Abstract

Reflection, transmission, and absorption of electromagnetic waves by periodic arrays of conducting or dielectric rectangular cylinders are studied by a finite-difference time-domain technique. Truncated gratings made of lossless and lossy conducting and dielectric elements are considered. Results for surface current density, transmission, and reflection coefficients are calculated and compared with corresponding results in the literature, which are obtained by approximate or rigorous methods applicable only to idealized infinite models. An excellent agreement is observed in all cases, which demonstrates the accuracy and efficacy of our proposed analysis technique. Additionally, this numerical method easily analyzes practical gratings that contain a finite number of elements made of lossless, lossy, or even inhomogeneous materials. The results rapidly approach those for the idealized infinite arrays as the number of elements is increased. The method can also solve nested gratings, stacked gratings, and holographic gratings with little analytical or computational effort.

© 2006 Optical Society of America

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References

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  1. W. P. Pinello, R. Lee, and A. C. Cangellaris, "Finite element modeling of electromagnetic wave interactions with periodic dielectric structures," IEEE Trans. Microwave Theory Tech. 42, 2294-2301 (1994).
    [CrossRef]
  2. H. A. Kalhor, "Diffraction of electromagnetic waves by plane metallic gratings," J. Opt. Soc. Am. 68, 1202-1205 (1978).
    [CrossRef]
  3. M. Neviere, M. Cadilhac, and R. Petit, "Applications of conformal mappings to the diffraction of electromagnetic waves by a grating," IEEE Trans. Antennas Propag. AP-21, 37-46 (1973).
    [CrossRef]
  4. R. C. Hall and R. Mittra, "Scattering from a periodic array of resistive arrays," IEEE Trans. Antennas Propag. AP-33, 1009-1011 (1985).
    [CrossRef]
  5. J. A. Kong, "Second-order coupled-mode equations for spatially periodic media," J. Opt. Soc. Am. 67, 825-829 (1977).
    [CrossRef]
  6. D. E. Tremain and K. K. Mei, "Application of the unimoment method to scattering from periodic dielectric structures," J. Opt. Soc. Am. 68, 775-783 (1978).
    [CrossRef]
  7. 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]
  8. T. L. Zienko, A. I. Nosieh, and Y. Okuno, "Plane wave scattering and absorption by resistive strip and dielectric-strip periodic gratings," IEEE Trans. Antennas Propag. 46, 498-505 (1998).
  9. C. Zuffada, T. Cwik, and C. Ditchman, "Synthesis of novel all-dielectric grating filters using genetic algorithms," IEEE Trans. Antennas Propag. 46, 657-663 (1998).
    [CrossRef]
  10. S. Cui and D. Weile, "Analysis of electromagnetic scattering from periodic structures by FEM truncated by anisotropic PML boundary condition," Microwave Opt. Technol. Lett. 35, 106-110 (2002).
    [CrossRef]
  11. W. Ya, S. Dey, and R. Mittra, "Modeling of periodic structures using the finite-difference time-domain (FDTD)," in Proceedings of 1999 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1999), Vol. 1, pp. 594-597.
  12. H. A. Kalhor, "Plane metallic gratings of finite number of strip," IEEE Trans. Antennas Propag. 37, 406-407 (1989).
    [CrossRef]
  13. H. Liu and R. Paknys, "Scattering from periodic array of grounded parallel strips, TE incidence," IEEE Trans. Antennas Propag. 50, 798-806 (2002).
    [CrossRef]
  14. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
    [CrossRef]
  15. J. Fang and K. K. Mei, "A super-absorbing boundary algorithm for solving electromagnetic problems by time-domain finite-difference method," in Proceedings of 1988 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1988), pp. 472-475.
  16. A. Peterson, "Integral equation computer program for periodic planar conducting arrays," Electrical and Computer Engineering Department, Georgia Institute of Technology, Atlanta, Ga. (personal communication, 2004).
  17. H. L. Bertoni, Li-H. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
    [CrossRef]

2002 (2)

S. Cui and D. Weile, "Analysis of electromagnetic scattering from periodic structures by FEM truncated by anisotropic PML boundary condition," Microwave Opt. Technol. Lett. 35, 106-110 (2002).
[CrossRef]

H. Liu and R. Paknys, "Scattering from periodic array of grounded parallel strips, TE incidence," IEEE Trans. Antennas Propag. 50, 798-806 (2002).
[CrossRef]

1998 (2)

T. L. Zienko, A. I. Nosieh, and Y. Okuno, "Plane wave scattering and absorption by resistive strip and dielectric-strip periodic gratings," IEEE Trans. Antennas Propag. 46, 498-505 (1998).

