Abstract

A numerical method in the frequency domain is developed for analyzing three-dimensional gratings using the concept of a double-periodic magnetodielectric layer. The method is based on the three-dimensional volume integral equations for the equivalent electric and magnetic polarization currents of the assumed periodic medium. The integral equations are solved by using the integral functionals related to the polarization current distributions and the technique of double Floquet–Fourier series expansion. Once the integral functionals are determined, the scattered fields outside the layer are calculated accordingly. The unit cell of the layer comprises several parallelepiped segments of materials characterized by the complex-valued relative permittivity and permeability of step function profiles. The arbitrary profiles of three-dimensional dielectric or metallic gratings can be flexibly modeled by adjusting the material parameters and sizes or locations of the parallelepiped segments in the unit cell. Numerical examples for various grating geometries and their comparisons with those presented in the literature demonstrate the accuracy and usefulness of the proposed method.

© 2007 Optical Society of America

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  1. P. de Maagt, R. Gonzalo, J. Vardaxoglou, and J.-M. Baracco, "Review of electromagnetic-bandgap technology and applications," Radio Sci. Bull. 309, 11-25 (2004).
  2. H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexopoulos, "Effective response functions for photonic band gap materials," J. Opt. Soc. Am. A 16, 1682-1699 (1999).
    [CrossRef]
  3. D. Maystre and M. Nevière, "Electromagnetic theory of crossed gratings," J. Opt. (Paris) 9, 301-306 (1978).
    [CrossRef]
  4. E. Popov and M. Nevière, "Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A 18, 2886-2894 (2001).
    [CrossRef]
  5. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  6. R. L. Kipp and C. H. 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. C. C. Chen, "Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes," IEEE Trans. Microwave Theory Tech. 19, 475-481 (1971).
    [CrossRef]
  8. M. Bozzi, L. Perregrini, J. Weinzierl, and C. Winnewisser, "Efficient analysis of quasi-optical filters by a hybrid MoM/BI-RME method," IEEE Trans. Antennas Propag. 49, 1054-1064 (2001).
    [CrossRef]
  9. 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]
  10. J. Elschner, R. Hinder, and G. Schmidt, "Finite element solution of conical diffraction problems," Adv. Comput. Math. 16, 139-156 (2002).
    [CrossRef]
  11. G. B. Antilla and N. G. Alexopoulos, "Scattering from complex three-dimensional geometries by a curvilinear hybrid finite-element-integral equation approach," J. Opt. Soc. Am. A 11, 1513-1527 (1994).
  12. T. F. Eibert and J. L. Volakis, "Fast spectral domain algorithm for hybrid finite element/boundary integral modeling of doubly periodic structures," IEE Proc., Part H: Microwaves, Antennas Propag. 147, 329-334 (2000).
    [CrossRef]
  13. J. G. Maloney and M. P. Kessler, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A.Taflove, ed. (Artech House, 1998), pp. 345-405.
  14. W. Yu, S. Dey, and R. Mittra, "On the modeling of periodic structures using the finite-difference time-domain algorithm," Microwave Opt. Technol. Lett. 24, 151-155 (2000).
    [CrossRef]
  15. R. T. Lee and G. S. Smith "A conceptually simple method for incorporating periodic boundary conditions into the FDTD method," Microwave Opt. Technol. Lett. 45, 472-476 (2005).
    [CrossRef]
  16. L. E. Peterson and J. M. Jin, "A three-dimensional time-domain finite element formulation for periodic structures," IEEE Trans. Antennas Propag. 54, 12-18 (2006).
    [CrossRef]
  17. L. K. Gal and N. A. Khizhnyak, "The scattering of oblique incident wave on thin metallic cylinder of elliptic cross-section," Izv. Vyzov Radiofiz. 14, 1596-1610 (1971) L. K. Gal and N. A. Khizhnyak[Radiophys. Quantum Electron. 14, 1596-1610 (1971)].
  18. V. V. Yachin and N. V. Ryazantseva, "The scattering of electromagnetic waves by rectangular-cell double-periodic magnetodielectric gratings," Microwave Opt. Technol. Lett. 23, 177-183 (1999).
    [CrossRef]
  19. V. Yachin, K. Yasumoto, and N. Sidorchuk, "Method of integral functionals in problems of electromagnetic wave scattering by doubly-periodic magnetodielectric structures," Proceedings of the 2004 Korea-Japan Joint Conference on AP/EMC/EMT (Korea Electromagnetic Engineering Society, 2004), pp. 223-226.
  20. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  21. V. V. Yachin, "Substantiation of the field functional method as applied to scattering by a doubly periodic magnetodielectric structure," J. Comput. Math. Math. Phys. 46, 1589-1594 (2006).
    [CrossRef]
  22. N. A. Khizhnyak, "Green's function of Maxwell's equations for inhomogeneous media," Zh. Tekh. Fiz. 28, 1592-1609 (1958).
  23. M. Born and E. Wolf, Principles of Optics (Pergamon, 1968), chap. 2.
  24. V. V. Yachin, "Elimination of a singularity in the problems of scattering on 3-D magnetodielectric structures with the chosen direction," Radiofiz. Elektron. Kharkov, 8, 197-200 (2003).
  25. N. Amitay, V. Galindo, and C. P. Wu, Theory and Analysis of Phased Array Antennas (Wiley, 1972).
  26. N. A. Khizhnyak, N. V. Ryazantseva, and V. V. Yachin, "The scattering of electromagnetic waves by a periodic magnetodielectric layer," J. Electromagn. Waves Appl. 10, 731-739 (1996).
    [CrossRef]
  27. H.-Y. D. Yang, R. Diaz, and N. G. Alexopolous, "Reflection and transmission of waves from multilayered structures with planar implanted periodic material blocks," J. Opt. Soc. Am. B 14, 2513-2521 (1997).
    [CrossRef]
  28. 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]
  29. G. Zarrillo and K. Agular, "Closed-form low frequency solution for electromagnetic waves through a frequency selective surface," IEEE Trans. Antennas Propag. 35, 1406-1418 (1987).
    [CrossRef]
  30. V. Twersky, "Rayleigh scattering," Appl. Opt. 3, 1152-1160 (1964).
    [CrossRef]
  31. B. J. Rubin and H. L. Bertoni, "Reflection from a periodically perforated plane using a subsectional current approximation," IEEE Trans. Antennas Propag. 31, 829-836 (1983).
    [CrossRef]
  32. S. L. Prosvirnin, S. A. Tretyakov, T. D. Vasilyeva, A. Fourrier-Lammer, and S. Zouhdi, "Analysis of reflection and transmission of electromagnetic waves in complex layered arrays," J. Electromagn. Waves Appl. 14, 807-826 (2000).
    [CrossRef]
  33. V. V. Yachin, N. V. Ryazantseva, and N. A. Khizhnyak, "The scattering of electromagnetic waves by a periodic magnetodielectric structures with arbitrary profiles and inhomogeneous media," J. Electromagn. Waves Appl. 11, 1349-1366 (1997).
    [CrossRef]

