Abstract

In this paper, we present the multilevel Green’s function interpolation method (MLGFIM) for analyses of three-dimensional doubly periodic structures consisting of dielectric media and conducting objects. The volume integral equation (VIE) and surface integral equation (SIE) are adopted, respectively, for the inhomogeneous dielectric and conducting objects in a unit cell. Conformal basis functions defined on curvilinear hexahedron and quadrilateral elements are used to solve the volume/surface integral equation (VSIE). Periodic boundary conditions are introduced at the boundaries of the unit cell. Computation of the space-domain Green’s function is accelerated by means of Ewald’s transformation. A periodic octary-cube-tree scheme is developed to allow adaptation of the MLGFIM for analyses of doubly periodic structures. The proposed algorithm is first validated by comparison with published data in the open literature. More complex periodic structures, such as dielectric coated conducting shells, folded dielectric structures, photonic bandgap structures, and split ring resonators (SRRs), are then simulated to illustrate that the MLGFIM has a computational complexity of O(N) when applied to periodic structures.

© 2010 Optical Society of America

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  1. R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces--a review,” IEEE Proc. 76, 1593-1615 (1988).
    [CrossRef]
  2. K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (Optical Engineering) (CRC Press, 2005).
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  3. 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]
  4. N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley-Interscience, 2006).
  5. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  6. J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications (IEEE Press, 1998).
  7. S. D. 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 Technol. 40, 363-370 (1992).
    [CrossRef]
  8. T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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. R. A. Kipp and C. H. Chan, “A numerically efficient technique for the method of moments solution for periodic structures in layered media,” IEEE Trans. Microwave Theory Technol. 42, 635-643 (1994).
    [CrossRef]
  10. L. C. Trintinalia and H. Ling, “Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme,” IEEE Trans. Antennas Propag. 52, 2253-2261 (2004).
    [CrossRef]
  11. 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]
  12. I. Stevanovic, P. C. Valero, K. Blagovic, F. Bongard, and J. R. Mosig, “Integral equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microwave Theory Technol. 54, 3688-3697 (2006).
    [CrossRef]
  13. M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-REM method,” IEEE Trans. Antennas Propag. 51, 2830-2836 (2003).
    [CrossRef]
  14. V. Rokhlin, “Rapid solution of integral equations of classic potential theory,” J. Comput. Phys. 60, 187-207 (1985).
    [CrossRef]
  15. W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).
  16. 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]
  17. F. Ling, C. F. Wang, and J. M. Jin, “An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex image method,” IEEE Trans. Microwave Theory Technol. 48, 832-837 (2000).
    [CrossRef]
  18. C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron. E80-C, 1354-1359 (1997).
  19. S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
    [CrossRef]
  20. J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 16, 1059-1072 (1997).
    [CrossRef]
  21. X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).
  22. H. G. Wang, C. H. Chan, and L. Tsang, “A new multilevel Green's function interpolation method for large-scale low-frequency EM simulations,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1427-1443 (2005).
    [CrossRef]
  23. H. G. Wang and C. H. Chan, “The implementation of multilevel Green's function interpolation method for full-wave electromagnetic problems,” IEEE Trans. Antennas Propag. 55, 1348-1358 (2007).
    [CrossRef]
  24. L. Li, H. G. Wang, and C. H. Chan, “An improved multilevel Green's function interpolation method with adaptive phase compensation for large-scale full-wave EM simulation,” IEEE Trans. Antennas Propag. 56, 1381-1393 (2008).
    [CrossRef]
  25. Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green's function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A 25, 2535-2548 (2008).
    [CrossRef]
  26. Y. Shi and C. H. Chan, “Solution to electromagnetic scattering by bi-isotropic media using multilevel Green's function interpolation method,” PIER 97, 259-274 (2009).
    [CrossRef]
  27. P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. 64, 253-287 (1921).
    [CrossRef]
  28. K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63, 222-235 (1986).
    [CrossRef]
  29. D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77-85 (1984).
    [CrossRef]
  30. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409-418 (1982).
    [CrossRef]
  31. F. S. Johansson, “Frequency-scanned gratings consisting of photo-etched arrays,” IEEE Trans. Antennas Propag. 37, 996-1002 (1989).
    [CrossRef]
  32. E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
    [CrossRef]
  33. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
    [CrossRef]

