Abstract

We present an improvement of the standard Fourier Modal Method (FMM) for the analysis of lamellar gratings that is based on the use of automatically generated adaptive coordinates for arbitrarily shaped material profiles in the lateral plane of periodicity. This allows for an accurate resolution of small geometric features and/or large material contrasts within the unit. For dielectric gratings, we obtain considerable convergence accelerations. Similarly, for metallic gratings, our approach allows efficient and accurate computations of transmittance and reflectance coefficients into various Bragg orders, the spectral positions of Rayleigh anomalies, and field enhancement values within the grating structures.

© 2010 Optical Society of America

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  1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
    [CrossRef]
  2. F. J. Garcia de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
    [CrossRef]
  3. M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
    [CrossRef] [PubMed]
  4. M. G. Moharam, and T. K. Gaylord, "Diffraction analysis fo dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982).
    [CrossRef]
  5. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation of stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  6. P. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  7. G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  8. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  9. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1034 (1996).
    [CrossRef]
  10. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
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  12. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
    [CrossRef]
  13. G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999).
    [CrossRef]
  14. J. Chandezon, M. T. Dupuis, G. Gornet, and D. Maystre, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982).
    [CrossRef]
  15. T. Vallius, and M. Honkanen, "Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles," Opt. Express 10, 24-34 (2002).
    [PubMed]
  16. G. Granet, and J.-P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, S145-S149 (2002).
  17. T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, "Matched coordinates and adaptive spatial resolution in the Fourier modal method," Opt. Express 17, 8051-8061 (2009).
    [CrossRef] [PubMed]
  18. U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006).
    [CrossRef]
  19. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009).
    [CrossRef]
  20. G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
    [CrossRef]
  21. F. Gygi, "Electronic-structure calculations in adaptive coordinates," Phys. Rev. B 48, 11692-11700 (1993).
    [CrossRef]
  22. P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express 16, 17295-17301 (2008).
    [CrossRef] [PubMed]
  23. L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
  24. L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003).
    [CrossRef]
  25. G. Granet, and B. Guizal, "Analysis of strip gratings using a parametric modal method by Fourier expansions," Opt. Commun. 255, 1-11 (2005).
    [CrossRef]
  26. http://www.gnu.org.
  27. A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
    [CrossRef]
  28. B. Bai, and L. Li, "Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry," J. Opt. A 7, 783-789 (2005).
  29. J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005).
    [CrossRef]
  30. H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007).
    [CrossRef]
  31. Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009).
    [CrossRef] [PubMed]

2009 (3)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009).
[CrossRef]

Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009).
[CrossRef] [PubMed]

T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, "Matched coordinates and adaptive spatial resolution in the Fourier modal method," Opt. Express 17, 8051-8061 (2009).
[CrossRef] [PubMed]

2008 (2)

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express 16, 17295-17301 (2008).
[CrossRef] [PubMed]

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

2007 (4)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

F. J. Garcia de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
[CrossRef]

H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007).
[CrossRef]

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
[CrossRef]

2006 (1)

U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006).
[CrossRef]

2005 (5)

G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
[CrossRef]

G. Granet, and B. Guizal, "Analysis of strip gratings using a parametric modal method by Fourier expansions," Opt. Commun. 255, 1-11 (2005).
[CrossRef]

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
[CrossRef]

B. Bai, and L. Li, "Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry," J. Opt. A 7, 783-789 (2005).

J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005).
[CrossRef]

2003 (2)

L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003).
[CrossRef]

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).

2002 (2)

G. Granet, and J.-P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, S145-S149 (2002).

T. Vallius, and M. Honkanen, "Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles," Opt. Express 10, 24-34 (2002).
[PubMed]

1999 (1)

1997 (1)

1996 (4)

1995 (1)

1993 (1)

F. Gygi, "Electronic-structure calculations in adaptive coordinates," Phys. Rev. B 48, 11692-11700 (1993).
[CrossRef]

1982 (2)

Anderton, C. R.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

Bai, B.

B. Bai, and L. Li, "Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry," J. Opt. A 7, 783-789 (2005).

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
[CrossRef]

Bird, D. M.

G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Bocchio, N.

H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007).
[CrossRef]

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Chandezon, J.

Choi, Y.

Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009).
[CrossRef] [PubMed]

Dupuis, M. T.

Frenner, K.

G¨otz, P.

Garcia de Abajo, F. J.

F. J. Garcia de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
[CrossRef]

Gaylord, T. K.

Giessen, H.

Gippius, N. A.

Gornet, G.

Granet, G.

Grann, E. B.

Gray, S. K.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009).
[CrossRef]

Grimault, A.-S.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
[CrossRef]

Guizal, B.

G. Granet, and B. Guizal, "Analysis of strip gratings using a parametric modal method by Fourier expansions," Opt. Commun. 255, 1-11 (2005).
[CrossRef]

G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

Gygi, F.

