Abstract

A new hybrid finite-element/rigorous coupled wave analysis formulation is presented for the modeling of electromagnetic wave interactions with doubly periodic structures. The structures under investigation are periodic in two dimensions and have a finite extent in the third dimension. The proposed model can handle structures that have material properties varying arbitrarily in any of the dimensions within the unit cell. Employment of Fourier series expansion and Floquet’s theory in one of the periodic dimensions helps to reduce the dimension of the mesh. Results obtained from alternative methods are used to verify the proposed method’s validity.

© 2012 Optical Society of America

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  1. E. Loewen and E. Popov, Diffraction Gratings and Applications, Optical Engineering (Marcel Dekker, 1997).
  2. H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexópoulos, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Am. A 16, 1682–1699 (1999).
    [CrossRef]
  3. M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  4. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  5. S. Gedney, J.-F. Lee, and R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
    [CrossRef]
  6. R. Kipp and C. Chan, “A numerically efficient technique for the method of moments solution for planar periodic structures in layered media,” IEEE Trans. Microwave Theory Tech. 42, 635–643 (1994).
    [CrossRef]
  7. W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials. I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
    [CrossRef]
  8. 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]
  9. Y.-C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A 23, 638–645 (2006).
    [CrossRef]
  10. W. Pinello, R. Lee, and A. Cangellaris, “Finite element modeling of electromagnetic wave interactions with periodic dielectric structures,” IEEE Trans. Microwave Theory Tech. 42, 2294–2301 (1994).
    [CrossRef]
  11. W.-J. Tsay and D. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microw. Guided Wave Lett. 3, 250–252 (1993).
    [CrossRef]
  12. P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag. 42, 1317–1324 (1994).
    [CrossRef]
  13. L. Petersson and J.-M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antennas Propag. 54, 12–19 (2006).
    [CrossRef]
  14. L. Li and C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  15. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  16. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  17. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  18. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  19. E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199–206 (2004).
    [CrossRef]
  20. 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]
  21. S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
    [CrossRef]
  22. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified s-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
    [CrossRef]
  23. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  24. Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
    [CrossRef]
  25. Ansoft Corporation, “High Frequency Structure Simulator (HFSS), ver. 13.0” (2011).
  26. T. A. Davis, “Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw. 30, 196–199 (2004).
    [CrossRef]
  27. J. R. Shewchuk, “Triangle: engineering a 2D quality mesh generator and Delaunay triangulator,” in Applied Computational Geometry: Towards Geometric Engineering, Vol. 1148 of Lecture Notes in Computer Science, M. C. Lin and D. Manocha, eds. (Springer-Verlag, 1996), pp. 203–222.
  28. A. M. Attiya and A. A. Kishk, “Modal analysis of a two-dimensional dielectric grating slab excited by an obliquely incident plane wave,” Prog. Electromagn. Res. 60, 221–243 (2006).
    [CrossRef]
  29. EM Software & Systems S.A. (Pty) Ltd, “FEKO Suite 6.0” (2011).
  30. L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced finite element method for nano-resonators,” Proc. SPIE 6115, 611515 (2006).
    [CrossRef]

2011 (1)

2008 (1)

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[CrossRef]

2007 (1)

2006 (4)

Y.-C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A 23, 638–645 (2006).
[CrossRef]

A. M. Attiya and A. A. Kishk, “Modal analysis of a two-dimensional dielectric grating slab excited by an obliquely incident plane wave,” Prog. Electromagn. Res. 60, 221–243 (2006).
[CrossRef]

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced finite element method for nano-resonators,” Proc. SPIE 6115, 611515 (2006).
[CrossRef]

L. Petersson and J.-M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antennas Propag. 54, 12–19 (2006).
[CrossRef]

2004 (2)

T. A. Davis, “Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw. 30, 196–199 (2004).
[CrossRef]

E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199–206 (2004).
[CrossRef]

1999 (3)

