Abstract

Uniform high-frequency solutions in closed form are derived for the diffraction of a plane wave normally impacting on a penetrable wedge having an obtuse apex angle and arbitrary dielectric permittivity. The approach used here takes advantage of a physical optics approximation for the electric and magnetic equivalent surface currents in the scattering integrals related to the inner region of the wedge and the surrounding space. Numerical tests and comparisons with finite-difference time-domain results demonstrate the accuracy and effectiveness of the proposed solutions.

© 2011 Optical Society of America

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  1. R. H. T. Bates, “Wavefunctions for prisms,” Int. J. Electron. 34, 81–95 (1973).
    [CrossRef]
  2. T. S. Yeo, D. J. Wall, and R. H. T. Bates, “Diffraction by a prism,” J. Opt. Soc. Am. A 2, 964–970 (1985).
    [CrossRef]
  3. S. Berntsen, “Diffraction of an electric polarized wave by a dielectric wedge,” SIAM J. Appl. Math. 43, 186–211 (1983).
    [CrossRef]
  4. S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part I. Physical optics approximation,” IEEE Trans. Antennas Propag. 39, 1272–1281 (1991).
    [CrossRef]
  5. S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part II. Correction to physical optics solution,” IEEE Trans. Antennas Propag. 39, 1282–1292 (1991).
    [CrossRef]
  6. S. Y. Kim, “Diffraction coefficients and field patterns of obtuse angle dielectric wedge illuminated by E-polarized plane wave,” IEEE Trans. Antennas Propag. 40, 1427–1431 (1992).
    [CrossRef]
  7. R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
    [CrossRef]
  8. A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. London A 455, 2655–2686 (1999).
    [CrossRef]
  9. M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in presence of an infinite dielectric wedge,” Proc. R. Soc. London A 462, 2503–2522 (2006).
    [CrossRef]
  10. R. J. Luebbers, “Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propag. 32, 70–76 (1984).
    [CrossRef]
  11. J. F. Rouviere, N. Douchin, and P. F. Combes, “Diffraction by lossy dielectric wedges using both heuristic UTD formulations and FDTD,” IEEE Trans. Antennas Propag. 47, 1702–1708(1999).
    [CrossRef]
  12. G. Stratis, V. Anantha, and A. Taflove, “Numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using FDTD,” IEEE Trans. Antennas Propag. 45, 1525–1529 (1997).
    [CrossRef]
  13. J. H. Chang and A. Taflove, “Three-dimensional diffraction by infinite conducting and dielectric wedges using a generalized total-field/scattered-field FDTD formulation,” IEEE Trans. Antennas Propag. 53, 1444–1454 (2005).
    [CrossRef]
  14. G. Gennarelli and G. Riccio, “A uniform asymptotic solution for diffraction by a right-angled dielectric wedge,” PIERS Online 6, 746–749 (2010).
    [CrossRef]
  15. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
    [CrossRef]
  16. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  17. G. Riccio, “Uniform asymptotic physical optics solutions for a set of diffraction problems,” in Wave Propagation in Materials for Modern Applications, A.Petrin, ed. (InTech, 2010), pp. 33–54.
  18. C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).
  19. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 2000).
  20. S. D. Gedney, “An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
    [CrossRef]

2010

G. Gennarelli and G. Riccio, “A uniform asymptotic solution for diffraction by a right-angled dielectric wedge,” PIERS Online 6, 746–749 (2010).
[CrossRef]

2006

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in presence of an infinite dielectric wedge,” Proc. R. Soc. London A 462, 2503–2522 (2006).
[CrossRef]

2005

J. H. Chang and A. Taflove, “Three-dimensional diffraction by infinite conducting and dielectric wedges using a generalized total-field/scattered-field FDTD formulation,” IEEE Trans. Antennas Propag. 53, 1444–1454 (2005).
[CrossRef]

1999

J. F. Rouviere, N. Douchin, and P. F. Combes, “Diffraction by lossy dielectric wedges using both heuristic UTD formulations and FDTD,” IEEE Trans. Antennas Propag. 47, 1702–1708(1999).
[CrossRef]

