Abstract

A surface integral formulation for light scattering on periodic structures is presented. Electric and magnetic field equations are derived on the scatterers’ surfaces in the unit cell with periodic boundary conditions. The solution is calculated with the method of moments and relies on the evaluation of the periodic Green’s function performed with Ewald’s method. The accuracy of this approach is assessed in detail. With this versatile boundary element formulation, a very large variety of geometries can be simulated, including doubly periodic structures on substrates and in multilayered media. The surface discretization shows a high flexibility, allowing the investigation of irregular shapes including fabrication accuracy. Deep insights into the extreme near-field of the scatterers as well as in the corresponding far-field are revealed. This method will find numerous applications for the design of realistic photonic nanostructures, in which light propagation is tailored to produce novel optical effects.

© 2010 Optical Society of America

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References

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  1. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  2. J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997).
    [CrossRef]
  3. C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
    [CrossRef] [PubMed]
  4. S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
    [CrossRef] [PubMed]
  5. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  6. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).
  7. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  8. D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
    [CrossRef]
  9. S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
    [CrossRef]
  10. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  11. P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
    [CrossRef]
  12. J. Jin, Finite Element Method in Electromagnetics (Wiley, 2002).
  13. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. Nouv. Rev. Opt. 11, 235–241 (1980).
  14. A. Wirgin and R. Deleuil, “Theoretical and experimental investigation of a new type of blazed grating,” J. Opt. Soc. Am. 59, 1348–1357 (1969).
    [CrossRef]
  15. T. Delort and D. Maystre, “Finite-element method for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
    [CrossRef]
  16. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  17. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
    [CrossRef]
  18. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
    [CrossRef]
  19. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A 25, 2693–2703 (2008).
    [CrossRef]
  20. P. C. Chaumet and A. Sentenac, “Simulation of light scattering by multilayer cross-gratings with the coupled dipole method,” J. Quant. Spectrosc. Radiat. Transf. 110, 409–414 (2009).
    [CrossRef]
  21. T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
    [CrossRef]
  22. M. S. Yeung and E. Barouch, “Three-dimensional nonplanar lithography simulation using a periodic fast multipole method,” Proc. SPIE 3051, 509–521 (1997).
    [CrossRef]
  23. M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999).
    [CrossRef]
  24. F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
    [CrossRef]
  25. A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
    [CrossRef]
  26. R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1968).
  27. N. Marly, D. De Zutter, and H. Pues, “A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures,” IEEE Trans. Electromagn. Compat. 36, 14–22 (1994).
    [CrossRef]
  28. N. Marly, B. Baekelandt, D. De Zutter, and H. Pues, “Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures,” IEEE Trans. Antennas Propag. 43, 1281–1287 (1995).
  29. L. Trintinalia and H. Ling, “Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme,” IEEE Trans. Antennas Propag. 52, 2253–2261 (2004).
    [CrossRef]
  30. I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
    [CrossRef]
  31. H. Fischer, A. Nesci, G. Leveque, and O. J. F. Martin, “Characterization of the polarization sensitivity anisotropy of a near-field probe using phase measurements,” J. Microsc. 230, 27–31 (2008).
    [CrossRef] [PubMed]
  32. C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, IEEE Series on Electromagnetic Waves, 2nd ed. (IEEE, 1994).
  33. W. Ludwig and C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems, Vol. 64 of Springer Series in Solid-State Sciences (Springer-Verlag, 1988).
  34. A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984–1009 (2006).
  35. F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys. 223, 250–261 (2007).
    [CrossRef]
  36. I. Stevanoviæ and J. R. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microwave Opt. Technol. Lett. 49, 1353–1357 (2007).
    [CrossRef]
  37. K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the green-function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63, 222–235 (1986).
    [CrossRef]
  38. S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
    [CrossRef]
  39. P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
    [CrossRef]
  40. G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng. 7, 405–408 (1973).
    [CrossRef]
  41. T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
    [CrossRef]
  42. X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
    [CrossRef]
  43. P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
    [CrossRef]
  44. P. J. Flatau, “Improvements in the discrete-dipole approximation method of computing scattering and absorption,” Opt. Lett. 22, 1205–1207 (1997).
    [CrossRef] [PubMed]
  45. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
    [CrossRef]
  46. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
    [CrossRef]
  47. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  48. B. Gallinet and O. J. F. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formation,” Photonics Nanostruct. Fund. Appl. 8, 278–284 (2010).
    [CrossRef]
  49. I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” PIER 63, 243–278 (2006).
    [CrossRef]

