Abstract

An exact solution to the two-dimensional scattering properties of an anisotropic elliptic cylinder for transverse electric polarization is presented. The internal field in an anisotropic elliptic cylinder is expressed as integral representations of Mathieu functions and Fourier series. The coefficients of the series expansion are obtained by imposing boundary conditions on the anisotropic–free-space interface. A matrix is developed to solve the nonorthogonality properties of Mathieu functions at the interface between two different media. Numerical results are given for the bistatic radar cross section and the amplitude of the total magnetic field along the x and y axes. The result is in agreement with that available as expected when an elliptic cylinder degenerates to a circular one.

© 2008 Optical Society of America

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References

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  1. R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, Part 1: General theory,” IEEE Trans. Antennas Propag. AP-32, 867-869 (1984).
    [CrossRef]
  2. R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750-760 (1989).
    [CrossRef]
  3. V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
    [CrossRef]
  4. J. C. Mozon, “Three-dimensional field expansion in the most general rotationally symmetric anisotropic material; application to scattering by a sphere,” IEEE Trans. Antennas Propag. 37, 728-735 (1989).
    [CrossRef]
  5. Spyros N. Papadakis, Nikolaos K. Uzunoglu, and Christos N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A 7, 991-997 (1990).
    [CrossRef]
  6. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
    [CrossRef]
  7. Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104/1-12 (2003).
    [CrossRef]
  8. Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
    [CrossRef]
  9. C. Monzon, D. W. Forester, and L. N. Medgyesi-Mitschang, “Scattering properties of an impedance-matched, ideal, homogeneous, causal 'left-handed' sphere,” J. Opt. Soc. Am. A 21, 2311-2319 (2004).
    [CrossRef]
  10. J. C. Monzon, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243-1249 (1986).
    [CrossRef]
  11. J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670-682 (1987).
    [CrossRef]
  12. W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664-673 (1993).
    [CrossRef]
  13. W. Ren, X. B. Wu, Z. Yi, and W. G. Lin, “Properties of wave functions in homogeneous anisotropic media,” Phys. Rev. E 51, 671-679 (1995).
    [CrossRef]
  14. W. Ren and X. B. Wu, “Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder,” J. Phys. D: Appl. Phys. 28, 1031-1039 (1995).
    [CrossRef]
  15. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65-71 (1963).
    [CrossRef]
  16. C. Yeh, “Backscattering cross section of a dielectric elliptic cylinder,” J. Opt. Soc. Am. 55, 309-314 (1965).
    [CrossRef]
  17. J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061-1066 (1988).
    [CrossRef]
  18. H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115-1122 (1988).
    [CrossRef]
  19. H. A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218-223 (1991).
    [CrossRef]
  20. R. Holland and V. P. Cable, “Mathieu functions and their applications to scattering by a coated strip,” IEEE Trans. Electromagn. Compat. 34, 9-16 (1992).
    [CrossRef]
  21. A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574-1580 (2000).
    [CrossRef]
  22. S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926-935 (1997).
    [CrossRef]
  23. S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinder,” IEEE Trans. Antennas Propag. 52, 189-196 (2004).
    [CrossRef]
  24. G. Blanch, in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1965), Chap. 20, pp. 721-746.
  25. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  26. N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford U. Press, 1947).
  27. P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw-Hill, 1953).

2004 (3)

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[CrossRef]

C. Monzon, D. W. Forester, and L. N. Medgyesi-Mitschang, “Scattering properties of an impedance-matched, ideal, homogeneous, causal 'left-handed' sphere,” J. Opt. Soc. Am. A 21, 2311-2319 (2004).
[CrossRef]

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinder,” IEEE Trans. Antennas Propag. 52, 189-196 (2004).
[CrossRef]

2003 (1)

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104/1-12 (2003).
[CrossRef]

2000 (1)

A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574-1580 (2000).
[CrossRef]

1997 (1)

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926-935 (1997).
[CrossRef]

1995 (2)

W. Ren, X. B. Wu, Z. Yi, and W. G. Lin, “Properties of wave functions in homogeneous anisotropic media,” Phys. Rev. E 51, 671-679 (1995).
[CrossRef]

W. Ren and X. B. Wu, “Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder,” J. Phys. D: Appl. Phys. 28, 1031-1039 (1995).
[CrossRef]

1993 (1)

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664-673 (1993).
[CrossRef]

1992 (1)

