Abstract

A fast coupled-integral-equation (CIE) technique is developed to compute the plane-TE-wave scattering by a wide class of periodic 2D inhomogeneous structures with curvilinear boundaries, which includes finite-thickness relief and rod gratings made of homogeneous material as special cases. The CIEs in the spectral domain are derived from the standard volume electric field integral equation. The kernel of the CIEs is of Picard type and offers therefore the possibility of deriving recursions, which allow the computation of the convolution integrals occurring in the CIEs with linear amounts of arithmetic complexity and memory. To utilize this advantage, the CIEs are solved iteratively. We apply the biconjugate gradient stabilized method. To make the iterative solution process faster, an efficient preconditioning operator (PO) is proposed that is based on a formal analytical inversion of the CIEs. The application of the PO also takes only linear complexity and memory. Numerical studies are carried out to demonstrate the potential and flexibility of the CIE technique proposed. Though the best efficiency and accuracy are observed at either low permittivity contrast or high conductivity, the technique can be used in a wide range of variation of material parameters of the structures including when they contain components made of both dielectrics with high permittivity and typical metals.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  36. S. H. Wei, Y. Y. Lu, “Application of Bi-CGSTAB to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
    [CrossRef]
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    [CrossRef]

2004 (2)

T. Magath, “Rapid solution of Fredholm integral equations of the second kind with Picard-kernel,” IEICE Trans. Electron. E87-C, 1548–1549 (2004).

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

2003 (1)

S. Menon, Q. Su, R. Grobe, “Iterative approach to Maxwell equations for dielectric media of spatially varying refractive index,” Phys. Rev. E 67, 046619 (2003).
[CrossRef]

2002 (9)

D.-Y. Jeong, Y. H. Ye, Q. M. Jang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. 92, 4194–4200 (2002).
[CrossRef]

F. Bilotti, A. Toscano, L. Vegni, “Design of inhomogeneous slabs for filtering applications via closed form solutions of the reflection coefficient,” J. Electromagn. Waves Appl. 16, 1233–1254 (2002).
[CrossRef]

S. H. Wei, Y. Y. Lu, “Application of Bi-CGSTAB to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[CrossRef]

E. Moreno, D. Erni, C. Hafner, “Band structure computations of metallic photonic crystals with the multiple mutipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Z. Q. Zhang, Q. H. Liu, “A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,” IEEE Antennas Wireless Propag. Lett. 1, 102–105 (2002).
[CrossRef]

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

L. Shen, S. He, “Analysis for the convergence problem of the plane-wave expansion method for photonic crystals,” J. Opt. Soc. Am. A 19, 1021–1024 (2002).
[CrossRef]

T. Vallius, “Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization,” J. Opt. Soc. Am. A 19, 1555–1562 (2002).
[CrossRef]

K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A 19, 2245–2252 (2002).
[CrossRef]

2001 (2)

2000 (3)

1999 (1)

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

1998 (1)

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

1997 (1)

1996 (3)

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

J. P. Donnelly, S. D. Lau, “Generalized effective index series solution analysis of waveguide structures with positionally varying refractive index profiles,” IEEE J. Quantum Electron. 32, 1070–1079 (1996).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

1995 (4)

1994 (2)

J. Xia, A. K. Jordan, J. A. Kong, “Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model,” J. Opt. Soc. Am. A 11, 1081–1086 (1994).
[CrossRef]

T. J. Cui, C. H. Liang, “Nonlinear differential equation for the reflection coefficient of an inhomogeneous lossy medium and its inverse scattering solutions,” IEEE Trans. Antennas Propag. 42, 621–626 (1994).
[CrossRef]

1993 (2)

1992 (1)

H. Van der Vorst, “Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 13, 631–644 (1992).
[CrossRef]

1991 (1)

1990 (1)

E. N. Glytsis, T. K. Gaylord, “Three-dimentional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1415 (1990).
[CrossRef]

Akmansoy, E.

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

Ammouche, A.

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

Atkinson, K. E.

K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind (Cambridge U. Press, 1997).
[CrossRef]

Awada, K. A.

Bilotti, F.

F. Bilotti, A. Toscano, L. Vegni, “Design of inhomogeneous slabs for filtering applications via closed form solutions of the reflection coefficient,” J. Electromagn. Waves Appl. 16, 1233–1254 (2002).
[CrossRef]

Bonod, N.

Brillat, T.

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

Buschman, R. G.

H. M. Srivastava, R. G. Buschman, Theory and Applications for Convolution Integral Equations (Kluwer Academic, 1992).
[CrossRef]

Chan, C. T.