2009

Y. Shi and C. H. Chan, “Solution to electromagnetic scattering by bi-isotropic media using multilevel Green's function interpolation method,” PIER 97, 259-274 (2009).
[CrossRef]

2008

Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green's function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A 25, 2535-2548 (2008).
[CrossRef]

L. Li, H. G. Wang, and C. H. Chan, “An improved multilevel Green's function interpolation method with adaptive phase compensation for large-scale full-wave EM simulation,” IEEE Trans. Antennas Propag. 56, 1381-1393 (2008).
[CrossRef]

2007

H. G. Wang and C. H. Chan, “The implementation of multilevel Green's function interpolation method for full-wave electromagnetic problems,” IEEE Trans. Antennas Propag. 55, 1348-1358 (2007).
[CrossRef]

2006

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

2005

X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).

H. G. Wang, C. H. Chan, and L. Tsang, “A new multilevel Green's function interpolation method for large-scale low-frequency EM simulations,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1427-1443 (2005).
[CrossRef]

2004

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

2003

M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-REM method,” IEEE Trans. Antennas Propag. 51, 2830-2836 (2003).
[CrossRef]

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

2001

S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
[CrossRef]

2000

F. Ling, C. F. Wang, and J. M. Jin, “An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex image method,” IEEE Trans. Microwave Theory Technol. 48, 832-837 (2000).
[CrossRef]

1999

T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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

C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron. E80-C, 1354-1359 (1997).

J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 16, 1059-1072 (1997).
[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]

1996

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]

1994

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

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

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

S. D. 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 Technol. 40, 363-370 (1992).
[CrossRef]

1989

F. S. Johansson, “Frequency-scanned gratings consisting of photo-etched arrays,” IEEE Trans. Antennas Propag. 37, 996-1002 (1989).
[CrossRef]

1988

R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces--a review,” IEEE Proc. 76, 1593-1615 (1988).
[CrossRef]

1986

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

1985

V. Rokhlin, “Rapid solution of integral equations of classic potential theory,” J. Comput. Phys. 60, 187-207 (1985).
[CrossRef]

1984

D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77-85 (1984).
[CrossRef]

1982

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

1921

P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. 64, 253-287 (1921).
[CrossRef]

Abeyta, A.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Alexopoulos, N. G.

Biswas, R.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Blagovic, K.

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

Bongard, F.

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

Bozzi, M.

M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-REM method,” IEEE Trans. Antennas Propag. 51, 2830-2836 (2003).
[CrossRef]

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.

Y. Shi and C. H. Chan, “Solution to electromagnetic scattering by bi-isotropic media using multilevel Green's function interpolation method,” PIER 97, 259-274 (2009).
[CrossRef]

Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green's function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A 25, 2535-2548 (2008).
[CrossRef]

L. Li, H. G. Wang, and C. H. Chan, “An improved multilevel Green's function interpolation method with adaptive phase compensation for large-scale full-wave EM simulation,” IEEE Trans. Antennas Propag. 56, 1381-1393 (2008).
[CrossRef]

H. G. Wang and C. H. Chan, “The implementation of multilevel Green's function interpolation method for full-wave electromagnetic problems,” IEEE Trans. Antennas Propag. 55, 1348-1358 (2007).
[CrossRef]

H. G. Wang, C. H. Chan, and L. Tsang, “A new multilevel Green's function interpolation method for large-scale low-frequency EM simulations,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1427-1443 (2005).
[CrossRef]

S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
[CrossRef]

C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron. E80-C, 1354-1359 (1997).

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

R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces--a review,” IEEE Proc. 76, 1593-1615 (1988).
[CrossRef]

Chan, C. T.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Chan, K. F.

S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
[CrossRef]

Chatterjee, A.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications (IEEE Press, 1998).

Chew, W. C.

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

Cwik, T.

R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces--a review,” IEEE Proc. 76, 1593-1615 (1988).
[CrossRef]

Diaz, R.

Eibert, T. F.

T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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]

Engheta, N.

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley-Interscience, 2006).

Ewald, P. P.

P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. 64, 253-287 (1921).
[CrossRef]

Gan, Y. B.

X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).

Gedney, S. D.

S. D. 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 Technol. 40, 363-370 (1992).
[CrossRef]

Glisson, A. W.

