Abstract

We present a novel semianalytical method using multipole expansion technique to solve the problem of scattering from multiple perfectly conducting cylinders placed above a perfectly conducting ground plane. The idea behind the formulation is based on the observation that an infinite flat ground plane can be approximated as a cylinder with a radius approaching infinity. Using Green’s representation of the electromagnetic fields and using addition theorem of Bessel function, we expand the fields in terms of multipoles. Applying the appropriate boundary condition on the surface of the cylinders and the ground plane based on the field polarization results in a set of linear systems of equations containing the multipole’s coefficients. The technique presented here is highly efficient in terms of computing resources, versatile, and accurate. The near fields are generated for single and multiple object examples.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
    [CrossRef]
  2. R. Holland and R. St. John, “EM pickup and scattering by a wire,” IEEE Trans. Electromagn. Compat. 42, 461–469 (2000).
    [CrossRef]
  3. M. S. Sarto, “Electromagnetic interference from carrier channels on finite-length power lines above a lossy ground in a wide frequency range,” IEEE Trans. Power Deliv. 13, 336–343 (1998).
    [CrossRef]
  4. P. J. Valle, F. Gonzalez, and F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat substrates: study by means of the extinction theorem,” Appl. Opt. 33, 512–523 (1994).
    [CrossRef] [PubMed]
  5. P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158–162 (1998).
    [CrossRef]
  6. P. J. Valle, F. Moreno, J. M. Saiz, and F. Gonzalez, “Near-field scattering from subwavelength metallic protuberances on conducting flat substrates,” Phys. Rev. B 51, 13681–13690 (1995).
    [CrossRef]
  7. J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
    [CrossRef]
  8. A. Madrazo and M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a cylinder in front of a conducting plane,” J. Opt. Soc. Am. A 12, 1298–1309 (1995).
    [CrossRef]
  9. M. A. Taubenblatt, “Light scattering from cylindrical structures on surfaces,” Opt. Lett. 15, 255–257 (1990).
    [CrossRef] [PubMed]
  10. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettin, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A 13, 483–493 (1996).
    [CrossRef]
  11. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettin, “Plane-wave scattering by a set of perfectly conducting circular cylinders in the presence of a plane surface,” J. Opt. Soc. Am. A 13, 2441–2452 (1996).
    [CrossRef]
  12. J. C. Chao, F. J. Rizzo, I. Elshafiey, Y. J. Liu, L. Upda, and P. A. Martin, “General formulation for light scattering by a dielectric body near a perfectly conducting surface,” J. Opt. Soc. Am. A 13, 338–344 (1996).
    [CrossRef]
  13. E. Arvas, R. F. Harrington, and J. R. Mautz, “Radiation and scattering from electrically small conducting bodies of arbitrary shape above an infinite ground plane,” IEEE Trans. Antennas Propag. 35, 378–383 (1987).
    [CrossRef]
  14. Y. Wang, Y. Zhang, M. He, and L. Guo, “Calculation of electromagnetic scattering from a two-dimensional target in the vicinity of a plane surface by a hybrid method,” J. Opt. Soc. Am. A 25, 1232–1239 (2008).
    [CrossRef]
  15. F. Baumgartner, J. Munk, and J. Daniels, “A geometric optics model for high-frequency electromagnetic scattering from dielectric cylinders,” Geophysics 66, 1130–1140 (2001).
    [CrossRef]
  16. J. T. Johnson, “A study of the Four-Path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30, 130–134 (2001).
    [CrossRef]
  17. B. Alavikia and O. M. Ramahi, “Fundamental limitations on the use of open-region boundary conditions and matched layers to solve the problem of gratings in metallic screens,” Applied Computational Electromagnetics Society Journal 25, 652–658 (2010).
  18. F. Saydou, T. Seppanen, and O. M. Ramahi, “Computation of the Helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique,” IEEE Trans. Antennas Propag. 57, 1169–1177 (2009).
    [CrossRef]
  19. R. F. Harrington, “Cylindrical wave functions,” in Time-Harmonic Electromagnetic Fields (IEEE, 2001), p. 232.
  20. “COMSOL Multiphysics Version 3.5a,” http://www.comsol.com/.

2010

B. Alavikia and O. M. Ramahi, “Fundamental limitations on the use of open-region boundary conditions and matched layers to solve the problem of gratings in metallic screens,” Applied Computational Electromagnetics Society Journal 25, 652–658 (2010).

