Abstract

a numerical matching method (NMM) based on the framework of the uniform geometrical theory of diffraction (UTD) is proposed to build the spectral functions for computing the diffraction field by anisotropic impedance wedge at an arbitrary skew incidence. The NMM starts from the coupled integral equations before they are converted into the coupled difference equations as the classic Maliuzhinets methods. Then, the spectral function in the Sommerfeld integral representation of the longitudinal components of the EM field is expanded by a series about the spectrum and the skew incident angle with unknown coefficients. With respect to the oblique incident angle based on normal to the edge incidence or grazing to the edge incidence, the spectral function is derived numerically by solving a system of algebraic equations constructed from the coupled integral equations, after choosing the numerical matching regions on the wedge faces and setting a Sommerfeld numerical integration path. On the basis of the sampled incidences, the asymptotic waveform evaluation (AWE) technique is employed to deduce the spectral function at any other skew incidence in the whole angle space (0°-90°) rapidly. Finally, the UTD solutions are provided far beyond the applicability of the perturbation approach and the numerical examples provide a uniform behavior of the field with respect to the observation angle.

©2011 Optical Society of America

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References

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  1. N. Y. Zhu and F. M. Landstofer, “Numerical study of diffraction and slope-diffraction at anisotropic impedance wedges by the method of parabolic equation: space waves,” IEEE Trans. Antenn. Propag. 45(5), 822–828 (1997).
    [Crossref]
  2. G. Pelosi, S. Selleri, and R. D. Graglia, “Numerical analysis of the diffraction at an anisotropic impedance wedge,” IEEE Trans. Antenn. Propag. 45(5), 767–771 (1997).
    [Crossref]
  3. B. V. Budaev and D. B. Bogy, “Diffraction of a plane skew electromagnetic wave by a wedge with general anisotropic impedance boundary conditions,” IEEE Trans. Antenn. Propag. 54(5), 1559–1567 (2006).
    [Crossref]
  4. F. Yuan and G. Q. Zhu, “Electromagnetic diffraction of a very obliquely incident plane wave field by a wedge with anisotropic impedance faces,” J. Electromagn. Waves Appl. 19(12), 1671–1685 (2005).
    [Crossref]
  5. G. Manara, P. Nepa, and G. Pelosi, “Electromagnetic scattering by a right angled anisotropic impedance wedge,” Electron. Lett. 32(13), 1179–1180 (1996).
    [Crossref]
  6. M. A. Lyalinov, “Diffraction by a wedge with anisotropic face impedances,” Ann. Telecommun. 49, 667–672 (1994).
  7. G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).
  8. F. Yuan and G. Q. Zhu, “Electromagnetic diffraction at skew incidence by a wedge with anisotropic impedance faces,” Radio Sci. 40(6), RS6014 (2005).
    [Crossref]
  9. G. Pelosi, G. Manara, and P. Nepa, “A UTD solution for the scattering by a wedge with anisotropic impedance faces: skew incidence case,” IEEE Trans. Antenn. Propag. 46(4), 579–588 (1998).
    [Crossref]
  10. R. G. Rojas, “Electromagnetic diffraction of an obliquely incident plane wave field by a wedge with impedance face,” IEEE Trans. Antenn. Propag. 36(7), 956–970 (1988).
    [Crossref]
  11. G. Pelosi, G. Manara, and P. Nepa, “Electromagnetic scattering by a wedge with anisotropic impedance faces,” IEEE Antenn. Propag. Mag. 40(6), 29–35 (1998).
    [Crossref]
  12. J. M. L. Bernard, “Diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory: a new class of canonical cases,” J. Phys. Math. Gen. 31(2), 595–613 (1998).
    [Crossref]
  13. M. A. Lyalinov and N. Y. Zhu, “Exact solution to diffraction problem by wedges with a class of anisotropic impedance faces: oblique incidence of a plane electromagnetic wave,” IEEE Trans. Antenn. Propag. 51(6), 1216–1220 (2003).
    [Crossref]
  14. G. Manara, P. Nepa, and G. Pelosi, “A UTD solution for plane wave diffraction at an edge in an artificially hard surface: oblique incidence case,” Electron. Lett. 31(19), 1649–1650 (1995).
    [Crossref]
  15. Z. Q. Gong, B. X. Xiao, G. Zhu, and H. Y. Ke, ““Improvements to the hybrid MM-PO technique for scattering of plane wave by an infinite wedge,” IEEE Trans. Antenn. Propag. 54(1), 251–255 (2006).
    [Crossref]
  16. Y. E. Erdemli, J. Gong, C. J. Reddy, and J. L. Volakis, “Fast RCS pattern fill using AWE technique,” IEEE Trans. Antenn. Propag. 46(11), 1752–1753 (1998).
    [Crossref]