D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77-85 (1984).
[CrossRef]

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

Ho, K. M.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Jackson, D. R.

T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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]

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]

Jin, J. M.

F. Ling, C. F. Wang, and J. M. Jin, “An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex image method,” IEEE Trans. Microwave Theory Technol. 48, 832-837 (2000).
[CrossRef]

J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

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

Johansson, F. S.

F. S. Johansson, “Frequency-scanned gratings consisting of photo-etched arrays,” IEEE Trans. Antennas Propag. 37, 996-1002 (1989).
[CrossRef]

Jordan, K. E.

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

Kempel, L. C.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications (IEEE Press, 1998).

Kipp, R. A.

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

Koschny, T.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

Lee, J. F.

S. D. 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 Technol. 40, 363-370 (1992).
[CrossRef]

Leung, Y. F.

C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron. E80-C, 1354-1359 (1997).

Li, L.

Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green's function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A 25, 2535-2548 (2008).
[CrossRef]

L. Li, H. G. Wang, and C. H. Chan, “An improved multilevel Green's function interpolation method with adaptive phase compensation for large-scale full-wave EM simulation,” IEEE Trans. Antennas Propag. 56, 1381-1393 (2008).
[CrossRef]

Li, L. W.

X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).

Li, S. Q.

S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
[CrossRef]

Lin, C. M.

C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron. E80-C, 1354-1359 (1997).

Ling, F.

F. Ling, C. F. Wang, and J. M. Jin, “An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex image method,” IEEE Trans. Microwave Theory Technol. 48, 832-837 (2000).
[CrossRef]

Ling, H.

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

Markos, P.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

Michielssen, E.

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

Mittra, R.

S. D. 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 Technol. 40, 363-370 (1992).
[CrossRef]

R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces--a review,” IEEE Proc. 76, 1593-1615 (1988).
[CrossRef]

Mosig, J. R.

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

Nie, X. C.

X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).

Ozbay, E.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Perregrini, L.

M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-REM method,” IEEE Trans. Antennas Propag. 51, 2830-2836 (2003).
[CrossRef]

Phillips, J. R.

J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 16, 1059-1072 (1997).
[CrossRef]

Rao, S. M.

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

Richter, G. R.

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

Rokhlin, V.

V. Rokhlin, “Rapid solution of integral equations of classic potential theory,” J. Comput. Phys. 60, 187-207 (1985).
[CrossRef]

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]

Schaubert, D. H.

D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77-85 (1984).
[CrossRef]

Sheng, P.

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

Shi, Y.

Y. Shi and C. H. Chan, “Solution to electromagnetic scattering by bi-isotropic media using multilevel Green's function interpolation method,” PIER 97, 259-274 (2009).
[CrossRef]

Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green's function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A 25, 2535-2548 (2008).
[CrossRef]

Sigalas, M.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Smith, D. R.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

Song, J. M.

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

Soukoulis, C. M.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Stevanovic, I.

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

Tringides, M. C.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Trintinalia, L. C.

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

Tsang, L.

H. G. Wang, C. H. Chan, and L. Tsang, “A new multilevel Green's function interpolation method for large-scale low-frequency EM simulations,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1427-1443 (2005).
[CrossRef]

S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
[CrossRef]

C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron. E80-C, 1354-1359 (1997).

Tuttle, G.

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Valero, P. C.

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

Volakis, J. L.

T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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]

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications (IEEE Press, 1998).

Wang, C. F.

F. Ling, C. F. Wang, and J. M. Jin, “An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex image method,” IEEE Trans. Microwave Theory Technol. 48, 832-837 (2000).
[CrossRef]

Wang, H. G.

L. Li, H. G. Wang, and C. H. Chan, “An improved multilevel Green's function interpolation method with adaptive phase compensation for large-scale full-wave EM simulation,” IEEE Trans. Antennas Propag. 56, 1381-1393 (2008).
[CrossRef]

Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green's function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A 25, 2535-2548 (2008).
[CrossRef]

H. G. Wang and C. H. Chan, “The implementation of multilevel Green's function interpolation method for full-wave electromagnetic problems,” IEEE Trans. Antennas Propag. 55, 1348-1358 (2007).
[CrossRef]

H. G. Wang, C. H. Chan, and L. Tsang, “A new multilevel Green's function interpolation method for large-scale low-frequency EM simulations,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1427-1443 (2005).
[CrossRef]

White, J. K.