2006

L. E. Peterson and J. M. Jin, "A three-dimensional time-domain finite element formulation for periodic structures," IEEE Trans. Antennas Propag. 54, 12-18 (2006).
[CrossRef]

V. V. Yachin, "Substantiation of the field functional method as applied to scattering by a doubly periodic magnetodielectric structure," J. Comput. Math. Math. Phys. 46, 1589-1594 (2006).
[CrossRef]

2005

R. T. Lee and G. S. Smith "A conceptually simple method for incorporating periodic boundary conditions into the FDTD method," Microwave Opt. Technol. Lett. 45, 472-476 (2005).
[CrossRef]

2004

P. de Maagt, R. Gonzalo, J. Vardaxoglou, and J.-M. Baracco, "Review of electromagnetic-bandgap technology and applications," Radio Sci. Bull. 309, 11-25 (2004).

2003

V. V. Yachin, "Elimination of a singularity in the problems of scattering on 3-D magnetodielectric structures with the chosen direction," Radiofiz. Elektron. Kharkov, 8, 197-200 (2003).

2002

J. Elschner, R. Hinder, and G. Schmidt, "Finite element solution of conical diffraction problems," Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

2001

M. Bozzi, L. Perregrini, J. Weinzierl, and C. Winnewisser, "Efficient analysis of quasi-optical filters by a hybrid MoM/BI-RME method," IEEE Trans. Antennas Propag. 49, 1054-1064 (2001).
[CrossRef]

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

2000

S. L. Prosvirnin, S. A. Tretyakov, T. D. Vasilyeva, A. Fourrier-Lammer, and S. Zouhdi, "Analysis of reflection and transmission of electromagnetic waves in complex layered arrays," J. Electromagn. Waves Appl. 14, 807-826 (2000).
[CrossRef]

T. F. Eibert and J. L. Volakis, "Fast spectral domain algorithm for hybrid finite element/boundary integral modeling of doubly periodic structures," IEE Proc., Part H: Microwaves, Antennas Propag. 147, 329-334 (2000).
[CrossRef]

W. Yu, S. Dey, and R. Mittra, "On the modeling of periodic structures using the finite-difference time-domain algorithm," Microwave Opt. Technol. Lett. 24, 151-155 (2000).
[CrossRef]

1999

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]

V. V. Yachin and N. V. Ryazantseva, "The scattering of electromagnetic waves by rectangular-cell double-periodic magnetodielectric gratings," Microwave Opt. Technol. Lett. 23, 177-183 (1999).
[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]

H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexopoulos, "Effective response functions for photonic band gap materials," J. Opt. Soc. Am. A 16, 1682-1699 (1999).
[CrossRef]

1997

1996

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

N. A. Khizhnyak, N. V. Ryazantseva, and V. V. Yachin, "The scattering of electromagnetic waves by a periodic magnetodielectric layer," J. Electromagn. Waves Appl. 10, 731-739 (1996).
[CrossRef]

1994

R. L. Kipp and C. H. 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]

G. B. Antilla and N. G. Alexopoulos, "Scattering from complex three-dimensional geometries by a curvilinear hybrid finite-element-integral equation approach," J. Opt. Soc. Am. A 11, 1513-1527 (1994).