2006 (2)

B. V. Budaev and D. B. Bogy, “Diffraction of a plane skew electromagnetic wave by a wedge with general anisotropic impedance boundary conditions,” IEEE Trans. Antenn. Propag. 54(5), 1559–1567 (2006).
[Crossref]

Z. Q. Gong, B. X. Xiao, G. Zhu, and H. Y. Ke, ““Improvements to the hybrid MM-PO technique for scattering of plane wave by an infinite wedge,” IEEE Trans. Antenn. Propag. 54(1), 251–255 (2006).
[Crossref]

2005 (2)

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction of a very obliquely incident plane wave field by a wedge with anisotropic impedance faces,” J. Electromagn. Waves Appl. 19(12), 1671–1685 (2005).
[Crossref]

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction at skew incidence by a wedge with anisotropic impedance faces,” Radio Sci. 40(6), RS6014 (2005).
[Crossref]

2003 (1)

M. A. Lyalinov and N. Y. Zhu, “Exact solution to diffraction problem by wedges with a class of anisotropic impedance faces: oblique incidence of a plane electromagnetic wave,” IEEE Trans. Antenn. Propag. 51(6), 1216–1220 (2003).
[Crossref]

1998 (4)

G. Pelosi, G. Manara, and P. Nepa, “Electromagnetic scattering by a wedge with anisotropic impedance faces,” IEEE Antenn. Propag. Mag. 40(6), 29–35 (1998).
[Crossref]

J. M. L. Bernard, “Diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory: a new class of canonical cases,” J. Phys. Math. Gen. 31(2), 595–613 (1998).
[Crossref]

Y. E. Erdemli, J. Gong, C. J. Reddy, and J. L. Volakis, “Fast RCS pattern fill using AWE technique,” IEEE Trans. Antenn. Propag. 46(11), 1752–1753 (1998).
[Crossref]

G. Pelosi, G. Manara, and P. Nepa, “A UTD solution for the scattering by a wedge with anisotropic impedance faces: skew incidence case,” IEEE Trans. Antenn. Propag. 46(4), 579–588 (1998).
[Crossref]

1997 (2)

N. Y. Zhu and F. M. Landstofer, “Numerical study of diffraction and slope-diffraction at anisotropic impedance wedges by the method of parabolic equation: space waves,” IEEE Trans. Antenn. Propag. 45(5), 822–828 (1997).
[Crossref]

G. Pelosi, S. Selleri, and R. D. Graglia, “Numerical analysis of the diffraction at an anisotropic impedance wedge,” IEEE Trans. Antenn. Propag. 45(5), 767–771 (1997).
[Crossref]

1996 (1)

G. Manara, P. Nepa, and G. Pelosi, “Electromagnetic scattering by a right angled anisotropic impedance wedge,” Electron. Lett. 32(13), 1179–1180 (1996).
[Crossref]

1995 (1)

G. Manara, P. Nepa, and G. Pelosi, “A UTD solution for plane wave diffraction at an edge in an artificially hard surface: oblique incidence case,” Electron. Lett. 31(19), 1649–1650 (1995).
[Crossref]

1994 (1)

M. A. Lyalinov, “Diffraction by a wedge with anisotropic face impedances,” Ann. Telecommun. 49, 667–672 (1994).

1988 (1)

R. G. Rojas, “Electromagnetic diffraction of an obliquely incident plane wave field by a wedge with impedance face,” IEEE Trans. Antenn. Propag. 36(7), 956–970 (1988).
[Crossref]

1958 (1)

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).

Bernard, J. M. L.

J. M. L. Bernard, “Diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory: a new class of canonical cases,” J. Phys. Math. Gen. 31(2), 595–613 (1998).
[Crossref]

Bogy, D. B.

B. V. Budaev and D. B. Bogy, “Diffraction of a plane skew electromagnetic wave by a wedge with general anisotropic impedance boundary conditions,” IEEE Trans. Antenn. Propag. 54(5), 1559–1567 (2006).
[Crossref]

Budaev, B. V.

