Abstract

We propose a novel formulation of the finite element method adapted to the calculation of the vector field diffracted by an arbitrarily shaped crossed-grating embedded in a multilayered stack and illuminated by an arbitrarily polarized plane wave under oblique incidence. A complete energy balance (transmitted and reflected diffraction efficiencies and losses) is deduced from field maps. The accuracy of the proposed formulation has been tested using classical cases computed with independent methods. Moreover, to illustrate the independence of our method with respect to the shape of the diffractive object, we present the global energy balance resulting from the diffraction of a plane wave by a lossy thin torus crossed-grating. Finally, computation time and convergence as a function of the mesh refinement are discussed. As far as integrated energy values are concerned, the presented method shows a remarkable convergence even for coarse meshes.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A  14, 2758–2767 (1997).
    [CrossRef]
  3. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A  11, 2494–2502 (1994).
    [CrossRef]
  4. 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]
  5. E. Popov and M. Nevière, “Maxwell’s equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A  18, 2886–2894 (2001).
    [CrossRef]
  6. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun.  26, 293–296 (1978).
    [CrossRef]
  7. D. Maystre and M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt.  9, 301–306 (1978).
    [CrossRef]
  8. G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: A theory and its applications,” Appl. Phys. B: Photophys. Laser Chem.  18, 39–52 (1979).
  9. R. C. McPhedran, G. H. Derrick, M. Neviere, and D. Maystre, “Metallic crossed gratings,” J. Opt.  13, 209–218 (1982).
    [CrossRef]
  10. G. Granet, “Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: Application to multilayer crossed gratings,” J. Opt. Soc. Am. A  15, 1121–1131 (1998).
    [CrossRef]
  11. J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, and R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A  13, 2041–2049 (1996).
    [CrossRef]
  12. J. J. Greffet, C. Baylard, and P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett.  17, 1740–1742 (1992).
    [CrossRef] [PubMed]
  13. O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A  10, 2551–2562 (1993).
    [CrossRef]
  14. O. P. Bruno, and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am.  104, 2579–2583 (1998).
    [CrossRef]
  15. 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]
  16. K. S. Yee and J. S. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell’s equations,” IEEE Trans. Antennas Propag.  45, 354–363 (1997).
    [CrossRef]
  17. J. L. Volakis, A. Chatterjee, and L. C. Kempel, “Review of the finite-element method for three-dimensional electromagnetic scattering,” J. Opt. Soc. Am. A  11, 1422–1422 (1994).
    [CrossRef]
  18. X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. A  24, 866–881 (2007).
    [CrossRef]
  19. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A  19, 33–42 (2002).
    [CrossRef]
  20. J. J. Greffet and Z. Maassarani, “Scattering of electromagnetic waves by a grating: a numerical evaluation of the iterative-series solution,” J. Opt. Soc. Am. A  7, 1483–1493 (1990).
    [CrossRef]
  21. G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express  15, 18089–18102 (2007).
    [CrossRef] [PubMed]
  22. G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
    [CrossRef]
  23. F. Zolla and R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A  13, 796–802 (1996).
    [CrossRef]
  24. A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math.  168, 321–329 (2004).
    [CrossRef]
  25. Y. Ould Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to gradient index MOF,” Compel  27, 95–109 (2008).
  26. P. Dular, A. Nicolet, A. Genon, and W. Legros, “A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials,” IEEE Trans. Magn.  31, 1356–1359 (1995).
    [CrossRef]
  27. P. Ingelstrom, “A new set of H (curl)-conforming hierarchical basis functions for tetrahedral meshes,” IEEE Trans. Microwave Theory Tech.  54, 106–114 (2006).
    [CrossRef]
  28. A. Bossavit and I. Mayergoyz, “Edge-elements for scattering problems,” IEEE Trans. Magn.  25, 2816–2821 (1989).
    [CrossRef]
  29. T. V. Yioultsis and T. D. Tsiboukis, “The Mystery and Magic of Whitney Elements—An Insight in their Properties and Construction,” ICS Newsletter  3, 1389–1392 (1996).
  30. T. V. Yioultsis and T. D. Tsiboukis, “Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions,” IEEE Trans. Magn.  32, 1389–1392 (1996).
    [CrossRef]
  31. R. Bräuer and O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun.  100, 1–5 (1993).
    [CrossRef]
  32. L. Arnaud, “Diffraction et diffusion de la lumière: modélisation tridimensionnelle et application à la métrologie de la microélectronique et aux techniques d’imagerie sélective en milieu diffusant,” Ph.D. thesis (Université Paul Cézanne, 2008).