H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexópoulos, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Am. A 16, 1682–1699 (1999).
[CrossRef]

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials. I. Scalar case,” J. Comput. Phys. 150, 468–481 (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]

1997 (1)

1996 (4)

1995 (1)

Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

1994 (4)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

W. Pinello, R. Lee, and A. Cangellaris, “Finite element modeling of electromagnetic wave interactions with periodic dielectric structures,” IEEE Trans. Microwave Theory Tech. 42, 2294–2301 (1994).
[CrossRef]

P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag. 42, 1317–1324 (1994).
[CrossRef]

R. Kipp and C. Chan, “A numerically efficient technique for the method of moments solution for planar periodic structures in layered media,” IEEE Trans. Microwave Theory Tech. 42, 635–643 (1994).
[CrossRef]

1993 (2)

W.-J. Tsay and D. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microw. Guided Wave Lett. 3, 250–252 (1993).
[CrossRef]

L. Li and C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

1992 (1)

S. Gedney, J.-F. Lee, and R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

1983 (1)

Alexópoulos, N. G.

Attiya, A. M.

A. M. Attiya and A. A. Kishk, “Modal analysis of a two-dimensional dielectric grating slab excited by an obliquely incident plane wave,” Prog. Electromagn. Res. 60, 221–243 (2006).
[CrossRef]

Axmann, W.

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials. I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[CrossRef]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bonod, N.

Burger, S.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced finite element method for nano-resonators,” Proc. SPIE 6115, 611515 (2006).
[CrossRef]

Cangellaris, A.

W. Pinello, R. Lee, and A. Cangellaris, “Finite element modeling of electromagnetic wave interactions with periodic dielectric structures,” IEEE Trans. Microwave Theory Tech. 42, 2294–2301 (1994).
[CrossRef]

Chan, C.

R. Kipp and C. Chan, “A numerically efficient technique for the method of moments solution for planar periodic structures in layered media,” IEEE Trans. Microwave Theory Tech. 42, 635–643 (1994).
[CrossRef]

Chang, Y.-C.

Chernov, B.

Chu, H.

Contopanagos, H. F.

Davis, T. A.

T. A. Davis, “Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw. 30, 196–199 (2004).
[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]

Frenner, K.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[CrossRef]

Gaylord, T. K.

Gedney, S.

S. Gedney, J.-F. Lee, and R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

Götz, P.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[CrossRef]

Granet, G.

Guizal, B.

Haggans, C. W.

Harms, P.

P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag. 42, 1317–1324 (1994).
[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.-M.

L. Petersson and J.-M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antennas Propag. 54, 12–19 (2006).
[CrossRef]

Kerwien, N.

Kettner, B.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced finite element method for nano-resonators,” Proc. SPIE 6115, 611515 (2006).
[CrossRef]

Kingsland, D.

Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Kipp, R.

R. Kipp and C. Chan, “A numerically efficient technique for the method of moments solution for planar periodic structures in layered media,” IEEE Trans. Microwave Theory Tech. 42, 635–643 (1994).
[CrossRef]

Kishk, A. A.

A. M. Attiya and A. A. Kishk, “Modal analysis of a two-dimensional dielectric grating slab excited by an obliquely incident plane wave,” Prog. Electromagn. Res. 60, 221–243 (2006).
[CrossRef]

Ko, W.

P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag. 42, 1317–1324 (1994).
[CrossRef]

Kuchment, P.

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials. I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[CrossRef]

Kyriazidou, C. A.

Lalanne, P.

Lee, J.-F.

Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

S. Gedney, J.-F. Lee, and R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

Lee, R.

Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

W. Pinello, R. Lee, and A. Cangellaris, “Finite element modeling of electromagnetic wave interactions with periodic dielectric structures,” IEEE Trans. Microwave Theory Tech. 42, 2294–2301 (1994).
[CrossRef]

Li, G.

Li, L.

Loewen, E.

