Abstract

An effective description is developed for a metalodielectric photonic bandgap (PBG) material far beyond the quasi-static limit of traditional effective-medium theories. An analytic approach, recently presented by the authors, is further advanced to provide the complete effective permittivity and permeability functions. Reflection and transmission coefficients are presented for both TM and TE oblique plane-wave incidence, based on the determination of the equivalent impedance for each lattice plane in the crystal and the transfer-matrix method for reconstructing the effect of successive lattice planes. An analysis of the semi-infinite and slab observables yields the anisotropic effective refractive index, effective permittivity, and effective permeability, thus completing the macroscopic description of the interaction of electromagnetic waves with the medium. Among the novel aspects of the analysis is the equivalence of our PBG system with a physically dispersive system at ultraviolet frequencies and the derivation and explanation of the development of high dispersive magnetization (permeability) for these media, independently of the microscopic magnetic properties of the metallic implants.

© 1999 Optical Society of America

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References

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  1. N. G. Alexopoulos, D. R. Jackson, “Fundamental superstrate effects on printed circuit antennas,” IEEE Trans. Antennas Propag. AP-32, 807–816 (1984).
    [CrossRef]
  2. N. G. Alexopoulos, D. R. Jackson, P. B. Katehi, “Criteria for nearly omnidirectional radiation patterns for printed antennas,” IEEE Trans. Antennas Propag. AP-33, 195–205 (1985).
    [CrossRef]
  3. H. Y. Yang, N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas through multiple superstrates,” IEEE Trans. Antennas Propag. AP-35, 860–863 (1987).
    [CrossRef]
  4. H. Y. Yang, N. G. Alexopoulos, E. Yablonovitch, “Photonic band-gap materials for high-gain printed circuit antennas,” IEEE Trans. Antennas Propag. 45, 185–187 (1997).
    [CrossRef]
  5. H. Y. Yang, R. E. Diaz, N. G. Alexopoulos, “Reflection and transmission of waves from multilayer structures with planar-implanted periodic material blocks,” J. Opt. Soc. Am. B 14, 2513–2519 (1997).
    [CrossRef]
  6. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metalodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
    [CrossRef]
  7. R. Coccioli, T. Itoh, G. Pelosi, “A finite element-generalized network analysis of finite thickness photonic crystals,” in 1997 IEEE MTT-S Digest, pp. 195–198.
  8. E. W. Lucas, T. P. Fontana, “A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays,” IEEE Trans. Antennas Propag. 43, 145–153 (1995).
    [CrossRef]
  9. S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Antennas Propag. 40, 363–370 (1992).
  10. H. Contopanagos, L. Zhang, N. G. Alexopoulos, “Thin frequency selective lattices integrated in novel compact MIC, MMIC and PCA architectures,” IEEE Trans. Microwave Theory Tech. 46, 1936–1948 (1998).
    [CrossRef]
  11. C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York1983), pp. 227–251 and references therein.
  12. C. A. Kyriazidou, H. Contopanagos, W. M. Merrill, N. G. Alexopoulos, “Artificial versus natural crystals: effective wave impedance for printed photonic band gap materials,” IEEE Trans. Antennas Propag. (to be published).
  13. R. E. Collin, Field Theory of Guided Waves, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 764–772 and references therein.
  14. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), pp. 66–70 and references therein.
  15. W. M. Merrill, C. A. Kyriazidou, H. Contopanagos, N. G. Alexopoulos, “Electromagnetic scattering from a PBG material excited by an electric line source,” IEEE Trans. Microwave Theory Tech. (to be published).
  16. H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
    [CrossRef]
  17. H. Contopanagos, E. Yablonovitch, N. G. Alexopoulos, “Electromagnetic properties of periodic multilayers of ultra-thin metallic films from DC to ultraviolet frequencies,” J. Opt. Soc. Am. A (to be published).