2009

F. Saydou, T. Seppanen, and O. M. Ramahi, “Computation of the Helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique,” IEEE Trans. Antennas Propag. 57, 1169–1177 (2009).
[CrossRef]

2008

Y. Wang, Y. Zhang, M. He, and L. Guo, “Calculation of electromagnetic scattering from a two-dimensional target in the vicinity of a plane surface by a hybrid method,” J. Opt. Soc. Am. A 25, 1232–1239 (2008).
[CrossRef]

M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
[CrossRef]

2001

F. Baumgartner, J. Munk, and J. Daniels, “A geometric optics model for high-frequency electromagnetic scattering from dielectric cylinders,” Geophysics 66, 1130–1140 (2001).
[CrossRef]

J. T. Johnson, “A study of the Four-Path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30, 130–134 (2001).
[CrossRef]

2000

R. Holland and R. St. John, “EM pickup and scattering by a wire,” IEEE Trans. Electromagn. Compat. 42, 461–469 (2000).
[CrossRef]

1998

M. S. Sarto, “Electromagnetic interference from carrier channels on finite-length power lines above a lossy ground in a wide frequency range,” IEEE Trans. Power Deliv. 13, 336–343 (1998).
[CrossRef]

P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158–162 (1998).
[CrossRef]

1996

1995

P. J. Valle, F. Moreno, J. M. Saiz, and F. Gonzalez, “Near-field scattering from subwavelength metallic protuberances on conducting flat substrates,” Phys. Rev. B 51, 13681–13690 (1995).
[CrossRef]

A. Madrazo and M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a cylinder in front of a conducting plane,” J. Opt. Soc. Am. A 12, 1298–1309 (1995).
[CrossRef]

1994

J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
[CrossRef]

P. J. Valle, F. Gonzalez, and F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat substrates: study by means of the extinction theorem,” Appl. Opt. 33, 512–523 (1994).
[CrossRef] [PubMed]

1990

1987

E. Arvas, R. F. Harrington, and J. R. Mautz, “Radiation and scattering from electrically small conducting bodies of arbitrary shape above an infinite ground plane,” IEEE Trans. Antennas Propag. 35, 378–383 (1987).
[CrossRef]

Alavikia, B.

B. Alavikia and O. M. Ramahi, “Fundamental limitations on the use of open-region boundary conditions and matched layers to solve the problem of gratings in metallic screens,” Applied Computational Electromagnetics Society Journal 25, 652–658 (2010).

Amore, M. D.

M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
[CrossRef]

Apra, M.

M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
[CrossRef]

Arvas, E.

E. Arvas, R. F. Harrington, and J. R. Mautz, “Radiation and scattering from electrically small conducting bodies of arbitrary shape above an infinite ground plane,” IEEE Trans. Antennas Propag. 35, 378–383 (1987).
[CrossRef]

Baumgartner, F.

F. Baumgartner, J. Munk, and J. Daniels, “A geometric optics model for high-frequency electromagnetic scattering from dielectric cylinders,” Geophysics 66, 1130–1140 (2001).
[CrossRef]

Borghi, R.

Chao, J. C.

Daniels, J.

F. Baumgartner, J. Munk, and J. Daniels, “A geometric optics model for high-frequency electromagnetic scattering from dielectric cylinders,” Geophysics 66, 1130–1140 (2001).
[CrossRef]

Elshafiey, I.

Frezza, F.

Gigliotti, K.

M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
[CrossRef]

Gonzalez, F.

P. J. Valle, F. Moreno, J. M. Saiz, and F. Gonzalez, “Near-field scattering from subwavelength metallic protuberances on conducting flat substrates,” Phys. Rev. B 51, 13681–13690 (1995).
[CrossRef]

P. J. Valle, F. Gonzalez, and F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat substrates: study by means of the extinction theorem,” Appl. Opt. 33, 512–523 (1994).
[CrossRef] [PubMed]

J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
[CrossRef]

Gori, F.

Guo, L.

Harrington, R. F.

E. Arvas, R. F. Harrington, and J. R. Mautz, “Radiation and scattering from electrically small conducting bodies of arbitrary shape above an infinite ground plane,” IEEE Trans. Antennas Propag. 35, 378–383 (1987).
[CrossRef]

R. F. Harrington, “Cylindrical wave functions,” in Time-Harmonic Electromagnetic Fields (IEEE, 2001), p. 232.

He, M.

