Abstract

A solution to the problem of three-dimensional scattering by an infinite homogeneous anisotropic elliptic cylinder for an obliquely incident plane wave of an arbitrary linear polarization is proposed. The axial components of the electromagnetic fields inside an anisotropic elliptic cylinder are represented as two coupled integrals of suitable eigenfunctions in elliptic coordinates in terms of Mathieu functions. Scattering by an anisotropic elliptic cylinder is different from scattering by a sphere or a circular cylinder because of the nonorthogonality properties of Mathieu functions at the interface between two different media. In order to solve this problem, Galerkin’s method is applied to the boundary conditions to solve the unknown coefficients. Numerical results are presented, discussed, and compared with available data.

© 2009 Optical Society of America

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  1. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195 (1955).
    [CrossRef]
  2. R.-B. Wu and C. H. Chen, “Variational reaction formulation of scattering problem for anisotropic dielectric cylinders,” IEEE Trans. Antennas Propag. 34, 640-645 (1986).
    [CrossRef]
  3. J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. 34, 1243-1249 (1986).
    [CrossRef]
  4. 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]
  5. X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996-2003 (1997).
    [CrossRef]
  6. N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for oblique incidence scattering by an infinite dielectric cylinder,” Electr. Eng. 89, 353-361 (2007).
    [CrossRef]
  7. S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A 7, 991-997 (1990).
    [CrossRef]
  8. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
    [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. R.-J. Tarento, K. H. Bennemann, P. Joyes, and J. V. d. Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606/026601-026605 (2004).
    [CrossRef]
  11. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic scattering by an inhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
    [CrossRef]
  12. K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325-1328 (1995).
  13. N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
    [CrossRef]
  14. A. Taflove and K. Umashankar, “Radar cross section of general three-dimensional scatterers,” IEEE Trans. Electromagn. Compat. 25, 433-440 (1983).
    [CrossRef]
  15. R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, Part 1: General theory,” IEEE Trans. Antennas Propag. 32, 867-869 (1984).
    [CrossRef]
  16. R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, part II: Computer code and numerical results in two dimensions,” IEEE Trans. Antennas Propag. 35, 225-232 (1987).
    [CrossRef]
  17. 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]
  18. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (1989).
    [CrossRef]
  19. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65-71 (1963).
    [CrossRef]
  20. C. Yeh, “Scattering of obliquely incident light waves by elliptical fibers,” J. Opt. Soc. Am. 54, 1227-1231 (1964).
    [CrossRef]
  21. C. Yeh, “Backscattering cross section of a dielectric elliptic cylinder,” J. Opt. Soc. Am. 55, 309-314 (1965).
    [CrossRef]
  22. P. M. V. D. Berg and H. J. V. Schaik, “Diffraction of a plane electromagnetic wave by a perfectly conducting elliptic cylinder,” Appl. Sci. Res. 28, 145-157 (1973).
  23. N. G. Alexopoulos and G. A. Tadler, “Electromagnetic scattering from an elliptic cylinder loaded by continuous and discontinuous surface impedances,” J. Appl. Phys. 46, 1128-1134 (1975).
    [CrossRef]
  24. J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061-1066 (1988).
    [CrossRef]
  25. H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115-1122 (1988).
    [CrossRef]
  26. 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]
  27. 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]
  28. M. Hussein, A. Sebak, and M. Hamid, “Scattering and coupling properties of a slotted elliptic cylinder,” IEEE Trans. Electromagn. Compat. 36, 76-81 (1994).
    [CrossRef]
  29. 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]
  30. A.-R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574-1580 (2000).
    [CrossRef]
  31. S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinder,” IEEE Trans. Antennas Propag. 52, 189-196 (2004).
    [CrossRef]
  32. S.-C. Mao and Z.-S. Wu, “Scattering by an infinite homogenous anisotropic elliptic cylinder in terms of Mathieu functions and Fourier series,” J. Opt. Soc. Am. A 25, 2925-2931 (2008).
    [CrossRef]
  33. S. I. Tsakiris and N. K. Uzunoglu, “Electromagnetic analysis of coupling and guiding phenomena in elliptical cross-section parallel waveguides with rotated symmetry planes,” J. Opt. Soc. Am. A 24, 470-481 (2007).
    [CrossRef]
  34. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  35. N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford Univ. Press, 1947).
  36. P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw-Hill, 1953).
  37. G. Blanch, in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1965).