C. Zuffada, T. Cwik, and C. Ditchman, "Synthesis of novel all-dielectric grating filters using genetic algorithms," IEEE Trans. Antennas Propag. 46, 657-663 (1998).
[CrossRef]

1996 (1)

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 (1)

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

1989 (2)

H. A. Kalhor, "Plane metallic gratings of finite number of strip," IEEE Trans. Antennas Propag. 37, 406-407 (1989).
[CrossRef]

H. L. Bertoni, Li-H. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

1985 (1)

R. C. Hall and R. Mittra, "Scattering from a periodic array of resistive arrays," IEEE Trans. Antennas Propag. AP-33, 1009-1011 (1985).
[CrossRef]

1978 (2)

1977 (1)

1973 (1)

M. Neviere, M. Cadilhac, and R. Petit, "Applications of conformal mappings to the diffraction of electromagnetic waves by a grating," IEEE Trans. Antennas Propag. AP-21, 37-46 (1973).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Bertoni, H. L.

H. L. Bertoni, Li-H. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

Cadilhac, M.

M. Neviere, M. Cadilhac, and R. Petit, "Applications of conformal mappings to the diffraction of electromagnetic waves by a grating," IEEE Trans. Antennas Propag. AP-21, 37-46 (1973).
[CrossRef]

Cangellaris, A. C.

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

Cheo, Li-H. S.

H. L. Bertoni, Li-H. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

Cui, S.

S. Cui and D. Weile, "Analysis of electromagnetic scattering from periodic structures by FEM truncated by anisotropic PML boundary condition," Microwave Opt. Technol. Lett. 35, 106-110 (2002).
[CrossRef]

Cwik, T.

C. Zuffada, T. Cwik, and C. Ditchman, "Synthesis of novel all-dielectric grating filters using genetic algorithms," IEEE Trans. Antennas Propag. 46, 657-663 (1998).
[CrossRef]

Dey, S.

W. Ya, S. Dey, and R. Mittra, "Modeling of periodic structures using the finite-difference time-domain (FDTD)," in Proceedings of 1999 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1999), Vol. 1, pp. 594-597.

Ditchman, C.

C. Zuffada, T. Cwik, and C. Ditchman, "Synthesis of novel all-dielectric grating filters using genetic algorithms," IEEE Trans. Antennas Propag. 46, 657-663 (1998).
[CrossRef]

Fang, J.

J. Fang and K. K. Mei, "A super-absorbing boundary algorithm for solving electromagnetic problems by time-domain finite-difference method," in Proceedings of 1988 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1988), pp. 472-475.

Hall, R. C.

R. C. Hall and R. Mittra, "Scattering from a periodic array of resistive arrays," IEEE Trans. Antennas Propag. AP-33, 1009-1011 (1985).
[CrossRef]

Kalhor, H. A.

H. A. Kalhor, "Plane metallic gratings of finite number of strip," IEEE Trans. Antennas Propag. 37, 406-407 (1989).
[CrossRef]

H. A. Kalhor, "Diffraction of electromagnetic waves by plane metallic gratings," J. Opt. Soc. Am. 68, 1202-1205 (1978).
[CrossRef]

Khizhnyak, N. A.

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]

Kong, J. A.

Lee, R.

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

Liu, H.

H. Liu and R. Paknys, "Scattering from periodic array of grounded parallel strips, TE incidence," IEEE Trans. Antennas Propag. 50, 798-806 (2002).
[CrossRef]

Mei, K. K.

D. E. Tremain and K. K. Mei, "Application of the unimoment method to scattering from periodic dielectric structures," J. Opt. Soc. Am. 68, 775-783 (1978).
[CrossRef]

J. Fang and K. K. Mei, "A super-absorbing boundary algorithm for solving electromagnetic problems by time-domain finite-difference method," in Proceedings of 1988 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1988), pp. 472-475.

Mittra, R.

R. C. Hall and R. Mittra, "Scattering from a periodic array of resistive arrays," IEEE Trans. Antennas Propag. AP-33, 1009-1011 (1985).
[CrossRef]

W. Ya, S. Dey, and R. Mittra, "Modeling of periodic structures using the finite-difference time-domain (FDTD)," in Proceedings of 1999 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1999), Vol. 1, pp. 594-597.

Neviere, M.

M. Neviere, M. Cadilhac, and R. Petit, "Applications of conformal mappings to the diffraction of electromagnetic waves by a grating," IEEE Trans. Antennas Propag. AP-21, 37-46 (1973).
[CrossRef]

Nosieh, A. I.