F. Gygi, "Electronic-structure calculations in adaptive coordinates," Phys. Rev. B 48, 11692-11700 (1993).
[CrossRef]

Hedley, T. D.

G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Hong, S.

Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009).
[CrossRef] [PubMed]

Honkanen, M.

Kerwien, N.

Kreiter, M.

H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007).
[CrossRef]

J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005).
[CrossRef]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009).
[CrossRef]

Lalanne, P.

Lamy de la Chapelle, M.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
[CrossRef]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009).
[CrossRef]

Lee, L. P.

Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006).
[CrossRef]

Li, L.

Linden, S.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Macías, D.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
[CrossRef]

Maria, J.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

Maystre, D.

Mingaleev, S. F.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Nuzzo, R. G.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

Osten, W.

Pearce, G. J.

G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
[CrossRef]

Philbin, T. G.

U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006).
[CrossRef]

Plumey, J.-P.

G. Granet, and J.-P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, S145-S149 (2002).

Pommet, D. A.

Rafler, S.

Rochholz, H.

H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007).
[CrossRef]

J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005).
[CrossRef]

Rogers, J. A.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

Ruoff, J.

Schuster, T.

Shumaker-Parry, J. S.

J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005).
[CrossRef]

Stewart, M. E.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

Thompson, L. B.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

Tikhodeev, S. G.

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009).
[CrossRef]

Vallius, T.

Vial, A.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
[CrossRef]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Weiss, T.

Adv. Mater. (1)

J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005).
[CrossRef]

Chem. Rev. (1)

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008).
[CrossRef] [PubMed]

J. Opt. A (3)

B. Bai, and L. Li, "Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry," J. Opt. A 7, 783-789 (2005).

G. Granet, and J.-P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, S145-S149 (2002).

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (9)

G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999).
[CrossRef]

L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[CrossRef]

P. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1034 (1996).
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation of stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003).
[CrossRef]

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
[CrossRef]

N. J. Phys. (2)

H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007).
[CrossRef]

U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006).
[CrossRef]

Nano Lett. (1)

Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009).
[CrossRef] [PubMed]

Opt. Commun. (1)

G. Granet, and B. Guizal, "Analysis of strip gratings using a parametric modal method by Fourier expansions," Opt. Commun. 255, 1-11 (2005).
[CrossRef]

Opt. Express (3)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Phys. Rev. B (3)

G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005).
[CrossRef]

F. Gygi, "Electronic-structure calculations in adaptive coordinates," Phys. Rev. B 48, 11692-11700 (1993).
[CrossRef]

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005).
[CrossRef]

Rev. Mod. Phys. (1)

F. J. Garcia de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007).
[CrossRef]

SIAM Rev. (1)

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009).
[CrossRef]

Other (2)

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Figures (11)

Fig. 3
Fig. 3

(a) Schematic top view of the square-disk test structure and (b) automatically generated ASR via Eq. 14. See text for details.

Fig. 5
Fig. 5

(a) Schematic top view of the circular disk structure and (b) automatically generated ASR via Eq. 14. See text for details.

Fig. 8
Fig. 8

(a) Schematic top view of the crescent-shape nano-antennas and (b) automatically generated ASR via Eq. 14. See text for details.

Fig. 1
Fig. 1

Illustration of a crossed-grating in the Cartesian coordinate system Ox̄123Oxyz. The system is periodic in the 1- and 2-directions with periodicities d1 and d2, respectively. The grating is illuminated by a plane wave with wave vector kin whose direction defines the polar angle φ and the azimuthal angle θ.

Fig. 2
Fig. 2

Illustration of the gradient of a smoothed structure function (ws = 25) for a circular-shaped patch (radius r) of guest material in the center of a quadratic unit cell (r/d1 = r/d2 = 0.25): Panel (a) displays the gradient field and panel (b) shows the relative magnitude of the gradient vectors with better spatial resolution.

Fig. 4
Fig. 4

Transmittance spectra (a) and convergence characteristics (b) for a square array of square gold disks. The computations have been carried out within standard FMM and three different ASR approaches within FMM: (a) Transmittance into the zeroth diffraction order within standard FMM (blue solid line), analytical ASR (green dashed line), numerical ASR with tangential term (red dash-dotted line), and numerical ASR without tangential term (dotted, cyan) using N = 317 plane waves; (b) Convergence characteristics for the transmittance into the zeroth diffraction order with ɛ = −122.03+12.85i (gold at λ = 1600nm) for standard FMM (blue open circles), analytical ASR (green open square), numerical ASR with tangential energy term (red open triangles), and numerical ASR without tangential term (cyan open diamonds).