1998 (2)

H. Contopanagos, L. Zhang, N. G. Alexopoulos, “Thin frequency selective lattices integrated in novel compact MIC, MMIC and PCA architectures,” IEEE Trans. Microwave Theory Tech. 46, 1936–1948 (1998).
[CrossRef]

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

1997 (2)

H. Y. Yang, N. G. Alexopoulos, E. Yablonovitch, “Photonic band-gap materials for high-gain printed circuit antennas,” IEEE Trans. Antennas Propag. 45, 185–187 (1997).
[CrossRef]

H. Y. Yang, R. E. Diaz, N. G. Alexopoulos, “Reflection and transmission of waves from multilayer structures with planar-implanted periodic material blocks,” J. Opt. Soc. Am. B 14, 2513–2519 (1997).
[CrossRef]

1996 (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metalodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

1995 (1)

E. W. Lucas, T. P. Fontana, “A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays,” IEEE Trans. Antennas Propag. 43, 145–153 (1995).
[CrossRef]

1992 (1)

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Antennas Propag. 40, 363–370 (1992).

1987 (1)

H. Y. Yang, N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas through multiple superstrates,” IEEE Trans. Antennas Propag. AP-35, 860–863 (1987).
[CrossRef]

1985 (1)

N. G. Alexopoulos, D. R. Jackson, P. B. Katehi, “Criteria for nearly omnidirectional radiation patterns for printed antennas,” IEEE Trans. Antennas Propag. AP-33, 195–205 (1985).
[CrossRef]

1984 (1)

N. G. Alexopoulos, D. R. Jackson, “Fundamental superstrate effects on printed circuit antennas,” IEEE Trans. Antennas Propag. AP-32, 807–816 (1984).
[CrossRef]

Alexopoulos, N. G.

H. Contopanagos, L. Zhang, N. G. Alexopoulos, “Thin frequency selective lattices integrated in novel compact MIC, MMIC and PCA architectures,” IEEE Trans. Microwave Theory Tech. 46, 1936–1948 (1998).
[CrossRef]

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

H. Y. Yang, N. G. Alexopoulos, E. Yablonovitch, “Photonic band-gap materials for high-gain printed circuit antennas,” IEEE Trans. Antennas Propag. 45, 185–187 (1997).
[CrossRef]

H. Y. Yang, R. E. Diaz, N. G. Alexopoulos, “Reflection and transmission of waves from multilayer structures with planar-implanted periodic material blocks,” J. Opt. Soc. Am. B 14, 2513–2519 (1997).
[CrossRef]

H. Y. Yang, N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas through multiple superstrates,” IEEE Trans. Antennas Propag. AP-35, 860–863 (1987).
[CrossRef]

N. G. Alexopoulos, D. R. Jackson, P. B. Katehi, “Criteria for nearly omnidirectional radiation patterns for printed antennas,” IEEE Trans. Antennas Propag. AP-33, 195–205 (1985).
[CrossRef]

N. G. Alexopoulos, D. R. Jackson, “Fundamental superstrate effects on printed circuit antennas,” IEEE Trans. Antennas Propag. AP-32, 807–816 (1984).
[CrossRef]

C. A. Kyriazidou, H. Contopanagos, W. M. Merrill, N. G. Alexopoulos, “Artificial versus natural crystals: effective wave impedance for printed photonic band gap materials,” IEEE Trans. Antennas Propag. (to be published).

W. M. Merrill, C. A. Kyriazidou, H. Contopanagos, N. G. Alexopoulos, “Electromagnetic scattering from a PBG material excited by an electric line source,” IEEE Trans. Microwave Theory Tech. (to be published).

H. Contopanagos, E. Yablonovitch, N. G. Alexopoulos, “Electromagnetic properties of periodic multilayers of ultra-thin metallic films from DC to ultraviolet frequencies,” J. Opt. Soc. Am. A (to be published).

Bohren, C.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York1983), pp. 227–251 and references therein.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), pp. 66–70 and references therein.

Coccioli, R.

R. Coccioli, T. Itoh, G. Pelosi, “A finite element-generalized network analysis of finite thickness photonic crystals,” in 1997 IEEE MTT-S Digest, pp. 195–198.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 764–772 and references therein.