2009 (1)

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

2008 (1)

Y. Ould Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to gradient index MOF,” Compel  27, 95–109 (2008).

2007 (3)

2006 (1)

P. Ingelstrom, “A new set of H (curl)-conforming hierarchical basis functions for tetrahedral meshes,” IEEE Trans. Microwave Theory Tech.  54, 106–114 (2006).
[CrossRef]

2004 (1)

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math.  168, 321–329 (2004).
[CrossRef]

2002 (1)

2001 (1)

1998 (2)

G. Granet, “Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: Application to multilayer crossed gratings,” J. Opt. Soc. Am. A  15, 1121–1131 (1998).
[CrossRef]

O. P. Bruno, and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am.  104, 2579–2583 (1998).
[CrossRef]

1997 (2)

K. S. Yee and J. S. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell’s equations,” IEEE Trans. Antennas Propag.  45, 354–363 (1997).
[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]

1996 (4)

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, and R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A  13, 2041–2049 (1996).
[CrossRef]

F. Zolla and R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A  13, 796–802 (1996).
[CrossRef]

T. V. Yioultsis and T. D. Tsiboukis, “The Mystery and Magic of Whitney Elements—An Insight in their Properties and Construction,” ICS Newsletter  3, 1389–1392 (1996).

T. V. Yioultsis and T. D. Tsiboukis, “Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions,” IEEE Trans. Magn.  32, 1389–1392 (1996).
[CrossRef]

1995 (2)

P. Dular, A. Nicolet, A. Genon, and W. Legros, “A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials,” IEEE Trans. Magn.  31, 1356–1359 (1995).
[CrossRef]

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 (2)

1993 (2)

1992 (1)

1990 (1)

1989 (1)

A. Bossavit and I. Mayergoyz, “Edge-elements for scattering problems,” IEEE Trans. Magn.  25, 2816–2821 (1989).
[CrossRef]

1982 (1)

R. C. McPhedran, G. H. Derrick, M. Neviere, and D. Maystre, “Metallic crossed gratings,” J. Opt.  13, 209–218 (1982).
[CrossRef]

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: A theory and its applications,” Appl. Phys. B: Photophys. Laser Chem.  18, 39–52 (1979).

1978 (2)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun.  26, 293–296 (1978).
[CrossRef]

D. Maystre and M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt.  9, 301–306 (1978).
[CrossRef]

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]

Arnaud, L.

L. Arnaud, “Diffraction et diffusion de la lumière: modélisation tridimensionnelle et application à la métrologie de la microélectronique et aux techniques d’imagerie sélective en milieu diffusant,” Ph.D. thesis (Université Paul Cézanne, 2008).

Baylard, C.

Bossavit, A.

A. Bossavit and I. Mayergoyz, “Edge-elements for scattering problems,” IEEE Trans. Magn.  25, 2816–2821 (1989).
[CrossRef]

Bräuer, R.

R. Bräuer and O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun.  100, 1–5 (1993).
[CrossRef]

Bruno, O. P.

O. P. Bruno, and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am.  104, 2579–2583 (1998).
[CrossRef]

O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A  10, 2551–2562 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer and O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun.  100, 1–5 (1993).
[CrossRef]

Chatterjee, A.

Chen, J. S.

K. S. Yee and J. S. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell’s equations,” IEEE Trans. Antennas Propag.  45, 354–363 (1997).
[CrossRef]

Commandré, M.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express  15, 18089–18102 (2007).
[CrossRef] [PubMed]

Demésy, G.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express  15, 18089–18102 (2007).
[CrossRef] [PubMed]

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, M. Neviere, and D. Maystre, “Metallic crossed gratings,” J. Opt.  13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: A theory and its applications,” Appl. Phys. B: Photophys. Laser Chem.  18, 39–52 (1979).