E. Loewen and E. Popov, Diffraction Gratings and Applications, Optical Engineering (Marcel Dekker, 1997).

Mattheij, R.

Maubach, J.

Merrill, W. M.

Mittra, R.

P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag. 42, 1317–1324 (1994).
[CrossRef]

S. Gedney, J.-F. Lee, and R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Nevière, M.

Opsal, J.

Osten, W.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[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]

Petersson, L.

L. Petersson and J.-M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antennas Propag. 54, 12–19 (2006).
[CrossRef]

Petschow, M.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[CrossRef]

Pinello, W.

W. Pinello, R. Lee, and A. Cangellaris, “Finite element modeling of electromagnetic wave interactions with periodic dielectric structures,” IEEE Trans. Microwave Theory Tech. 42, 2294–2301 (1994).
[CrossRef]

Pisarenco, M.

Popov, E.

Pozar, D.

W.-J. Tsay and D. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microw. Guided Wave Lett. 3, 250–252 (1993).
[CrossRef]

Rafler, S.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[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]

Ruoff, J.

Sacks, Z.

Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Schmidt, F.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced finite element method for nano-resonators,” Proc. SPIE 6115, 611515 (2006).
[CrossRef]

Schuster, T.

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[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]

Setija, I.

Shewchuk, J. R.

J. R. Shewchuk, “Triangle: engineering a 2D quality mesh generator and Delaunay triangulator,” in Applied Computational Geometry: Towards Geometric Engineering, Vol. 1148 of Lecture Notes in Computer Science, M. C. Lin and D. Manocha, eds. (Springer-Verlag, 1996), pp. 203–222.

Tsay, W.-J.

W.-J. Tsay and D. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microw. Guided Wave Lett. 3, 250–252 (1993).
[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]

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]

Zschiedrich, L.

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced finite element method for nano-resonators,” Proc. SPIE 6115, 611515 (2006).
[CrossRef]

ACM Trans. Math. Softw. (1)

T. A. Davis, “Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw. 30, 196–199 (2004).
[CrossRef]

IEEE Microw. Guided Wave Lett. (1)

W.-J. Tsay and D. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microw. Guided Wave Lett. 3, 250–252 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag. 42, 1317–1324 (1994).
[CrossRef]

L. Petersson and J.-M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antennas Propag. 54, 12–19 (2006).
[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]

Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

W. Pinello, R. Lee, and A. Cangellaris, “Finite element modeling of electromagnetic wave interactions with periodic dielectric structures,” IEEE Trans. Microwave Theory Tech. 42, 2294–2301 (1994).
[CrossRef]

S. Gedney, J.-F. Lee, and R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

R. Kipp and C. Chan, “A numerically efficient technique for the method of moments solution for planar periodic structures in layered media,” IEEE Trans. Microwave Theory Tech. 42, 635–643 (1994).
[CrossRef]

J. Comput. Phys. (2)

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials. I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[CrossRef]

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199–206 (2004).
[CrossRef]

Y.-C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A 23, 638–645 (2006).
[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]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified s-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
[CrossRef]

H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexópoulos, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Am. A 16, 1682–1699 (1999).
[CrossRef]

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

L. Li and C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

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

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

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

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

Proc. SPIE (2)

L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced finite element method for nano-resonators,” Proc. SPIE 6115, 611515 (2006).
[CrossRef]

S. Rafler, P. Götz, M. Petschow, T. Schuster, K. Frenner, and W. Osten, “Investigation of methods to set up the normal vector field for the differential method,” Proc. SPIE 6995, 69950Y (2008).
[CrossRef]

Prog. Electromagn. Res. (1)

A. M. Attiya and A. A. Kishk, “Modal analysis of a two-dimensional dielectric grating slab excited by an obliquely incident plane wave,” Prog. Electromagn. Res. 60, 221–243 (2006).
[CrossRef]

Other (4)

EM Software & Systems S.A. (Pty) Ltd, “FEKO Suite 6.0” (2011).

