Abstract

We present a modal method for the fast analysis of 2D-layered gratings. It combines exact discrete formulations of Maxwell equations in 2D space with polynomial approximations of the constitutive equations, and provides a sparse formulation of the eigenvalue equations. In specific cases, the use of sparse matrices allows us to calculate the electromagnetic response while solving only a small fraction of the eigenmodes. This significantly increases computational speed up to 100×, as shown on numerical examples of both dielectric and metallic subwavelength gratings.

© 2013 Optical Society of America

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  1. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  2. D. Maystre and M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. 9, 301–306 (1978).
    [CrossRef]
  3. E. Popov and M. Nevière, “Maxwell 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]
  4. R. Bruer and O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5(1993).
    [CrossRef]
  5. 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]
  6. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [CrossRef]
  7. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  8. C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
    [CrossRef]
  9. D. C. Dobson and J. A. Cox, “An integral equation method for biperiodic diffraction structures,” Proc. SPIE 1545, 106–113 (1991).
    [CrossRef]
  10. O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems—a method of variation of boundaries. 3: Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  11. 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]
  12. 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, 2444 (1998).
    [CrossRef]
  13. 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]
  14. G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “All-purpose finite element formulation for arbitrarily shaped crossed-gratings embedded in a multilayered stack,” J. Opt. Soc. Am. A 27, 878–889 (2010).
    [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).
  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. A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7, 1425–1449 (1998).
    [CrossRef]
  18. A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transfer 113, 158–171 (2012).
    [CrossRef]
  19. T. Weiland, “A discretization model for the solution of Maxwell’s equations for six-component fields,” Archiv Elektronik und Uebertragungstechnik 31, 116–120 (1977).
  20. M. Clemens and T. Weiland, “Discrete electromagnetism with the finite integration technique,” Prog. Electromagn. Res. 32, 65–87 (2001).
    [CrossRef]
  21. E. Tonti, “Finite formulation of electromagnetic field,” IEEE Trans. Magn. 38, 333–336 (2002).
    [CrossRef]
  22. M. Marrone, V. Rodriguez-Esquerre, and H. Hernandez-Figueroa, “Novel numerical method for the analysis of 2D photonic crystals: the cell method,” Opt. Express 10, 1299–1304 (2002).
    [CrossRef]
  23. D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  24. P. Bouchon, F. Pardo, R. Haïdar, G. Vincent, and J.-L. Pelouard, “Reduced scattering-matrix algorithm for high-density plasmonic structures,” Opt. Lett. 35, 3222–3224 (2010).
    [CrossRef]
  25. R. B. Lehoucq, C.-C. Yang, and D. C. Sorensen, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).
  26. J.-P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).
  27. P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
    [CrossRef]

2012 (1)

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transfer 113, 158–171 (2012).
[CrossRef]

2011 (1)

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

2010 (2)

2004 (1)

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
[CrossRef]

2002 (2)

2001 (2)

1998 (3)

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7, 1425–1449 (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]

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, 2444 (1998).
[CrossRef]

1997 (3)

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]

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (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]

1994 (1)

1993 (2)

1992 (1)

1991 (1)

D. C. Dobson and J. A. Cox, “An integral equation method for biperiodic diffraction structures,” Proc. SPIE 1545, 106–113 (1991).
[CrossRef]

1988 (1)

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]

1977 (1)

T. Weiland, “A discretization model for the solution of Maxwell’s equations for six-component fields,” Archiv Elektronik und Uebertragungstechnik 31, 116–120 (1977).

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).

Bardou, N.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Baylard, C.

Bouchon, P.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, G. Vincent, and J.-L. Pelouard, “Reduced scattering-matrix algorithm for high-density plasmonic structures,” Opt. Lett. 35, 3222–3224 (2010).
[CrossRef]

Bruer, R.

R. Bruer 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. 3: Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

Bryngdahl, O.

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

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]

Clemens, M.

M. Clemens and T. Weiland, “Discrete electromagnetism with the finite integration technique,” Prog. Electromagn. Res. 32, 65–87 (2001).
[CrossRef]

Commandré, M.

Cox, J. A.

D. C. Dobson and J. A. Cox, “An integral equation method for biperiodic diffraction structures,” Proc. SPIE 1545, 106–113 (1991).
[CrossRef]

Dagher, G.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Demésy, G.

Dobson, D. C.

D. C. Dobson and J. A. Cox, “An integral equation method for biperiodic diffraction structures,” Proc. SPIE 1545, 106–113 (1991).
[CrossRef]

Dupuis, C.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Ferlazzo, L.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Ghenuche, P.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Granet, G.

Greffet, J.-J.

Haïdar, R.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, G. Vincent, and J.-L. Pelouard, “Reduced scattering-matrix algorithm for high-density plasmonic structures,” Opt. Lett. 35, 3222–3224 (2010).
[CrossRef]

Hernandez-Figueroa, H.