Contopanagos, H.

H. Contopanagos, L. Zhang, N. G. Alexopoulos, “Thin frequency selective lattices integrated in novel compact MIC, MMIC and PCA architectures,” IEEE Trans. Microwave Theory Tech. 46, 1936–1948 (1998).
[CrossRef]

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

W. M. Merrill, C. A. Kyriazidou, H. Contopanagos, N. G. Alexopoulos, “Electromagnetic scattering from a PBG material excited by an electric line source,” IEEE Trans. Microwave Theory Tech. (to be published).

H. Contopanagos, E. Yablonovitch, N. G. Alexopoulos, “Electromagnetic properties of periodic multilayers of ultra-thin metallic films from DC to ultraviolet frequencies,” J. Opt. Soc. Am. A (to be published).

C. A. Kyriazidou, H. Contopanagos, W. M. Merrill, N. G. Alexopoulos, “Artificial versus natural crystals: effective wave impedance for printed photonic band gap materials,” IEEE Trans. Antennas Propag. (to be published).

Diaz, R. E.

Fan, S.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metalodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Fontana, T. P.

E. W. Lucas, T. P. Fontana, “A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays,” IEEE Trans. Antennas Propag. 43, 145–153 (1995).
[CrossRef]

Gedney, S. D.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Antennas Propag. 40, 363–370 (1992).

Huffman, D.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York1983), pp. 227–251 and references therein.

Itoh, T.

R. Coccioli, T. Itoh, G. Pelosi, “A finite element-generalized network analysis of finite thickness photonic crystals,” in 1997 IEEE MTT-S Digest, pp. 195–198.

Jackson, D. R.

N. G. Alexopoulos, D. R. Jackson, P. B. Katehi, “Criteria for nearly omnidirectional radiation patterns for printed antennas,” IEEE Trans. Antennas Propag. AP-33, 195–205 (1985).
[CrossRef]

N. G. Alexopoulos, D. R. Jackson, “Fundamental superstrate effects on printed circuit antennas,” IEEE Trans. Antennas Propag. AP-32, 807–816 (1984).
[CrossRef]

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metalodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Katehi, P. B.

N. G. Alexopoulos, D. R. Jackson, P. B. Katehi, “Criteria for nearly omnidirectional radiation patterns for printed antennas,” IEEE Trans. Antennas Propag. AP-33, 195–205 (1985).
[CrossRef]

Kyriazidou, C. A.

C. A. Kyriazidou, H. Contopanagos, W. M. Merrill, N. G. Alexopoulos, “Artificial versus natural crystals: effective wave impedance for printed photonic band gap materials,” IEEE Trans. Antennas Propag. (to be published).

W. M. Merrill, C. A. Kyriazidou, H. Contopanagos, N. G. Alexopoulos, “Electromagnetic scattering from a PBG material excited by an electric line source,” IEEE Trans. Microwave Theory Tech. (to be published).

Lee, J. F.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Antennas Propag. 40, 363–370 (1992).

Lucas, E. W.

E. W. Lucas, T. P. Fontana, “A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays,” IEEE Trans. Antennas Propag. 43, 145–153 (1995).
[CrossRef]

Merrill, W. M.

W. M. Merrill, C. A. Kyriazidou, H. Contopanagos, N. G. Alexopoulos, “Electromagnetic scattering from a PBG material excited by an electric line source,” IEEE Trans. Microwave Theory Tech. (to be published).

C. A. Kyriazidou, H. Contopanagos, W. M. Merrill, N. G. Alexopoulos, “Artificial versus natural crystals: effective wave impedance for printed photonic band gap materials,” IEEE Trans. Antennas Propag. (to be published).

Mittra, R.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Antennas Propag. 40, 363–370 (1992).

Pelosi, G.

R. Coccioli, T. Itoh, G. Pelosi, “A finite element-generalized network analysis of finite thickness photonic crystals,” in 1997 IEEE MTT-S Digest, pp. 195–198.