Dular, P.

P. Dular, A. Nicolet, A. Genon, and W. Legros, “A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials,” IEEE Trans. Magn.  31, 1356–1359 (1995).
[CrossRef]

Dunne, B.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

Fossati, C.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express  15, 18089–18102 (2007).
[CrossRef] [PubMed]

Gagliano, O.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

Gaylord, T. K.

Genon, A.

P. Dular, A. Nicolet, A. Genon, and W. Legros, “A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials,” IEEE Trans. Magn.  31, 1356–1359 (1995).
[CrossRef]

Geuzaine, C.

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math.  168, 321–329 (2004).
[CrossRef]

Gralak, B.

Granet, G.

Grann, E. B.

Greffet, J. J.

Guenneau, S.

Y. Ould Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to gradient index MOF,” Compel  27, 95–109 (2008).

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math.  168, 321–329 (2004).
[CrossRef]

Harris, J. B.

Ingelstrom, P.

P. Ingelstrom, “A new set of H (curl)-conforming hierarchical basis functions for tetrahedral meshes,” IEEE Trans. Microwave Theory Tech.  54, 106–114 (2006).
[CrossRef]

Kempel, L. C.

Kerwien, N.

Legros, W.

P. Dular, A. Nicolet, A. Genon, and W. Legros, “A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials,” IEEE Trans. Magn.  31, 1356–1359 (1995).
[CrossRef]

Li, L.

Maassarani, Z.

Mayergoyz, I.

A. Bossavit and I. Mayergoyz, “Edge-elements for scattering problems,” IEEE Trans. Magn.  25, 2816–2821 (1989).
[CrossRef]

Maystre, D.

R. C. McPhedran, G. H. Derrick, M. Neviere, and D. Maystre, “Metallic crossed gratings,” J. Opt.  13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: A theory and its applications,” Appl. Phys. B: Photophys. Laser Chem.  18, 39–52 (1979).

D. Maystre and M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt.  9, 301–306 (1978).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, M. Neviere, and D. Maystre, “Metallic crossed gratings,” J. Opt.  13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: A theory and its applications,” Appl. Phys. B: Photophys. Laser Chem.  18, 39–52 (1979).

Moharam, M. G.

Neviere, M.

R. C. McPhedran, G. H. Derrick, M. Neviere, and D. Maystre, “Metallic crossed gratings,” J. Opt.  13, 209–218 (1982).
[CrossRef]

Nevière, M.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A  19, 33–42 (2002).
[CrossRef]

E. Popov and M. Nevière, “Maxwell’s equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A  18, 2886–2894 (2001).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: A theory and its applications,” Appl. Phys. B: Photophys. Laser Chem.  18, 39–52 (1979).

D. Maystre and M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt.  9, 301–306 (1978).
[CrossRef]

Nicolet, A.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

Y. Ould Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to gradient index MOF,” Compel  27, 95–109 (2008).

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express  15, 18089–18102 (2007).
[CrossRef] [PubMed]

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math.  168, 321–329 (2004).
[CrossRef]

P. Dular, A. Nicolet, A. Genon, and W. Legros, “A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials,” IEEE Trans. Magn.  31, 1356–1359 (1995).
[CrossRef]

Noponen, E.

Osten, W.

Ould Agha, Y.

Y. Ould Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to gradient index MOF,” Compel  27, 95–109 (2008).

Petit, R.

Pommet, D. A.

Popov, E.

Preist, T. W.

Rafler, S.

Reitich, F.

O. P. Bruno, and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am.  104, 2579–2583 (1998).
[CrossRef]

O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A  10, 2551–2562 (1993).
[CrossRef]

Ricq, S.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

Ruoff, J.

Sambles, J. R.

Schuster, T.

Tayeb, G.

Thorpe, R. N.

Tsiboukis, T. D.