2010 (1)

B. Gallinet and O. J. F. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formation,” Photonics Nanostruct. Fund. Appl. 8, 278–284 (2010).
[CrossRef]

2009 (2)

P. C. Chaumet and A. Sentenac, “Simulation of light scattering by multilayer cross-gratings with the coupled dipole method,” J. Quant. Spectrosc. Radiat. Transf. 110, 409–414 (2009).
[CrossRef]

A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[CrossRef]

2008 (2)

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A 25, 2693–2703 (2008).
[CrossRef]

H. Fischer, A. Nesci, G. Leveque, and O. J. F. Martin, “Characterization of the polarization sensitivity anisotropy of a near-field probe using phase measurements,” J. Microsc. 230, 27–31 (2008).
[CrossRef] [PubMed]

2007 (4)

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys. 223, 250–261 (2007).
[CrossRef]

I. Stevanoviæ and J. R. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microwave Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

2006 (3)

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
[CrossRef]

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” PIER 63, 243–278 (2006).
[CrossRef]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984–1009 (2006).

2005 (3)

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
[CrossRef]

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).

2004 (1)

L. Trintinalia and H. Ling, “Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme,” IEEE Trans. Antennas Propag. 52, 2253–2261 (2004).
[CrossRef]

2003 (2)

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
[CrossRef]

2002 (4)

J. Jin, Finite Element Method in Electromagnetics (Wiley, 2002).

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

2001 (1)

S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

1999 (4)

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999).
[CrossRef]

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

1998 (2)

X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

1997 (4)

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

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

M. S. Yeung and E. Barouch, “Three-dimensional nonplanar lithography simulation using a periodic fast multipole method,” Proc. SPIE 3051, 509–521 (1997).
[CrossRef]

P. J. Flatau, “Improvements in the discrete-dipole approximation method of computing scattering and absorption,” Opt. Lett. 22, 1205–1207 (1997).
[CrossRef] [PubMed]

1995 (2)

N. Marly, B. Baekelandt, D. De Zutter, and H. Pues, “Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures,” IEEE Trans. Antennas Propag. 43, 1281–1287 (1995).

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

1994 (3)

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

N. Marly, D. De Zutter, and H. Pues, “A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures,” IEEE Trans. Electromagn. Compat. 36, 14–22 (1994).
[CrossRef]

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, IEEE Series on Electromagnetic Waves, 2nd ed. (IEEE, 1994).

1993 (1)

1988 (1)

W. Ludwig and C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems, Vol. 64 of Springer Series in Solid-State Sciences (Springer-Verlag, 1988).

1986 (1)

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the green-function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63, 222–235 (1986).
[CrossRef]

1982 (1)

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

1980 (2)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. Nouv. Rev. Opt. 11, 235–241 (1980).

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

1977 (1)

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

1973 (1)

G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng. 7, 405–408 (1973).
[CrossRef]

1969 (1)

1968 (1)

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1968).

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Baekelandt, B.

N. Marly, B. Baekelandt, D. De Zutter, and H. Pues, “Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures,” IEEE Trans. Antennas Propag. 43, 1281–1287 (1995).

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Barouch, E.

M. S. Yeung and E. Barouch, “Three-dimensional nonplanar lithography simulation using a periodic fast multipole method,” Proc. SPIE 3051, 509–521 (1997).
[CrossRef]

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Blackford, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Blagovic, K.

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
[CrossRef]

Bongard, F.

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
[CrossRef]

Bozhevolnyi, S.

S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Bryant, G. W.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

Capolino, F.

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys. 223, 250–261 (2007).
[CrossRef]

Chandezon, J.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. Nouv. Rev. Opt. 11, 235–241 (1980).

Chaumet, P. C.

