ) for a variety of subpixel smoothing techniques. Straight lines for perfect linear (dashed) and perfect quadratic (solid) convergence are shown for reference. Most curves are for the first eigenvalue band (left inset shows E z in unit cell), with vacuum wavelength λ = 4.85 a . Hollow squares show new method for band 15 (middle inset), with λ = 1.7 a . Maximum resolution for all curves is 100   pixels a .

Fig. 3
Fig. 3

Relative error Δω/ω for an eigenmode calculation with a cubic lattice (period a) of 3D anisotropic ellipsoids (right inset) versus spatial resolution (units of pixels per vacuum wavelength λ), for a variety of subpixel smoothing techniques. Straight lines for perfect linear (dashed) and perfect quadratic (solid) convergence are shown for reference. Most curves are for the first eigenvalue band (left inset shows E x in x y cross-section of unit cell), with vacuum wavelength λ = 5.15 a . Hollow squares show new method for band 13 (middle inset), with λ = 2.52 a . New method for bands 1 and 13 is shown for resolution up to 100  pixels a .

Equations (4)

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

ε ̃ = ( ε 1 1 0 0 0 ε 0 0 0 ε ) ,
ε ̃ = τ 1 [ τ ( ε ) ] ,
τ ( ε ) = ( 1 ε 11 ε 12 ε 11 ε 13 ε 11 ε 21 ε 11 ε 22 ε 21 ε 12 ε 11 ε 23 ε 21 ε 13 ε 11 ε 31 ε 11 ε 32 ε 31 ε 12 ε 11 ε 33 ε 31 ε 13 ε 11 ) ,
τ 1 [ τ ] = ( 1 τ 11 τ 12 τ 11 τ 13 τ 11 τ 21 τ 11 τ 22 τ 21 τ 12 τ 11 τ 23 τ 21 τ 13 τ 11 τ 31 τ 11 τ 32 τ 31 τ 12 τ 11 τ 33 τ 31 τ 13 τ 11 ) .

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