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. London A 455, 2655–2686 (1999).
[CrossRef]

1997

G. Stratis, V. Anantha, and A. Taflove, “Numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using FDTD,” IEEE Trans. Antennas Propag. 45, 1525–1529 (1997).
[CrossRef]

1996

S. D. Gedney, “An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

1992

S. Y. Kim, “Diffraction coefficients and field patterns of obtuse angle dielectric wedge illuminated by E-polarized plane wave,” IEEE Trans. Antennas Propag. 40, 1427–1431 (1992).
[CrossRef]

1991

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part I. Physical optics approximation,” IEEE Trans. Antennas Propag. 39, 1272–1281 (1991).
[CrossRef]

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part II. Correction to physical optics solution,” IEEE Trans. Antennas Propag. 39, 1282–1292 (1991).
[CrossRef]

1985

1984

R. J. Luebbers, “Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propag. 32, 70–76 (1984).
[CrossRef]

1983

S. Berntsen, “Diffraction of an electric polarized wave by a dielectric wedge,” SIAM J. Appl. Math. 43, 186–211 (1983).
[CrossRef]

1974

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

1973

R. H. T. Bates, “Wavefunctions for prisms,” Int. J. Electron. 34, 81–95 (1973).
[CrossRef]

1962

Anantha, V.

G. Stratis, V. Anantha, and A. Taflove, “Numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using FDTD,” IEEE Trans. Antennas Propag. 45, 1525–1529 (1997).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

Bates, R. H. T.

Berntsen, S.

S. Berntsen, “Diffraction of an electric polarized wave by a dielectric wedge,” SIAM J. Appl. Math. 43, 186–211 (1983).
[CrossRef]

Burge, R. E.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

Carroll, B. D.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

Chang, J. H.

J. H. Chang and A. Taflove, “Three-dimensional diffraction by infinite conducting and dielectric wedges using a generalized total-field/scattered-field FDTD formulation,” IEEE Trans. Antennas Propag. 53, 1444–1454 (2005).
[CrossRef]

Combes, P. F.

J. F. Rouviere, N. Douchin, and P. F. Combes, “Diffraction by lossy dielectric wedges using both heuristic UTD formulations and FDTD,” IEEE Trans. Antennas Propag. 47, 1702–1708(1999).
[CrossRef]

Douchin, N.

J. F. Rouviere, N. Douchin, and P. F. Combes, “Diffraction by lossy dielectric wedges using both heuristic UTD formulations and FDTD,” IEEE Trans. Antennas Propag. 47, 1702–1708(1999).
[CrossRef]

Fisher, N. E.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Gennarelli, G.

G. Gennarelli and G. Riccio, “A uniform asymptotic solution for diffraction by a right-angled dielectric wedge,” PIERS Online 6, 746–749 (2010).
[CrossRef]

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 2000).

Hall, T. J.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

Kamel, A. H.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in presence of an infinite dielectric wedge,” Proc. R. Soc. London A 462, 2503–2522 (2006).
[CrossRef]

Keller, J. B.

Kim, S. Y.

S. Y. Kim, “Diffraction coefficients and field patterns of obtuse angle dielectric wedge illuminated by E-polarized plane wave,” IEEE Trans. Antennas Propag. 40, 1427–1431 (1992).
[CrossRef]

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part I. Physical optics approximation,” IEEE Trans. Antennas Propag. 39, 1272–1281 (1991).
[CrossRef]

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part II. Correction to physical optics solution,” IEEE Trans. Antennas Propag. 39, 1282–1292 (1991).
[CrossRef]

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Lester, G. A.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

Luebbers, R. J.

R. J. Luebbers, “Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propag. 32, 70–76 (1984).
[CrossRef]

Oliver, C. J.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

Osipov, A. V.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in presence of an infinite dielectric wedge,” Proc. R. Soc. London A 462, 2503–2522 (2006).
[CrossRef]

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Ra, J. W.