2008 (1)

2007 (2)

S. I. Tsakiris and N. K. Uzunoglu, “Electromagnetic analysis of coupling and guiding phenomena in elliptical cross-section parallel waveguides with rotated symmetry planes,” J. Opt. Soc. Am. A 24, 470-481 (2007).
[CrossRef]

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for oblique incidence scattering by an infinite dielectric cylinder,” Electr. Eng. 89, 353-361 (2007).
[CrossRef]

2006 (1)

N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
[CrossRef]

2005 (1)

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic scattering by an inhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

2004 (3)

R.-J. Tarento, K. H. Bennemann, P. Joyes, and J. V. d. Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606/026601-026605 (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]

2000 (1)

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

1997 (2)

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]

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996-2003 (1997).
[CrossRef]

1995 (1)

K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325-1328 (1995).

1994 (1)

M. Hussein, A. Sebak, and M. Hamid, “Scattering and coupling properties of a slotted elliptic cylinder,” IEEE Trans. Electromagn. Compat. 36, 76-81 (1994).
[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 (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]

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by three-dimensional anisotropic scatterers,” IEEE Trans. Antennas Propag. 37, 800-802 (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 (2)

R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic scattering from anisotropic materials, part II: Computer code and numerical results in two dimensions,” IEEE Trans. Antennas Propag. 35, 225-232 (1987).
[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]

1986 (2)

R.-B. Wu and C. H. Chen, “Variational reaction formulation of scattering problem for anisotropic dielectric cylinders,” IEEE Trans. Antennas Propag. 34, 640-645 (1986).
[CrossRef]

J. C. Monzon and N. J. Damaskos, “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. 32, 867-869 (1984).
[CrossRef]

1983 (1)

A. Taflove and K. Umashankar, “Radar cross section of general three-dimensional scatterers,” IEEE Trans. Electromagn. Compat. 25, 433-440 (1983).
[CrossRef]

1975 (1)

N. G. Alexopoulos and G. A. Tadler, “Electromagnetic scattering from an elliptic cylinder loaded by continuous and discontinuous surface impedances,” J. Appl. Phys. 46, 1128-1134 (1975).
[CrossRef]

1973 (1)

P. M. V. D. Berg and H. J. V. Schaik, “Diffraction of a plane electromagnetic wave by a perfectly conducting elliptic cylinder,” Appl. Sci. Res. 28, 145-157 (1973).

1965 (1)

1964 (1)

1963 (1)

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

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195 (1955).
[CrossRef]

Alexopoulos, N. G.

N. G. Alexopoulos and G. A. Tadler, “Electromagnetic scattering from an elliptic cylinder loaded by continuous and discontinuous surface impedances,” J. Appl. Phys. 46, 1128-1134 (1975).
[CrossRef]

Alivizatos, E. G.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for oblique incidence scattering by an infinite dielectric cylinder,” Electr. Eng. 89, 353-361 (2007).
[CrossRef]

Anastassiu, H. T.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for oblique incidence scattering by an infinite dielectric cylinder,” Electr. Eng. 89, 353-361 (2007).
[CrossRef]

Athanasiadis, C.

N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
[CrossRef]

Bennemann, K. H.

R.-J. Tarento, K. H. Bennemann, P. Joyes, and J. V. d. Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606/026601-026605 (2004).
[CrossRef]

Berg, P. M. V. D.

P. M. V. D. Berg and H. J. V. Schaik, “Diffraction of a plane electromagnetic wave by a perfectly conducting elliptic cylinder,” Appl. Sci. Res. 28, 145-157 (1973).

Blanch, G.

G. Blanch, in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1965).

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, C. N.