P. C. Chaumet and A. Sentenac, “Simulation of light scattering by multilayer cross-gratings with the coupled dipole method,” J. Quant. Spectrosc. Radiat. Transf. 110, 409–414 (2009).
[CrossRef]

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

Chew, W. C.

X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Cowper, G. R.

G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng. 7, 405–408 (1973).
[CrossRef]

Crespo-Valero, P.

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
[CrossRef]

Culshaw, I.

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

De Zutter, D.

N. Marly, B. Baekelandt, D. De Zutter, and H. Pues, “Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures,” IEEE Trans. Antennas Propag. 43, 1281–1287 (1995).

N. Marly, D. De Zutter, and H. Pues, “A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures,” IEEE Trans. Electromagn. Compat. 36, 14–22 (1994).
[CrossRef]

Deleuil, R.

Delort, T.

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Dolling, G.

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Draine, B. T.

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Eibert, T.

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

Erland, J.

S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Falter, C.

W. Ludwig and C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems, Vol. 64 of Springer Series in Solid-State Sciences (Springer-Verlag, 1988).

Fan, S. H.

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Fischer, H.

H. Fischer, A. Nesci, G. Leveque, and O. J. F. Martin, “Characterization of the polarization sensitivity anisotropy of a near-field probe using phase measurements,” J. Microsc. 230, 27–31 (2008).
[CrossRef] [PubMed]

Flatau, P. J.

Gallinet, B.

B. Gallinet and O. J. F. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formation,” Photonics Nanostruct. Fund. Appl. 8, 278–284 (2010).
[CrossRef]

García de Abajo, F. J.

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Gaylord, T. K.

Gippius, N.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Glisson, A.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Grann, E. B.

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Hanninen, I.

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” PIER 63, 243–278 (2006).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1968).

Howie, A.

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Hvam, J.

S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Ishihara, T.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Jackson, D.

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

Jin, J.

J. Jin, Finite Element Method in Electromagnetics (Wiley, 2002).

Jin, J. M.

X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Johnson, W. A.

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys. 223, 250–261 (2007).
[CrossRef]

Jordan, K. E.

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the green-function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63, 222–235 (1986).
[CrossRef]

Kern, A. M.

Kleemann, B. H.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984–1009 (2006).

Leosson, K.

S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Leveque, G.

H. Fischer, A. Nesci, G. Leveque, and O. J. F. Martin, “Characterization of the polarization sensitivity anisotropy of a near-field probe using phase measurements,” J. Microsc. 230, 27–31 (2008).
[CrossRef] [PubMed]

Li, L.

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

Ling, H.

L. Trintinalia and H. Ling, “Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme,” IEEE Trans. Antennas Propag. 52, 2253–2261 (2004).
[CrossRef]

Lu, C. C.

X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Ludwig, W.

W. Ludwig and C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems, Vol. 64 of Springer Series in Solid-State Sciences (Springer-Verlag, 1988).

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Marly, N.

N. Marly, B. Baekelandt, D. De Zutter, and H. Pues, “Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures,” IEEE Trans. Antennas Propag. 43, 1281–1287 (1995).

N. Marly, D. De Zutter, and H. Pues, “A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures,” IEEE Trans. Electromagn. Compat. 36, 14–22 (1994).
[CrossRef]

Martin, O. J. F.

B. Gallinet and O. J. F. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formation,” Photonics Nanostruct. Fund. Appl. 8, 278–284 (2010).
[CrossRef]

A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[CrossRef]

H. Fischer, A. Nesci, G. Leveque, and O. J. F. Martin, “Characterization of the polarization sensitivity anisotropy of a near-field probe using phase measurements,” J. Microsc. 230, 27–31 (2008).
[CrossRef] [PubMed]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

Maystre, D.

T. Delort and D. Maystre, “Finite-element method for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. Nouv. Rev. Opt. 11, 235–241 (1980).

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Moharam, M. G.

Monk, P.

P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
[CrossRef]

Mosig, J. R.

I. Stevanoviæ and J. R. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microwave Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
[CrossRef]

Muljarov, E.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Nesci, A.

H. Fischer, A. Nesci, G. Leveque, and O. J. F. Martin, “Characterization of the polarization sensitivity anisotropy of a near-field probe using phase measurements,” J. Microsc. 230, 27–31 (2008).
[CrossRef] [PubMed]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

Piller, N. B.