Hugonin, J.-P.

J.-P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

Ko, D. Y. K.

Lalanne, P.

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
[CrossRef]

J.-P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

Lehoucq, R. B.

R. B. Lehoucq, C.-C. Yang, and D. C. Sorensen, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Li, L.

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
[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]

Marrone, M.

Maystre, D.

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

Nevière, M.

Nicolet, A.

Noponen, E.

Pardo, F.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, G. Vincent, and J.-L. Pelouard, “Reduced scattering-matrix algorithm for high-density plasmonic structures,” Opt. Lett. 35, 3222–3224 (2010).
[CrossRef]

Pelouard, J.-L.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, G. Vincent, and J.-L. Pelouard, “Reduced scattering-matrix algorithm for high-density plasmonic structures,” Opt. Lett. 35, 3222–3224 (2010).
[CrossRef]

Popov, E.

Portier, B.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

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. 3: Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

Rodriguez-Esquerre, V.

Sambles, J. R.

Shcherbakov, A. A.

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transfer 113, 158–171 (2012).
[CrossRef]

Sorensen, D. C.

R. B. Lehoucq, C.-C. Yang, and D. C. Sorensen, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Tishchenko, A. V.

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transfer 113, 158–171 (2012).
[CrossRef]

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7, 1425–1449 (1998).
[CrossRef]

Tonti, E.

E. Tonti, “Finite formulation of electromagnetic field,” IEEE Trans. Magn. 38, 333–336 (2002).
[CrossRef]

Turunen, J.

Versaevel, P.

Vincent, G.

Vincent, P.

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

Weiland, T.

M. Clemens and T. Weiland, “Discrete electromagnetism with the finite integration technique,” Prog. Electromagn. Res. 32, 65–87 (2001).
[CrossRef]

T. Weiland, “A discretization model for the solution of Maxwell’s equations for six-component fields,” Archiv Elektronik und Uebertragungstechnik 31, 116–120 (1977).

Yang, C.-C.

R. B. Lehoucq, C.-C. Yang, and D. C. Sorensen, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

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).

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]

Zhou, C.

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
[CrossRef]

Zolla, F.

Appl. Phys. Lett. (1)

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Archiv Elektronik und Uebertragungstechnik (1)

T. Weiland, “A discretization model for the solution of Maxwell’s equations for six-component fields,” Archiv Elektronik und Uebertragungstechnik 31, 116–120 (1977).

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).

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. (1)

E. Tonti, “Finite formulation of electromagnetic field,” IEEE Trans. Magn. 38, 333–336 (2002).
[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. Opt. (1)

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

J. Opt. A (1)

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
[CrossRef]

J. Opt. Soc. Am. A (8)

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]

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, 2444 (1998).
[CrossRef]

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (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]

D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
[CrossRef]

O. P. Bruno and F. Reitich, “Numerical solution of diffraction problems—a method of variation of boundaries. 3: Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

E. Popov and M. Nevière, “Maxwell 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. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “All-purpose finite element formulation for arbitrarily shaped crossed-gratings embedded in a multilayered stack,” J. Opt. Soc. Am. A 27, 878–889 (2010).
[CrossRef]

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

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transfer 113, 158–171 (2012).
[CrossRef]

Opt. Commun. (2)

R. Bruer 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. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

D. C. Dobson and J. A. Cox, “An integral equation method for biperiodic diffraction structures,” Proc. SPIE 1545, 106–113 (1991).
[CrossRef]

Prog. Electromagn. Res. (1)

M. Clemens and T. Weiland, “Discrete electromagnetism with the finite integration technique,” Prog. Electromagn. Res. 32, 65–87 (2001).
[CrossRef]

Pure Appl. Opt. (1)

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7, 1425–1449 (1998).
[CrossRef]

Other (2)

R. B. Lehoucq, C.-C. Yang, and D. C. Sorensen, ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

J.-P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic of a layered-crossed grating, periodic along the x and y axes. The layers are stacked along the vertical z axis. An incident monochromatic wave arrives on the structure with a polar angle θ to the z axis and in an incidence plane, making an azimuthal angle φ with the (xz) plane.

Fig. 2.
Fig. 2.

Detail of the 2D rectangular lattices G and G in the (xy) plane. The mesh points (circles), segments, and surface elements of G (respectively G) are labeled pi, li and si (respectively pj, lj and sj). The intersecting points between two segments of G and G (crosses) are labeled qk.

Fig. 3.
Fig. 3.

For a z-invariant layer, the field variables on grid G can be related to each other by writing the Maxwell–Faraday equations on the (a) horizontal or (b) vertical facets of an infinitely flat elementary brick.

Fig. 4.
Fig. 4.