B. V. Budaev and D. B. Bogy, “Diffraction of a plane skew electromagnetic wave by a wedge with general anisotropic impedance boundary conditions,” IEEE Trans. Antenn. Propag. 54(5), 1559–1567 (2006).
[Crossref]

Erdemli, Y. E.

Y. E. Erdemli, J. Gong, C. J. Reddy, and J. L. Volakis, “Fast RCS pattern fill using AWE technique,” IEEE Trans. Antenn. Propag. 46(11), 1752–1753 (1998).
[Crossref]

Gong, J.

Y. E. Erdemli, J. Gong, C. J. Reddy, and J. L. Volakis, “Fast RCS pattern fill using AWE technique,” IEEE Trans. Antenn. Propag. 46(11), 1752–1753 (1998).
[Crossref]

Gong, Z. Q.

Z. Q. Gong, B. X. Xiao, G. Zhu, and H. Y. Ke, ““Improvements to the hybrid MM-PO technique for scattering of plane wave by an infinite wedge,” IEEE Trans. Antenn. Propag. 54(1), 251–255 (2006).
[Crossref]

Graglia, R. D.

G. Pelosi, S. Selleri, and R. D. Graglia, “Numerical analysis of the diffraction at an anisotropic impedance wedge,” IEEE Trans. Antenn. Propag. 45(5), 767–771 (1997).
[Crossref]

Ke, H. Y.

Z. Q. Gong, B. X. Xiao, G. Zhu, and H. Y. Ke, ““Improvements to the hybrid MM-PO technique for scattering of plane wave by an infinite wedge,” IEEE Trans. Antenn. Propag. 54(1), 251–255 (2006).
[Crossref]

Landstofer, F. M.

N. Y. Zhu and F. M. Landstofer, “Numerical study of diffraction and slope-diffraction at anisotropic impedance wedges by the method of parabolic equation: space waves,” IEEE Trans. Antenn. Propag. 45(5), 822–828 (1997).
[Crossref]

Lyalinov, M. A.

M. A. Lyalinov and N. Y. Zhu, “Exact solution to diffraction problem by wedges with a class of anisotropic impedance faces: oblique incidence of a plane electromagnetic wave,” IEEE Trans. Antenn. Propag. 51(6), 1216–1220 (2003).
[Crossref]

M. A. Lyalinov, “Diffraction by a wedge with anisotropic face impedances,” Ann. Telecommun. 49, 667–672 (1994).

Maliuzhinets, G. D.

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).

Manara, G.

G. Pelosi, G. Manara, and P. Nepa, “A UTD solution for the scattering by a wedge with anisotropic impedance faces: skew incidence case,” IEEE Trans. Antenn. Propag. 46(4), 579–588 (1998).
[Crossref]

G. Pelosi, G. Manara, and P. Nepa, “Electromagnetic scattering by a wedge with anisotropic impedance faces,” IEEE Antenn. Propag. Mag. 40(6), 29–35 (1998).
[Crossref]

G. Manara, P. Nepa, and G. Pelosi, “Electromagnetic scattering by a right angled anisotropic impedance wedge,” Electron. Lett. 32(13), 1179–1180 (1996).
[Crossref]

G. Manara, P. Nepa, and G. Pelosi, “A UTD solution for plane wave diffraction at an edge in an artificially hard surface: oblique incidence case,” Electron. Lett. 31(19), 1649–1650 (1995).
[Crossref]

Nepa, P.

G. Pelosi, G. Manara, and P. Nepa, “Electromagnetic scattering by a wedge with anisotropic impedance faces,” IEEE Antenn. Propag. Mag. 40(6), 29–35 (1998).
[Crossref]

G. Pelosi, G. Manara, and P. Nepa, “A UTD solution for the scattering by a wedge with anisotropic impedance faces: skew incidence case,” IEEE Trans. Antenn. Propag. 46(4), 579–588 (1998).
[Crossref]

G. Manara, P. Nepa, and G. Pelosi, “Electromagnetic scattering by a right angled anisotropic impedance wedge,” Electron. Lett. 32(13), 1179–1180 (1996).
[Crossref]

G. Manara, P. Nepa, and G. Pelosi, “A UTD solution for plane wave diffraction at an edge in an artificially hard surface: oblique incidence case,” Electron. Lett. 31(19), 1649–1650 (1995).
[Crossref]

Pelosi, G.