T. V. Yioultsis and T. D. Tsiboukis, “Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions,” IEEE Trans. Magn.  32, 1389–1392 (1996).
[CrossRef]

T. V. Yioultsis and T. D. Tsiboukis, “The Mystery and Magic of Whitney Elements—An Insight in their Properties and Construction,” ICS Newsletter  3, 1389–1392 (1996).

Turunen, J.

Urbach, H. P.

Versaevel, P.

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun.  26, 293–296 (1978).
[CrossRef]

Volakis, J. L.

Wachters, A. J.

Watts, R. A.

Wei, X.

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]

Yee, K. S.

K. S. Yee and J. S. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell’s equations,” IEEE Trans. Antennas Propag.  45, 354–363 (1997).
[CrossRef]

Yioultsis, T. V.

T. V. Yioultsis and T. D. Tsiboukis, “Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions,” IEEE Trans. Magn.  32, 1389–1392 (1996).
[CrossRef]

T. V. Yioultsis and T. D. Tsiboukis, “The Mystery and Magic of Whitney Elements—An Insight in their Properties and Construction,” ICS Newsletter  3, 1389–1392 (1996).

Zolla, F.

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

Y. Ould Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to gradient index MOF,” Compel  27, 95–109 (2008).

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express  15, 18089–18102 (2007).
[CrossRef] [PubMed]

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math.  168, 321–329 (2004).
[CrossRef]

F. Zolla and R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A  13, 796–802 (1996).
[CrossRef]

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

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: A theory and its applications,” Appl. Phys. B: Photophys. Laser Chem.  18, 39–52 (1979).

Compel (1)

Y. Ould Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes: an application to gradient index MOF,” Compel  27, 95–109 (2008).

ICS Newsletter (1)

T. V. Yioultsis and T. D. Tsiboukis, “The Mystery and Magic of Whitney Elements—An Insight in their Properties and Construction,” ICS Newsletter  3, 1389–1392 (1996).

IEEE Trans. Antennas Propag. (2)

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]

K. S. Yee and J. S. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell’s equations,” IEEE Trans. Antennas Propag.  45, 354–363 (1997).
[CrossRef]

IEEE Trans. Magn. (3)

T. V. Yioultsis and T. D. Tsiboukis, “Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions,” IEEE Trans. Magn.  32, 1389–1392 (1996).
[CrossRef]

P. Dular, A. Nicolet, A. Genon, and W. Legros, “A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials,” IEEE Trans. Magn.  31, 1356–1359 (1995).
[CrossRef]

A. Bossavit and I. Mayergoyz, “Edge-elements for scattering problems,” IEEE Trans. Magn.  25, 2816–2821 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. Ingelstrom, “A new set of H (curl)-conforming hierarchical basis functions for tetrahedral meshes,” IEEE Trans. Microwave Theory Tech.  54, 106–114 (2006).
[CrossRef]

J. Acoust. Soc. Am. (1)

O. P. Bruno, and F. Reitich, “Boundary-variation solutions for bounded-obstacle scattering problems in three dimensions,” J. Acoust. Soc. Am.  104, 2579–2583 (1998).
[CrossRef]

J. Comput. Appl. Math. (1)

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, “Modelling of electromagnetic waves in periodic media with finite elements,” J. Comput. Appl. Math.  168, 321–329 (2004).
[CrossRef]

J. Opt. (2)

D. Maystre and M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt.  9, 301–306 (1978).
[CrossRef]

R. C. McPhedran, G. H. Derrick, M. Neviere, and D. Maystre, “Metallic crossed gratings,” J. Opt.  13, 209–218 (1982).
[CrossRef]

J. Opt. Soc. Am. A (13)

G. Granet, “Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: Application to multilayer crossed gratings,” J. Opt. Soc. Am. A  15, 1121–1131 (1998).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, and R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A  13, 2041–2049 (1996).
[CrossRef]

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]

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

E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A  11, 2494–2502 (1994).
[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]

E. Popov and M. Nevière, “Maxwell’s equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A  18, 2886–2894 (2001).
[CrossRef]

O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A  10, 2551–2562 (1993).
[CrossRef]