M. M. Sigalas, C. T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

Chernov, B.

Cui, T. J.

T. J. Cui, C. H. Liang, “Nonlinear differential equation for the reflection coefficient of an inhomogeneous lossy medium and its inverse scattering solutions,” IEEE Trans. Antennas Propag. 42, 621–626 (1994).
[CrossRef]

de Lustrac, A.

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

Donnelly, J. P.

J. P. Donnelly, S. D. Lau, “Generalized effective index series solution analysis of waveguide structures with positionally varying refractive index profiles,” IEEE J. Quantum Electron. 32, 1070–1079 (1996).
[CrossRef]

Erdogan, T.

Erni, D.

E. Moreno, D. Erni, C. Hafner, “Band structure computations of metallic photonic crystals with the multiple mutipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Gadot, F.

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, D. Q. Pommet, E. B. Grann, T. K. Gaylord, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Three-dimentional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1415 (1990).
[CrossRef]

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, “Three-dimentional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1394–1415 (1990).
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

Grann, E. B.

Grobe, R.

S. Menon, Q. Su, R. Grobe, “Iterative approach to Maxwell equations for dielectric media of spatially varying refractive index,” Phys. Rev. E 67, 046619 (2003).
[CrossRef]

Hafner, C.

E. Moreno, D. Erni, C. Hafner, “Band structure computations of metallic photonic crystals with the multiple mutipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

He, S.

Ho, K. M.

M. M. Sigalas, C. T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

Jang, Q. M.

D.-Y. Jeong, Y. H. Ye, Q. M. Jang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. 92, 4194–4200 (2002).
[CrossRef]

Jeong, D.-Y.

D.-Y. Jeong, Y. H. Ye, Q. M. Jang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. 92, 4194–4200 (2002).
[CrossRef]

Jiang, M.

Jordan, A. K.

Kong, J. A.

Lau, S. D.

J. P. Donnelly, S. D. Lau, “Generalized effective index series solution analysis of waveguide structures with positionally varying refractive index profiles,” IEEE J. Quantum Electron. 32, 1070–1079 (1996).
[CrossRef]

Li, L.

Liang, C. H.

T. J. Cui, C. H. Liang, “Nonlinear differential equation for the reflection coefficient of an inhomogeneous lossy medium and its inverse scattering solutions,” IEEE Trans. Antennas Propag. 42, 621–626 (1994).
[CrossRef]

Liu, Q. H.

Z. Q. Zhang, Q. H. Liu, “A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,” IEEE Antennas Wireless Propag. Lett. 1, 102–105 (2002).
[CrossRef]

Lourtioz, J.-M.

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

Lu, Y. Y.

S. H. Wei, Y. Y. Lu, “Application of Bi-CGSTAB to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[CrossRef]

Magath, T.

T. Magath, “Rapid solution of Fredholm integral equations of the second kind with Picard-kernel,” IEICE Trans. Electron. E87-C, 1548–1549 (2004).

Matsumoto, K.

Maystre, D.

D. Maystre, “Integral Methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, 1980), Chap. 3.
[CrossRef]

McPhedran, R. C.

N. A. Nicorovici, R. C. McPhedran, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135–1145 (1995).
[CrossRef]

Menon, S.

S. Menon, Q. Su, R. Grobe, “Iterative approach to Maxwell equations for dielectric media of spatially varying refractive index,” Phys. Rev. E 67, 046619 (2003).
[CrossRef]

Mittra, R.

A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, 1998).

Moharam, M. G.

Moreno, E.

E. Moreno, D. Erni, C. Hafner, “Band structure computations of metallic photonic crystals with the multiple mutipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Morris, G. M.

Nevière, M.

Nicorovici, N. A.

N. A. Nicorovici, R. C. McPhedran, “Photonic band gaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135–1145 (1995).
[CrossRef]

Norton, S. M.

Pai, D. M.

Peterson, A. F.

A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, 1998).

Petit, R.

Pommet, D. Q.

Popov, E.

Qiu, M.

M. Qiu, S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[CrossRef]

M. Qiu, S. He, “Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional bandgap,” J. Opt. Soc. Am. B 17, 1027–1030 (2000).
[CrossRef]

Ray, S. L.

A. F. Peterson, S. L. Ray, R. Mittra, Computational Methods for Electromagnetics (IEEE Press, 1998).

Rokushima, K.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
[CrossRef]

Shen, L.

Sigalas, M. M.

M. M. Sigalas, C. T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

Soukoulis, C. M.

M. M. Sigalas, C. T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

Srivastava, H. M.