R. Holland and V. P. Cable, “Mathieu functions and their applications to scattering by a coated strip,” IEEE Trans. Electromagn. Compat. 34, 9-16 (1992).
[CrossRef]

1991 (2)

H. A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218-223 (1991).
[CrossRef]

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

1990 (1)

1989 (3)

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750-760 (1989).
[CrossRef]

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

J. C. Mozon, “Three-dimensional field expansion in the most general rotationally symmetric anisotropic material; application to scattering by a sphere,” IEEE Trans. Antennas Propag. 37, 728-735 (1989).
[CrossRef]

1988 (2)

J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061-1066 (1988).
[CrossRef]

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115-1122 (1988).
[CrossRef]

1987 (1)

J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670-682 (1987).
[CrossRef]

1986 (1)

J. C. Monzon, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243-1249 (1986).
[CrossRef]

1984 (1)

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, Part 1: General theory,” IEEE Trans. Antennas Propag. AP-32, 867-869 (1984).
[CrossRef]

1965 (1)

1963 (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65-71 (1963).
[CrossRef]

Blanch, G.

G. Blanch, in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1965), Chap. 20, pp. 721-746.

Cable, V. P.

R. Holland and V. P. Cable, “Mathieu functions and their applications to scattering by a coated strip,” IEEE Trans. Electromagn. Compat. 34, 9-16 (1992).
[CrossRef]

Caorsi, S.

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinder,” IEEE Trans. Antennas Propag. 52, 189-196 (2004).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926-935 (1997).
[CrossRef]

Capsalis, Christos N.

Feshback, H.

P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw-Hill, 1953).

Forester, D. W.

Geng, Y. L.

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104/1-12 (2003).
[CrossRef]

Graglia, R. D.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750-760 (1989).
[CrossRef]

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, Part 1: General theory,” IEEE Trans. Antennas Propag. AP-32, 867-869 (1984).
[CrossRef]

Hamid, M.

H. A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218-223 (1991).
[CrossRef]

Holland, R.

R. Holland and V. P. Cable, “Mathieu functions and their applications to scattering by a coated strip,” IEEE Trans. Electromagn. Compat. 34, 9-16 (1992).
[CrossRef]

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

Li, L. W.

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104/1-12 (2003).
[CrossRef]

Lin, W. G.

W. Ren, X. B. Wu, Z. Yi, and W. G. Lin, “Properties of wave functions in homogeneous anisotropic media,” Phys. Rev. E 51, 671-679 (1995).
[CrossRef]

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford U. Press, 1947).

Medgyesi-Mitschang, L. N.

Monzon, C.

Monzon, J. C.

J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670-682 (1987).
[CrossRef]

J. C. Monzon, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243-1249 (1986).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw-Hill, 1953).

Mozon, J. C.

J. C. Mozon, “Three-dimensional field expansion in the most general rotationally symmetric anisotropic material; application to scattering by a sphere,” IEEE Trans. Antennas Propag. 37, 728-735 (1989).
[CrossRef]

Papadakis, Spyros N.

Pastorino, M.

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinder,” IEEE Trans. Antennas Propag. 52, 189-196 (2004).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926-935 (1997).
[CrossRef]

Raffetto, M.

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926-935 (1997).
[CrossRef]

Ragheb, H. A.

H. A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218-223 (1991).
[CrossRef]

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115-1122 (1988).
[CrossRef]

Ren, W.

W. Ren and X. B. Wu, “Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder,” J. Phys. D: Appl. Phys. 28, 1031-1039 (1995).
[CrossRef]

W. Ren, X. B. Wu, Z. Yi, and W. G. Lin, “Properties of wave functions in homogeneous anisotropic media,” Phys. Rev. E 51, 671-679 (1995).
[CrossRef]

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664-673 (1993).
[CrossRef]

Richmond, J. H.

J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061-1066 (1988).
[CrossRef]

Sebak, A. R.

A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574-1580 (2000).
[CrossRef]

Shafai, L.

H. A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218-223 (1991).
[CrossRef]

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115-1122 (1988).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Uslenghi, P. L. E.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750-760 (1989).
[CrossRef]

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, Part 1: General theory,” IEEE Trans. Antennas Propag. AP-32, 867-869 (1984).
[CrossRef]

Uzunoglu, Nikolaos K.

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

Varadran, V. K.

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

Wang, Y. P.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Wu, X. B.