T. L. Zienko, A. I. Nosieh, and Y. Okuno, "Plane wave scattering and absorption by resistive strip and dielectric-strip periodic gratings," IEEE Trans. Antennas Propag. 46, 498-505 (1998).

Okuno, Y.

T. L. Zienko, A. I. Nosieh, and Y. Okuno, "Plane wave scattering and absorption by resistive strip and dielectric-strip periodic gratings," IEEE Trans. Antennas Propag. 46, 498-505 (1998).

Paknys, R.

H. Liu and R. Paknys, "Scattering from periodic array of grounded parallel strips, TE incidence," IEEE Trans. Antennas Propag. 50, 798-806 (2002).
[CrossRef]

Peterson, A.

A. Peterson, "Integral equation computer program for periodic planar conducting arrays," Electrical and Computer Engineering Department, Georgia Institute of Technology, Atlanta, Ga. (personal communication, 2004).

Petit, R.

M. Neviere, M. Cadilhac, and R. Petit, "Applications of conformal mappings to the diffraction of electromagnetic waves by a grating," IEEE Trans. Antennas Propag. AP-21, 37-46 (1973).
[CrossRef]

Pinello, W. P.

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

Ryazantseva, N. V.

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]

Tamir, T.

H. L. Bertoni, Li-H. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

Tremain, D. E.

Weile, D.

S. Cui and D. Weile, "Analysis of electromagnetic scattering from periodic structures by FEM truncated by anisotropic PML boundary condition," Microwave Opt. Technol. Lett. 35, 106-110 (2002).
[CrossRef]

Ya, W.

W. Ya, S. Dey, and R. Mittra, "Modeling of periodic structures using the finite-difference time-domain (FDTD)," in Proceedings of 1999 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1999), Vol. 1, pp. 594-597.

Yachin, V. V.

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]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Zienko, T. L.

T. L. Zienko, A. I. Nosieh, and Y. Okuno, "Plane wave scattering and absorption by resistive strip and dielectric-strip periodic gratings," IEEE Trans. Antennas Propag. 46, 498-505 (1998).

Zuffada, C.

C. Zuffada, T. Cwik, and C. Ditchman, "Synthesis of novel all-dielectric grating filters using genetic algorithms," IEEE Trans. Antennas Propag. 46, 657-663 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (8)

T. L. Zienko, A. I. Nosieh, and Y. Okuno, "Plane wave scattering and absorption by resistive strip and dielectric-strip periodic gratings," IEEE Trans. Antennas Propag. 46, 498-505 (1998).

C. Zuffada, T. Cwik, and C. Ditchman, "Synthesis of novel all-dielectric grating filters using genetic algorithms," IEEE Trans. Antennas Propag. 46, 657-663 (1998).
[CrossRef]

H. A. Kalhor, "Plane metallic gratings of finite number of strip," IEEE Trans. Antennas Propag. 37, 406-407 (1989).
[CrossRef]

H. Liu and R. Paknys, "Scattering from periodic array of grounded parallel strips, TE incidence," IEEE Trans. Antennas Propag. 50, 798-806 (2002).
[CrossRef]

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

H. L. Bertoni, Li-H. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

M. Neviere, M. Cadilhac, and R. Petit, "Applications of conformal mappings to the diffraction of electromagnetic waves by a grating," IEEE Trans. Antennas Propag. AP-21, 37-46 (1973).
[CrossRef]

R. C. Hall and R. Mittra, "Scattering from a periodic array of resistive arrays," IEEE Trans. Antennas Propag. AP-33, 1009-1011 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

J. Electromagn. Waves Appl. (1)

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]

J. Opt. Soc. Am. (3)

Microwave Opt. Technol. Lett. (1)

S. Cui and D. Weile, "Analysis of electromagnetic scattering from periodic structures by FEM truncated by anisotropic PML boundary condition," Microwave Opt. Technol. Lett. 35, 106-110 (2002).
[CrossRef]

Other (3)

W. Ya, S. Dey, and R. Mittra, "Modeling of periodic structures using the finite-difference time-domain (FDTD)," in Proceedings of 1999 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1999), Vol. 1, pp. 594-597.

J. Fang and K. K. Mei, "A super-absorbing boundary algorithm for solving electromagnetic problems by time-domain finite-difference method," in Proceedings of 1988 IEEE Antennas and Propagation Society International Symposium (IEEE Press, 1988), pp. 472-475.

A. Peterson, "Integral equation computer program for periodic planar conducting arrays," Electrical and Computer Engineering Department, Georgia Institute of Technology, Atlanta, Ga. (personal communication, 2004).