H. M. Srivastava, R. G. Buschman, Theory and Applications for Convolution Integral Equations (Kluwer Academic, 1992).
[CrossRef]

Su, Q.

S. Menon, Q. Su, R. Grobe, “Iterative approach to Maxwell equations for dielectric media of spatially varying refractive index,” Phys. Rev. E 67, 046619 (2003).
[CrossRef]

Tamir, T.

M. Jiang, T. Tamir, S. Zhang, “Modal theory of diffraction by multilayered gratings containing dielectric and metallic components,” J. Opt. Soc. Am. A 18, 807–820 (2001).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

Toscano, A.

F. Bilotti, A. Toscano, L. Vegni, “Design of inhomogeneous slabs for filtering applications via closed form solutions of the reflection coefficient,” J. Electromagn. Waves Appl. 16, 1233–1254 (2002).
[CrossRef]

Vallius, T.

Van der Vorst, H.

H. Van der Vorst, “Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 13, 631–644 (1992).
[CrossRef]

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

Vegni, L.

F. Bilotti, A. Toscano, L. Vegni, “Design of inhomogeneous slabs for filtering applications via closed form solutions of the reflection coefficient,” J. Electromagn. Waves Appl. 16, 1233–1254 (2002).
[CrossRef]

Watanabe, K.

Wei, S. H.

S. H. Wei, Y. Y. Lu, “Application of Bi-CGSTAB to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[CrossRef]

Xia, J.

Ye, Y. H.

D.-Y. Jeong, Y. H. Ye, Q. M. Jang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. 92, 4194–4200 (2002).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Zhang, S.

M. Jiang, T. Tamir, S. Zhang, “Modal theory of diffraction by multilayered gratings containing dielectric and metallic components,” J. Opt. Soc. Am. A 18, 807–820 (2001).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

Zhang, Z. Q.

Z. Q. Zhang, Q. H. Liu, “A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,” IEEE Antennas Wireless Propag. Lett. 1, 102–105 (2002).
[CrossRef]

IEEE Antennas Wireless Propag. Lett. (1)

Z. Q. Zhang, Q. H. Liu, “A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,” IEEE Antennas Wireless Propag. Lett. 1, 102–105 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. P. Donnelly, S. D. Lau, “Generalized effective index series solution analysis of waveguide structures with positionally varying refractive index profiles,” IEEE J. Quantum Electron. 32, 1070–1079 (1996).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. H. Wei, Y. Y. Lu, “Application of Bi-CGSTAB to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T. J. Cui, C. H. Liang, “Nonlinear differential equation for the reflection coefficient of an inhomogeneous lossy medium and its inverse scattering solutions,” IEEE Trans. Antennas Propag. 42, 621–626 (1994).
[CrossRef]

IEICE Trans. Electron. (1)

T. Magath, “Rapid solution of Fredholm integral equations of the second kind with Picard-kernel,” IEICE Trans. Electron. E87-C, 1548–1549 (2004).

J. Appl. Phys. (3)

D.-Y. Jeong, Y. H. Ye, Q. M. Jang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. 92, 4194–4200 (2002).
[CrossRef]

M. Qiu, S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[CrossRef]

F. Gadot, A. de Lustrac, J.-M. Lourtioz, T. Brillat, A. Ammouche, E. Akmansoy, “High-transmission defect modes in two-dimensional metallic photonic crystals,” J. Appl. Phys. 85, 8499–8501 (1999).
[CrossRef]

J. Electromagn. Waves Appl. (1)

F. Bilotti, A. Toscano, L. Vegni, “Design of inhomogeneous slabs for filtering applications via closed form solutions of the reflection coefficient,” J. Electromagn. Waves Appl. 16, 1233–1254 (2002).
[CrossRef]

J. Lightwave Technol. (1)

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Figures (18)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Comparison of CIE with Riccati equation applied to calculation of reflection coefficient for 1D inhomogeneous slab. CIE: Q = 400 , solid curve; Q = 70 , dashed–dotted curve; Q = 20 , dashed curve; Riccati equation, dotted curve. The solid and dotted curves coincide.

Fig. 3
Fig. 3

Near-field pattern of a rectangular-bar dielectric grating calculated using CIE.

Fig. 4
Fig. 4

Relative difference between field values calculated using CIE and EMM in units of 20 log δ for the same parameters as in Fig. 3.

Fig. 5
Fig. 5

Near-field pattern of a finite-thickness sinusoidal grating made of PTFE with ϵ 2 r = 2.1 .

Fig. 6
Fig. 6

Near-field pattern of a finite-thickness sinusoidal grating made of GaAs with ϵ 2 r = 11.4 .

