Abstract

For block-shaped dielectric gratings with two-dimensional periodicity, a spectral-domain volume integral equation is derived in which explicit Fourier factorization rules are employed. The Fourier factorization rules are derived from a projection-operator framework and enhance the numerical accuracy of the method, while maintaining a low computational complexity of O(NlogN) or better and a low memory demand of O(N).

© 2011 Optical Society of America

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  1. C.-F. Yang, W. D. Burnside, and R. C. Rudduck, “A doubly periodic moment method solution for the analysis and design of an absorber covered wall,” IEEE Trans. Antennas Propag. 41, 600–609 (1993).
    [CrossRef]
  2. H.-Y. D. Yang and J. Wang, “Surface waves of printed antennas on planar artificial periodic dielectric structures,” IEEE Trans. Antennas Propag. 49, 444–450 (2001).
    [CrossRef]
  3. M. C. van Beurden and B. P. de Hon, “Electromagnetic modelling of antennas mounted on a band-gap slab—discretisation issues and domain and boundary integral equations,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications ICEAA ’03 (Politecnico di Torino, 2003), pp. 637–640.
    [PubMed]
  4. B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.
  5. 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]
  6. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392(1982).
    [CrossRef]
  7. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  8. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  9. T.K.Sarkar, ed., Application of Conjugate Gradient Method to Electromagnetics and Signal Analysis, Vol.  5 of Progress in Electromagnetics Research (Elsevier, 1991).
  10. P. Zwamborn and P. M. van den Berg, “The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems,” IEEE Trans. Microwave Theory Tech. 40, 1757–1766 (1992).
    [CrossRef]
  11. E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251(1996).
    [CrossRef]
  12. W.C.Chew, J.Jin, and E.Michielssen, eds., Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).
  13. H.-Y. D. Yang, R. Diaz, and N. G. Alexopoulos, “Reflection and transmission of waves from multilayer structures with planar-implanted periodic material blocks,” J. Opt. Soc. Am. B 14, 2513–2521 (1997).
    [CrossRef]
  14. Y. Shi and C. H. Chan, “Multilevel Green’s function interpolation method for analysis of 3-D frequency selective structures using volume/surface integral equation,” J. Opt. Soc. Am. A 27, 308–318 (2010).
    [CrossRef]
  15. G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
    [CrossRef]
  16. T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22, 2405–2418 (2005).
    [CrossRef]
  17. T. Magath, “Coupled integral equations for diffraction by profiled, anisotropic, periodic structures,” IEEE Trans. Antennas Propag. 54, 681–686 (2006).
    [CrossRef]
  18. 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]
  19. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598(1997).
    [CrossRef]
  20. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, 2nd ed. (IEEE, 1994).
    [CrossRef]
  21. A. G. Tijhuis and A. Rubio Bretones, “Transient excitation of a layered dielectric medium by a pulsed electric dipole: a spectral representation,” in Ultra-Wideband Short-Pulse Electromagnetics (Kluwer Academic/Plenum, 2000), pp. 167–174.
  22. 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]
  23. I. Gohberg and I. Koltracht, “Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation,” Numer. Math. 47, 237–288 (1985).
    [CrossRef]
  24. G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).
  25. G. L. G. Sleijpen, H. A. van der Vorst, and D. R. Fokkema, “BiCGstab(ℓ) and other hybrid Bi-CG methods,” Numer. Algorithms 7, 75–109 (1994).
    [CrossRef]
  26. 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]

2010 (1)

2009 (1)

G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
[CrossRef]

2007 (1)

2006 (2)

2005 (1)

2004 (1)

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

2003 (1)

M. C. van Beurden and B. P. de Hon, “Electromagnetic modelling of antennas mounted on a band-gap slab—discretisation issues and domain and boundary integral equations,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications ICEAA ’03 (Politecnico di Torino, 2003), pp. 637–640.
[PubMed]

2001 (2)

H.-Y. D. Yang and J. Wang, “Surface waves of printed antennas on planar artificial periodic dielectric structures,” IEEE Trans. Antennas Propag. 49, 444–450 (2001).
[CrossRef]

W.C.Chew, J.Jin, and E.Michielssen, eds., Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

2000 (1)

A. G. Tijhuis and A. Rubio Bretones, “Transient excitation of a layered dielectric medium by a pulsed electric dipole: a spectral representation,” in Ultra-Wideband Short-Pulse Electromagnetics (Kluwer Academic/Plenum, 2000), pp. 167–174.