1987

G. Zarrillo and K. Agular, "Closed-form low frequency solution for electromagnetic waves through a frequency selective surface," IEEE Trans. Antennas Propag. 35, 1406-1418 (1987).
[CrossRef]

1983

B. J. Rubin and H. L. Bertoni, "Reflection from a periodically perforated plane using a subsectional current approximation," IEEE Trans. Antennas Propag. 31, 829-836 (1983).
[CrossRef]

1978

D. Maystre and M. Nevière, "Electromagnetic theory of crossed gratings," J. Opt. (Paris) 9, 301-306 (1978).
[CrossRef]

1971

C. C. Chen, "Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes," IEEE Trans. Microwave Theory Tech. 19, 475-481 (1971).
[CrossRef]

L. K. Gal and N. A. Khizhnyak, "The scattering of oblique incident wave on thin metallic cylinder of elliptic cross-section," Izv. Vyzov Radiofiz. 14, 1596-1610 (1971) L. K. Gal and N. A. Khizhnyak[Radiophys. Quantum Electron. 14, 1596-1610 (1971)].

1964

V. Twersky, "Rayleigh scattering," Appl. Opt. 3, 1152-1160 (1964).
[CrossRef]

1958

N. A. Khizhnyak, "Green's function of Maxwell's equations for inhomogeneous media," Zh. Tekh. Fiz. 28, 1592-1609 (1958).

Adv. Comput. Math.

J. Elschner, R. Hinder, and G. Schmidt, "Finite element solution of conical diffraction problems," Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

Appl. Opt.

V. Twersky, "Rayleigh scattering," Appl. Opt. 3, 1152-1160 (1964).
[CrossRef]

IEE Proc., Part H: Microwaves, Antennas Propag.

T. F. Eibert and J. L. Volakis, "Fast spectral domain algorithm for hybrid finite element/boundary integral modeling of doubly periodic structures," IEE Proc., Part H: Microwaves, Antennas Propag. 147, 329-334 (2000).
[CrossRef]

IEEE Trans. Antennas Propag.

M. Bozzi, L. Perregrini, J. Weinzierl, and C. Winnewisser, "Efficient analysis of quasi-optical filters by a hybrid MoM/BI-RME method," IEEE Trans. Antennas Propag. 49, 1054-1064 (2001).
[CrossRef]

L. E. Peterson and J. M. Jin, "A three-dimensional time-domain finite element formulation for periodic structures," IEEE Trans. Antennas Propag. 54, 12-18 (2006).
[CrossRef]

B. J. Rubin and H. L. Bertoni, "Reflection from a periodically perforated plane using a subsectional current approximation," IEEE Trans. Antennas Propag. 31, 829-836 (1983).
[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]

G. Zarrillo and K. Agular, "Closed-form low frequency solution for electromagnetic waves through a frequency selective surface," IEEE Trans. Antennas Propag. 35, 1406-1418 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

R. L. Kipp and C. H. 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]

C. C. Chen, "Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes," IEEE Trans. Microwave Theory Tech. 19, 475-481 (1971).
[CrossRef]

Izv. Vyzov Radiofiz.

L. K. Gal and N. A. Khizhnyak, "The scattering of oblique incident wave on thin metallic cylinder of elliptic cross-section," Izv. Vyzov Radiofiz. 14, 1596-1610 (1971) L. K. Gal and N. A. Khizhnyak[Radiophys. Quantum Electron. 14, 1596-1610 (1971)].

J. Comput. Math. Math. Phys.

V. V. Yachin, "Substantiation of the field functional method as applied to scattering by a doubly periodic magnetodielectric structure," J. Comput. Math. Math. Phys. 46, 1589-1594 (2006).
[CrossRef]

J. Comput. Phys.

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. Electromagn. Waves Appl.

N. A. Khizhnyak, N. V. Ryazantseva, and V. V. Yachin, "The scattering of electromagnetic waves by a periodic magnetodielectric layer," J. Electromagn. Waves Appl. 10, 731-739 (1996).
[CrossRef]

S. L. Prosvirnin, S. A. Tretyakov, T. D. Vasilyeva, A. Fourrier-Lammer, and S. Zouhdi, "Analysis of reflection and transmission of electromagnetic waves in complex layered arrays," J. Electromagn. Waves Appl. 14, 807-826 (2000).
[CrossRef]

V. V. Yachin, N. V. Ryazantseva, and N. A. Khizhnyak, "The scattering of electromagnetic waves by a periodic magnetodielectric structures with arbitrary profiles and inhomogeneous media," J. Electromagn. Waves Appl. 11, 1349-1366 (1997).
[CrossRef]

J. Opt. (Paris)

D. Maystre and M. Nevière, "Electromagnetic theory of crossed gratings," J. Opt. (Paris) 9, 301-306 (1978).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Microwave Opt. Technol. Lett.