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

Pommet, D. A.

Pues, H.

N. Marly, B. Baekelandt, D. De Zutter, and H. Pues, “Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures,” IEEE Trans. Antennas Propag. 43, 1281–1287 (1995).

N. Marly, D. De Zutter, and H. Pues, “A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures,” IEEE Trans. Electromagn. Compat. 36, 14–22 (1994).
[CrossRef]

Rahmani, A.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

Rao, S.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. Nouv. Rev. Opt. 11, 235–241 (1980).

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984–1009 (2006).

Richter, G. R.

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the green-function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63, 222–235 (1986).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).

Sarvas, J.

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” PIER 63, 243–278 (2006).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
[CrossRef]

Schmidt, G.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984–1009 (2006).

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sentenac, A.

P. C. Chaumet and A. Sentenac, “Simulation of light scattering by multilayer cross-gratings with the coupled dipole method,” J. Quant. Spectrosc. Radiat. Transf. 110, 409–414 (2009).
[CrossRef]

Sheng, P.

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the green-function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63, 222–235 (1986).
[CrossRef]

Sheng, X. Q.

X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Skovgaard, P.

S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Song, J. M.

X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Soukoulis, C. M.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Stevanoviæ, I.

I. Stevanoviæ and J. R. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microwave Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

Stevanovic, I.

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
[CrossRef]

Tai, C. -T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, IEEE Series on Electromagnetic Waves, 2nd ed. (IEEE, 1994).

Taskinen, M.

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” PIER 63, 243–278 (2006).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
[CrossRef]

Tikhodeev, S.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Trintinalia, L.

L. Trintinalia and H. Ling, “Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme,” IEEE Trans. Antennas Propag. 52, 2253–2261 (2004).
[CrossRef]

Tsai, L. L.

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Volakis, J.

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

Wegener, M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

Whittaker, D.

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

Wilton, D.

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

Wilton, D. R.

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys. 223, 250–261 (2007).
[CrossRef]

Wirgin, A.

Wu, T. K.

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

Yablonskii, A.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Yeung, M. S.

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999).
[CrossRef]

M. S. Yeung and E. Barouch, “Three-dimensional nonplanar lithography simulation using a periodic fast multipole method,” Proc. SPIE 3051, 509–521 (1997).
[CrossRef]

Ylä-Oijala, P.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

Comm. Comp. Phys. (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984–1009 (2006).

IEEE Trans. Antennas Propag. (8)

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[CrossRef]

N. Marly, B. Baekelandt, D. De Zutter, and H. Pues, “Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures,” IEEE Trans. Antennas Propag. 43, 1281–1287 (1995).

L. Trintinalia and H. Ling, “Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme,” IEEE Trans. Antennas Propag. 52, 2253–2261 (2004).
[CrossRef]

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999).
[CrossRef]

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718–1726 (1998).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

N. Marly, D. De Zutter, and H. Pues, “A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures,” IEEE Trans. Electromagn. Compat. 36, 14–22 (1994).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

I. Stevanovic, P. Crespo-Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Tech. 54, 3688–3697 (2006).
[CrossRef]

Int. J. Numer. Methods Eng. (1)

G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng. 7, 405–408 (1973).
[CrossRef]

J. Comput. Phys. (2)

K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the green-function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63, 222–235 (1986).
[CrossRef]

F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys. 223, 250–261 (2007).
[CrossRef]

J. Microsc. (1)

H. Fischer, A. Nesci, G. Leveque, and O. J. F. Martin, “Characterization of the polarization sensitivity anisotropy of a near-field probe using phase measurements,” J. Microsc. 230, 27–31 (2008).
[CrossRef] [PubMed]

J. Opt. Nouv. Rev. Opt. (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. Nouv. Rev. Opt. 11, 235–241 (1980).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Quant. Spectrosc. Radiat. Transf. (1)

P. C. Chaumet and A. Sentenac, “Simulation of light scattering by multilayer cross-gratings with the coupled dipole method,” J. Quant. Spectrosc. Radiat. Transf. 110, 409–414 (2009).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