1997 (3)

1996 (3)

1994 (2)

G. L. G. Sleijpen, H. A. van der Vorst, and D. R. Fokkema, “BiCGstab(ℓ) and other hybrid Bi-CG methods,” Numer. Algorithms 7, 75–109 (1994).
[CrossRef]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, 2nd ed. (IEEE, 1994).
[CrossRef]

1993 (2)

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

C.-F. Yang, W. D. Burnside, and R. C. Rudduck, “A doubly periodic moment method solution for the analysis and design of an absorber covered wall,” IEEE Trans. Antennas Propag. 41, 600–609 (1993).
[CrossRef]

1992 (1)

P. Zwamborn and P. M. van den Berg, “The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems,” IEEE Trans. Microwave Theory Tech. 40, 1757–1766 (1992).
[CrossRef]

1991 (1)

T.K.Sarkar, ed., Application of Conjugate Gradient Method to Electromagnetics and Signal Analysis, Vol.  5 of Progress in Electromagnetics Research (Elsevier, 1991).

1988 (1)

1985 (1)

I. Gohberg and I. Koltracht, “Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation,” Numer. Math. 47, 237–288 (1985).
[CrossRef]

1982 (1)

Alderman-B, B. E. J.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Alexopoulos, N. G.

Azcona, L.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Baccarelli, P.

G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
[CrossRef]

Bleszynski, E.

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251(1996).
[CrossRef]

Bleszynski, M.

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251(1996).
[CrossRef]

Bretones, A. Rubio

A. G. Tijhuis and A. Rubio Bretones, “Transient excitation of a layered dielectric medium by a pulsed electric dipole: a spectral representation,” in Ultra-Wideband Short-Pulse Electromagnetics (Kluwer Academic/Plenum, 2000), pp. 167–174.

Burnside, W. D.

C.-F. Yang, W. D. Burnside, and R. C. Rudduck, “A doubly periodic moment method solution for the analysis and design of an absorber covered wall,” IEEE Trans. Antennas Propag. 41, 600–609 (1993).
[CrossRef]

Chan, C. H.

Chang, Y.-C.

Chu, H.

de Hon, B. P.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

M. C. van Beurden and B. P. de Hon, “Electromagnetic modelling of antennas mounted on a band-gap slab—discretisation issues and domain and boundary integral equations,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications ICEAA ’03 (Politecnico di Torino, 2003), pp. 637–640.
[PubMed]

de Maagt, P. J. I.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

del Rio, C.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Diaz, R.

Ederra, I.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, 2nd ed. (IEEE, 1994).
[CrossRef]

Fokkema, D. R.

G. L. G. Sleijpen, H. A. van der Vorst, and D. R. Fokkema, “BiCGstab(ℓ) and other hybrid Bi-CG methods,” Numer. Algorithms 7, 75–109 (1994).
[CrossRef]

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

Frezza, F.

G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
[CrossRef]

Galli, A.

G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
[CrossRef]

Gaylord, T. K.

Gohberg, I.

I. Gohberg and I. Koltracht, “Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation,” Numer. Math. 47, 237–288 (1985).
[CrossRef]

Gonzalo, R.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Huggard, P. G.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Jaroszewicz, T.

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251(1996).
[CrossRef]

Kerwien, N.

Ko, D. Y. K.

Koltracht, I.

I. Gohberg and I. Koltracht, “Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation,” Numer. Math. 47, 237–288 (1985).
[CrossRef]

Lalanne, P.

Li, G.

Li, L.

Magath, T.

T. Magath, “Coupled integral equations for diffraction by profiled, anisotropic, periodic structures,” IEEE Trans. Antennas Propag. 54, 681–686 (2006).
[CrossRef]

T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22, 2405–2418 (2005).
[CrossRef]

Marchand, L.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, 2nd ed. (IEEE, 1994).
[CrossRef]

Martinez, B.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

Moharam, M. G.

Morris, G. M.

Opsal, J.

Osten, W.

Paulotto, S.

G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
[CrossRef]

Rafler, S.

Rudduck, R. C.