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metalodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), pp. 66–70 and references therein.

Yablonovitch, E.

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

H. Y. Yang, N. G. Alexopoulos, E. Yablonovitch, “Photonic band-gap materials for high-gain printed circuit antennas,” IEEE Trans. Antennas Propag. 45, 185–187 (1997).
[CrossRef]

H. Contopanagos, E. Yablonovitch, N. G. Alexopoulos, “Electromagnetic properties of periodic multilayers of ultra-thin metallic films from DC to ultraviolet frequencies,” J. Opt. Soc. Am. A (to be published).

Yang, H. Y.

H. Y. Yang, R. E. Diaz, N. G. Alexopoulos, “Reflection and transmission of waves from multilayer structures with planar-implanted periodic material blocks,” J. Opt. Soc. Am. B 14, 2513–2519 (1997).
[CrossRef]

H. Y. Yang, N. G. Alexopoulos, E. Yablonovitch, “Photonic band-gap materials for high-gain printed circuit antennas,” IEEE Trans. Antennas Propag. 45, 185–187 (1997).
[CrossRef]

H. Y. Yang, N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas through multiple superstrates,” IEEE Trans. Antennas Propag. AP-35, 860–863 (1987).
[CrossRef]

Zhang, L.

H. Contopanagos, L. Zhang, N. G. Alexopoulos, “Thin frequency selective lattices integrated in novel compact MIC, MMIC and PCA architectures,” IEEE Trans. Microwave Theory Tech. 46, 1936–1948 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (6)

E. W. Lucas, T. P. Fontana, “A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays,” IEEE Trans. Antennas Propag. 43, 145–153 (1995).
[CrossRef]

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Antennas Propag. 40, 363–370 (1992).

N. G. Alexopoulos, D. R. Jackson, “Fundamental superstrate effects on printed circuit antennas,” IEEE Trans. Antennas Propag. AP-32, 807–816 (1984).
[CrossRef]

N. G. Alexopoulos, D. R. Jackson, P. B. Katehi, “Criteria for nearly omnidirectional radiation patterns for printed antennas,” IEEE Trans. Antennas Propag. AP-33, 195–205 (1985).
[CrossRef]

H. Y. Yang, N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas through multiple superstrates,” IEEE Trans. Antennas Propag. AP-35, 860–863 (1987).
[CrossRef]

H. Y. Yang, N. G. Alexopoulos, E. Yablonovitch, “Photonic band-gap materials for high-gain printed circuit antennas,” IEEE Trans. Antennas Propag. 45, 185–187 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

H. Contopanagos, N. G. Alexopoulos, E. Yablonovitch, “High-Q radio frequency structures using one-dimensionally periodic metallic films,” IEEE Trans. Microwave Theory Tech. 46, 1310–1312 (1998).
[CrossRef]

H. Contopanagos, L. Zhang, N. G. Alexopoulos, “Thin frequency selective lattices integrated in novel compact MIC, MMIC and PCA architectures,” IEEE Trans. Microwave Theory Tech. 46, 1936–1948 (1998).
[CrossRef]

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

Phys. Rev. B (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “Large omnidirectional band gaps in metalodielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[CrossRef]

Other (7)

R. Coccioli, T. Itoh, G. Pelosi, “A finite element-generalized network analysis of finite thickness photonic crystals,” in 1997 IEEE MTT-S Digest, pp. 195–198.

H. Contopanagos, E. Yablonovitch, N. G. Alexopoulos, “Electromagnetic properties of periodic multilayers of ultra-thin metallic films from DC to ultraviolet frequencies,” J. Opt. Soc. Am. A (to be published).

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York1983), pp. 227–251 and references therein.

C. A. Kyriazidou, H. Contopanagos, W. M. Merrill, N. G. Alexopoulos, “Artificial versus natural crystals: effective wave impedance for printed photonic band gap materials,” IEEE Trans. Antennas Propag. (to be published).

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 764–772 and references therein.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, UK, 1970), pp. 66–70 and references therein.