I. Stevanoviæ and J. R. Mosig, “Periodic Green’s function for skewed 3-D lattices using the Ewald transformation,” Microwave Opt. Technol. Lett. 49, 1353–1357 (2007).
[CrossRef]

Nature (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: Putting a new twist on light,” Nature 386, 143–149 (1997).
[CrossRef]

Opt. Lett. (2)

Photonics Nanostruct. Fund. Appl. (1)

B. Gallinet and O. J. F. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formation,” Photonics Nanostruct. Fund. Appl. 8, 278–284 (2010).
[CrossRef]

Phys. Rev. B (5)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Phys. Rev. E (1)

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

S. Bozhevolnyi, J. Erland, K. Leosson, P. Skovgaard, and J. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86, 3008–3011 (2001).
[CrossRef] [PubMed]

PIER (2)

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
[CrossRef]

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” PIER 63, 243–278 (2006).
[CrossRef]

Proc. SPIE (1)

M. S. Yeung and E. Barouch, “Three-dimensional nonplanar lithography simulation using a periodic fast multipole method,” Proc. SPIE 3051, 509–521 (1997).
[CrossRef]

Radio Sci. (1)

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

Science (1)

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

Other (8)

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).

P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
[CrossRef]

J. Jin, Finite Element Method in Electromagnetics (Wiley, 2002).

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1968).

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, IEEE Series on Electromagnetic Waves, 2nd ed. (IEEE, 1994).

W. Ludwig and C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems, Vol. 64 of Springer Series in Solid-State Sciences (Springer-Verlag, 1988).

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Space division into regions V n with dielectric permittivity ϵ n and magnetic permeability μ n . The regions have the symmetry of a lattice with unit cell Ω and primitive lattice vectors a i . Equivalent surface currents J and M flow on the scatterer’s surfaces.

Fig. 2
Fig. 2

(a) Conservation of current across the boundary between regions V 1 and V 2 implies for the RWG functions f i 1 and f i 2 associated with the same edge i to have opposite signs. (b) This property can be generalized to an arbitrary number of boundaries associated with the same edge. (c) If the discretization reaches opposite ends of the unit cell, currents on these borders are not linearly independent. The mesh has to be translation symmetric and one border is removed from the calculations.

Fig. 3
Fig. 3

Relative error on the transmittance through a planar infinite interface between air and a material with a relative dielectric permittivity of 2. The incident field in air is an s-polarized plane wave with a 45° incidence angle. Results are compared to the analytical solution and a relative error is computed as a function of the DOFs for different wavelengths λ = 200   nm and λ = 700   nm , and different numbers of terms N in Ewald’s sum. The unit cell has dimensions 500   nm × 500   nm .

Fig. 4
Fig. 4

Plane-wave scattering at normal incidence on a planar infinite interface between two dielectric media. (a) Discretization of the interface and incidence conditions. (b) Air–metal interface [ ϵ = ( 17 + i ) ϵ 0 ] . (c),(d) Air–high-permittivity dielectric interface ( ϵ = 14 ϵ 0 ) . The tangential component of the instantaneous electric field is calculated along the incidence direction and compared to the analytical solution. The lattice period is a = 50   nm . Insets of (b)–(d): discretized objects used for the calculations.

Fig. 5
Fig. 5

Reflectance of a square array of pillars with a refractive index of 3.36 and dimensions w = 100   nm and h = 200   nm . The lattice period is a = 200   nm . Two cases are compared for normal (solid black curve), 45° p-polarized (solid gray curve), and 45° s-polarized (dashed curve) plane-wave incidences: (a) without substrate (758 mesh triangles) and (b) with a substrate of the same material (968 mesh triangles).

Fig. 6
Fig. 6

Photonic crystal made with an infinite square array of pillars with a refractive index of 3.36. The real part of the total ( incident + scattered ) instantaneous electric field is calculated in planes at 500 nm above, 500 nm below, and in the array for a 45° p-polarized plane wave incident from above. The scale is normalized in each frame. The arrow length is proportional to the electric field. (a),(b) No substrate [cf. Fig. 5a]; (c),(d) with substrate [cf. Fig. 5b]. Different illumination wavelengths λ are considered: (a) λ = 350   nm , (b) λ = 700   nm , (c) λ = 340   nm , (d) λ = 700   nm .