C.-F. Yang, W. D. Burnside, and R. C. Rudduck, “A doubly periodic moment method solution for the analysis and design of an absorber covered wall,” IEEE Trans. Antennas Propag. 41, 600–609 (1993).
[CrossRef]

Ruoff, J.

Sambles, J. R.

Schuster, T.

Serebryannikov, A. E.

Shi, Y.

Sleijpen, G. L. G.

G. L. G. Sleijpen, H. A. van der Vorst, and D. R. Fokkema, “BiCGstab(ℓ) and other hybrid Bi-CG methods,” Numer. Algorithms 7, 75–109 (1994).
[CrossRef]

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

Tijhuis, A. G.

A. G. Tijhuis and A. Rubio Bretones, “Transient excitation of a layered dielectric medium by a pulsed electric dipole: a spectral representation,” in Ultra-Wideband Short-Pulse Electromagnetics (Kluwer Academic/Plenum, 2000), pp. 167–174.

Valerio, G.

G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
[CrossRef]

van Beurden, M. C.

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

M. C. van Beurden and B. P. de Hon, “Electromagnetic modelling of antennas mounted on a band-gap slab—discretisation issues and domain and boundary integral equations,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications ICEAA ’03 (Politecnico di Torino, 2003), pp. 637–640.
[PubMed]

van den Berg, P. M.

P. Zwamborn and P. M. van den Berg, “The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems,” IEEE Trans. Microwave Theory Tech. 40, 1757–1766 (1992).
[CrossRef]

van der Vorst, H. A.

G. L. G. Sleijpen, H. A. van der Vorst, and D. R. Fokkema, “BiCGstab(ℓ) and other hybrid Bi-CG methods,” Numer. Algorithms 7, 75–109 (1994).
[CrossRef]

Wang, J.

H.-Y. D. Yang and J. Wang, “Surface waves of printed antennas on planar artificial periodic dielectric structures,” IEEE Trans. Antennas Propag. 49, 444–450 (2001).
[CrossRef]

Yang, C.-F.

C.-F. Yang, W. D. Burnside, and R. C. Rudduck, “A doubly periodic moment method solution for the analysis and design of an absorber covered wall,” IEEE Trans. Antennas Propag. 41, 600–609 (1993).
[CrossRef]

Yang, H.-Y. D.

H.-Y. D. Yang and J. Wang, “Surface waves of printed antennas on planar artificial periodic dielectric structures,” IEEE Trans. Antennas Propag. 49, 444–450 (2001).
[CrossRef]

H.-Y. D. Yang, R. Diaz, and N. G. Alexopoulos, “Reflection and transmission of waves from multilayer structures with planar-implanted periodic material blocks,” J. Opt. Soc. Am. B 14, 2513–2521 (1997).
[CrossRef]

Zwamborn, P.

P. Zwamborn and P. M. van den Berg, “The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems,” IEEE Trans. Microwave Theory Tech. 40, 1757–1766 (1992).
[CrossRef]

Electron. Trans. Numer. Anal. (1)

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal. 1, 11–32 (1993).

IEEE Trans. Antennas Propag. (4)

G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag. 57, 122–134 (2009).
[CrossRef]

C.-F. Yang, W. D. Burnside, and R. C. Rudduck, “A doubly periodic moment method solution for the analysis and design of an absorber covered wall,” IEEE Trans. Antennas Propag. 41, 600–609 (1993).
[CrossRef]

H.-Y. D. Yang and J. Wang, “Surface waves of printed antennas on planar artificial periodic dielectric structures,” IEEE Trans. Antennas Propag. 49, 444–450 (2001).
[CrossRef]

T. Magath, “Coupled integral equations for diffraction by profiled, anisotropic, periodic structures,” IEEE Trans. Antennas Propag. 54, 681–686 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. Zwamborn and P. M. van den Berg, “The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems,” IEEE Trans. Microwave Theory Tech. 40, 1757–1766 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

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]

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]

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

T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22, 2405–2418 (2005).
[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]

Y. Shi and C. H. Chan, “Multilevel Green’s function interpolation method for analysis of 3-D frequency selective structures using volume/surface integral equation,” J. Opt. Soc. Am. A 27, 308–318 (2010).
[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]

J. Opt. Soc. Am. B (1)

Numer. Algorithms (1)