W. M. Merrill, C. A. Kyriazidou, H. Contopanagos, N. G. Alexopoulos, “Electromagnetic scattering from a PBG material excited by an electric line source,” IEEE Trans. Microwave Theory Tech. (to be published).

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Figures (15)

Fig. 1
Fig. 1

Metalodielectric layer and corresponding artificial crystal under TE and TM plane-wave incidence.

Fig. 2
Fig. 2

Reflectivity of semi-infinite PBG medium at oblique incidence for TE (solid curves) and TM (dotted curves) polarizations. The bandgaps for TE/TM polarizations are the regions marked with the lower/upper thick solid bars.

Fig. 3
Fig. 3

Lorentzian generator p at normal incidence (dashed curves) and at θ=60° incidence for TE (solid curves) and TM (dotted curves) polarizations: (a) real part, (b) imaginary part.

Fig. 4
Fig. 4

Microwave–UV connection: (a) 3-D metalodielectric PBG with dispersionless PEC implants within a dispersionless dielectric host (or air), (b) 1-D periodic structure of thin dispersive metal films at plasma frequency spaced by a dispersionless dielectric (or air).

Fig. 5
Fig. 5

Dispersion diagrams for the PBG system (solid curves) and the UV-equivalent system (dashed curve) in the first Brillouin zone. The design for the PBG is =1, r/a=0.4, and a/c=0.35, while for the UV-equivalent system ω=ωp and dm=0.3δ0.

Fig. 6
Fig. 6

Reflection coefficient for semi-infinite medium at normal incidence: power (dashed curves), real part (solid curves), and imaginary part (dotted curves). (a) PBG medium, (b) UV-equivalent system (ω=ωp, c varies).

Fig. 7
Fig. 7

Same as Fig. 6, but for the ten-layer medium.

Fig. 8
Fig. 8

Normalized electric field distribution, inside the semi-infinite UV-equivalent structure (z>0), for normal incidence and various air-gap thicknesses: (a) inside the bandgap, the system being a maximum magnetic wall (solid curves) or a maximum electric wall (dotted curve), (b) outside the bandgap, where k0da=π/3 (solid curve), π/2 (dashed curve), and 2π/3 (dotted curve).

Fig. 9
Fig. 9

Normalized electric field distribution, inside the semi-infinite PBG structure (z>0), for normal incidence and various k0c: (a) inside the bandgap, the system being a maximum magnetic wall (solid curve) or a maximum electric wall (dotted curve); (b) outside the bandgap, where k0c=π/3 (solid curve), π/2 (dashed curve), and 2π/3 (dotted curve).

Fig. 10
Fig. 10

Refractive index at normal incidence (dashed curves) and at θ=30° for TE (solid curves) and TM (dotted curves) polarizations: (a) real part, (b) imaginary part.

Fig. 11
Fig. 11

Same as Fig. 10, but for effective wave impedance.

Fig. 12
Fig. 12

Same as Fig. 10, but for effective permittivity function.

Fig. 13
Fig. 13

Same as Fig. 10, but for effective permeability function.

Fig. 14
Fig. 14

Electric–magnetic wall duality in the first bandgap for normal incidence: (a) electric conductor behavior, (b) magnetic conductor behavior.

Fig. 15
Fig. 15

Evolution of the PBG effective permittivity as a function of inclusion size: (a) real part, (b) imaginary part.

Equations (118)