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part II. Correction to physical optics solution,” IEEE Trans. Antennas Propag. 39, 1282–1292 (1991).
[CrossRef]

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part I. Physical optics approximation,” IEEE Trans. Antennas Propag. 39, 1272–1281 (1991).
[CrossRef]

Rawlins, A. D.

A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. London A 455, 2655–2686 (1999).
[CrossRef]

Riccio, G.

G. Gennarelli and G. Riccio, “A uniform asymptotic solution for diffraction by a right-angled dielectric wedge,” PIERS Online 6, 746–749 (2010).
[CrossRef]

G. Riccio, “Uniform asymptotic physical optics solutions for a set of diffraction problems,” in Wave Propagation in Materials for Modern Applications, A.Petrin, ed. (InTech, 2010), pp. 33–54.

Rouviere, J. F.

J. F. Rouviere, N. Douchin, and P. F. Combes, “Diffraction by lossy dielectric wedges using both heuristic UTD formulations and FDTD,” IEEE Trans. Antennas Propag. 47, 1702–1708(1999).
[CrossRef]

Salem, M. A.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in presence of an infinite dielectric wedge,” Proc. R. Soc. London A 462, 2503–2522 (2006).
[CrossRef]

Shin, S. Y.

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part I. Physical optics approximation,” IEEE Trans. Antennas Propag. 39, 1272–1281 (1991).
[CrossRef]

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part II. Correction to physical optics solution,” IEEE Trans. Antennas Propag. 39, 1282–1292 (1991).
[CrossRef]

Stratis, G.

G. Stratis, V. Anantha, and A. Taflove, “Numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using FDTD,” IEEE Trans. Antennas Propag. 45, 1525–1529 (1997).
[CrossRef]

Ta?ove, A.

J. H. Chang and A. Taflove, “Three-dimensional diffraction by infinite conducting and dielectric wedges using a generalized total-field/scattered-field FDTD formulation,” IEEE Trans. Antennas Propag. 53, 1444–1454 (2005).
[CrossRef]

G. Stratis, V. Anantha, and A. Taflove, “Numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using FDTD,” IEEE Trans. Antennas Propag. 45, 1525–1529 (1997).
[CrossRef]

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 2000).

Taket, N. D.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

Wall, D. J.

Yeo, T. S.

Yuan, X. C.

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

IEEE Trans. Antennas Propag.

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part I. Physical optics approximation,” IEEE Trans. Antennas Propag. 39, 1272–1281 (1991).
[CrossRef]

S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: part II. Correction to physical optics solution,” IEEE Trans. Antennas Propag. 39, 1282–1292 (1991).
[CrossRef]

S. Y. Kim, “Diffraction coefficients and field patterns of obtuse angle dielectric wedge illuminated by E-polarized plane wave,” IEEE Trans. Antennas Propag. 40, 1427–1431 (1992).
[CrossRef]

R. E. Burge, X. C. Yuan, B. D. Carroll, N. E. Fisher, T. J. Hall, G. A. Lester, N. D. Taket, and C. J. Oliver, “Microwave scattering from dielectric wedges with planar surfaces: a diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag. 47, 1515–1527 (1999).
[CrossRef]

R. J. Luebbers, “Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propag. 32, 70–76 (1984).
[CrossRef]

J. F. Rouviere, N. Douchin, and P. F. Combes, “Diffraction by lossy dielectric wedges using both heuristic UTD formulations and FDTD,” IEEE Trans. Antennas Propag. 47, 1702–1708(1999).
[CrossRef]

G. Stratis, V. Anantha, and A. Taflove, “Numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using FDTD,” IEEE Trans. Antennas Propag. 45, 1525–1529 (1997).
[CrossRef]

J. H. Chang and A. Taflove, “Three-dimensional diffraction by infinite conducting and dielectric wedges using a generalized total-field/scattered-field FDTD formulation,” IEEE Trans. Antennas Propag. 53, 1444–1454 (2005).
[CrossRef]

S. D. Gedney, “An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Int. J. Electron.

R. H. T. Bates, “Wavefunctions for prisms,” Int. J. Electron. 34, 81–95 (1973).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

PIERS Online

G. Gennarelli and G. Riccio, “A uniform asymptotic solution for diffraction by a right-angled dielectric wedge,” PIERS Online 6, 746–749 (2010).
[CrossRef]

Proc. IEEE

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Proc. R. Soc. London A

A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. London A 455, 2655–2686 (1999).
[CrossRef]

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in presence of an infinite dielectric wedge,” Proc. R. Soc. London A 462, 2503–2522 (2006).
[CrossRef]

SIAM J. Appl. Math.