G. Pelosi, G. Manara, and P. Nepa, “Electromagnetic scattering by a wedge with anisotropic impedance faces,” IEEE Antenn. Propag. Mag. 40(6), 29–35 (1998).
[Crossref]

G. Pelosi, G. Manara, and P. Nepa, “A UTD solution for the scattering by a wedge with anisotropic impedance faces: skew incidence case,” IEEE Trans. Antenn. Propag. 46(4), 579–588 (1998).
[Crossref]

G. Pelosi, S. Selleri, and R. D. Graglia, “Numerical analysis of the diffraction at an anisotropic impedance wedge,” IEEE Trans. Antenn. Propag. 45(5), 767–771 (1997).
[Crossref]

G. Manara, P. Nepa, and G. Pelosi, “Electromagnetic scattering by a right angled anisotropic impedance wedge,” Electron. Lett. 32(13), 1179–1180 (1996).
[Crossref]

G. Manara, P. Nepa, and G. Pelosi, “A UTD solution for plane wave diffraction at an edge in an artificially hard surface: oblique incidence case,” Electron. Lett. 31(19), 1649–1650 (1995).
[Crossref]

Reddy, C. J.

Y. E. Erdemli, J. Gong, C. J. Reddy, and J. L. Volakis, “Fast RCS pattern fill using AWE technique,” IEEE Trans. Antenn. Propag. 46(11), 1752–1753 (1998).
[Crossref]

Rojas, R. G.

R. G. Rojas, “Electromagnetic diffraction of an obliquely incident plane wave field by a wedge with impedance face,” IEEE Trans. Antenn. Propag. 36(7), 956–970 (1988).
[Crossref]

Selleri, S.

G. Pelosi, S. Selleri, and R. D. Graglia, “Numerical analysis of the diffraction at an anisotropic impedance wedge,” IEEE Trans. Antenn. Propag. 45(5), 767–771 (1997).
[Crossref]

Volakis, J. L.

Y. E. Erdemli, J. Gong, C. J. Reddy, and J. L. Volakis, “Fast RCS pattern fill using AWE technique,” IEEE Trans. Antenn. Propag. 46(11), 1752–1753 (1998).
[Crossref]

Xiao, B. X.

Z. Q. Gong, B. X. Xiao, G. Zhu, and H. Y. Ke, ““Improvements to the hybrid MM-PO technique for scattering of plane wave by an infinite wedge,” IEEE Trans. Antenn. Propag. 54(1), 251–255 (2006).
[Crossref]

Yuan, F.

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction of a very obliquely incident plane wave field by a wedge with anisotropic impedance faces,” J. Electromagn. Waves Appl. 19(12), 1671–1685 (2005).
[Crossref]

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction at skew incidence by a wedge with anisotropic impedance faces,” Radio Sci. 40(6), RS6014 (2005).
[Crossref]

Zhu, G.

Z. Q. Gong, B. X. Xiao, G. Zhu, and H. Y. Ke, ““Improvements to the hybrid MM-PO technique for scattering of plane wave by an infinite wedge,” IEEE Trans. Antenn. Propag. 54(1), 251–255 (2006).
[Crossref]

Zhu, G. Q.

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction at skew incidence by a wedge with anisotropic impedance faces,” Radio Sci. 40(6), RS6014 (2005).
[Crossref]

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction of a very obliquely incident plane wave field by a wedge with anisotropic impedance faces,” J. Electromagn. Waves Appl. 19(12), 1671–1685 (2005).
[Crossref]

Zhu, N. Y.

M. A. Lyalinov and N. Y. Zhu, “Exact solution to diffraction problem by wedges with a class of anisotropic impedance faces: oblique incidence of a plane electromagnetic wave,” IEEE Trans. Antenn. Propag. 51(6), 1216–1220 (2003).
[Crossref]

N. Y. Zhu and F. M. Landstofer, “Numerical study of diffraction and slope-diffraction at anisotropic impedance wedges by the method of parabolic equation: space waves,” IEEE Trans. Antenn. Propag. 45(5), 822–828 (1997).
[Crossref]

Ann. Telecommun. (1)

M. A. Lyalinov, “Diffraction by a wedge with anisotropic face impedances,” Ann. Telecommun. 49, 667–672 (1994).

Electron. Lett. (2)