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

E{0,r}={R0, L0}yˆ exp(-γ0xxγ0zz)(TE){R0, L0}(xˆ cos θzˆ sin θ)×exp(-γ0xxγ0zz)(TM),
R0L0Ta,dR1L1.
R0L0SRoutLout,
S=Ta,dUNTd,a
U=Pc/2TYPc/2,
n×E=continuous,
n×[H(z0-)-H(z0+)]=YEtan(z0),
Ta,d=12 1+ηaηd1-ηaηd1-ηaηd1+ηaηd,
Ta,d=12 cos θdcos θ 1+ηaηd1-ηaηd1-ηaηd1+ηaηd,
TY(i)=12 2+Y(i)Y(i)-Y(i)2-Y(i),
Pc/2(i)=exp(γdc/2)00exp(-γdc/2).
γα=jk0nα cos θα,
cos θα=1-1nα2 sin2 θ1/2,
nα=α.
ηα(i)ηαcosi θα,i{1, -1},
ηαμαα=1nα.
U(i)=12 exp(γdc)[2+Y(i)]Y(i)-Y(i)exp(-γdc)[2-Y(i)].
ΓN(i)L0R0=S21S11=1-[ξ(i)]N21-Y(i)2Ψ(i)21/2 ηd(i)ηa(i)-ηa(i)ηd(i)-ηa(i)ηd(i)+ηd(i)ηa(i) Y(i)2Ψ(i)1+[ξ(i)]N+1-[ξ(i)]N21-Y(i)2Ψ(i)21/2 ηd(i)ηa(i)+ηa(i)ηd(i)+ηa(i)ηd(i)-ηd(i)ηa(i) Y(i)2Ψ(i),
TN(i)RoutR0=1S11=2{[1+ζ(i)]τ(i)}N 1+[ξ(i)]N+1-[ξ(i)]N21-Y(i)2Ψ(i)21/2 ηd(i)ηa(i)+ηa(i)ηd(i)+ηa(i)ηd(i)-ηd(i)ηa(i) Y(i)2Ψ(i)-1,
Ψ(i)j sin(k0cn cos θd)+cos(k0cn cos θd)[Y(i)/2],
τ(i)cos(k0cn cos θd)+sin(k0cn cos θd)[jY(i)/2],
ζ(i)Ψ(i)τ(i) 1-Y(i)2Ψ(i)21/2,ξ(i)=1-ζ(i)1+ζ(i).
cos[kF(i)c]=Tr[U(i)]2=cos(k0cn cos θd)+sin(k0cn cos θd)[jY(i)/2]τ(i).
Γlm=2πla+k0n sin θd2+mπb2-(k0n)21/2,
l=0,±1,,m=0,1,,
k0csin2 θ+[(nr)2-sin2 θ]mπca2-1+2lπca2(nr)21/2+2lπca sin θ(nr)2-sin2 θ.
k0c1nr πca2-11/2,
Y(i)jB(i)=j[BC(i)-BL(i)],
BC=163 ra3 ab ac nk0ccos θd 11-αeCe,
BL=83 ra3 ab ac nk0c1cos θd-cos θd 11-αmCm,
B=163 ra3 ab ac k0cn(cos θd) 11-αeCe,
BL=0.
αeCe=163 ra31.2π ab3-8πab3K02πab,
αmCm=163 ra31.22π ab3+1.22π-4πab3×K02πab-4πK02πba,
Γ(i)limN ΓN(i)=121-[p(i)]2 ηd(i)ηa(i)-ηa(i)ηd(i)-ηa(i)ηd(i)+ηd(i)ηa(i)p(i)1+121-[p(i)]2 ηd(i)ηa(i)+ηa(i)ηd(i)+ηa(i)ηd(i)-ηd(i)ηa(i)p(i),
p(i)Y(i)2Ψ(i)=B(i)/2sin(k0cn cos θd)+cos(k0cn cos θd)[B(i)/2].
L=1+L(ω),
L(ω)=(ωp/ωn)2(ω0/ωn)2-(ω/ωn)2+j(ω/ωn)(γ/ωn),
m=1-ωp2ω(ω-jγ)1,