G. L. G. Sleijpen, H. A. van der Vorst, and D. R. Fokkema, “BiCGstab(ℓ) and other hybrid Bi-CG methods,” Numer. Algorithms 7, 75–109 (1994).
[CrossRef]

Numer. Math. (1)

I. Gohberg and I. Koltracht, “Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation,” Numer. Math. 47, 237–288 (1985).
[CrossRef]

Radio Sci. (1)

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251(1996).
[CrossRef]

Other (6)

W.C.Chew, J.Jin, and E.Michielssen, eds., Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, 2nd ed. (IEEE, 1994).
[CrossRef]

A. G. Tijhuis and A. Rubio Bretones, “Transient excitation of a layered dielectric medium by a pulsed electric dipole: a spectral representation,” in Ultra-Wideband Short-Pulse Electromagnetics (Kluwer Academic/Plenum, 2000), pp. 167–174.

T.K.Sarkar, ed., Application of Conjugate Gradient Method to Electromagnetics and Signal Analysis, Vol.  5 of Progress in Electromagnetics Research (Elsevier, 1991).

M. C. van Beurden and B. P. de Hon, “Electromagnetic modelling of antennas mounted on a band-gap slab—discretisation issues and domain and boundary integral equations,” in Proceedings of the International Conference on Electromagnetics in Advanced Applications ICEAA ’03 (Politecnico di Torino, 2003), pp. 637–640.
[PubMed]

B. P. de Hon, M. C. van Beurden, R. Gonzalo, B. Martinez, I. Ederra, C. del Rio, L. Azcona, B. E. J. Alderman-B, P. G. Huggard, P. J. I. de Maagt, and L. Marchand, “Domain integral equations for electromagnetic bandgap slab simulations,” in URSI EMTS International Symposium on Electromagnetic Theory (University of Pisa, 2004), pp. 1239–1241.

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Figures (6)

Fig. 1
Fig. 1

Side and top views of the unit cell with a block-shaped grating on top of a layered medium.

Fig. 2
Fig. 2

(a) Top and side views of the array of holes and (b) the corresponding convergence of the zeroth-order diffraction efficiency for an array of rectangular holes compared to the RCWA reference solution [26] versus the number of Fourier modes per direction.

Fig. 3
Fig. 3

Convergence of the first-order parallel reflection coefficient obtained via (a) the classical spectral volume integral formulation and (b) the spectral volume integral formulation with explicit Fourier factorization rules for an increasing number of Fourier modes, compared to the RCWA reference solution. The number of samples in the z direction is indicated in the legend.

Fig. 4
Fig. 4

Self-consistent relative error in the (a) zeroth-order and (b) first-order parallel reflection coefficients obtained via the spectral volume integral formulation with explicit Fourier factorization rules and via the classical volume integral formulation, when the number of samples in the z direction is given by Eq. (38).

Fig. 5
Fig. 5

(a) Peak-memory usage and (b) total CPU time versus the total number of unknowns, when the number of samples in the z direction is given by Eq. (38). The dashed curves indicate the lines (a)  3 · 10 6 N log N and (b)  3 · 10 4 N .

Fig. 6
Fig. 6

(a) Two side views of a four-layer woodpile structure of aluminum bars and (b) the transmission coefficients versus frequency for the corresponding woodpile structure consisting of multiple layers together with the solution in [14] indicated by stars for the case of 20 layers.

Tables (1)

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Table 1 Computation Times for the Woodpile Structure Consisting of Several Layers

Equations (39)