G. Manara, P. Nepa, and G. Pelosi, “Electromagnetic scattering by a right angled anisotropic impedance wedge,” Electron. Lett. 32(13), 1179–1180 (1996).
[Crossref]

G. Manara, P. Nepa, and G. Pelosi, “A UTD solution for plane wave diffraction at an edge in an artificially hard surface: oblique incidence case,” Electron. Lett. 31(19), 1649–1650 (1995).
[Crossref]

IEEE Antenn. Propag. Mag. (1)

G. Pelosi, G. Manara, and P. Nepa, “Electromagnetic scattering by a wedge with anisotropic impedance faces,” IEEE Antenn. Propag. Mag. 40(6), 29–35 (1998).
[Crossref]

IEEE Trans. Antenn. Propag. (8)

G. Pelosi, G. Manara, and P. Nepa, “A UTD solution for the scattering by a wedge with anisotropic impedance faces: skew incidence case,” IEEE Trans. Antenn. Propag. 46(4), 579–588 (1998).
[Crossref]

R. G. Rojas, “Electromagnetic diffraction of an obliquely incident plane wave field by a wedge with impedance face,” IEEE Trans. Antenn. Propag. 36(7), 956–970 (1988).
[Crossref]

Z. Q. Gong, B. X. Xiao, G. Zhu, and H. Y. Ke, ““Improvements to the hybrid MM-PO technique for scattering of plane wave by an infinite wedge,” IEEE Trans. Antenn. Propag. 54(1), 251–255 (2006).
[Crossref]

Y. E. Erdemli, J. Gong, C. J. Reddy, and J. L. Volakis, “Fast RCS pattern fill using AWE technique,” IEEE Trans. Antenn. Propag. 46(11), 1752–1753 (1998).
[Crossref]

M. A. Lyalinov and N. Y. Zhu, “Exact solution to diffraction problem by wedges with a class of anisotropic impedance faces: oblique incidence of a plane electromagnetic wave,” IEEE Trans. Antenn. Propag. 51(6), 1216–1220 (2003).
[Crossref]

N. Y. Zhu and F. M. Landstofer, “Numerical study of diffraction and slope-diffraction at anisotropic impedance wedges by the method of parabolic equation: space waves,” IEEE Trans. Antenn. Propag. 45(5), 822–828 (1997).
[Crossref]

G. Pelosi, S. Selleri, and R. D. Graglia, “Numerical analysis of the diffraction at an anisotropic impedance wedge,” IEEE Trans. Antenn. Propag. 45(5), 767–771 (1997).
[Crossref]

B. V. Budaev and D. B. Bogy, “Diffraction of a plane skew electromagnetic wave by a wedge with general anisotropic impedance boundary conditions,” IEEE Trans. Antenn. Propag. 54(5), 1559–1567 (2006).
[Crossref]

J. Electromagn. Waves Appl. (1)

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction of a very obliquely incident plane wave field by a wedge with anisotropic impedance faces,” J. Electromagn. Waves Appl. 19(12), 1671–1685 (2005).
[Crossref]

J. Phys. Math. Gen. (1)

J. M. L. Bernard, “Diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory: a new class of canonical cases,” J. Phys. Math. Gen. 31(2), 595–613 (1998).
[Crossref]

Radio Sci. (1)

F. Yuan and G. Q. Zhu, “Electromagnetic diffraction at skew incidence by a wedge with anisotropic impedance faces,” Radio Sci. 40(6), RS6014 (2005).
[Crossref]

Sov. Phys. Dokl. (1)

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).

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Figures (15)

Fig. 1
Fig. 1 Scattering of a plane wave by a wedge at skew incidence.

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Fig. 2
Fig. 2 Integration paths in the α -complex plane.

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Fig. 3
Fig. 3 Outline of the solution.

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Fig. 4
Fig. 4 Numerical matching regions.

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Fig. 5
Fig. 5 Numerical integration path γ 0 (dashed lines).

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Fig. 6
Fig. 6 Validation based on normal incidence: copolar (a) and cross-polar (b). The impedance tensors are Z zz 0,n / η 0 =1, Z ρρ 0,n / η 0 =2. The incident plane wave is TM polarized E β 0 i =1,  E ϕ 0 i =0, and the geometrical parameters: ϕ 0 = 45 ∘ and β 0 = 30 o , 50 o , 70 o , 90 o . The proposed solution: copolar and cross-polar components, respectively (solid lines); parabolic equation method in [2]: β 0 = 70 o (marked by *), β 0 = 50 o (marked by o); the MM-PO method when β 0 = 30 o (marked by + ).