S. Berntsen, “Diffraction of an electric polarized wave by a dielectric wedge,” SIAM J. Appl. Math. 43, 186–211 (1983).
[CrossRef]

Other

G. Riccio, “Uniform asymptotic physical optics solutions for a set of diffraction problems,” in Wave Propagation in Materials for Modern Applications, A.Petrin, ed. (InTech, 2010), pp. 33–54.

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method (Artech, 2000).

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Figures (13)

Fig. 1
Fig. 1

Geometry of the scattering problem. The source point is assumed on S 0 .

Fig. 2
Fig. 2

Integration path C.

Fig. 3
Fig. 3

Case 1: ( n 1 ) π < ϕ < π .

Fig. 4
Fig. 4

GO and UAPO diffraction contributions of E z when ε r = 1.25 , γ = 110 ° , ϕ = 160 ° , and ρ = 5 λ 0 .

Fig. 5
Fig. 5

Relative magnitude of E z when ε r = 1.25 , γ = 110 ° , ϕ = 160 ° , and ρ = 5 λ 0 .

Fig. 6
Fig. 6

Relative magnitude of E ϕ when ε r = 1.25 , γ = 110 ° , ϕ = 160 ° , and ρ = 5 λ 0 .

Fig. 7
Fig. 7

Relative magnitude of E z when ε r = 10 , γ = 110 ° , ϕ = 160 ° , and ρ = 5 λ 0 .

Fig. 8
Fig. 8

Relative magnitude of E z when ε r = 10 , γ = 110 ° , ϕ = 160 ° , and ρ = λ 0 .

Fig. 9
Fig. 9

Case 2.1: cos ϕ < ε r 1 / 2 | cos ( n π ) | .

Fig. 10
Fig. 10

Relative magnitude of E z when ε r = 2.5 , γ = 110 ° , ϕ = 60 ° , and ρ = 5 λ 0 .

Fig. 11
Fig. 11

Case 2.2: cos ϕ > ε r 1 / 2 | cos ( n π ) | .

Fig. 12
Fig. 12

Relative magnitude of E z when ε r = 1.25 , γ = 110 ° , ϕ = 30 ° , and ρ = 5 λ 0 .

Fig. 13
Fig. 13

Relative magnitude of E z when ε r = 1.25 , γ = 110 ° , ϕ = 40 ° , and ρ = 5 λ 0 .

Equations (21)