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× H = j ω D = j ω ε E ,
× H = j ω ( ε ε b ) E + j ω ε b E = J + j ω ε b E .
E ( x , y , z ) = m 1 = m 2 = e ( m 1 , m 2 , z ) exp ( j k T m · r T ) , H ( x , y , z ) = m 1 = m 2 = h ( m 1 , m 2 , z ) exp ( j k T m · r T ) , J ( x , y , z ) = m 1 = m 2 = j ( m 1 , m 2 , z ) exp ( j k T m · r T ) ,
j k T m × h ( m 1 , m 2 , z ) + u z × d d z h ( m 1 , m 2 , z ) = j ( m 1 , m 2 , z ) + j ω ε 0 ε r ( z ) e ( m 1 , m 2 , z ) , j k T m × e ( m 1 , m 2 , z ) + u z × d d z e ( m 1 , m 2 , z ) = j ω μ 0 h ( m 1 , m 2 , z ) ,
e T = j u k v e j ( u z × u k ) v h , h T = j u k i h + j ( u z × u k ) i e ,
j ω ε ( z ) d d z v e = γ 2 i e + k T j z ,
d d z i e = j ω ε ( z ) v e + j u k · j T ,
d d z v h = j ω μ 0 i h ,
j ω μ 0 d d z i h = γ 2 v h + ω μ 0 ( u z × u k ) · j T ,
e z = k T j ω ε ( z ) i e 1 j ω ε j z ,
h z = k T j ω μ 0 v h .
e = j u k v e j ( u z × u k ) v h + [ k T j ω ε ( z ) i e 1 j ω ε ( z ) j z ] u z .
e ( m 1 , m 2 , z ) = e i ( m 1 , m 2 , z ) + z R G ¯ ¯ ( m 1 , m 2 , z , z ) j ( m 1 , m 2 , z ) d z ,
G ¯ ¯ h = ( k T k T j k T u z d d z j u z k T d d z + u z u z d 2 d z 2 ) 1 2 j γ ω ε 1 exp ( γ | z z | ) ,
Π Δ ( x ) = { 1 x [ Δ / 2 , Δ / 2 ] 0 elsewhere .
ε / ε b = 1 + χ c Π Δ x ( x x 0 ) Π Δ y ( y y 0 ) ,
( ε / ε b ) 1 = 1 + χ ^ c Π Δ x ( x x 0 ) Π Δ y ( y y 0 ) ,
χ ^ c = χ c 1 + χ c ,
D x = j ω ε b [ 1 + χ c Π Δ x ( x x 0 ) Π Δ y ( y y 0 ) ] E x .
J = j ω ( ε ε b ) E = j ω ( D ε b E ) ,
j ω ε b E x = [ 1 + χ ^ c Π Δ x ( x x 0 ) Π Δ y ( y y 0 ) ] D x ,
( I + B Π Δ y ) ( I + A Π Δ y ) = I + ( B + A + B A ) Π Δ y ,
j ω ε b [ I χ ^ c Π Δ x Π Δ y ( I + χ ^ c Π Δ x ) 1 ] E x = D x ,
d x = j ω ε b [ I χ ^ c P x P y ( I + χ ^ c P x ) 1 ] e x ,
d y = j ω ε b [ I χ ^ c P x P y ( I + χ ^ c P y ) 1 ] e y ,
d z = j ω ε b χ c P x P y e z ,
j x = d x j ω ε b e x = j ω ε b χ c P x P y [ 1 1 + χ c ( I + χ ^ c P x ) 1 ] e x , j y = d y j ω ε b e y = j ω ε b χ c P x P y [ 1 1 + χ c ( I + χ ^ c P y ) 1 ] e y , j z = d z j ω ε b e z = j ω ε b χ c P x P y e z ,
j x = j ω ε b χ c P x P y e x ,
j y = j ω ε b χ c P x P y e y ,
j z = j ω ε b χ c P x P y e z .
e x = ( 1 + χ ^ c ) ( I + χ ^ c P x ) f x ,
e y = ( 1 + χ ^ c ) ( I + χ ^ c P y ) f y ,
e z = f z .
e ( m 1 , m 2 , z ) = [ L f ] ( m 1 , m 2 , z ) .
j = j ω ε b χ c P x P y f = M f ,
e i ( m 1 , m 2 , z ) = [ L f ] ( m 1 , m 2 , z ) z R G ¯ ¯ ( m 1 , m 2 , z , z ) [ M f ] ( m 1 , m 2 , z ) d z .
G ¯ ¯ ( z , z ) = G ¯ ¯ h ( | z z | ) + R ¯ ¯ u exp [ γ ( z + z ) ] + R ¯ ¯ l exp [ γ ( z + z 2 h ) ] ,
G ¯ ¯ h ( | z z | ) = { g ¯ ¯ 1 ( z ) g ¯ ¯ 2 ( z ) z < z g ¯ ¯ 3 ( z ) g ¯ ¯ 4 ( z ) z > z ,
N z = 2 M h max { a 1 , a 2 } .

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