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Fig. 7
Fig. 7 Validation based on normal incidence. The impedance tensors are Z zz 0,n / η 0 =4.1+3.5i, Z ρρ 0,n / η 0 =4.1-3.5i. The incident plane wave is TE polarized E β 0 i =0, E ϕ 0 i =1, and the geometrical parameters: ϕ 0 = 45 ∘ and β 0 = 50 ∘ . The proposed solution: the total fields (solid line) and its diffracted component (dashed line); the probabilistic random walk method solution in [3]: the total fields (marked by *) and its diffracted component (marked by o).

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Fig. 8
Fig. 8 Validation based on normal incidence: copolar (a) and cross-polar (b). The impedance tensors are Z zz 0 / η 0 = Z ρρ 0 / η 0 =j/2, Z zz n / η 0 = Z ρρ n / η 0 =0. The incident plane wave is TE polarized E β 0 i =0, E ϕ 0 i =1 , and the geometrical parameters: ϕ 0 = 45 ∘ and β 0 = 30 o , 50 o , 70 o , 90 o . The proposed solution: copolar and cross-polar components, respectively (solid lines); the solution in [10]: β 0 = 70 o (marked by *), β 0 = 50 o (marked by o); the MM-PO method in [15] β 0 = 30 o (marked by + ).

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Fig. 9
Fig. 9 Validation based on normal incidence. The impedance tensors are Z zz 0 / η 0 = Z zz n / η 0 =0, Z ρρ 0 / η 0 =(1+j)/2, Z ρρ n / η 0 =(1−j)/2. The incident plane wave is TE polarized E β 0 i =0,  E ϕ 0 i =1, and the geometrical parameters: ϕ 0 = 30 ∘ and β 0 = 30 o , 50 o , 70 o , 90 o . The proposed solution: copolar components, respectively (solid lines); UTD solutions in the published papers [9]: β 0 = 70 o (marked by *), β 0 = 50 o (marked by o); the MM-PO method in [15] β 0 = 30 o (marked by +).

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Fig. 10
Fig. 10 Validation based on grazing to the edge incidence: copolar (a) and cross-polar (b) . The impedance tensors are Z zz 0,n / η 0 =1, Z ρρ 0,n / η 0 =2. The incident plane wave is TE polarized E β 0 i =0,  E ϕ 0 i =1, and the geometrical parameters: ϕ 0 = 45 ∘ and β 0 = 1 o , 5 o , 15 o , 20 o , 25 o , 30 o . The proposed solution: based on grazing to the edge incidence (solid lines); the MM-PO method (marked by +).

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Fig. 11
Fig. 11 Validation based on based on grazing to the edge incidence: copolar (a) and cross-polar (b). The impedance tensors are Z zz 0 / η 0 =1, Z ρρ 0 / η 0 =0, Z zz n / η 0 = Z ρρ n / η 0 =j/2. The incident plane wave is TM polarized E β 0 i =1,  E ϕ 0 i =0, and the geometrical parameters: ϕ 0 = 30 ∘ and β 0 = 1 o , 5 o , 15 o , 20 o , 25 o , 30 o . This solution: based on grazing to the edge incidence (solid lines); the MM-PO method (marked by +).

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Fig. 12
Fig. 12 Examination based on two integral equations. The impedance tensors are Z zz 0,n / η 0 =1, Z ρρ 0,n / η 0 =2. The incident plane wave is TE polarized E β 0 i =0, E ϕ 0 i =1, and the geometrical parameters: ϕ 0 = 45 ∘ and β 0 = 25 o , 30 o . The proposed solution: based on the normal incidence case (solid lines), based on grazing to the edge incidence case (dashed lines); the MM-PO method (marked by +).

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Fig. 13
Fig. 13 Examination based on two integral equations. The impedance tensors are Z zz 0 / η 0 =1, Z ρρ 0 / η 0 =0, Z zz n / η 0 = Z ρρ n / η 0 =j/2. The incident plane wave is TM polarized E β 0 i =1, E ϕ 0 i =0. and the geometrical parameters: ϕ 0 = 30 ∘ and β 0 = 25 o , 30 o . The proposed solution: based on the normal incidence case (solid lines), based on grazing to the edge incidence case (dashed lines); the MM-PO method (marked by +).

