Abstract

Finite-difference time-domain (FDTD) methods suffer from reduced accuracy when modeling discontinuous dielectric materials, due to the inhererent discretization (pixelization). We show that accuracy can be significantly improved by using a subpixel smoothing of the dielectric function, but only if the smoothing scheme is properly designed. We develop such a scheme based on a simple criterion taken from perturbation theory and compare it with other published FDTD smoothing methods. In addition to consistently achieving the smallest errors, our scheme is the only one that attains quadratic convergence with resolution for arbitrarily sloped interfaces. Finally, we discuss additional difficulties that arise for sharp dielectric corners.

© 2006 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. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 48, 8434 (1993).
    [CrossRef]
  3. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
    [CrossRef]
  4. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
    [CrossRef] [PubMed]
  5. N. Kaneda, B. Houshmand, and T. Itoh, IEEE Trans. Microwave Theory Tech. 45, 1645 (1997).
    [CrossRef]
  6. S. Dey and R. Mittra, IEEE Trans. Microwave Theory Tech. 47, 1737 (1999).
    [CrossRef]
  7. A. Mohammadi, H. Nadgaran, and M. Agio, Opt. Express 13, 10367 (2005).
    [CrossRef] [PubMed]
  8. A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comput. Phys. 170, 39 (2001).
    [CrossRef]
  9. "Meep FDTD package," http://jdj.mit.edu/meep.
  10. J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
    [CrossRef]
  11. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  12. S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
    [CrossRef]
  13. I. A. Zagorodnov, R. Schuhmann, and T. Weiland, Int. J. Numer. Model. 16, 127 (2003).
    [CrossRef]
  14. S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).
  15. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  16. V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys. 107, 6756 (1997).
    [CrossRef]
  17. J. B. Andersen and V. Solodukhov, IEEE Trans. Antennas Propag. 26, 589 (1978).
    [CrossRef]
  18. W. W. Lui, C.-L. Xu, W.-P. Huang, K. Yokoyama, and S. Seki, J. Lightwave Technol. 17, 1509 (1999).
    [CrossRef]
  19. G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002).
    [CrossRef]

2005 (2)

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

A. Mohammadi, H. Nadgaran, and M. Agio, Opt. Express 13, 10367 (2005).
[CrossRef] [PubMed]

2003 (2)

I. A. Zagorodnov, R. Schuhmann, and T. Weiland, Int. J. Numer. Model. 16, 127 (2003).
[CrossRef]

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
[CrossRef]

2002 (2)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002).
[CrossRef]

2001 (2)

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
[CrossRef] [PubMed]

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comput. Phys. 170, 39 (2001).
[CrossRef]

1999 (3)

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).

W. W. Lui, C.-L. Xu, W.-P. Huang, K. Yokoyama, and S. Seki, J. Lightwave Technol. 17, 1509 (1999).
[CrossRef]

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

1997 (3)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
[CrossRef]

N. Kaneda, B. Houshmand, and T. Itoh, IEEE Trans. Microwave Theory Tech. 45, 1645 (1997).
[CrossRef]

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

1978 (1)

J. B. Andersen and V. Solodukhov, IEEE Trans. Antennas Propag. 26, 589 (1978).
[CrossRef]

Agio, M.

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
[CrossRef]

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

Andersen, J. B.

J. B. Andersen and V. Solodukhov, IEEE Trans. Antennas Propag. 26, 589 (1978).
[CrossRef]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
[CrossRef]

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

Davidycheva, S.

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).

Deuflhard, P.

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
[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. Comput. Phys. 170, 39 (2001).
[CrossRef]

Dridi, K.

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comput. Phys. 170, 39 (2001).
[CrossRef]

Druskin, V.

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Habashy, T.

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).

Hadley, G. R.

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. Comput. Phys. 170, 39 (2001).
[CrossRef]

Houshmand, B.

N. Kaneda, B. Houshmand, and T. Itoh, IEEE Trans. Microwave Theory Tech. 45, 1645 (1997).
[CrossRef]

Huang, W.-P.

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Itoh, T.

N. Kaneda, B. Houshmand, and T. Itoh, IEEE Trans. Microwave Theory Tech. 45, 1645 (1997).
[CrossRef]

Jacobs, S.

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
[CrossRef]

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

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Johnson, S. G.

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
[CrossRef] [PubMed]

Kaneda, N.

N. Kaneda, B. Houshmand, and T. Itoh, IEEE Trans. Microwave Theory Tech. 45, 1645 (1997).
[CrossRef]

Karalis, A.

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

Lee, P.

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).

Lui, W. W.

Mandelshtam, V. A.

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

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
[CrossRef]

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

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mittra, R.

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

Mohammadi, A.

Moskow, S.

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).

Nadgaran, H.

Nadobny, J.

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
[CrossRef]

Povinelli, M. L.

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
[CrossRef]

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

Schuhmann, R.

I. A. Zagorodnov, R. Schuhmann, and T. Weiland, Int. J. Numer. Model. 16, 127 (2003).
[CrossRef]

Seki, S.

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Soljacic, M.

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

Solodukhov, V.

J. B. Andersen and V. Solodukhov, IEEE Trans. Antennas Propag. 26, 589 (1978).
[CrossRef]

Sullivan, D.

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
[CrossRef]

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]

Weiland, T.

I. A. Zagorodnov, R. Schuhmann, and T. Weiland, Int. J. Numer. Model. 16, 127 (2003).
[CrossRef]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wlodarczyk, W.

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
[CrossRef]

Wust, P.

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
[CrossRef]

Xu, C.-L.

Yokoyama, K.

Zagorodnov, I. A.

I. A. Zagorodnov, R. Schuhmann, and T. Weiland, Int. J. Numer. Model. 16, 127 (2003).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem. (1)

S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Photophys. Laser Chem. 81, 283 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, IEEE Trans. Antennas Propag. 51, 1760 (2003).
[CrossRef]

J. B. Andersen and V. Solodukhov, IEEE Trans. Antennas Propag. 26, 589 (1978).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

N. Kaneda, B. Houshmand, and T. Itoh, IEEE Trans. Microwave Theory Tech. 45, 1645 (1997).
[CrossRef]

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

Int. J. Numer. Model. (1)

I. A. Zagorodnov, R. Schuhmann, and T. Weiland, Int. J. Numer. Model. 16, 127 (2003).
[CrossRef]

J. Chem. Phys. (1)

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

J. Comput. Phys. (1)

A. Ditkowski, K. Dridi, and J. S. Hesthaven, J. Comput. Phys. 170, 39 (2001).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (2)

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]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Phys. Rev. B 55, 15942 (1997), erratum of Ref. .
[CrossRef]

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 (2002).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davidycheva, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 442 (1999).

Other (3)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

"Meep FDTD package," http://jdj.mit.edu/meep.

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

TE eigenfrequency error versus resolution for a square lattice of elliptical air holes in ϵ = 12 (inset).

Fig. 2
Fig. 2

Eigenfrequency error versus resolution for a cubic lattice of ϵ = 12 ellipsoids in air (inset).

Fig. 3
Fig. 3

Degraded accuracy due to field singularities at sharp corners: TE eigenfrequency error versus resolution for a square lattice of tilted-square air holes in ϵ = 12 (inset).

Equations (1)

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ϵ ̃ 1 = P ϵ 1 + ( 1 P ) ϵ 1 ,

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