J. R. Shewchuk, “Triangle: engineering a 2D quality mesh generator and Delaunay triangulator,” in Applied Computational Geometry: Towards Geometric Engineering, Vol. 1148 of Lecture Notes in Computer Science, M. C. Lin and D. Manocha, eds. (Springer-Verlag, 1996), pp. 203–222.

Ansoft Corporation, “High Frequency Structure Simulator (HFSS), ver. 13.0” (2011).

E. Loewen and E. Popov, Diffraction Gratings and Applications, Optical Engineering (Marcel Dekker, 1997).

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Figures (12)

Fig. 1.
Fig. 1.

Unit cells of typical 3D doubly periodic material blocks. The structure is periodic in the x and z directions with periodicities a and d, respectively.

Fig. 2.
Fig. 2.

Illustrative fill profile for a typical FE/RCWA matrix (N2D=43, B=11).

Fig. 3.
Fig. 3.

UMFPACK reported computational cost dependence (a) on the size of Fourier modes included in the solution (i.e., B) when N2D=10552 and (b) on the size of finite element unknowns when B=19.

Fig. 4.
Fig. 4.

Unit cell of the 3D structure solved for results verification. Slab height=0.1d; period, d×d; block side lengths, 0.5d×0.5d.

Fig. 5.
Fig. 5.

Normal incidence (i.e., kx=0, ky=0) TM plane wave reflection (|Γ00|) from a dielectric slab (ϵr1=4) with periodically embedded material blocks (ϵr2=10) (see Fig. 4) compared to reference values from [8].

Fig. 6.
Fig. 6.

Convergence rate of the solution as number of modes included changes for the structure given in Fig. 4.

Fig. 7.
Fig. 7.

Unit cell of the four-layer FSS. The structure consists of four layers of square-shaped dielectric rings (ϵr1=12) supported by three supporting dielectric slabs (ϵr2=2.2).

Fig. 8.
Fig. 8.

Normal incidence (i.e., kx=0, ky=0) TM plane wave reflection (|Γ00|) from the FSS depicted in Fig. 7. Results are compared to solutions from HFSS.

Fig. 9.
Fig. 9.

Unit cell of periodically placed spheres in free space. Structure is composed of dielectric spheres (ϵr1=12) with radius r=0.1d. Unit cell dimensions in the x and z dimensions are given as d.

Fig. 10.
Fig. 10.

Normal incidence (i.e., kx=0, ky=0) comparison of the transmission coefficients for the structure given in Fig. 9. TE and TM mode solutions of the presented method are compared with FEKO [29]. L is given as 10.

Fig. 11.
Fig. 11.

TE and TM mode convergence of the FE/RCWA method. Fundamental mode transmission coefficient change with respect to L.

Fig. 12.
Fig. 12.

PML distance (tair) effect on fundamental mode reflection coefficient around resonance frequencies for the structure given in Fig. 4. L is fixed to be 2.

Tables (1)

Tables Icon

Table 1. Performance Comparison Between FE-RCWA and 3D-FEM on the Test Case

Equations (38)