Fig. 6
Fig. 6

Transmittance spectra into the zeroth diffraction order for a square array of circular disks disks. The computations have been carried out within standard FMM (blue solid line) with N = 1257 plane waves, analytical ASR within FMM (green dashed line) with N = 317 plane waves, and numerical ASR within FMM (red dash-dotted line) with N = 317 plane waves.

Fig. 7
Fig. 7

Convergence characteristics of the transmittance into the zeroth transmittance order for a square array of cicular metallic disks (ɛ = −110.9 + 11.24i; gold at λ = 1530 nm). The computations have been carried out using standard FMM (blue open circles), analytical ASR (green open squares), and numerical ASR within FMM (red open triangles). The results depicted in panel (b) represent a close up of the results shown in panel (a). See text for details.

Fig. 9
Fig. 9

Transmittance spectrum into the zeroth diffraction order for a square array of gold crescents that have been computed within standard FMM (blue solid line), numerical ASR with tangential term (red dash-dotted line) and numerical ASR without tangential term (cyan dashed line). The calculations have been carried out for y-polarized excitation (curves with pronounced resonances near λ ≈ 1900 nm) as well as for x-polarized excitations (curves with pronounced resonances near λ ≈ 1100 nm).

Fig. 10
Fig. 10

Convergence characteristics of the transmittance into the zeroth transmittance order for a square array of metallic crescents made from gold. The computations have been carried out using standard FMM (blue open circles), numerical ASR with tangential energy term (red open triangles), and numerical ASR without tangential term (cyan open diamond) within FMM. Panel (a) depicts the case of y-polarized excitation at λ = 1900 nm with ɛ = −175.08 + 21.43i (gold at λ = 1900 nm) and panel (b) depicts the case of x-polarized excitation at λ = 1100 nm with ɛ = −53.21 + 4.20i (gold at λ = 1100 nm).

Fig. 11
Fig. 11

Magnitude of the electric field enhancement for a square array of gold crescent-shaped nanoparticles. Panel (a) depicts the field enhancement at the magnetic resonance at λ = 1900nm (y-polarized excitation) and panel (b) depicts the field enhancement at the electric resonance at λ = 1100nm (x-polarized excitation).

Tables (1)

Tables Icon

Table 1 Convergence of first propagation constant k3d1 (see Eq. (11) for a square array of square dielectric disks (ɛ = 12) as a function of the number of plane waves (modes). The results for the largest purely real eigenvalue at λ = 1600nm are shown. The eigenvalues, which are normalized to the lattice constant d1, have been calculated with standard FMM (Li), with analytical ASR (analytical), and numerically determined ASR via (i) minimization with tangential energy term (minT, Fig. 3 (b)) and (ii) without this (min). See the text for details about the system setup

Equations (20)

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x ¯ 1 = x ¯ 1 ( x 1 , x 2 ) ,
x ¯ 2 = x ¯ 2 ( x 1 , x 2 ) ,
x ¯ 3 = x 3 .
g kl = n = 1 3 x k x ¯ n x l x ¯ n .
ξ klm l E m = i k 0 g g kl H l ,
ξ klm l H m = i k 0 ɛ ¯ g g kl E l .
ɛ kl = ɛ ¯ g g kl ,
μ kl = g g kl ,
F σ ( x 1 , x 2 , x 3 ) = m , n f σ mn ( x 3 ) exp ( i α m x 1 + i β n x 2 ) .
ɛ ^ = ( L 1 L 2 ( ɛ ) + L 2 L 1 ( ɛ ) ) / 2 .
f σ mn ( x 3 ) = f σ mn exp ( i k 3 x 3 ) .
x ¯ 1 ( x 1 , x 2 ) = x 1 + m , n x mn 1 exp ( i m 2 π / d 1 x 1 + i n 2 π / d 2 x 2 ) ,
x ¯ 2 ( x 1 , x 2 ) = x 2 + m , n x mn 2 exp ( i m 2 π / d 1 x 1 + i n 2 π / d 2 x 2 ) .
= dx 1 dx 2 ( c ( x 1 , x 2 ) + s ( x 1 , x 2 ) + g ( x 1 , x 2 ) + t ( x 1 , x 2 ) ) ,
c ( x 1 , x 2 ) = η c det ( g mn ) ,
s ( x 1 , x 2 ) = η s tr ( g mn ) .
S sm ( x ¯ 1 , x ¯ 2 ) = p , q S sm , pq e i p 2 π / d 1 x ¯ 1 + i q 2 π / d 2 x ¯ 2 ,
S sm , pq = exp ( ( ( 2 π / d 1 p ) 2 + ( 2 π / d 2 q ) 2 ) / 2 w s 2 ) S pq
g ( x 1 , x 2 ) = | ɛ s [ x ¯ 1 ( x 1 , x 2 ) , x ¯ 2 ( x 1 , x 2 ) ] | .
t ( x 1 , x 2 ) = η t ( | ɛ s b 1 | 2 + | ɛ s b 2 | 2 ) .

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