hωp2π=7eVωp=1016Hz,γ=10-2ωp.
m=mr-j σ0ω,mr=1-ωp2ω2+γ2,
σ=0γ ωp2ω2+γ2.
ΦF(k0c)kFc(k0c).
2ρ<zn<2ρ+1z
=ρ(dm+da)+(zn-2ρ)dm,
2ρ+1<zn<2(ρ+1)z
=ρ(dm+da)+dm+(zn-2ρ-1)da.
Jv(2k+ρ)=σE(2k+ρ)=0γ ωp2ωp2+γ2 E(2k+ρ)0γE(2k+ρ)yˆ(-1)ρ-1Jv(2k+ρ)yˆ,
Js(2k+ρ)=dm2(2k+ρ-1)2(2k+ρ-1)+1Jv(2k+ρ)(zn)dzn.
Jv(2k+ρ)(r)=(-1)ρ-1δ(r-c(2k+ρ)zˆ)Js(2k+ρ)yˆ
m(k+1)=12 ρ=12vr×Jv(2k+ρ) d3r=ρ=12[-xˆc(2k+ρ)+zˆx](-1)ρ-1×Js(2k+ρ) dxdy,
M(k+1)=dm(k+1)c dxdy=12 Js(2k+1)xˆ
Γeq;slab(i)=Γeq;(i){1-exp[-2jβeq(i)b]}1-[Γeq;(i)]2 exp[-2jβeq(i)b],
Teq;slab(i)=4[1+ηeq(i)/ηa(i)][1+ηa(i)/ηeq(i)]exp[jβeq(i)b]+[1-ηeq(i)/ηa(i)][1-ηa(i)/ηeq(i)]exp[-jβeq(i)b],
Γeq;(i)=ηeq(i)-ηa(i)ηeq(i)+ηa(i),ηeq(i)=ηeqcosi θeq,
ΓN(i)=F(i)1+G(i) {1-[ξ(i)]N}1-G(i)-1G(i)+1[ξ(i)]N,
F(i)=121-Y(i)2Ψ(i)21/2×ηd(i)ηa(i)-ηa(i)ηd(i)-ηa(i)ηd(i)+ηd(i)ηa(i) Y(i)2Ψ(i),
G(i)=121-Y(i)2Ψ(i)21/2×ηd(i)ηa(i)+ηa(i)ηd(i)+ηa(i)ηd(i)-ηd(i)ηa(i) Y(i)2Ψ(i).
[τ(i)]2+-jΨ(i)1-Y(i)2Ψ(i)21/22=1,
ξ(i)=τ(i)-Ψ(i)1-Y(i)2Ψ(i)21/2τ(i)+Ψ(i)1-Y(i)2Ψ(i)21/2=cos[kF(i)c]-j sin[kF(i)c]cos[kF(i)c]+j sin[kF(i)c]=exp[-j2kF(i)c].
Γeq;(i)F(i)1+G(i)=Γ(i),
[Γ(i)]2=G(i)-1G(i)+1,
βeq(i)kF(i).
neff,z(i)=kF(i)ck0c=arccos τ(i)k0c.
sin θ=neff,x(i),
neff(i)={sin2 θ+[neff,z(i)]2}1/2,cos θeff(i)=neff,z(i)neff(i).
d[arccos τ(i)]d(k0c)=n cos θd{1-[p(i)]2}1/2×1+p(i) sin(k0cn cos θd)k0cn cos θd
neff,z(i)=n cos θdk0c 0k0c 1{1-[p(i)(y)]2}1/2×1+p(i)(y) sin(yn cos θd)yn cos θddy.
[1+ζ(i)]τ(i)=τ(i)+Ψ(i)1-Y(i)2Ψ(i)21/2=exp[jkF(i)c],
TN
=2[1+G(i)]exp[jkF(i)Nc]+[1-G(i)]exp[-jkF(i)Nc],
Γ(i)=ηd(i) 1-p(i){1-[p(i)]2}1/2-ηa(i)ηd(i) 1-p(i){1-[p(i)]2}1/2+ηa(i).
Γeff;(i)=ηeff(i)cosi θeff(i)-ηa(i)ηeff(i)cosi θeff(i)+ηa(i),
ηeq(i)ηeff(i)cosi θeff(i)=ηd(i) 1-p(i){1-[p(i)]2}1/2,
ηa(i)ηeq(i)+ηeq(i)ηa(i)=2G(i)
eff(i)=neff(i)ηeff(i),μeff(i)=neff(i)ηeff(i),