Holland, R.

R. Holland and R. St. John, “EM pickup and scattering by a wire,” IEEE Trans. Electromagn. Compat. 42, 461–469 (2000).
[CrossRef]

Johnson, J. T.

J. T. Johnson, “A study of the Four-Path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30, 130–134 (2001).
[CrossRef]

Jordan, D. L.

J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
[CrossRef]

Liu, Y. J.

Madrazo, A.

Martin, P. A.

Mautz, J. R.

E. Arvas, R. F. Harrington, and J. R. Mautz, “Radiation and scattering from electrically small conducting bodies of arbitrary shape above an infinite ground plane,” IEEE Trans. Antennas Propag. 35, 378–383 (1987).
[CrossRef]

Moreno, F.

P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158–162 (1998).
[CrossRef]

P. J. Valle, F. Moreno, J. M. Saiz, and F. Gonzalez, “Near-field scattering from subwavelength metallic protuberances on conducting flat substrates,” Phys. Rev. B 51, 13681–13690 (1995).
[CrossRef]

P. J. Valle, F. Gonzalez, and F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat substrates: study by means of the extinction theorem,” Appl. Opt. 33, 512–523 (1994).
[CrossRef] [PubMed]

J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
[CrossRef]

Munk, J.

F. Baumgartner, J. Munk, and J. Daniels, “A geometric optics model for high-frequency electromagnetic scattering from dielectric cylinders,” Geophysics 66, 1130–1140 (2001).
[CrossRef]

Nieto-Vesperinas, M.

Ramahi, O. M.

B. Alavikia and O. M. Ramahi, “Fundamental limitations on the use of open-region boundary conditions and matched layers to solve the problem of gratings in metallic screens,” Applied Computational Electromagnetics Society Journal 25, 652–658 (2010).

F. Saydou, T. Seppanen, and O. M. Ramahi, “Computation of the Helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique,” IEEE Trans. Antennas Propag. 57, 1169–1177 (2009).
[CrossRef]

Rizzo, F. J.

Saiz, J. M.

P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158–162 (1998).
[CrossRef]

P. J. Valle, F. Moreno, J. M. Saiz, and F. Gonzalez, “Near-field scattering from subwavelength metallic protuberances on conducting flat substrates,” Phys. Rev. B 51, 13681–13690 (1995).
[CrossRef]

J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
[CrossRef]

Santarsiero, M.

Sarto, M. S.

M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
[CrossRef]

M. S. Sarto, “Electromagnetic interference from carrier channels on finite-length power lines above a lossy ground in a wide frequency range,” IEEE Trans. Power Deliv. 13, 336–343 (1998).
[CrossRef]

Saydou, F.

F. Saydou, T. Seppanen, and O. M. Ramahi, “Computation of the Helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique,” IEEE Trans. Antennas Propag. 57, 1169–1177 (2009).
[CrossRef]

Schettin, G.

Seppanen, T.

F. Saydou, T. Seppanen, and O. M. Ramahi, “Computation of the Helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique,” IEEE Trans. Antennas Propag. 57, 1169–1177 (2009).
[CrossRef]

St. John, R.

R. Holland and R. St. John, “EM pickup and scattering by a wire,” IEEE Trans. Electromagn. Compat. 42, 461–469 (2000).
[CrossRef]

Taubenblatt, M. A.

Upda, L.

Valle, P. J.

P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158–162 (1998).
[CrossRef]

P. J. Valle, F. Moreno, J. M. Saiz, and F. Gonzalez, “Near-field scattering from subwavelength metallic protuberances on conducting flat substrates,” Phys. Rev. B 51, 13681–13690 (1995).
[CrossRef]

J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
[CrossRef]

P. J. Valle, F. Gonzalez, and F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat substrates: study by means of the extinction theorem,” Appl. Opt. 33, 512–523 (1994).
[CrossRef] [PubMed]

Volpi, V.

M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
[CrossRef]

Wang, Y.

Zhang, Y.

Appl. Opt.

Applied Computational Electromagnetics Society Journal

B. Alavikia and O. M. Ramahi, “Fundamental limitations on the use of open-region boundary conditions and matched layers to solve the problem of gratings in metallic screens,” Applied Computational Electromagnetics Society Journal 25, 652–658 (2010).