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Figures (10)

Fig. 1
Fig. 1

Configuration of the array of rectangular cylinders and the incident plane wave.

Fig. 2
Fig. 2

Convergence study of reflection coefficients for self-complementary conducting strip gratings.

Fig. 3
Fig. 3

Magnitude of surface current density on the center strip of a 31-strip perfectly conducting grating for θ i = 15 ° , 30°, and 45°. Asterisks are results obtained by the software supplied by Peterson.[16]

Fig. 4
Fig. 4

Magnitude of the reflection coefficient versus b λ for perfectly conducting strip gratings and normal incidence. Asterisks are results reported in Ref. [4].

Fig. 5
Fig. 5

Magnitude of the reflection coefficient versus b λ for R = 500 Ω square and normal incidence. Asterisks are results reported in Ref. [4].

Fig. 6
Fig. 6

Magnitude of the reflection coefficient of a 31-strip grating for θ i = 30 ° and R = 500 Ω square . Asterisks are results reported in Ref. [4].

Fig. 7
Fig. 7

Configuration of the nested dielectric grating.

Fig. 8
Fig. 8

Variation of the reflection coefficient with b λ for E-polarized incidence at θ i = 45 ° with μ 1 , , 4 = 1 , ϵ 1 = 2.56 , ϵ 2 = 2.56 , ϵ 3 = 1.44 , ϵ 4 = 1.44 , a 0 = 0.0 , a 1 = 0.5 , a 2 = 0.5 , a 3 = 1.0 , a 4 = 1.0 with h b = 1 . Asterisks are results reported in Ref. [17]. Note that k 0 = 2 π λ .

Fig. 9
Fig. 9

Variation of the transmission coefficient with b λ for E-polarized normal incidence with μ 1 , , 4 = 1 , ϵ 1 = 5 , ϵ 2 = 1 , ϵ 3 = 2.44 , ϵ 4 = 1 , a 0 = 0.0 , a 1 = 0.3 , a 2 = 0.3 , a 3 = 0.5 , a 4 = 1.0 with h b = 1 . Asterisks are results reported in Ref. [7].

Fig. 10
Fig. 10

(a) Variation of the transmission coefficient with b λ for E-polarized normal incidence with μ 1 , , 4 = 1 , ϵ 1 = 5 , ϵ 2 , , 4 = 1 , a 0 = 0.0 , a 1 = 0.3 , a 2 , , 4 = 1.0 with h b = 1 . (b) Variation of the transmission coefficient with b λ for E-polarized normal incidence with μ 1 , , 4 = 1 , ϵ 1 = 5 , ϵ 2 = 2.44 , ϵ 3 = 1.0 , ϵ 4 = 1 , a 0 = 0.0 , a 1 = 0.3 , a 2 = 0.3 , a 3 = 0.4 , a 4 = 1.0 with h b = 1 . (c) Variation of the transmission coefficient with b λ for E-polarized normal incidence with μ 1 , , 4 = 1 , ϵ 1 = 5 , ϵ 2 = 1.0 , ϵ 3 = 2.44 , ϵ 4 = 1 , a 0 = 0.0 , a 1 = 0.3 , a 2 = 0.4 , a 3 = 0.5 , a 4 = 1.0 with h b = 1 . (d). Variation of the transmission coefficient with b λ for E-polarized normal incidence with μ 1 , , 4 = 1 , ϵ 1 = 5 , ϵ 2 = 1.0 , ϵ 3 = 2.44 , ϵ 4 = 1 , a 0 = 0.0 , a 1 = 0.3 , a 2 = 0.5 , a 3 = 0.6 , a 4 = 1.0 with h b = 1 . (e) Variation of the transmission coefficient with b λ for E-polarized normal incidence with μ 1 , , 4 = 1 , ϵ 1 = 5 , ϵ 2 = 1.0 , ϵ 3 = 2.44 , ϵ 4 = 1 , a 0 = 0.0 , a 1 = 0.3 , a 2 = 0.6 , a 3 = 0.7 , a 4 = 1.0 with h b = 1 .

Equations (6)

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H x t = 1 μ ( E y z ) ,
H z t = 1 μ ( E y x ) , E y t = 1 ϵ ( H x z H z x ) .
F n ( i , j ) = F ( i Δ x , j Δ z , n Δ t ) ,
E i = y ̂ E 0 f ( t ) , H i = E 0 η 0 ( x ̂ cos θ i + z ̂ sin θ i ) f ( t ) ,
T ( f ) = i = 1 N E total ( i , j ) E inc ( i , j ) ,
R ( f ) = E ave scat ( f ) E inc ( f ) .

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