J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 16, 1059-1072 (1997).
[CrossRef]

Wilton, D. R.

T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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]

D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77-85 (1984).
[CrossRef]

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

Yasumoto, K.

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (Optical Engineering) (CRC Press, 2005).
[CrossRef]

Yeo, T. S.

X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).

Yu, Y. X.

S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
[CrossRef]

Yuan, N.

X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).

Ziolkowski, R. W.

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley-Interscience, 2006).

Ann. Phys.

P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. 64, 253-287 (1921).
[CrossRef]

IEEE Proc.

R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces--a review,” IEEE Proc. 76, 1593-1615 (1988).
[CrossRef]

IEEE Trans. Antennas Propag.

H. G. Wang and C. H. Chan, “The implementation of multilevel Green's function interpolation method for full-wave electromagnetic problems,” IEEE Trans. Antennas Propag. 55, 1348-1358 (2007).
[CrossRef]

L. Li, H. G. Wang, and C. H. Chan, “An improved multilevel Green's function interpolation method with adaptive phase compensation for large-scale full-wave EM simulation,” IEEE Trans. Antennas Propag. 56, 1381-1393 (2008).
[CrossRef]

D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77-85 (1984).
[CrossRef]

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

F. S. Johansson, “Frequency-scanned gratings consisting of photo-etched arrays,” IEEE Trans. Antennas Propag. 37, 996-1002 (1989).
[CrossRef]

T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. 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]

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

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]

M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-REM method,” IEEE Trans. Antennas Propag. 51, 2830-2836 (2003).
[CrossRef]

X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag. 52, 818-824 (2005).

IEEE Trans. Comput. Aided Des. Integr. Circuits Syst.

H. G. Wang, C. H. Chan, and L. Tsang, “A new multilevel Green's function interpolation method for large-scale low-frequency EM simulations,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24, 1427-1443 (2005).
[CrossRef]

J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 16, 1059-1072 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Technol.

S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparse-matrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microwave Theory Technol. 49, 1221-1228 (2001).
[CrossRef]

S. D. 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 Technol. 40, 363-370 (1992).
[CrossRef]

F. Ling, C. F. Wang, and J. M. Jin, “An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex image method,” IEEE Trans. Microwave Theory Technol. 48, 832-837 (2000).
[CrossRef]

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

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

IEICE Trans. Electron.

C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron. E80-C, 1354-1359 (1997).

J. Comput. Phys.

V. Rokhlin, “Rapid solution of integral equations of classic potential theory,” J. Comput. Phys. 60, 187-207 (1985).
[CrossRef]

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Phys. Rev. B

E. Ozbay, A. Abeyta, G. Tuttle, M. C. Tringides, R. Biswas, M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B 50, 1945-1948 (1994).
[CrossRef]

Phys. Rev. E

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602 (2003).
[CrossRef]

PIER

Y. Shi and C. H. Chan, “Solution to electromagnetic scattering by bi-isotropic media using multilevel Green's function interpolation method,” PIER 97, 259-274 (2009).
[CrossRef]

Radio Sci.

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

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

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley-Interscience, 2006).

J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications (IEEE Press, 1998).

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (Optical Engineering) (CRC Press, 2005).
[CrossRef]

Cited By

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Figures (13)

Fig. 1
Fig. 1

Infinite periodic structures.

Fig. 2
Fig. 2

Hexahedral elements and quadrilateral elements.

Fig. 3
Fig. 3

Two-dimensional pictorial representation of periodic octary-cube-tree: (a) second level in octary-cube-tree; (b) third level in octary-cube-tree.

Fig. 4
Fig. 4

Computational range of periodic Green’s function.

Fig. 5
Fig. 5

Frequency selective surface with dielectric backing: (a) geometry in a unit cell; (b) TM power reflection coefficient.

Fig. 6
Fig. 6

Cross-patches on a dielectric substrate: (a) geometry in a unit cell; (b) TM reflection coefficient.

Fig. 7
Fig. 7

Dielectric slab with planarly embedded periodic material block: (a) geometry in a unit cell; (b) TM reflection coefficient.

Fig. 8
Fig. 8

Dielectric slab with drilled square holes: (a) geometrical configuration; (b) TM reflection coefficient.