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104/1-12 (2003).
[CrossRef]

W. Ren and X. B. Wu, “Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder,” J. Phys. D: Appl. Phys. 28, 1031-1039 (1995).
[CrossRef]

W. Ren, X. B. Wu, Z. Yi, and W. G. Lin, “Properties of wave functions in homogeneous anisotropic media,” Phys. Rev. E 51, 671-679 (1995).
[CrossRef]

Wu, Z. S.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Yeh, C.

C. Yeh, “Backscattering cross section of a dielectric elliptic cylinder,” J. Opt. Soc. Am. 55, 309-314 (1965).
[CrossRef]

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65-71 (1963).
[CrossRef]

Yi, Z.

W. Ren, X. B. Wu, Z. Yi, and W. G. Lin, “Properties of wave functions in homogeneous anisotropic media,” Phys. Rev. E 51, 671-679 (1995).
[CrossRef]

Zich, R. S.

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750-760 (1989).
[CrossRef]

Can. J. Phys. (1)

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115-1122 (1988).
[CrossRef]

IEEE Antennas Wireless Propag. Lett. (1)

Y. L. Geng, X. B. Wu, and L. W. Li, “Characterization of electromagnetic scattering by a plasma anisotropic spherical shell,” IEEE Antennas Wireless Propag. Lett. 3, 100-103 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (9)

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, Part 1: General theory,” IEEE Trans. Antennas Propag. AP-32, 867-869 (1984).
[CrossRef]

V. V. Varadan, A. Lakhtakia, and V. K. Varadran, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
[CrossRef]

J. C. Mozon, “Three-dimensional field expansion in the most general rotationally symmetric anisotropic material; application to scattering by a sphere,” IEEE Trans. Antennas Propag. 37, 728-735 (1989).
[CrossRef]

H. A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218-223 (1991).
[CrossRef]

J. C. Monzon, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243-1249 (1986).
[CrossRef]

J. C. Monzon, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: a spectral approach,” IEEE Trans. Antennas Propag. 35, 670-682 (1987).
[CrossRef]

A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574-1580 (2000).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926-935 (1997).
[CrossRef]

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinder,” IEEE Trans. Antennas Propag. 52, 189-196 (2004).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

R. Holland and V. P. Cable, “Mathieu functions and their applications to scattering by a coated strip,” IEEE Trans. Electromagn. Compat. 34, 9-16 (1992).
[CrossRef]

J. Math. Phys. (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65-71 (1963).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D: Appl. Phys. (1)

W. Ren and X. B. Wu, “Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder,” J. Phys. D: Appl. Phys. 28, 1031-1039 (1995).
[CrossRef]

Phys. Rev. E (2)

W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E 47, 664-673 (1993).
[CrossRef]

W. Ren, X. B. Wu, Z. Yi, and W. G. Lin, “Properties of wave functions in homogeneous anisotropic media,” Phys. Rev. E 51, 671-679 (1995).
[CrossRef]

Proc. IEEE (1)

R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE 77, 750-760 (1989).
[CrossRef]

Radio Sci. (3)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci. 38, 1104/1-12 (2003).
[CrossRef]

J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061-1066 (1988).
[CrossRef]

Other (4)

G. Blanch, in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1965), Chap. 20, pp. 721-746.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford U. Press, 1947).

P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw-Hill, 1953).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

H polarization, bistatic RCS σ λ dB, ε x x = 4 ε y y = 4 ε 0 , ε x y = ε y x = 0 , μ z z = 2 μ 0 , u 0 = 6.0 , θ i = 0 ° , d λ = 1.239 E 3 , N = 5 . The circular marks are the solution taken from [10] [Fig. 3a].

Fig. 3
Fig. 3

Bistatic RCS. Scattering by an elliptic cylinder, d λ = 0.4 , ε x x = 4 ε 0 , ε y y = 2 ε 0 , ε x y = ε y x = 0 , μ z z = 2 μ 0 , N = 5 .

Fig. 4
Fig. 4

(a) Amplitude of the total magnetic field along the x axis. (b) Amplitude of the total magnetic field along the y axis. The parameters are same as in Fig. 3.