W. Yu, S. Dey, and R. Mittra, "On the modeling of periodic structures using the finite-difference time-domain algorithm," Microwave Opt. Technol. Lett. 24, 151-155 (2000).
[CrossRef]

R. T. Lee and G. S. Smith "A conceptually simple method for incorporating periodic boundary conditions into the FDTD method," Microwave Opt. Technol. Lett. 45, 472-476 (2005).
[CrossRef]

V. V. Yachin and N. V. Ryazantseva, "The scattering of electromagnetic waves by rectangular-cell double-periodic magnetodielectric gratings," Microwave Opt. Technol. Lett. 23, 177-183 (1999).
[CrossRef]

Radio Sci. Bull.

P. de Maagt, R. Gonzalo, J. Vardaxoglou, and J.-M. Baracco, "Review of electromagnetic-bandgap technology and applications," Radio Sci. Bull. 309, 11-25 (2004).

Zh. Tekh. Fiz.

N. A. Khizhnyak, "Green's function of Maxwell's equations for inhomogeneous media," Zh. Tekh. Fiz. 28, 1592-1609 (1958).

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968), chap. 2.

V. V. Yachin, "Elimination of a singularity in the problems of scattering on 3-D magnetodielectric structures with the chosen direction," Radiofiz. Elektron. Kharkov, 8, 197-200 (2003).

N. Amitay, V. Galindo, and C. P. Wu, Theory and Analysis of Phased Array Antennas (Wiley, 1972).

V. Yachin, K. Yasumoto, and N. Sidorchuk, "Method of integral functionals in problems of electromagnetic wave scattering by doubly-periodic magnetodielectric structures," Proceedings of the 2004 Korea-Japan Joint Conference on AP/EMC/EMT (Korea Electromagnetic Engineering Society, 2004), pp. 223-226.

J. G. Maloney and M. P. Kessler, in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A.Taflove, ed. (Artech House, 1998), pp. 345-405.

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Figures (10)

Fig. 1
Fig. 1

Double-periodic magnetodielectric layer of thickness h illuminated by a plane wave. The unit cell with periods L x 1 and L y 1 along the x 1 and y 1 axes contains several parallelepiped segments characterized by the complex-valued relative permittivity ϵ i j and permeability μ i j of step function profiles. The x 1 axis is parallel to the x axis, whereas the y 1 axis forms an angle β with respect to the x axis.

Fig. 2
Fig. 2

Reflection coefficient R 00 of the zeroth-order diffraction calculated for a dielectric layer of ϵ = 4.0 and h L = 0.1 with implanted double-periodic square-shaped dielectric blocks of ϵ = 10.0 . The truncation number of spatial harmonics is ( 2 N + 1 ) 2 = 121 . The results for three different situations of the incident wave are compared with those reported in the literature [27].

Fig. 3
Fig. 3

Power transmission coefficient T 00 2 of the zeroth-order diffraction calculated for a free-standing array of square-shaped apertures on a magnetodielectric layer with material parameters ϵ = 250 + i 2500000 and μ = 1 ϵ . The thickness of the layer is h L = 0.005 and the aperture size is 0.8 L × 0.8 L . The truncation number of spatial harmonics is ( 2 N + 1 ) 2 = 441 . The results for three different situations of the incident wave are compared with those of the same aperture array on the PEC layer reported in the literature [29].

Fig. 4
Fig. 4

Reflection coefficient R 00 calculated for the normal incidence of the plane wave on a skewed grating of circular holes on a layer with three different material parameter models. The thickness of the layer is h = 0.15 L , and the radius of the hole is r = 0.6 L in the unit cell defined by L x 1 = 1.7 L , L y 1 = 1.73 L , and β = 60 ° in Fig. 1. The dotted curve is obtained for ( 2 N + 1 ) 2 = 529 when the layer is made of gold with ϵ = 157.92 + i 21.414 and μ = 1 , whereas the solid curve is obtained for ( 2 N + 1 ) 2 = 169 when the layer is made of a PEC-like magnetodielectric with ϵ = 157.92 + i 21.414 and μ = 1 ϵ . The results are compared with those of the same skew grating on the PEC layer reported in the literature [7].

Fig. 5
Fig. 5

Convergence of the reflection coefficient R 00 of the skewed grating in Fig. 4 as functions of the truncation number ( 2 N + 1 ) 2 of the spatial harmonic expansions for two different material parameters (a) ϵ = 157.92 + i 21.414 and μ = 1 , and (b) ϵ = 157.92 + i 21.414 and μ = 1 ϵ . The results for four different cell sizes L λ are compared.