Geophysics

F. Baumgartner, J. Munk, and J. Daniels, “A geometric optics model for high-frequency electromagnetic scattering from dielectric cylinders,” Geophysics 66, 1130–1140 (2001).
[CrossRef]

IEEE Trans. Antennas Propag.

E. Arvas, R. F. Harrington, and J. R. Mautz, “Radiation and scattering from electrically small conducting bodies of arbitrary shape above an infinite ground plane,” IEEE Trans. Antennas Propag. 35, 378–383 (1987).
[CrossRef]

F. Saydou, T. Seppanen, and O. M. Ramahi, “Computation of the Helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique,” IEEE Trans. Antennas Propag. 57, 1169–1177 (2009).
[CrossRef]

IEEE Trans. Electromagn Compat.

M. Apra, M. D. Amore, K. Gigliotti, M. S. Sarto, and V. Volpi, “Lightning indirect effects certification of a transport aircraft by numerical simulation,” IEEE Trans. Electromagn Compat. 50, 513–523 (2008).
[CrossRef]

IEEE Trans. Electromagn. Compat.

R. Holland and R. St. John, “EM pickup and scattering by a wire,” IEEE Trans. Electromagn. Compat. 42, 461–469 (2000).
[CrossRef]

IEEE Trans. Power Deliv.

M. S. Sarto, “Electromagnetic interference from carrier channels on finite-length power lines above a lossy ground in a wide frequency range,” IEEE Trans. Power Deliv. 13, 336–343 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Microw. Opt. Technol. Lett.

J. T. Johnson, “A study of the Four-Path model for scattering from an object above a half space,” Microw. Opt. Technol. Lett. 30, 130–134 (2001).
[CrossRef]

Opt. Eng.

J. M. Saiz, P. J. Valle, F. Gonzalez, F. Moreno, and D. L. Jordan, “Backscattering from particulate surfaces: experiment and theoretical modeling,” Opt. Eng. 33, 1261–1270 (1994).
[CrossRef]

Opt. Lett.

Phys. Rev. B

P. J. Valle, F. Moreno, J. M. Saiz, and F. Gonzalez, “Near-field scattering from subwavelength metallic protuberances on conducting flat substrates,” Phys. Rev. B 51, 13681–13690 (1995).
[CrossRef]

Other

R. F. Harrington, “Cylindrical wave functions,” in Time-Harmonic Electromagnetic Fields (IEEE, 2001), p. 232.

“COMSOL Multiphysics Version 3.5a,” http://www.comsol.com/.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Schematic of the scattering problem from 2D cylindrical objects.

Fig. 2
Fig. 2

Amplitude of the total E-field on the x axis for two 2D PEC cylinders, TM case, oblique incident θ = 30 ° . a 1 = 0.25 λ , ( x 1 , y 1 ) = ( 0 , 0.5 λ ) , a 2 = 0.65 λ , ( x 2 , y 2 ) = ( 0 , 1.0 λ ) .

Fig. 3
Fig. 3

Schematic of the scattering problem of two PEC cylinders when the radius of one of them approaches infinity to approximate the flat ground plane. The dashed line depicts the contour around the smaller cylinder where field values are calculated.

Fig. 4
Fig. 4

Amplitude of the total E-field at 0.01 λ around the smaller cylinder shown in Fig. 3, TM case and oblique incidence of θ = 30 ° , calculated using the semianalytical method presented in this work, FE-BIM, and FE-ABC. The cylinder has a radius of 0.65 λ and is located above a cylinder with a radius of a 2 = 100 λ in the semianalytical case and a 2 = (flat ground plane) in the FE-BIM and FE-ABC cases. ( x 1 , y 1 ) = ( 0 , 1.15 λ ) and ( x 2 , y 2 ) = ( 0 , 100 λ ) .

Fig. 5
Fig. 5

Amplitude of the total H-field at 0.01 λ around the smaller cylinder shown in Fig. 3, TE case and oblique incidence of θ = 30 ° , calculated using semianalytical method presented in this work, FE-BIM, and FE-ABC. The cylinder has a radius of 0.65 λ and is located above a cylinder with a radius of a 2 = 100 λ in the semianalytical case and a 2 = (flat ground plane) in the FE-BIM and FE-ABC cases. ( x 1 , y 1 ) = ( 0 , 1.15 λ ) , and ( x 2 , y 2 ) = ( 0 , 100 λ ) .