Equations (45)

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

x = d cosh u cos v ,
y = d sinh u sin v .
ε ̿ = [ ε x x ε x y 0 ε y x ε y y 0 0 0 ε z z ] , μ ̿ = [ μ x x μ x y 0 μ y x μ y y 0 0 0 μ z z ] .
ε x x 2 H x 2 + ε y y 2 H y 2 + ( ε x y + ε y x ) 2 H x y + ω 2 μ z z γ H H = 0 ,
γ H = ε x x ε y y ε x y ε y x .
H ( x , y ) = C α d α f ( α , β ( α ) ) e j ( α x + β ( α ) y ) .
ε x x α 2 + ε y y β 2 + ( ε x y + ε y x ) α β = ω 2 μ z z γ H .
α = k cos θ , β = k sin θ ,
k ( θ ) = [ n H 2 ε + + ε cos 2 θ + σ + sin 2 θ ] 1 2 ,
H ( x , y ) = C α d α f ( α , β ( α ) ) e j ( α x + β ( α ) y ) = C θ d θ s ( θ ) e j k ( θ ) ( x cos θ + y sin θ ) ,
H ( u , v ) = C n d θ g ( θ ) m = 0 j m M c m ( 1 ) ( q ( θ ) , u ) c e m ( q ( θ ) , v ) c e m ( q ( θ ) , θ ) + C θ d θ h ( θ ) m = 1 j m M s m ( 1 ) ( q ( θ ) , u ) s e m ( q ( θ ) , v ) s e m ( q ( θ ) , θ ) ,
g ( θ ) = l = g l e j l θ , h ( θ ) = p = h p e j p θ .
H ( u , v ) = l = g l 2 π d θ e j l θ m = 0 j m M c m ( 1 ) ( q ( θ ) , u ) c e m ( q ( θ ) , v ) c e m ( q ( θ ) , θ ) + p = h p 2 π d θ e j p θ m = 1 j m M s m ( 1 ) ( q ( θ ) , u ) s e m ( q ( θ ) , v ) s e m ( q ( θ ) , θ ) .
H i = E 0 e j k 0 ( x cos θ i + y sin θ i ) = 2 E 0 m = 0 j m M c m ( 1 ) ( q 0 , u ) c e m ( q 0 , v ) c e m ( q 0 , θ i ) + 2 E 0 m = 1 j m M s m ( 1 ) ( q 0 , u ) s e m ( q 0 , v ) s e m ( q 0 , θ i ) ,
H s = m = 0 e m M c m ( 4 ) ( q 0 , u ) c e m ( q 0 , v ) + m = 1 o m M s m ( 4 ) ( q 0 , u ) s e m ( q 0 , v ) ,
j ω γ h E v = { ε u ( u , v ) H u + ε v ( u , v ) H v } ,
ε u ( u , v ) = 1 a 2 + b 2 [ a 2 ε x x + a b ( ε x y + ε y x ) + b 2 ε y y ] ,
ε v ( u , v ) = 1 a 2 + b 2 [ a 2 ε y x + a b ( ε y y ε x x ) b 2 ε x y ] ,
a = sinh u cos v , b = cosh u sin v ,
γ = ε x x ε y y ε x y ε y x , h = d cosh 2 u cos 2 v ,
E v = j ω ε 0 h H u ,
H z = H z i + H z s , E v = E v i + E v s , u = u 0 ,
E v i = j ω ε 0 h H i u = 2 j E 0 ω ε 0 h [ m = 0 j m D M c m ( 1 ) ( q 0 , u ) c e m ( q 0 , v ) c e m ( q 0 , θ i ) + m = 1 j m D M s m ( 1 ) ( q 0 , u ) s e m ( q 0 , v ) s e m ( q 0 , θ i ) ] ,
E v s = j ω ε 0 h H s u = j ω ε 0 h [ m = 0 e m D M c m ( 4 ) ( q 0 , u ) c e m ( q 0 , v ) + m = 1 o m D M s m ( 4 ) ( q 0 , u ) s e m ( q 0 , v ) ] ,
l = g l C θ d θ G n ( 1 ) ( θ ) + p = h p C θ d θ H n ( 1 ) ( θ ) = F n ( 1 ) ,
n = 0 , 1 , 2 , ,
l = g l C θ d θ G n ( 2 ) ( θ ) + p = h p C θ d θ H n ( 2 ) ( θ ) = F n ( 2 ) ,
n = 1 , 2 , 3 , ,