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

E s j k S 0 S n [ ( I ̲ ̲ u R u R ) ( ζ J s ) + J m s × u R ] G ( r , r ) d S .
E s = E 0 s + E n s = j k [ ( I ̲ ̲ u s u s ) ( ζ J ˜ s 0 ) + J ˜ m s 0 × u s ] S 0 exp ( j φ 0 ) G ( r , r ) d S 0 j k [ ( I ̲ ̲ u s u s ) ( ζ J ˜ s n ) + J ˜ m s n × u s ] S n exp ( j φ n ) G ( r , r ) d S n = [ M ̲ ̲ 0 I 0 + M ̲ ̲ n I n ] E i .
I ( Ω ) = 1 j 2 π C g ( α ) exp [ Ω f ( α ) ] d α ,
( E z d E ϕ d ) = E d = [ M ̲ ̲ 0 I 0 d + M ̲ ̲ n I n d ] E i = [ M ̲ ̲ 0 h 0 ( ρ , ϕ , ϕ ) + M ̲ ̲ n h n ( ρ , ϕ , ϕ ) ] E i exp ( j k ρ ) ρ 1 / 2 = [ D ̲ ̲ 0 + D ̲ ̲ n ] E i exp ( j k ρ ) ρ 1 / 2 = D ̲ ̲ E i exp ( j k ρ ) ρ 1 / 2 = ( D z z 0 0 D ϕ ϕ ) ( E z i E ϕ i ) exp ( j k ρ ) ρ 1 / 2 ,
F t ( η ) = 2 j η exp ( j η ) η + exp ( j ξ 2 ) d ξ
M ̲ ̲ 0 out = ( ( 1 R 0 ) sin ϕ ( 1 + R 0 ) sin ϕ 0 0 ( 1 R 0 ) sin ϕ + ( 1 + R 0 ) sin ϕ ) ,
h 0 out = exp ( j π / 4 ) 2 ( 2 π k 0 ) 1 / 2 F t [ 2 k 0 ρ cos 2 ( ϕ ± ϕ 2 ) ] cos ϕ + cos ϕ ,
M ̲ ̲ n out = ( ( 1 R n ) sin ( n π ϕ ) ( 1 + R n ) sin ( n π ϕ ) 0 0 ( 1 R n ) sin ( n π ϕ ) + ( 1 + R n ) sin ( n π ϕ ) ) ,
h n out = exp ( j π / 4 ) 2 ( 2 π k 0 ) 1 / 2 F t [ 2 k 0 ρ cos 2 ( ( n π ϕ ) ± ( n π ϕ ) 2 ) ] cos ( n π ϕ ) + cos ( n π ϕ ) ,
M ̲ ̲ 0 in = [ ε r 1 / 2 sin ϕ ( ε r cos 2 ϕ ) 1 / 2 ] ( T 0 0 0 T 0 ) ,
h 0 in = exp ( j π / 4 ) 2 ( 2 π k ) 1 / 2 F t { 2 k ρ cos 2 [ ϕ cos 1 ( cos ϕ / ε r 1 / 2 ) 2 ] } ε r 1 / 2 cos ϕ + cos ϕ ,
M ̲ ̲ n in = { ε r 1 / 2 sin ( n π ϕ ) [ ε r cos 2 ( n π ϕ ) ] 1 / 2 } ( T n 0 0 T n ) ,
h n in = exp ( j π / 4 ) 2 ( 2 π k ) 1 / 2 F t { 2 k ρ cos 2 [ n π ϕ cos 1 ( cos ( n π ϕ ) / ε r 1 / 2 ) 2 ] } ε r 1 / 2 cos ( n π ϕ ) + cos ( n π ϕ ) .
M ̲ ̲ n out = [ cos θ n t + sin ( n π ϕ ) ] ( T 0 T 0 n 0 0 T 0 T n ) ,
h n out = exp ( j π / 4 ) 2 ( 2 π k 0 ) 1 / 2 F t [ 2 k 0 ρ cos 2 ( cos 1 ( η ) ± ( ϕ + ( 1 n ) π ) 2 ) ] cos ( n π ϕ ) + η ,
θ n t = sin 1 { ε r 1 / 2 sin [ ( 2 n ) π sin 1 ( cos ϕ / ε r 1 / 2 ) ] } ,
η = cos ( n π ) cos ϕ + sin ( n π ) ( ε r cos 2 ϕ ) 1 / 2 ,
M n 11 in = ε r 1 / 2 T 0 [ ( 1 R 0 n ) cos ( n π + θ 0 t ) + ( 1 + R 0 n ) sin ( n π ϕ ) ] ,
M n 12 in = M n 21 in = 0 ,
M n 22 in = ε r 1 / 2 T 0 [ ( 1 R 0 n ) cos ( n π + θ 0 t ) + ( 1 + R 0 n ) sin ( n π ϕ ) ] ,
h n in = exp ( j π / 4 ) 2 ( 2 π k ) 1 / 2 F t [ 2 k ρ cos 2 ( cos 1 ( η / ε r 1 / 2 ) ( ϕ + ( 1 n ) π ) 2 ) ] ε r 1 / 2 cos ( n π ϕ ) + η ,

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