Fig. 6
Fig. 6

(a) Power reflection coefficient R 00 2 and (b) phase of the reflection coefficient R 00 calculated for the normal incidence of the plane wave on a PEC-like magnetodielectric layer periodically perforated with cross-shaped apertures. The electric field of the incident wave is parallel to the x axis. The width and length of the cross in the unit cell are w = L 7 and s = 5 L 7 , and the material parameters of the PEC-like layer are assumed to be ϵ = 250 + i 2500000 and μ = 1 ϵ . The truncation number of the spatial harmonics is ( 2 N + 1 ) 2 = 121 . The results for three different thicknesses h L of the layer are compared.

Fig. 7
Fig. 7

Comparison of (a) the power transmission coefficient T 00 2 and (b) the phase of the transmission coefficient T 00 calculated by the PEC-like model of a magnetodielectric layer with the measured data[8] for a free-standing copper foil of a thickness 10 μ m periodically perforated with cross-shaped apertures. The unit cell and dimensions of the aperture used in the experiment are shown in the inset. The material parameters of a PEC-like layer are the same as those in Fig. 6. The truncation number of the spatial harmonics is ( 2 N + 1 ) 2 = 121 .

Fig. 8
Fig. 8

(a) Power reflection coefficient R 00 2 and (b) phase of the reflection coefficient R 00 calculated for the normal incidence of the plane wave on a PEC-like magnetodielectric layer periodically perforated with gammadion-shaped apertures. The electric field of the incident wave is parallel to the x axis. The aperture length and width in the unit cell are s = 11 L 7 and w = L 7 , respectively. The material parameters of the PEC-like layer are the same as those in Fig. 6. The truncation number of the spatial harmonic expansion is ( 2 N + 1 ) 2 = 529 for the case of h L = 0.01 and ( 2 N + 1 ) 2 = 361 for h L = 1.0 and h L = 5.0 .

Fig. 9
Fig. 9

Power reflection coefficient R 00 2 calculated for the normal incidence of the plane wave on a PEC-like magnetodielectric layer periodically perforated with C-shaped apertures. The electric field of the incident wave is parallel to the y axis. The width and length of C-shaped aperture are w L = 1 7 and s L = 18 7 , respectively. The material parameters of the PEC-like layer and the truncation number of the spatial harmonics are the same as those in Fig. 8.

Fig. 10
Fig. 10

Power reflection coefficient R 00 2 for the normal incidence of the plane wave with electric field parallel to the x axis. The structural parameters of the layer and the truncation number of the spatial harmonics are the same as those in Fig. 9.

Equations (69)