G n ( 1 ) ( θ ) = e j l θ m = 0 j m c e m ( q ( θ ) , θ ) { D M c m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π c e m ( q ( θ ) , v ) c e n ( q 0 , v ) ε u ( u 0 , v ) d v + M c m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π D c e m ( q ( θ ) , v ) c e n ( q 0 , v ) ε v ( u 0 , v ) d v γ ε 0 D M c n ( 4 ) ( q 0 , u 0 ) M c m ( 1 ) ( q ( θ ) , u 0 ) M c n ( 4 ) ( q 0 , u 0 ) 0 2 π c e m ( q ( θ ) , v ) c e n ( q 0 , v ) d v } ,
G n ( 2 ) ( θ ) = e j l θ m = 0 j m c e m ( q ( θ ) , θ ) { D M c m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π c e m ( q ( θ ) , v ) s e n ( q 0 , v ) ε u ( u 0 , v ) d v + M c m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π D c e m ( q ( θ ) , v ) s e n ( q 0 , v ) ε v ( u 0 , v ) d v } ,
H n ( 1 ) ( θ ) = e j p θ m = 1 j m s e m ( q ( θ ) , θ ) { D M s m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π s e m ( q ( θ ) , v ) c e n ( q 0 , v ) ε u ( u 0 , v ) d v + M s m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π D s e m ( q ( θ ) , v ) c e n ( q 0 , v ) ε v ( u 0 , v ) d v } ,
H n ( 2 ) ( θ ) = e j p θ m = 0 j m s e m ( q ( θ ) , θ ) { D M s m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π s e m ( q ( θ ) , v ) s e n ( q 0 , v ) ε u ( u 0 , v ) d v + M s m ( 1 ) ( q ( θ ) , u 0 ) 0 2 π D s e m ( q ( θ ) , v ) s e n ( q 0 , v ) ε v ( u 0 , v ) d v γ ε 0 D M s n ( 4 ) ( q 0 , u 0 ) M s m ( 1 ) ( q ( θ ) , u 0 ) M s n ( 4 ) ( q 0 , u 0 ) 0 2 π s e m ( q ( θ ) , v ) s e n ( q 0 , v ) d v } ,
F n ( 1 ) = 2 π E 0 γ j n ε 0 c e n ( q 0 , θ i ) × [ D M c n ( 1 ) ( q 0 , u 0 ) D M c n ( 4 ) ( q 0 , u 0 ) M c n ( 1 ) ( q 0 , u 0 ) M c n ( 4 ) ( q 0 , u 0 ) ] ,
F n ( 2 ) = 2 π E 0 γ j n ε 0 s e n ( q 0 , θ i ) × [ D M s n ( 1 ) ( q 0 , u 0 ) D M s n ( 4 ) ( q 0 , u 0 ) M s n ( 1 ) ( q 0 , u 0 ) M s n ( 4 ) ( q 0 , u 0 ) ] .
l = N N G n l ( 1 ) g l + q = N N H n q ( 1 ) h q = F n ( 1 ) , n = 0 , 1 , 2 , ,
l = N N G n l ( 2 ) g l + q = N N H n q ( 2 ) h q = F n ( 2 ) , n = 1 , 2 , 3 , ,
G n l ( 1 ) = π ( 1 + N ) ( N + 1 2 ) π ( l + 1 + N ) ( N + 1 2 ) G n ( 1 ) ( θ ) d θ ,
H n q ( 1 ) = π ( q + N ) ( N + 1 2 ) π ( q + 1 + N ) ( N + 1 2 ) H n ( 1 ) ( θ ) d θ ,
G n l ( 2 ) = π ( 1 + N ) ( N + 1 2 ) π ( l + 1 + N ) ( N + 1 2 ) G n ( 2 ) ( θ ) d θ ,
H n q ( 2 ) = π ( q + N ) ( N + 1 2 ) π ( q + 1 + N ) ( N + 1 2 ) H n ( 2 ) ( θ ) d θ ,
G n l ( 1 ) g 1 + H n p ( 1 ) h p = F n ( 1 ) ,
G n l ( 2 ) g 1 + H n p ( 2 ) h p = F n ( 2 ) .
σ = lim r [ 2 π r H s 2 H i 2 ] .
M c m ( 4 ) ( q , u ) 2 π d k 0 cosh u e j { d k 0 cosh u [ ( 2 m + 1 ) 4 ] π } M s m ( 4 ) ( q , u ) .
σ λ ( v , θ i ) = 2 π m = 0 2 N e m c e m ( q 0 , v ) e j ( m 2 ) π + m = 1 2 N + 1 o m s e m ( q 0 , v ) e j ( m 2 ) π 2 .

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