Fig. 6
Fig. 6

Schematic of the scattering problem from multiple electrically small PEC circular cylinders in close vicinity of the large cylinder with a radius approaching to infinity to approximate the flat ground plane. The dashed line depicts the contour around the cylinder no. 1 where the field values are calculated.

Fig. 7
Fig. 7

Amplitude of the total E-field at 0.005 λ around the wire no. 1 shown in Fig. 6, TM case and oblique incidence of θ = 30 ° , calculated using the semianalytical method presented in this work, FE-BIM, and FE-ABC. The wires have a radius of 0.05 λ and are located above a cylinder with a radius of a 4 = 100 λ in the semianalytical case and a 4 = (flat ground plane) in the FE-BIM and FE-ABC cases. ( x 1 , y 1 ) = ( 0.075 λ , 0.15 λ ) , ( x 2 , y 2 ) = ( 0.075 λ , 0.15 λ ) , ( x 3 , y 3 ) = ( 0.0 , 0.25 λ ) , and ( x 4 , y 4 ) = ( 0 , 100 λ ) .

Fig. 8
Fig. 8

Amplitude of the total H-field at 0.005 λ around the wire no. 1 shown in Fig. 6, TE case and oblique incidence of θ = 30 ° , calculated using the semianalytical method presented in this work, FE-BIM, and FE-ABC. The wires have radius of 0.05 λ and are located above a cylinder with a radius of a 4 = 100 λ in the semianalytical case and a 4 = (flat ground plane) in the FE-BIM and FE-ABC cases. ( x 1 , y 1 ) = ( 0.075 λ , 0.15 λ ) , ( x 2 , y 2 ) = ( 0.075 λ , 0.15 λ ) , ( x 3 , y 3 ) = ( 0.0 , 0.25 λ ) , and ( x 4 , y 4 ) = ( 0 , 100 λ ) .

Fig. 9
Fig. 9

Schematic of the scattering problem from multiple electrically small PEC circular cylinders in close vicinity of the large cylinder with a radius approaching to infinity to approximate the flat ground plane. The dashed line represents the points located along a line parallel to the x axis where the field values are calculated.

Fig. 10
Fig. 10

Amplitude of the total E-field at the points located along a line parallel to the x axis shown in Fig. 9, TM case and oblique incidence of θ = 30 ° , calculated using the semianalytical method presented in this work, FE-BIM, and FE-ABC. The wires have radius of 0.05 λ and are located above a cylinder with a radius of a 4 = 200 λ in the semianalytical case and a 4 = (flat ground plane) in the FE-BIM and FE-ABC cases. ( x 1 , y 1 ) = ( 0.075 λ , 0.15 λ ) , ( x 2 , y 2 ) = ( 0.075 λ , 0.15 λ ) , ( x 3 , y 3 ) = ( 0.0 , 0.25 λ ) , and ( x 4 , y 4 ) = ( 0 , 200 λ ) . The field points are located at 0.35 λ above the x axis.

Fig. 11
Fig. 11

Amplitude of the total H-field at the points located along a line parallel to the x axis shown in Fig. 9, TE case and oblique incidence of θ = 30 ° , calculated using the semianalytical method presented in this work, FE-BIM, and FE-ABC. The wires have radius of 0.05 λ and are located above a cylinder with a radius of a 4 = 200 λ in the semianalytical case and a 4 = (flat ground plane) in the FE-BIM and FE-ABC cases. ( x 1 , y 1 ) = ( 0.075 λ , 0.15 λ ) , ( x 2 , y 2 ) = ( 0.075 λ , 0.15 λ ) , ( x 3 , y 3 ) = ( 0.0 , 0.25 λ ) , and ( x 4 , y 4 ) = ( 0 , 200 λ ) . The field points are located at 0.35 λ above the x axis.

Fig. 12
Fig. 12

Schematic of the 2D scattering problem from a bundle of PEC wires in close vicinity of ground planes with different curvatures.