First-order (a) surface-to-point and (b) segment-to-segment interpolation schemes used to write the constitutive equations relating the local/integrated components of the electric field E and the displacement field D. The crosses in (b) indicate the intersecting points used in the interpolations.

Fig. 5.
Fig. 5.

Comparison of calculation speed versus matrix size when solving all the eigenmodes with the eig function in the full eigenproblem at the second interpolation order (FF-PA_2_full), or 200 modes with the eigs function in the sparse eigenproblem at the mth interpolation order (FF-PA_m_sp, where m=0, 1, 2).

Fig. 6.
Fig. 6.

Reflectivity of a plane wave at normal incidence on a gold grating with high aspect ratio crossed slits (width wx=wy=150nm, depth h=2μm, and period dx=dy=2.5μm) (see inset). Solid line, RCWA code; circles, full implementation of the FF-PA code at second interpolation order; crosses, sparse implementation, 500 eigenmodes solved per layer (air, grating, and gold substrate).

Fig. 7.
Fig. 7.

Convergence of the transmitted diffraction efficiency of the (0,0)th-order for the RCWA method and for various implementations of the FF-PA code for the dielectric checkerboard grating (top view in inset).

Fig. 8.
Fig. 8.

Computation times for the RCWA code and for various implementations of the FF-PA code when calculating the diffraction efficiencies of the checkerboard grating.

Fig. 9.
Fig. 9.

Coordinates (xi,yi) (respectively (xjyj)) of the mesh points of lattice G (respectively G) for a typical crossed grating unit cell.

Fig. 10.
Fig. 10.

Detail of the set of discrete values selected in the point-to-segment (top) and the point-to-surface (bottom) interpolations at orders m=02 for the lattice G (dashed lines). The dotted lines correspond to the lattice G. The values are either defined on the intersecting points between the segments of both lattices (crosses), or on the mesh points of G (circles).

Fig. 11.
Fig. 11.

Typical interpolation cases for which the set of discrete interpolating values is truncated. The truncated sets for a1, a2, b1, b2, c1, and d1 are shown at interpolation orders m=02. In cases a1 and c1, the interpolations have to be carried independently on each side of the boundary, and include the discrete values located on the boundary.

Tables (4)

Tables Icon

Table 1. Diffraction Efficiencies (in Percentage) of the Transmitted Orders for the Dielectric Checkerboard Grating under Normal Incidence, Calculated with the FF-PA Code at Second Interpolation Ordera

Tables Icon

Table 2. Detail of the Monomials Used in the Point-to-Surface Interpolation, in Function of the Interpolation Order m and the Number of Selected Discrete Values

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Table 3. Detail of the Monomials Used in the Point-to-Segment Interpolation, in Function of the Interpolation Order m and the Number of Selected Discrete Values

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Table 4. Monomials (if Changed) for the Interpolation Cases Shown in Fig. 11, in Function of the Interpolation Order m

Equations (30)

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f(x,y,z,t)=f(x,y)exp(ikzziωt),
AE·dl=ABt·dS.
l1Exdx+l2Eydyl3Exdxl4Eydy=iωsnBzdz.
(0+1+1110)ΣG(Et(l1)Et(l2)Et(l3)Et(l4))=iω(Bz(sn)),
[Bz]siG=i1k0cΣG[Et]liG,
ikzl2EydyEz(p1)+Ez(p2)=iωl2Bxdy.
ikz(Et(l2))+(01+10)ΔG(Ez(p1)Ez(p2))=iω(Bn(l2)),
c[Bn]liGi1k0ΔG[Ez]piG=kzk0[Et]liG,
[Dz]sjG=+i1k0cΣG[Ht]ljG,
c[Dn]ljGi1k0ΔG[Hz]pjG=kzk0[Ht]ljG.
D=ε0εrE,
B=μ0μrH,
[Dz]sjG=1μ0c2[εz]int[Ez]piG,
[Ez]piG=μ0c2[1εz]int[Dz]sjG.
[Hz]pjG=1μ0[1μz]int[Bz]siG.
Um=i=0mui,
v0=u01,
v1=v0u1v0,
v2=v0(u1v1+u2v0),etc.
[Dn]ljG=Upl[D]qk,
[Et]liG=Upl[D]qk,
[Dn]ljG=1μ0c2[ε]int[Et]liG,
[Bn]liG=μ0[μ]int[Ht]ljG.
μ0c([μ]int+1k02ΔG[1εz]intΣG)[Ht]ljG=kzk0[Et]liG,
1μ0c([ε]int1k02ΔG[1μz]intΣG)[Et]liG=kzk0[Ht]ljG.
Mh[Ht]ljG=kzN2[Ht]ljG,
HAU+HASAAU=HBSBAU,
EAUEASAAU=EBSBAU,
SAA=12(1+HA1HBEB1EA)1,
SBA=EB1EA(1SAA).

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