eff(i)={sin2 θ+[neff,z(i)]2}(i+1)/2[neff,z(i)]iηd(i) 1+p(i)1-p(i)1/2,
μeff(i)={sin2 θ+[neff,z(i)]2}(-i+1)/2[neff,z(i)]iηd(i)1-p(i)1+p(i)1/2,
neff,z(i)=n cos θdk0c 0k0c 1{1-[p(i)(y)]2}1/2×1+p(i)(y) sin(yn cos θd)yn cos θddy.
Hinc=yˆa0 exp(-jβxx-jβzz),
Einc=a0ωd (xˆβz-zˆβx)exp(-jβxx-jβzz),
Hrefl=-yˆRa0 exp(-jβxx+jβzz),
Erefl=Ra0ωd (xˆβz+zˆβx)exp(-jβxx+jβzz),
R=Re+Ro2.
Hrefl=-yˆb0 exp(-jβxx+jβzz)-yˆa0 exp(-jβxx+jβzz),
Erefl=b0ωd (xˆβz+zˆβx)exp(-jβxx+jβzz)+a0ωd (xˆβz+zˆβx)exp(-jβxx+jβzz).
H1=-2ja0yˆ exp(-jβxx)sin(βzz),
E1=-2ja0jωd ×yˆ exp(-jβxx)sin(βzz).
H2=-yˆb0 exp(-jβxx+jβzz),
E2=b0ωd (xˆβz+zˆβx)exp(-jβxx+jβzz),
Hs=H2+lm00blmHlm-,
Es=E2+lm00blmElm-.
Elm=(elmezlm)exp(Γlmz),
Hlm=(hlm+hzlm)exp(±Γlmz)
Hlm=hlm[exp(-Γlmz)-exp(Γlmz)]+hzlm[exp(-Γlmz)+exp(Γlmz)],
Elm=elm[exp(-Γlmz)+exp(Γlmz)]+ezlm[exp(-Γlmz)-exp(Γlmz)].
Γlm2=βx+2lπa2+mπb2-β2,
S(Elm*×Hs-Es×Hlm*)nˆ dS=0.
blm-b/2b/2-a/2a/2elm*×hlmzˆ dxdy=-Seelm*J dxdy,
SeE*J dxdy=α=13SeJαEα(0)+α=13rβ Eαrβ (0)+dxdy.
E(0)SeJ dxdy+Se14 α,β(Jαrβ-Jβrα)×Eαrβ (0)-Eβrα (0)+12 α,β×(Jαrβ+Jβrα) Eαrβ (0)+
=jωpE(0)-jωμ0mH(0)+jω6 α,βQαβ Eαrβ (0)
-jω6 α,βQαβM Bαrβ (0)+ ,
p=1jω SeJ dxdy,
m=12 Ser×J dxdy,
Qαβ=(3rαrβ-r2δαβ)ρ(r)dxdy,
QαβM=(3rαrβ-r2δαβ)ρM(r)dxdy.
blm-b/2b/2-a/2a/2elm*×hlmzˆ dxdy=-12 jωpElm*(0)-12 jωμ0mHlm*(0).
αex=αey=83 r3,αez=0,
αmx=αmy=0,αmz=-43 r3.
p=αed|E1|x=z=0=83 r3 2|Γ0|a0ω xˆ.
E00=exp(-jβxx)jωd {xˆΓ0[exp(-Γ0z)+exp(Γ0z)]-zˆjβx[exp(-Γ0z)-exp(Γ0z)]},
H00=yˆ exp(-jβxx)[exp(-Γ0z)-exp(Γ0z)],
b0=-jωab xˆp 11-αeCe=-j 163 |Γ0|ab r3a0 11-αeCe.
Re=a0+b0a0=1-j 163 |Γ0|ab r3 11-αeCe.
R=Ro+Re2=b02a0=-j 83 |Γ0|ab r3 11-αeCe.
R=-jB2+jB-jB2.
B=163 |Γ0|ab r3=163 k0nd cos θdab r3 11-αeCe

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