Fig. 13
Fig. 13

Amplitude of the total E-field at the points located along a line parallel to the x axis shown in Fig. 12, TM case and oblique incidence of θ = 30 ° , calculated using the semianalytical method presented in this work and FE-BIM. The wires have radius of 0.05 λ and are located above a cylinder with different curvatures ( 100 λ < a 4 < 1000 λ ) in the semianalytical case and a 4 = (flat ground plane) in the FE-BIM case. ( x 1 , y 1 ) = ( 0.075 λ , 0.15 λ ) , ( x 2 , y 2 ) = ( 0.075 λ , 0.15 λ ) , ( x 3 , y 3 ) = ( 0.0 , 0.25 λ ) , and ( x 4 , y 4 ) = ( 0 , a 4 ) . The field points are located at 0.35 λ above the x axis.

Tables (1)

Tables Icon

Table 1 Induced Voltage by the TE-Polarized Plane Wave Excitation for the Bundle of Wires Shown in Fig. 6 Calculated Using the Method Introduced in This Work (Semianalytic), and Those Obtained by FE-BIM and FE-ABC for Flat Ground Plane

Equations (15)

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

u z ( X g ) = u z inc ( X g ) j = 1 J Γ j u z ( X g ) G ( X g , X g ) n j d Γ j + j = 1 J Γ j G ( X g , X g ) u z ( X g ) n j d Γ j ,
G ( X g , X g ) = j 4 H 0 2 ( k | X g X g | ) .
H 0 2 ( k R ) = n = { H n 2 ( k ρ ) J n ( k ρ ) e ( j n ( ϕ ϕ ) , ρ > ρ J n ( k ρ ) H n 2 ( k ρ ) e ( j n ( ϕ ϕ ) , ρ < ρ ,
u z ( X g ) = u z inc ( X g ) + j = 1 J ( n = H n 2 ( k ρ j ) e j n ϕ j b n j ) ,
b n j = Γ j u z ( ρ j ) n j ( j 4 J n ( k ρ j ) e j n ϕ j ) d Γ j + Γ j ( j 4 J n ( k ρ j ) e j n ϕ j ) u z ( ρ j ) n j d Γ j .
u z inc ( a l ) = ( n = H n 2 ( k a l ) e j n ϕ l ) b n l + j l J { n = J n ( k a l ) e j n ϕ l × ( Γ j u z ( ρ j l ) n j ( j 4 H n 2 ( k ρ j l ) e j n ϕ j l ) d Γ j + Γ j ( j 4 H n 2 ( k ρ j l ) e j n ϕ j l ) u z ( ρ j l ) n j d Γ j ) } ,
u z inc ( a l ) = ( n = H n 2 ( k a l ) e j n ϕ l ) b n l + j l J { n = J n ( k a l ) e j n ϕ l ( q = H n q 2 ( k O l j ) e j ( n q ) ϕ l j ) } × b q j .
[ F l ] M × 1 = [ S l ] M × M [ b l ] M × 1 + j l J [ T l j ] M × Q [ b j ] Q × 1 .
F m l = ( j ) m e ( j k O x l ) J m ( k a l ) S m n l = H m 2 ( k a l ) δ m , n T m q l j = q = J m ( k a l ) H m q 2 ( k O l j ) e j ( m q ) ϕ l j .
[ S ( 1 ) ] [ T ( 12 ) ] [ T ( 1 J ) ] [ T ( 21 ) ] [ S ( 2 ) ] [ T ( 2 J ) ] [ T ( J 1 ) ] [ T ( J 2 ) ] [ S ( J ) ] [ b ( 1 ) ] [ b ( 2 ) ] [ b ( J ) ] = [ F ( 1 ) ] [ F ( 2 ) ] [ F ( J ) ] .
u z inc ( a l ) n l = ( n = n l ( H n 2 ( k a l ) e j n ϕ l ) ) b n l + j l J { n = n l ( J n ( k a l ) e j n ϕ l ) × ( q = H n q 2 ( k O l j ) e j ( n q ) ϕ l j ) } b q j .
F m l = ( j ) m e ( j k O x l ) ( J m + 1 ( k a l ) + m k a l J m ( k a l ) ) S m n l = ( H m + 1 2 ( k a l ) + m k a l H m 2 ( k a l ) ) δ m , n T m q l j = q = ( J m + 1 ( k a l ) + m k a l J m ( k a l ) ) × H m q 2 ( k O l j ) e j ( m q ) ϕ l j .
V = | E y d y | = | 1 j ω ε H z x d y | .
2 × 10 308 < H M j 2 ( k a j ) < 2 × 10 308
2 × 10 308 < H M j + 1 2 ( k a j ) ( M j k a j ) H M j 2 ( k a j ) < 2 × 10 308

Metrics