Fig. 7
Fig. 7

Near-field pattern of a finite-thickness deep sinusoidal grating made of GaAs with ϵ 2 r = 11.4 .

Fig. 8
Fig. 8

Near-field pattern of a finite-thickness deep sinusoidal grating made of Al with ϵ 2 r = ( 1.3 i 7.6 ) 2 .

Fig. 9
Fig. 9

Near-field pattern of a deep grating made of Al with ϵ 2 r = ( 1.3 i 7.6 ) 2 and with four superposed sinusoids as surface relief.

Fig. 10
Fig. 10

Effect of PO on the convergence of the iterative solution procedure. Geometrical, frequency, and material parameters are the same as in Figs. 7, 8. Solid and dotted curves correspond to GaAs, whereas dashed and dashed–dotted curves correspond to Al; p and np stand for the cases with and without PO applied, respectively.

Fig. 11
Fig. 11

Effect of preconditioning on the convergence of the iterative procedure for sinusoidal gratings made of (a) PTFE and (b) Al; solid, dashed, and dotted curves correspond to k h = 4 π , 8.0, and 3.8, respectively; p and np stand for the cases with and without PO applied, respectively.

Fig. 12
Fig. 12

Computational complexity: effect of (a) Q and (b) N on CPU time T needed to solve the CIE for deep sinusoidal gratings made of various materials.

Fig. 13
Fig. 13

Near-field pattern of periodic inhomogeneous slab with intermediate value of maximal contrast and curvilinear boundaries; d h = π and k h = 4 π .

Fig. 14
Fig. 14

Same as Fig. 13, but for k h = 3 π .

Fig. 15
Fig. 15

Near-field pattern of periodic inhomogeneous slab with large values of maximal contrast and curvilinear boundaries; k h = 4 π 2 and d h = 1 .

Fig. 16
Fig. 16

Same as Fig. 15, but for k h = 4 π and d h = π . The metallic insert is located at d 2 x d .

Fig. 17
Fig. 17

Transmission spectrum in the Γ X direction for a square lattice composed of eight layers of circular rods made of metal whose permittivity is given by Eq. (38). The filling factor f = π r 2 a 2 = 0.126 , φ i = 0 , γ ω p = 0.01 ; solid, dashed, dotted, and dashed–dotted curves correspond to ξ = 0.5 , 1, 2, and 6, respectively.

Fig. 18
Fig. 18

Demonstration of the effect of defects on the transmission spectrum in the Γ X direction for a dielectric square lattice composed of eight layers with filling factor f = π r 2 a 2 = 0 . 126 and φ i = 0 . Solid curve, ϵ r = 11.4 for all rods; dotted curve, ϵ r = 2.1 for the 3rd and 6th rods, 11.4 otherwise; dashed curve, ϵ r = 1 for the 2nd, 5th, and 7th rods, 11.4 otherwise. Rods are numbered from the side of the wave incidence.

Tables (7)

Tables Icon

Table 1 Comparison of DE Values Calculated Using Different Methods

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Table 2 Effect of k h , Q, and N on the DE Values for a Lossless Sinusoidal Grating a

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Table 3 Effect of k h , Q, and N on the DE Values for a Lossless Sinusoidal Grating with High Permittivity Contrast a

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Table 4 Effect of k h , Q, and N on the DE Values for a Metallic Sinusoidal Grating a

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Table 5 Effect of k h , Q, and N on the DE Values for Deep Sinusoidal Gratings Made of Various Materials a

Tables Icon

Table 6 Effect of Q and N on the DE Values for an Inhomogeneous Slab with Curvilinear Boundaries and an Intermediate Permittivity Contrast a

Tables Icon

Table 7 Effect of k h , Q, and N on the DE Values for Inhomogeneous Slabs with Curvilinear Boundaries and High Permittivity Contrast a

Equations (38)