Fig. 7
Fig. 7

(a) Geometry of the metamaterial’s unit cell (1352 mesh triangles): period a = 300   nm , Ag layer thickness t = 40   nm , MgF 2 layer thickness s = 17   nm , deviation from rectangular shape e = 8   nm , width w x = 102   nm , and w y = 102   nm . (b) Metamaterial’s effective refractive index n as a function of the wavelength: real and imaginary parts and figure of merit FOM = Re ( n ) / Im ( n ) . The FOM is set to zero if Re ( n ) is positive.

Tables (1)

Tables Icon

Table 1 Energy Balance E and Reciprocity R for the Square Array of Pillars (See Text) a

Equations (44)

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× × E ( r ) k n 2 E ( r ) = i ω μ n j ( r ) ,     r V n ,
× × G ͇ n ( r , r ) k n 2 G ͇ n ( r , r ) = 1 ͇ δ ( r r ) .
[ × × E ( r ) ] G ͇ n ( r , r ) E ( r ) [ × × G ͇ n ( r , r ) ] = i ω μ n j ( r ) G ͇ n ( r , r ) E ( r ) δ ( r r ) .
V n d V ( [ × E ( r ) ] × G ͇ n ( r , r ) + E ( r ) × [ × G ͇ n ( r , r ) ] ) = i ω μ n V n d V j ( r ) G ͇ n ( r , r ) { E ( r ) : r V n 0 : otherwise . }
i ω μ n V n d V j ( r ) G ͇ n ( r , r ) = i ω μ n V n d V G ͇ n ( r , r ) j ( r ) = E n inc ( r ) ,
V n d S n ̂ n ( r ) ( [ × E ( r ) ] × G ͇ n ( r , r ) + E ( r ) × [ × G ͇ n ( r , r ) ] ) = E n inc ( r ) { E ( r ) : r V n 0 : otherwise , }
n ̂ n ( r ) ( [ × E ( r ) ] × G ͇ n ( r , r ) ) = ( n ̂ n ( r ) × [ × E ( r ) ] ) G ͇ n ( r , r ) = i ω μ n G ͇ n ( r , r ) [ n ̂ n ( r ) × H ( r ) ] .
n ̂ n ( r ) ( E ( r ) × [ × G ͇ n ( r , r ) ] ) = [ n ̂ n ( r ) × E ( r ) ] [ × G ͇ n ( r , r ) ] = [ × G ͇ n ( r , r ) ] [ n ̂ n ( r ) × E ( r ) ] .
i ω μ n V n d S G ͇ n ( r , r ) J n ( r ) + V n d S [ × G ͇ n ( r , r ) ] M n ( r ) = E n inc ( r ) { E ( r ) : r V n 0 : otherwise . }
( i ω μ n V n d S G ͇ n ( r , r ) J n ( r ) + V n d S [ × G ͇ n ( r , r ) ] M n ( r ) ) tan = ( E n inc ( r ) ) tan ,     r V n ,
× × H ( r ) k n 2 H ( r ) = × j ( r ) ,     r V n ,
H n inc ( r ) = V n d V [ × j ( r ) ] G ͇ n ( r , r ) ,
i ω ϵ n V n d S G ͇ n ( r , r ) M n ( r ) V n d S [ × G ͇ n ( r , r ) ] J n ( r ) = H n inc ( r ) { H ( r ) : r V n 0 : otherwise , }
( i ω ϵ n V n d S G ͇ n ( r , r ) M n ( r ) V n d S [ × G ͇ n ( r , r ) ] J n ( r ) ) tan = ( H n inc ( r ) ) tan ,     r V n .
U k ( r t ) = e i k t U k ( r ) .
( i ω μ n V n Ω d S G ͇ n , k ( r , r ) J n , k ( r ) + V n Ω d S [ × G ͇ n , k ( r , r ) ] M n , k ( r ) ) tan = ( E n , k inc ( r ) ) tan ,     r V n Ω .