J. L. Volakis, A. Chatterjee, and L. C. Kempel, “Review of the finite-element method for three-dimensional electromagnetic scattering,” J. Opt. Soc. Am. A  11, 1422–1422 (1994).
[CrossRef]

X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. A  24, 866–881 (2007).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A  19, 33–42 (2002).
[CrossRef]

J. J. Greffet and Z. Maassarani, “Scattering of electromagnetic waves by a grating: a numerical evaluation of the iterative-series solution,” J. Opt. Soc. Am. A  7, 1483–1493 (1990).
[CrossRef]

F. Zolla and R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A  13, 796–802 (1996).
[CrossRef]

Opt. Commun. (2)

R. Bräuer and O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun.  100, 1–5 (1993).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun.  26, 293–296 (1978).
[CrossRef]

Opt. Eng. (Bellingham) (1)

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, C. Fossati, O. Gagliano, S. Ricq, and B. Dunne, “Finite element method as applied to the study of gratings embedded in complementary metal-oxide semiconductor image sensors,” Opt. Eng. (Bellingham)  48, 058002 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (1)

L. Arnaud, “Diffraction et diffusion de la lumière: modélisation tridimensionnelle et application à la métrologie de la microélectronique et aux techniques d’imagerie sélective en milieu diffusant,” Ph.D. thesis (Université Paul Cézanne, 2008).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Scheme and notation of the studied bi-gratings.

Fig. 2
Fig. 2

DOF of a second-order tetrahedral element.

Fig. 3
Fig. 3

Configuration of the studied cases.

Fig. 4
Fig. 4

(a) Diffractive element with vertical edges. (b) R e { E x } in V m .

Fig. 5
Fig. 5

(a) Diffractive element with oblique edges. (b) R e { E y } in V m .

Fig. 6
Fig. 6

(a) Diffractive element with oblique edges. (b) R e { E z } in V m .

Fig. 7
Fig. 7

(a) Lossy diffractive element with vertical edges. (b) R e { E y } in V m .

Fig. 8
Fig. 8

(a) Torus parameters. (b) Coarse mesh of the computational domain.

Fig. 9
Fig. 9

Convergence of R 0 , 0 as function of N m (circular apertures crossed-grating).

Fig. 10
Fig. 10

Computation time and number of DOF as functions of N M .

Tables (6)

Tables Icon

Table 1 Energy Balance [2]

Tables Icon

Table 2 Comparison with the Results Given in [8, 10, 12, 31]

Tables Icon

Table 3 Energy Balance [13]

Tables Icon

Table 4 Comparison with [1, 2, 4] and Energy Balance

Tables Icon

Table 5 Energy Balances at Normal and Oblique Incidence

Tables Icon

Table 6 Computation Time Variations from Solver to Solver

Equations (38)