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E i ( x , y ) = E 0 exp ( i α 0 x + i β 0 y ) ,
E i ( x , y ) = E ( x , y ) k 2 0 h G ( x x , y y ) χ ( x , y ) E ( x , y ) d x d y ,
χ ( x , y ) = m = χ m ( y ) exp ( i m K x ) ,
χ m ( y ) = 1 d 0 d χ ( x , y ) exp ( i m K x ) d x .
E ( x , y ) = n = E n ( y ) exp ( i α n x ) ,
E m i = E m ( y ) k 2 2 i β m n = 0 h exp ( i β m y y ) × χ m n ( y ) E n ( y ) d y , y [ 0 , h ] , m = 0 , ± 1 , ± 2 , ,
( G ̂ [ Φ ] ) ( y ) = 1 2 i β 0 h exp ( i β y y ) [ Φ ] ( y ) d y
[ E i ] ( y ) = [ E ] ( y ) k 2 ( G ̂ ( χ [ E ] ) ) ( y ) , y [ 0 , h ] .
B ( z ) = { 1 z , z < 1 0 , z 1 } .
χ ( x , y ) p = 0 P q = 0 Q χ ( x p , y q ) B ( x Δ x p ) B ( y Δ q ) .
χ n ( y ) 2 P 1 cos ( 2 π n P ) ( 2 π n P ) 2 p = 0 P 1 q = 0 Q χ ( x p , y q ) exp ( i 2 π n p P ) B ( y Δ q ) , n = 0 , ± 1 , , ± 2 N ,
χ ( y ) q = 0 Q χ q B ( y Δ q ) ,
[ E ] ( y ) q = 0 Q [ E ] q B ( y Δ q ) .
[ E i ] = [ E ] k 2 G X [ E ] ,
2 i β ( G ̂ ( χ [ E ] ) ) ( y ) = [ L ] ( y ) + [ R ] ( y ) ,
[ L ] ( y ) = 0 y exp ( i β ( y y ) ) χ ( y ) [ E ] ( y ) d y
[ R ] ( y ) = y h exp ( i β ( y y ) ) χ ( y ) [ E ] ( y ) d y .
[ L ] ( y + Δ ) = exp ( i β Δ ) [ L ] ( y ) + y y + Δ exp ( i β ( y + Δ y ) ) χ ( y ) [ E ] ( y ) d y ,
[ R ] ( y Δ ) = exp ( i β Δ ) [ R ] ( y ) + y Δ y exp ( i β ( y + Δ y ) ) χ ( y ) [ E ] ( y ) d y .
[ L ] q + 1 = exp ( δ ) [ L ] q + Δ { exp ( δ ) t 1 ( δ ) χ q [ E ] q + t 1 ( δ ) χ q + 1 [ E ] q + 1 + t 2 ( δ ) ( χ q [ E ] q + 1 + χ q + 1 [ E ] q ) } , q = 0 , , Q 1 ,
[ R ] q 1 = exp ( δ ) [ R ] q + Δ { exp ( δ ) t 1 ( δ ) χ q [ E ] q + t 1 ( δ ) χ q 1 [ E ] q 1 + t 2 ( δ ) ( χ q [ E ] q 1 + χ q 1 [ E ] q ) } , q = Q , , 1 ,
t 1 ( δ ) = 0 1 exp ( δ y ) B 2 ( y ) d y = 2 + δ ( δ 2 ) 2 exp ( δ ) δ 3 ,
t 2 ( δ ) = 0 1 exp ( δ y ) B ( y ) B ( y 1 ) d y = δ 2 + ( δ + 2 ) exp ( δ ) δ 3 .
[ E i ] = [ E ] k 2 2 i β ( [ L ] + [ R ] )
( G ̂ 1 [ Ψ ] ) ( y ) = d 2 [ Ψ ] d y 2 ( y ) β 2 [ Ψ ] ( y ) , y [ 0 , h ] ,
[ E ] ( y ) = ( P ̂ [ E i ] ) ( y ) , y [ 0 , h ] ,
P ̂ = [ G ̂ 1 k 2 χ ( y ) ] 1 G ̂ 1 .
( G ̂ 1 [ Ψ ] ) ( y n ) 1 Δ 2 ( [ Ψ ] n + 1 ( 2 + δ 2 ) [ Ψ ] n + [ Ψ ] n 1 ) .
r j [ E i ] [ E i ] j 2 [ E i ] 2 ,
[ E i ] j [ E ] j k 2 2 i β ( [ L ] j + [ R ] j ) .
ϵ 2 r ( y ) = ϵ 1 r + b exp [ ( y h 2 ) 2 η 2 ] .
h + ( x ) = h [ a 0 + a 1 sin ( K x ) ] ,
h ( x ) = 0 .
h + ( x ) = h m = 0 M a m sin ( m K x + ϕ m ) .
ϵ 2 r ( x , y ) = 2.84 0.74 sin ( K x ) sin ( 2 π y h ) ,
h ( x ) = h [ 0.24 0.16 sin ( K x ) ] ,
ϵ 2 r ( x , y ) = 9.2 5.2 sin ( K x ) sin ( 2 π y h ) .
ϵ r ( ω ) = 1 ω p 2 ω ( ω i γ ) ,

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