Fig. 9
Fig. 9

Dipole etched on a dielectric substrate over a ground plane: (a) geometry in a unit cell; (b) TE power reflection coefficient.

Fig. 10
Fig. 10

Periodic array of perfect conducting shell covered by dielectric: (a) geometry; (b) TM reflection coefficient.

Fig. 11
Fig. 11

Periodic folded dielectric structure: (a) geometry in a unit cell; (b) TM reflection coefficient; (c) computational complexity; (d) memory complexity.

Fig. 12
Fig. 12

Three-dimensional photonic bandgap structure: (a) geometry of photonic bandgap structure; (b) geometry in a unit cell; (c) magnitude of TM transmission coefficient; (d) phase of TM transmission coefficient.

Fig. 13
Fig. 13

Three-dimensional split ring resonators: (a) periodic conducting SRRs; (b) periodic conducting SRRs backed by dielectric; (c) reflection of conducting SRRs; (d) reflection of conducting SRRs backed by dielectric; (e) computational complexity; (f) memory complexity.

Tables (1)

Tables Icon

Table 1 Performance of MLFGIM

Equations (30)

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

ρ m n = m ρ a + n ρ b .
k = k t 00 + k z 00 ,
k t 00 = k 0 k ̂ t 00 , k z 00 = k 0 k ̂ z 00
k ̂ t 00 = sin θ i cos φ i x ̂ sin θ i sin φ i y ̂ , k ̂ z 00 = cos θ i z ̂ .
E s ( r ) = E S s ( r ) + E V s ( r ) ,
E α s ( r ) = i ω A α ( r ) φ α ,     ( r ) ( α = S , V ) ,
A α ( r ) = μ 0 4 π α G ( r , r ) J α ( r ) d α ,
φ α ( r ) = 1 i ω 4 π ε 0 α G ( r , r ) J α ( r ) d α ,
G 0 ( r , r ) = e i k 0 | r r | | r r | ,     ( k 0 = ω μ 0 ε 0 ) .
J V ( r ) = i ω [ ε ̃ ( r ) ε 0 ] E ( r ) = [ ε ̃ ( r ) ε 0 ε ̃ ( r ) ] [ i ω ε ̃ ( r ) E ( r ) ] .
J V ( r ) = x ( r ) C ( r ) ,
x ( r ) = ε ̃ ( r ) ε 0 ε ̃ ( r ) .
C ( r ) i ω ε ̃ ( r ) = E i ( r ) + E s ( r ) ,       ( r V ) .
n ̂ × [ E i ( r ) + E s ( r ) ] = 0 ,       ( r S ) .
J s ( r + ρ m n ) = J s ( r ) e j k t 00 ρ m n ,
C ( r + ρ m n ) = C ( r ) e j k t 00 ρ m n .
G p ( r , r ) = m = n = e j k t 00 ρ m n e j k 0 R m n 4 π R m n ,
R m n = | r r ρ m n | .
k 1 = 2 π ρ b × ( ρ a × ρ b ) | ρ a × ρ b | 2 ,
k 2 = 2 π ρ a × ( ρ b × ρ a ) | ρ a × ρ b | 2 .
G p ( r , r ) = m = n = e j k t m n ( ρ ρ ) | ρ a × ρ b | 2 j k z m n e j k z m n | z z | ,
k z m n = k 0 2 k t m n k t m n ,
k t m n = k t 00 + k p m n ,
k p m n = m k 1 + n k 2 ,
G p ( r , r ) = G p 1 ( r , r ) + G p 2 ( r , r ) .
G p 1 ( r , r ) = m = n = e j k t m n ( ρ ρ ) 4 j A k z m n [ e j k z m n | z z | erfc ( j k z m n 2 E | z z | E ) + e j k z m n | z z | erfc ( j k z m n 2 E + | z z | E ) ] ,
G p 2 ( r , r ) = m = n = e j k t 00 ρ m n 8 π R m n [ e j k 0 R m n erfc ( R m n E j k 2 E ) + e j k 0 R m n erfc ( R m n E + j k 2 E ) ] ,
k 2 4 E 2 < H 2 ,
J f = J b e j k t 00 ρ a .
G p ( r , r s ρ a t ρ b ) = G p ( r , r ) e j k t 00 ( s ρ a + t ρ b ) ,     ( s , t = 0 , ± 1 ) .

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