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

J i j e ( x 1 , y 1 , z ) = { ( ϵ i j 1 ) E i j ( x 1 , y 1 , z ) if a i 1 x 1 a i and b j 1 y 1 b j , 0 elsewhere on the unit cell } ,
J i j h ( x 1 , y 1 , z ) = { ( μ i j 1 ) H i j ( x 1 , y 1 , z ) if a i 1 x 1 a i and b j 1 y 1 b j 0 elsewhere on the unit cell . } .
E ( r ) = E 0 ( r ) + 1 4 π ( + k 2 ) V G ( r , r ) J e ( r ) d r + i k 4 π × V G ( r , r ) J h ( r ) d r ,
H ( r ) = H 0 ( r ) + 1 4 π ( + k 2 ) V G ( r , r ) J h ( r ) d r i k 4 π × V G ( r , r ) J e ( r ) d r ,
k 2 G + 2 G x 2 = 4 π δ ( r r ) 2 G y 2 2 G z 2 ,
k 2 G + 2 G y 2 = 4 π δ ( r r ) 2 G x 2 2 G z 2 ,
( ϵ i j E x , i j ( r ) ϵ i j E y , i j ( r ) E z , i j ( r ) ) = ( E 0 x ( r ) E 0 y ( r ) E 0 z ( r ) ) + i k 4 π V ( 0 G z G y G z 0 G x G y G x 0 ) ( J x h ( r ) J y h ( r ) J z h ( r ) ) d r + 1 4 π V ( 2 G y 2 2 G z 2 2 G x y 2 G x z 2 G y x 2 G x 2 2 G z 2 2 G y z 2 G z x 2 G z y k 2 G + 2 G z 2 ) ( J x e ( r ) J y e ( r ) J z e ( r ) ) d r ,
( μ i j H x , i j ( r ) μ i j H y , i j ( r ) H z ( r ) ) = ( H 0 x ( r ) H 0 y ( r ) H 0 z ( r ) ) i k 4 π V ( 0 G z G y G z 0 G x G y G x 0 ) ( J x e ( r ) J y e ( r ) J z e ( r ) ) d r + 1 4 π V ( 2 G y 2 2 G z 2 2 G x y 2 G x z 2 G y x 2 G x 2 2 G z 2 2 G y z 2 G z x 2 G z y k 2 G + 2 G z 2 ) ( J x h ( r ) J y h ( r ) J z h ( r ) ) d r .
J i j e ( h ) ( x 1 , y 1 , z ) = q = p = F i j , p q e ( h ) ( z ) exp [ i ( k x 1 + 2 π p L x 1 ) x 1 ] exp [ i ( k y 1 + 2 π q L y 1 ) y 1 ] ,
F i j , p q e ( h ) ( z ) = 1 L x 1 L y 1 b j 1 b j a i 1 a i exp [ i ( k x 1 + 2 π p L x 1 ) x 1 ] exp [ i ( k y 1 + 2 π q L y 1 ) y 1 ] J i j e ( h ) ( x 1 , y 1 , z ) d x 1 d y 1 ,
J e ( h ) ( x 1 , y 1 , z ) = j = 1 M y i = 1 M x J i j e ( h ) ( x 1 , y 1 , z ) = j = 1 M y i = 1 M x q = p = F i j , p q e ( h ) ( z ) exp [ i ( k x 1 + 2 π p L x 1 ) x 1 ] exp [ i ( k y 1 + 2 π q L y 1 ) y 1 ] ,
G ( r , r ) = i 2 π e i ς ( x x ) + i η ( y y ) + i z z k 2 η 2 ς 2 k 2 η 2 ς 2 d η d ς
x 1 = x y cot β , y 1 = y cosec β ,
V J e ( h ) ( r ) G ( r , r ) d r = 2 π i j = 1 M y i = 1 M x q = p = e i ξ p x e i γ p q y I i j , p q e ( h ) ( z ) ,
I i j , p q e ( h ) ( z ) = 1 κ p q 0 h F i j , p q e ( h ) ( z ) e i z z κ p q d z ,
ξ p = k x + 2 π p L x 1 , γ p q = k y + 2 π sin β ( q L y 1 p L x 1 cos β ) ,
κ p q = k 2 ξ p 2 γ p q 2 .
F i j , p q e ( h ) ( z ) = 1 2 i [ 2 z 2 I i j , p q e ( h ) ( z ) + κ p q 2 I i j , p q e ( h ) ( z ) ] .
( ϵ i j ( ϵ i j 1 ) ( 2 z 2 + κ p q 2 ) I x , i j , p q e ( z ) ϵ i j ( ϵ i j 1 ) ( 2 z 2 + κ p q 2 ) I y , i j , p q e ( z ) 1 ( ϵ i j 1 ) ( 2 z 2 + κ p q 2 ) I z , i j , p q e ( z ) ) = 2 i α i , p α j , q ( E 0 x e i k z z E 0 y e i k z z E 0 z e i k z z ) s = r = α i , p r α j , q s M r s e ( z ) ,
( μ i j ( μ i j 1 ) ( 2 z 2 + κ p q 2 ) I x , i j , p q h ( z ) μ i j ( μ i j 1 ) ( 2 z 2 + κ p q 2 ) I y , i j , p q h ( z ) 1 ( μ i j 1 ) ( 2 z 2 + κ p q 2 ) I z , i j , p q h ( z ) ) = 2 i α i , p α j , q ( H 0 x e i k z z H 0 y e i k z z H 0 z e i k z z ) s = r = α i , p r α j , q s M r s h ( z ) ,