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E⃗i=E⃗0ej(kxx+kyy+kzz).
×E⃗=jωμrμ0H⃗,
×H⃗=jωϵrϵ0E⃗.
×1μr(×E⃗)k02ϵrE⃗=0,
E⃗n=LLE⃗n(x,y)ejβnz,
×1μr(×n=LLE⃗n(x,y)ejβnz)k02ϵrn=LLE⃗n(x,y)ejβnz=0.
ϵr(x,y,z)=ϵr(x,y,z+s·d).
k02ϵr(x,y,z)m=MMfm(x,y)ej2mπdz.
×(×n=LLE⃗n(x,y)ejβnz)m=MMfm(x,y)ej2mπdzn=LLE⃗n(x,y)ejβnz=0.
n=LL(×(×E⃗n(x,y)ejβnz)m=MMfm(x,y)ej(βn+2mπd)zE⃗n(x,y))=0.
×(×E⃗q(x,y))n=LLfq-n(x,y)E⃗n(x,y)=0,q=L,,L.
E⃗q=E⃗tq+z^Ezq.
=x^x+y^yz^jβq=tz^jβq,
×E⃗q=t×E⃗tq+t×z^Ezqjβqz^×E⃗tq.
××E⃗q=(tz^jβq)×(t×E⃗tq+t×z^Ezqjβqz^×E⃗tq)=t×t×E⃗tq+t×(t×z^Ezq)t×(jβqz^×E⃗tq)jβqz^×(t×z^Ezq)+jβqz^×(jβqz^×E⃗tq).
t×z^Ezq=z^×tEzq.
××E⃗q=t×t×E⃗tqt2Ezqz^jβqz^(t·E⃗tq)jβqtEzq+βq2E⃗tq.
t×t×E⃗tq+βq2E⃗tqjβqtEzqn=LLfq-nE⃗tn=0,
t2Ezqjβq(t·E⃗tq)n=LLfq-nEzn=0.
(t×t×E⃗tq+βq2E⃗tqjβqtEzqn=LLfq-nE⃗tn)·W⃗tdS=0.
((t×E⃗tq)·(t×W⃗t)+βq2E⃗tq·W⃗tjβq(tEzq)·W⃗tn=LLfqnE⃗tn·W⃗t)dS=0.
(t2Ezqjβq(t·E⃗tq)n=LLfq-nEzn)WzdS=0.
Wzt2EzqdS=((tWz)·(tEzq)+t·(WztEzq))dS=(tWz)·(tEzq)dS+Wz(n^·tEzq)dl.
H⃗=1jωμ0×E⃗whereH⃗=H⃗t+z^Hz.
n^×H⃗=n^×H⃗t+n^×z^Hz=n^×(1jωμ0×E⃗).
n^×z^Hz=1jωμ0n^×(t×E⃗t),
n^×H⃗t=1jωμ0n^×(z^×(tEz+jβqE⃗t))=1jωμ0z^(n^·(tEz+jβqE⃗t))+transverse components,
Wzt2EzqdS=(tWz)·(tEzq)dS+Wz(n^·(tEzq+jβqE⃗tq))dlWz(n^·(jβqE⃗tq))dl.
Wzt2EzqdS=(tWz)·(tEzq)dSjβq(t·(WzE⃗tq))dS=((tWz)·(tEzq)jβq[Wz(t·E⃗tq)+E⃗tq·tWz])dS=((tWz)·(tEzq)jβqWz(t·E⃗tq)jβqE⃗tq·tWz)dS.
[(tWz)·(tEzq)jβqWz(t·E⃗tq)jβqE⃗tq·tWz+jβqWz(t·E⃗tq)+n=LLfq-nEznWz]dS=0.
((tEzq)·(tWz)+jβqE⃗tq·tWzn=LLfq-nEznWz)dS=0.
×(μ¯¯r1×E⃗)k02ϵ¯¯rE⃗=0.
μ¯¯r=[μxx000μyy000μzz],ϵ¯¯r=[ϵxx000ϵyy000ϵzz].
k02ϵ¯¯r(x,y,z)=m=MM[fmxx000fmyy000fmzz]ej2mπdz=m=MMf¯¯m(x,y)ej2mπdz.
(μzz1(t×E⃗tq)·(t×W⃗t)(jβqμ˜˜·tEzq)·W⃗t+(βq2μ˜˜·E⃗tq)·W⃗tn=LLf^^q-nE⃗tn·W⃗t)dS=0,
[(tWz)·(μyy1xEzqx^+μxx1yEzqy^)+jβq(tWz)·(x^μyy1Exq+y^μxx1Eyq)n=LLf(q-n)zzEznWz]dS=0.
μ˜˜=[μyy100μxx1],f^^q-n=[fq-nxx00fq-nyy].
Relative Error=E⃗0BE⃗0B=57E⃗0B=57.

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