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

E inc = A 0 e exp ( i k + r ) ,
k + = [ α 0 β 0 γ 0 ] = k + [ sin θ 0 cos φ 0 sin θ 0 sin φ 0 cos θ 0 ]
A 0 e = [ E x 0 E y 0 E z 0 ] = A e [ cos ψ 0 cos θ 0 cos φ 0 sin ψ 0 sin φ 0 cos ψ 0 cos θ 0 sin φ 0 + sin ψ 0 cos φ 0 cos ψ 0 sin θ 0 ] ,
curl E = i ω μ 0 μ H ,
curl H = i ω ϵ 0 ϵ E ,
M ϵ , v curl ( v curl E ) + k 0 2 ϵ E = 0 ,
κ ( x , y , z ) { κ + for z > z 0 κ n for z n 1 > z > z n with 1 n < g κ g ( x , y , z ) for z g 1 > z > z g κ n for z n 1 > z > z n with g < n N κ for z < z N } ,
κ 1 ( x , y , z ) { κ + for z > 0 κ n for z n 1 > z > z n with 1 n N κ for z < z N } ,
E 0 { E inc for z > z 0 0 for z z 0 } .
M ϵ , v ( E ) = 0 , such that E d E E 0 satisfies an OWC .
M ϵ 1 , v 1 ( E 1 ) = 0 such that E 1 d E 1 E 0 satisfies an OWC .
E 2 d E E 1 = E d E 1 d .
M ϵ , v ( E 2 d ) = M ϵ , v ( E 1 ) ,
R ϵ , v ( E , E ) = Ω curl ( v curl E ) E ¯ + k 0 2 ϵ E E ¯ d Ω .
R ϵ , v ( E , E ) = Ω v curl E curl E ¯ + k 0 2 ϵ E E ¯ d Ω Ω ( n × ( v curl E ) ) E ¯ d S ,
E L 2 ( curl , d x , d y , k ) , R ϵ , v ( E 2 d , E ) = R ϵ ϵ 1 , v v 1 ( E 1 , E ) .
{ ϑ i j = i j E 2 d t i j λ i d l ϑ j i = j i E 2 d t j i λ j d l } ,
{ ϑ i j k = f ( E 2 d × n i j k + ) grad λ j d s ϑ i k j = f ( E 2 d × n i j k ) grad λ k d s } .
{ w i j = ( 8 λ i 2 4 λ i ) grad λ j + ( 8 λ i λ j + 2 λ j ) grad λ i w i j k = 16 λ i λ j grad λ k 8 λ j λ k grad λ i 8 λ k λ i grad λ j } .
E 2 d , m = e E ϑ e w e + f F ϑ f w f .
E x d ( x , y , z ) = ( n , m ) Z 2 u n , m d , x ( z ) e i ( α n x + β m y ) ,
u n , m d , x ( z ) = 1 d x d y d x 2 d x 2 d y 2 d y 2 E x d ( x , y , z ) e i ( α n x + β m y ) d x d y .
u n , m d , x ( z ) = e n , m x , p e i γ n , m + z + e n , m x , c e i γ n , m + z ,
( n , m ) Z 2 { e n , m x , p = 0 for z > z 0 e n , m x , c = 0 for z < z N } .
{ R n , m = 1 A e 2 γ n , m + γ 0 e n , m c ( z c ) e n , m c ( z c ) ¯ , for z c > z 0 T n , m = 1 A e 2 γ n , m γ 0 e n , m p ( z c ) e n , m p ( z c ) ¯ , for z c < z N } ,
Q = V 1 2 ω ϵ 0 I m ( ϵ g ) E E ¯ d V S 1 2 R e { E 0 × H 0 ¯ } n d S .
Q + ( n , m ) Z 2 R e { R n , m } + ( n , m ) Z 2 R e { T n , m } ,
f ( x , y ) = h 4 [ cos ( 2 π x d ) + cos ( 2 π y d ) ] .
E 1 ( x , y , z ) = [ E 1 x , j , + E 1 y , j , + E 1 z , j , + ] exp ( j ( α 0 x + β 0 y + γ j z ) ) + [ E 1 x , j , E 1 y , j , E 1 z , j , ] exp ( j ( α 0 x + β 0 y γ j z ) ) ,
γ j 2 = k j 2 α 0 2 β 0 2 .
Ψ = [ E 1 x E 1 y i H 1 x i H 1 y ] .
[ i β 0 H 1 z H 1 y d z H 1 x d z i α 0 H 1 z i α 0 H 1 y i β 0 H 1 x ] = i ω ϵ [ E 1 x E 1 y E 1 z ]
[ i β 0 E 1 z E 1 y z E 1 x z i α 0 E 1 z i α 0 E 1 y i β 0 E 1 x ] = i ω μ [ H 1 x H 1 y H 1 z ] .
{ U x j , ± = E 1 x , j , ± exp ( ± i γ j z ) U y j , ± = E 1 y , j , ± exp ( ± i γ j z ) } ,
Φ j = [ U x + , j U x , j U y + , j U y , j ] .
Φ j + 1 ( z j ) = Π j + 1 1 Π j Φ j ( z j ) .
Φ j + 1 ( z j + 1 ) = T j + 1 Π j + 1 1 Π j Φ j ( z j ) .
Φ N + 1 ( z N ) = ( Π N + 1 ) 1 Π N j = 0 N 1 T N j ( Π N j ) 1 Π N j 1 Φ 0 ( z 0 ) .

Metrics