M r s e ( h ) ( z ) = ( γ r s 2 2 z 2 ξ r γ r s i ξ r z γ r s ξ r ξ r 2 2 z 2 i γ r s z i ξ r z i γ r s z k 2 + 2 z 2 ) ( I x , r s e ( h ) ( z ) I y , r s e ( h ) ( z ) I z , r s e ( h ) ( z ) ) ± i k ( 0 z i γ r s z 0 i ξ r i γ r s i ξ r 0 ) ( I x , r s h ( e ) ( z ) I y , r s h ( e ) ( z ) I z , r s h ( e ) ( z ) ) ,
I ν , p q e ( h ) ( z ) = j = 1 M y i = 1 M x I ν , i j , p q e ( h ) ( ν = x , y , z ) ,
α i , p r = 1 L x 1 a i 1 a i exp [ i 2 π ( p r ) L x 1 x 1 ] d x 1 , α j , q s = 1 L y 1 b j 1 b j exp [ i 2 π ( q s ) L y 1 y 1 ] d y 1 .
F ̃ ν , i j e ( h ) ( z ) = [ F ν , i j N , N e ( h ) F ν , i j , N , N e ( h ) F ν , i j , 00 e ( h ) F ν , i j , N , N e ( h ) F ν , i j , N , N e ( h ) ] T ,
I ̃ ν , i j e ( h ) ( z ) = [ I ν , i j N , N e ( h ) I ν , i j , N , N e ( h ) I ν , i j , 00 e ( h ) I ν , i j , N , N e ( h ) I ν , i j , N , N e ( h ) ] T ,
F ̃ x ( y ) , i j e ( z ) = ϵ k l ( ϵ i j 1 ) ϵ i j ( ϵ k l 1 ) A i j A k l 1 F ̃ x ( y ) , k l e ( z )
F ̃ z , i j e ( z ) = ϵ i j 1 ϵ k l 1 A i j A k l 1 F ̃ z , k l e ( z ) ,
A i j = [ α i , p r α j , q s ] .
I x ( y ) , p q e = 2 i k 2 E 0 x ( 0 y ) δ p 0 δ q 0 e i k z z , I z , p q e = 0 ,
I x , p q h = I y , p q h = I z , p q h = 0 .
( V ̃ x , i j e ( h ) V ̃ y , i j e ( h ) V ̃ z , i j e ( h ) ) = ( Γ 2 χ 2 I Ξ Γ i χ Ξ Ξ Γ Ξ 2 χ 2 I i χ Γ i χ Ξ i χ Γ ( 1 + χ 2 ) I ) ( W ̃ x e ( h ) W ̃ y e ( h ) W ̃ z e ( h ) ) i k ( 0 χ I i Γ χ I 0 i Ξ i Γ i Ξ 0 ) ( W ̃ x h ( e ) W ̃ y h ( e ) W ̃ z h ( e ) ) ,
Ξ = diag ( ξ p δ p r δ q s ) , Γ = diag ( γ p q δ p r δ q s ) , K = diag ( κ p q δ p r δ q s ) ,
V ̃ x ( y ) , i j e = A i j 1 ( K 2 + χ 2 I ) I ̃ x ( y ) , i j e ϵ i j ϵ i j 1 ,
V ̃ z , i j e = A i j 1 ( K 2 + χ 2 I ) I ̃ z , i j e 1 ϵ i j 1 ,
V ̃ x ( y ) , i j h = A i j 1 ( K 2 + χ 2 I ) I ̃ x ( y ) , i j h μ i j μ i j 1 ,
V ̃ z , i j h = A i j 1 ( K 2 + χ 2 I ) I ̃ z , i j h 1 μ i j 1 ,
F ν , i j , q p e ( h ) ( z ) = 1 2 i ( χ 2 + κ p q 2 ) I ν , i j , q p e ( h ) ( z ) ( ν = x , y , z ) ,
F ν , p q e ( h ) ( z ) = j = 1 M y i = 1 M x F ν , i j , p q e ( h ) = 1 2 i ( χ 2 + κ p q 2 ) I ν , q p e ( h ) ( z ) ( ν = x , y , z ) .
I ̃ x ( y ) , i j e = ϵ i j 1 ϵ i j ( K 2 + χ 2 I ) 1 A i j V ̃ x ( y ) , k l e ,
I ̃ z , i j e = ( ϵ i j 1 ) ( K 2 + χ 2 I ) 1 A i j V ̃ z , k l e .
I ̃ x ( y ) , i j h = μ i j 1 μ i j ( K 2 + χ 2 I ) 1 A i j V ̃ x ( y ) , k l h ,
I ̃ z , i j h = ( μ i j 1 ) ( K 2 + χ 2 I ) 1 A i j V ̃ z , k l h .
( D x e Ξ Γ i χ Ξ 0 i χ I Γ Ξ Γ D y e i χ Γ i χ I 0 Ξ i χ Ξ i χ Γ D z e Γ Ξ 0 0 i χ I Γ D x h Ξ Γ i χ Ξ i χ I 0 Ξ Ξ Γ D y h i χ Γ Γ Ξ 0 i χ Ξ i χ Γ D z h ) ( W ̃ x e W ̃ y e W ̃ z e W ̃ x h W ̃ y h W ̃ z h ) = 0 ,
D x e ( h ) = B x e ( h ) ( K 2 + χ 2 I ) + ( Γ 2 χ 2 I ) ,
D y e ( h ) = B y e ( h ) ( K 2 + χ 2 I ) + ( Ξ 2 χ 2 I ) ,
D z e ( h ) = B z e ( h ) ( K 2 + χ 2 I ) + ( 1 + χ 2 ) I ,
B x e = B y e = ( j = 1 M y i = 1 M x ( ϵ i j 1 ) ϵ i j A i j ) 1 ,
B z e = ( j = 1 M y i = 1 M x ( ϵ i j 1 ) A i j ) 1 ,
B x h = B y h = ( j = 1 M y i = 1 M x ( μ i j 1 ) μ i j A i j ) 1 ,
B z h = ( j = 1 M y i = 1 M x ( μ i j 1 ) A i j ) 1 .
[ D ] [ W ̃ x e W ̃ y e ] = [ D 1 D 2 D 3 D 4 ] [ W ̃ x e W ̃ y e ] = χ 2 [ W ̃ x e W ̃ y e ] .
I x ( y ) , p q e ( z ) = m = 1 2 ( 2 N + 1 ) 2 ( c m + e χ m z + c m e χ m z ) W x ( y ) , p q , m e + 2 i k 2 E 0 x ( 0 y ) e i k z z ,