( i ω ϵ n V n Ω d S G ͇ n , k ( r , r ) M n , k ( r ) V n Ω d S [ × G ͇ n , k ( r , r ) ] J n , k ( r ) ) tan = ( H n , k inc ( r ) ) tan ,     r V n Ω .
G ͇ n , k ( r , r ) = t e i k t G ͇ n ( r t , r ) .
E n , k scat ( r ) = i ω μ n V n Ω d S G ͇ n , k ( r , r ) J n , k ( r ) V n Ω d S [ × G ͇ n , k ( r , r ) ] M n , k ( r ) ,
H n , k scat ( r ) = i ω ϵ n V n Ω d S G ͇ n , k ( r , r ) M n , k ( r ) + V n Ω d S [ × G ͇ n , k ( r , r ) ] J n , k ( r ) .
G ͇ n ( r , r ) = ( 1 ͇ + k n 2 ) G n ( r , r ) ,
G n ( r , r ) = t e i k n | R t | 4 π | R t | e i k t .
G n = G n ( 1 ) + G n ( 2 ) ,
G n ( 2 ) ( r , r ) = 1 8 π t e i k t ± e ± i k n | R t | | R t | erfc ( | R t | E ± i k n 2 E ) .
b 1 = 2 π a 2 × ( a 1 × a 2 ) | a 1 × a 2 | 2 ,     b 2 = 2 π a 1 × ( a 2 × a 1 ) | a 1 × a 2 | 2 .
G n ( 1 ) ( r , r ) = 1 4 | Ω | u e i ( k u ) R ± e ± γ n , k , u R γ n , k , u erfc ( γ n , k , u 2 E ± R E ) ,
J n = i α i f i n ,
M n = i β i f i n ,
[ i ω μ n D n K n ] [ { α } { β } ] = q ( E ) , n ,
D i j n = V n d S f i n ( r ) V n d S G ͇ n ( r , r ) f j n ( r ) ,
K i j n = V n d S f i n ( r ) V n d S [ × G ͇ n ( r , r ) ] f j n ( r ) ,
q i ( E ) , n = V n d S f i n ( r ) E n inc ( r ) .
[ K n i ω ϵ n D n ] [ { α } { β } ] = q ( H ) , n ,
q i ( H ) , n = V n d S f i n ( r ) H n inc ( r ) .
[ n i ω μ n D n n K n n K n n i ω ϵ n D n ] [ { α } { β } ] = n [ q ( E ) , n q ( H ) , n ] .
E n scat ( r ) = i i ω μ n V n d S G ͇ n ( r , r ) α i f i n ( r ) V n d S [ × G ͇ n ( r , r ) ] β i f i n ( r ) ,
H n scat ( r ) = i i ω ϵ n V n d S G ͇ n ( r , r ) β i f i n ( r ) + V n d S [ × G ͇ n ( r , r ) ] α i f i n ( r ) .
J n ( r t ) = e i k t J n ( r ) ,
M n ( r t ) = e i k t M n ( r ) .
D i j n = V n d S f i n ( r ) ( 1 ͇ + k n 2 ) V n d S G n ( r , r ) f j n ( r ) = 1 k n 2 V n d S f i n ( r ) V n d S G n ( r , r ) f j n ( r ) + V n d S f i n ( r ) V n d S G n ( r , r ) f j n ( r ) = 1 k n 2 V n d S f i n ( r ) V n d S G n ( r , r ) f j n ( r ) + V n d S f i n ( r ) V n d S G n ( r , r ) f j n ( r ) .
K i j n = V n d S f i n ( r ) V n d S [ × G ͇ n , k ( r , r ) ] f j n ( r ) = V n d S f i n ( r ) V n d S [ G n ( r , r ) ] × f j n ( r ) .
G n ( r , r ) = G n ( s ) ( R ) + 1 4 π ( 1 | R | k n 2 | R | 2 ) ,
lim R 0 [ ± e ± i k R R erfc ( R E ± i k 2 E ) 2 R ] = 4 E π e k 2 / 4 E 2 + 2 i k [ erfc ( i k 2 E 1 ) ] ,
lim R 0 [ 4 E R π e R 2 E 2 + k 2 / 4 E 2 + ± 1 ± i k R R 2 e ± i k R   erfc ( R E ± i k 2 E ) + 2 R 2 ] = 2 k 2 .

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