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Fig. 14
Fig. 14 Validation based on AWE technique. The impedance tensors are Z zz 0,n / η 0 =1, Z ρρ 0,n / η 0 =2. The incident plane wave is TE polarized E β 0 i =0, E ϕ 0 i =1, and the geometrical parameters: ϕ 0 = 45 ∘ and β 0 = 35 o , 65 o . The proposed solution: by AWE (dashed lines); the MM-PO method (marked by +).

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Fig. 15
Fig. 15 Validation based two different interpolation techniques. The impedance tensors are Z zz 0 / η 0 =1, Z ρρ 0 / η 0 =0, Z zz n / η 0 = Z ρρ n / η 0 =j/2. The incident plane wave is TM polarized E β 0 i =1, E ϕ 0 i =0, and the geometrical parameters: ϕ 0 = 30 ∘ and β 0 =2 2 o , 28 o . The proposed solution: AWE (dashed lines); the MM-PO method (marked by +).

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Tables (4)

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Table 1 Numerical integration of A 1 h 0 (1)

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Table 2 Coefficients of spectral functions based on normal to the edge incidence case

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Table 3 Coefficient of spectral functions based on grazing to the edge incidence case

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Table 4 Coefficients of the spectral function with AWE

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Equations (37)

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[ E z i η 0 H z i ]=[ U e i U h i ] e jkρsin β 0 cos(ϕ− ϕ 0 ) e −jkzcos β 0
[ E z η 0 H z ]= e −jkzcos β 0 2πj ∫ γ S e,h (α−ϕ+ nπ 2 ) e j k t ρcosα dα
[ E ρ E z ]=±[ 0 Z ρρ 0,n − Z zz 0,n 0 ] [ H ρ H z ] ϕ=0,  nπ
1 ρ ∂( η 0 H z ) ∂ϕ ∓j k t sin θ h 0,n ( η 0 H z )+cos β 0 ∂ E z ∂ρ =0 ϕ=0, nπ
1 ρ ∂ E z ∂ϕ ∓j k t sin θ e 0,n E z −cos β 0 ∂( η 0 H z ) ∂ρ =0  ϕ=0,nπ
∫ γ e j k t ρcosα [(sinα±sin θ h 0,n ) S h (α±nπ/2) +cos β 0 cosα S e (α±nπ/2)]dα=0
∫ γ e j k t ρcosα [(sinα±sin θ e 0,n ) S e (α±nπ/2) −cos β 0 cosα S h (α±nπ/2)]dα=0 
(sinα±sin θ h 0,n ) S h (α±nπ/2)+(sinα∓sin θ h 0,n ) S h (−α±nπ/2) =−cos β 0 cosα[ S e (α±nπ/2)− S e (−α±nπ/2)]
(sinα±sin θ e 0,n ) S e (α±nπ/2)+(sinα∓sin θ e 0,n ) S e (−α±nπ/2) =cos β 0 cosα[ S h (α±nπ/2)− S h (−α±nπ/2)]
S e,h (α)= Ψ e,h (α) σ ϕ 0 (α) ∑ m=0 M a e,h m tan m (α/8) cos m ( β 0 )
σ ϕ 0 (α)=  sin( ϕ 0 /n) / [ nsin(α/n)−ncos( ϕ 0 /n) ]