I z , p q e ( z ) = m = 1 2 ( 2 N + 1 ) 2 ( c m + e χ m z + c m e χ m z ) W z , p q , m e ,
I ν , p q h ( z ) = m = 1 2 ( 2 N + 1 ) 2 ( c m + e χ m z + c m e χ m z ) W ν , p q , m h ( ν = x , y , z ) ,
V ( 2 G y 2 2 G z 2 2 G x y 2 G x z 2 G y x 2 G x 2 2 G z 2 2 G y z ) ( J x e ( r ) J y e ( r ) J z e ( r ) ) d r + i k V ( 0 G z G y G z 0 G x ) ( J x h ( r ) J y h ( r ) J z h ( r ) ) d r = 0 .
m = 1 2 ( N + 1 ) 2 u x , p q , m + c m + + m = 1 2 ( N + 1 ) 2 u x , p q , m c m = 4 k z k 2 E 0 x δ q 0 δ p 0 ,
m = 1 2 ( N + 1 ) 2 u y , p q , m + c m + + m = 1 2 ( N + 1 ) 2 u y , p q , m c m = 4 k z k 2 E 0 y δ q 0 δ p 0 ,
m = 1 2 ( N + 1 ) 2 v x , p q , m + c m + e χ m h + m = 1 2 ( N + 1 ) 2 v x , p q , m c m e χ m h = 0 ,
m = 1 2 ( N + 1 ) 2 v y , p q , m + c m + e χ m h + m = 1 2 ( N + 1 ) 2 v y , p q , m c m e χ m h = 0 ,
u x , p q , m ± = ( ± χ m + i κ p q ) [ ( γ p q 2 + κ p q 2 ) W x , p q , m e ξ p γ p q W y , p q , m e ξ p κ p q W z , p q , m e + κ p q W y , p q , m h γ p q W z , p q , m h ] ,
u y , p q , m ± = ( ± χ m + i κ p q ) [ ξ p γ p q W x , p q , m e + ( ξ p 2 + κ p q 2 ) W y , p q , m e γ p q κ p q W z , p q , m e κ p q W x , p q , m h + ξ p W z , p q , m h ] ,
v x , p q , m ± = ( ± χ m i κ p q ) [ ( γ p q 2 + κ p q 2 ) W x , p q , m e ξ p γ p q W y , p q , m e + ξ p κ p q W z , p q , m e κ p q W y , p q , m h γ p q W z , p q , m h ] ,
v y , p q , m ± = ( ± χ m i κ p q ) [ ξ p γ p q W x , p q , m e + ( ξ p 2 + κ p q 2 ) W y , p q , m e + γ p q κ p q W z , p q , m e + κ p q W x , p q , m h + ξ p W z , p q , m h ] .
0 h F ν , p q e ( h ) ( z ) e i κ p q z z d z = e i κ p q z 2 i m = 1 2 ( 2 N + 1 ) 2 [ c m + ( e ( χ m + i κ p q ) h 1 ) ( χ m i κ p q ) c m ( e ( χ m i κ p q ) h 1 ) ( χ m + i κ p q ) ] W ν , p q , m e ( h )
0 h F ν , p q e ( z ) e i κ p q z z d z = e i κ p q z 2 i m = 1 2 ( 2 N + 1 ) 2 [ c m + ( e ( χ m i κ p q ) h 1 ) ( χ m + i κ p q ) c m ( e ( χ m + i κ p q ) h 1 ) ( χ m i κ p q ) ] W ν , p q , m e + i k 2 ( κ 00 + k z ) ( δ ν x E 0 x + δ ν y E 0 y ) δ p 0 δ q 0 e i κ 00 z [ e i h ( k z κ 00 ) 1 ] ,
0 h F ν , p q h ( z ) e i κ p q z z d z = e i κ p q z 2 i m = 1 2 ( 2 N + 1 ) 2 [ c m + ( e ( χ m i κ p q ) h 1 ) ( χ m + i κ p q ) c m ( e ( χ m + i κ p q ) h 1 ) ( χ m i κ p q ) ] W ν , p q , m h
E x r ( x , y , z ) = 1 4 p = N N q = N N e i ( ξ p x + γ p q y κ p q z ) κ p q m = 0 2 ( 2 N + 1 ) 2 [ c m + ( χ m i κ p q ) c m ( χ m + i κ p q ) ] [ ( 1 ξ p 2 ) W x , p q , m e ξ p γ p q W y , p q , m e + ξ p κ p q W z , p q , m e γ p q W z , p q , m h κ p q W y , p q , m h ] ,
E y r ( x , y , z ) = 1 4 p = N N q = N N e i ( ξ p x + γ p q y κ p q z ) κ p q m = 0 2 ( 2 N + 1 ) 2 [ c m + ( χ m i κ p q ) c m ( χ m + i κ p q ) ] [ ( 1 γ p q 2 ) W y , p q , m e ξ p γ p q W x , p q , m e + γ p q κ p q W z , p q , m e + ξ p W z , p q , m h + κ p q W x , p q , m h ] ,
E z r ( x , y , z ) = 1 4 p = N N q = N N e i ( ξ p x + γ p q y κ p q z ) κ p q m = 0 2 ( 2 N + 1 ) 2 [ c m + ( χ m i κ p q ) c m ( χ m + i κ p q ) ] [ ( 1 κ p q 2 ) W z , p q , m e + ξ p κ p q W x , p q , m e + γ p q κ p q W y , p q , m e ξ p W y , p q , m h + γ p q W x , p q , m h ] .

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