S e,h (α)= σ ϕ 0 (α) Π e,h (α)
∑ m=0 4 A 1 h 0,n (m) a h m + ∑ m=0 4 B 1 e 0,n (m) a e m =0
∑ m=0 4 A 2 h 0,n (m) a h m + ∑ m=0 4 B 2 e 0,n (m) a e m =0
∑ m=0 4 A 1 e 0,n (m) a e m − ∑ m=0 4 B 1 h 0,n (m) a h m =0
∑ m=0 4 A 2 e 0,n (m) a e m − ∑ m=0 4 B 2 h 0,n (m) a h m =0
A 1 e,h 0,n (m)= ∫ γ 0 ∫ 0 ρ 1 [ e j k t ρcosα (sinα±sin θ e,h 0,n ) σ ϕ 0 (α±nπ/2) × Ψ e,h (α±nπ/2) tan m [(α±nπ/2)/8] cos m β 0 ]dρdα
A 2 e,h 0,n (m)= ∫ γ 0 ∫ ρ 1 ρ 2 [ e j k t ρcosα (sinα±sin θ e,h 0,n ) σ ϕ 0 (α±nπ/2) × Ψ e,h (α±nπ/2) tan m [(α±nπ/2)/8] cos m β 0 ]dρdα
B 1 e,h 0,n (m)= ∫ γ 0 ∫ 0 ρ 1 [ e j k t ρcosα cos β 0 cosα σ ϕ 0 (α±nπ/2) × Ψ e,h (α±nπ/2) tan m [(α±nπ/2)/8] cos m β 0 ]dρdα
B 2 e,h 0,n (m)= ∫ γ 0 ∫ ρ 1 ρ 2 [ e j k t ρcosα cos β 0 cosα σ ϕ 0 (α±nπ/2) × Ψ e,h (α±nπ/2) tan m [(α±nπ/2)/8] cos m β 0 ]dρdα
P= e −jkzcos β 0 2πj ∫ γ e jkρsin β 0 cosα S P (α+ nπ 2 −ϕ)dα
T= e −jkzcos β 0 2πj ∫ γ e jkρsin β 0 cosα S T (α+ nπ 2 −ϕ)dα
∫ γ e jkρsin β 0 cosα [(sinα−jcos β 0 cosα±sin β 0 K 1 0,n ) × S P (α±nπ/2)±sin β 0 K 2 0,n S T (α±nπ/2)]dα=0
∫ γ e jkρsin β 0 cosα [(sinα+jcos β 0 cosα±sin β 0 K 1 0,n ) × S T (α±nπ/2)∓sin β 0 K 2 0,n S P (α±nπ/2)]dα=0
S P,T (α)= Φ P,T (α) σ ϕ 0 (α) ∑ m=0 M a P,T m tan m (α/8) sin m ( β 0 )
Φ P,T (α)= χ n (α+nπ/2+π±Θ− σ 0 ) χ n (α+nπ/2−π∓Θ+ σ 0 ) χ n (α+nπ/2±Θ+ σ 0 ) χ n (α+nπ/2∓Θ− σ 0    × χ n (α−nπ/2+π∓Θ− σ n ) χ n (α−nπ/2−π±Θ+ σ n ) χ n (α−nπ/2∓Θ+ σ n ) χ n (α−nπ/2±Θ− σ n )
χ n (α)=exp{ 1 2 ∫ 0 ∞ 1 s sinh(πs) [ α nπ − sinh(αs) sinh(nπs) ]ds }
S e,h (α, β 0 )≈ ∑ i=0 L c e,h i tan i (α/8) cos i β 0 1+ ∑ j=1 M d e,h j tan j (α/8) cos j β 0 Ψ e,h (α) σ ϕ 0 (α)
[ λ e,h ( β k ) ]⋅[ x ]=[ γ e,h ( β k ) ]
γ e,h m ( β k )= ∑ i=0 m ( a e,h m−i cos m−i β k )
λ e,h mn ( β k )={ e mn ( β k ), 0≤n≤L − θ mn ( β k ), L<n≤L+M
e mn ( β k )=n!/(n−m)! cos n−m β k u(n−m)
θ mn ( β k )= ∑ i=0 m ( a e,h m−i cos m−i β k )· e in ( β k )
S e,h (α)= Π e,h (α) σ ϕ 0 (α)
S P,T (α, β 0 )≈ ∑ i=0 L c P,T i tan i (α/8) sin i β 0 1+ ∑ j=1 M d P,T j tan j (α/8) sin j β 0 Ψ e,h (α) σ ϕ 0 (α)
S e,h (α)= Π e,h (α) σ ϕ 0 (α)
Π e,h (α)= 1 2 [ Φ T,P (α) c T,P 0 + c T,P 1 tan(α/8)sin β 0 + c T,P 2 tan 2 (α/8) sin 2 β 0 1+ d T,P 1 tan(α/8)sin β 0 + d T,P 2 tan 2 (α/8) sin 2 β 0 +j c P,T 0 + c P,T 1 tan(α/8)sin β 0 + c P,T 2 tan 2 (α/8) sin 2 β 0 1+ d P,T 1 tan(α/8)sin β 0 + d P,T 2 tan 2 (α/8) sin 2 β 0 ]

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