Chen, C. H.

R.-B. Wu and C. H. Chen, “Variational reaction formulation of scattering problem for anisotropic dielectric cylinders,” IEEE Trans. Antennas Propag. 34, 640-645 (1986).
[CrossRef]

Damaskos, N. J.

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

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, L. W. Li, and B. R. Guan, “Electromagnetic scattering by an inhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[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 II: Computer code and numerical results in two dimensions,” IEEE Trans. Antennas Propag. 35, 225-232 (1987).
[CrossRef]

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

Guan, B. R.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic scattering by an inhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

Hamid, M.

M. Hussein, A. Sebak, and M. Hamid, “Scattering and coupling properties of a slotted elliptic cylinder,” IEEE Trans. Electromagn. Compat. 36, 76-81 (1994).
[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]

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]

Hussein, M.

M. Hussein, A. Sebak, and M. Hamid, “Scattering and coupling properties of a slotted elliptic cylinder,” IEEE Trans. Electromagn. Compat. 36, 76-81 (1994).
[CrossRef]

Joyes, P.

R.-J. Tarento, K. H. Bennemann, P. Joyes, and J. V. d. Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606/026601-026605 (2004).
[CrossRef]

Kaklamani, D. I.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for oblique incidence scattering by an infinite dielectric cylinder,” Electr. Eng. 89, 353-361 (2007).
[CrossRef]

Lakhtakia, A.

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

Lee, S. S.

K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325-1328 (1995).

Li, L. W.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic scattering by an inhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

Lim, K.

K. Lim and S. S. Lee, “Analysis of electromagnetic scattering from an eccentric multilayered sphere,” IEEE Trans. Antennas Propag. 43, 1325-1328 (1995).

Mao, S.-C.

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford Univ. 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 and N. J. Damaskos, “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).

Papadakis, S. 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]

Richmond, J. H.

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

Schaik, H. J. V.

P. M. V. D. Berg and H. J. V. Schaik, “Diffraction of a plane electromagnetic wave by a perfectly conducting elliptic cylinder,” Appl. Sci. Res. 28, 145-157 (1973).

Sebak, A.

M. Hussein, A. Sebak, and M. Hamid, “Scattering and coupling properties of a slotted elliptic cylinder,” IEEE Trans. Electromagn. Compat. 36, 76-81 (1994).
[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).

Tadler, G. A.

N. G. Alexopoulos and G. A. Tadler, “Electromagnetic scattering from an elliptic cylinder loaded by continuous and discontinuous surface impedances,” J. Appl. Phys. 46, 1128-1134 (1975).
[CrossRef]

Taflove, A.

A. Taflove and K. Umashankar, “Radar cross section of general three-dimensional scatterers,” IEEE Trans. Electromagn. Compat. 25, 433-440 (1983).
[CrossRef]

Tarento, R.-J.

R.-J. Tarento, K. H. Bennemann, P. Joyes, and J. V. d. Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606/026601-026605 (2004).
[CrossRef]

Tsakiris, S. I.

Tsitsas, N. L.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for oblique incidence scattering by an infinite dielectric cylinder,” Electr. Eng. 89, 353-361 (2007).
[CrossRef]

N. L. Tsitsas and C. Athanasiadis, “On the scattering of spherical electromagnetic waves by a layered sphere,” Q. J. Mech. Appl. Math. 59, 55-74 (2006).
[CrossRef]

Umashankar, K.

A. Taflove and K. Umashankar, “Radar cross section of general three-dimensional scatterers,” IEEE Trans. Electromagn. Compat. 25, 433-440 (1983).
[CrossRef]

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 II: Computer code and numerical results in two dimensions,” IEEE Trans. Antennas Propag. 35, 225-232 (1987).
[CrossRef]

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

Uzunoglu, N. K.

Varadan, V. K.

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

Varadan, V. V.

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

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189-195 (1955).
[CrossRef]

Walle, J. V. d.

R.-J. Tarento, K. H. Bennemann, P. Joyes, and J. V. d. Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606/026601-026605 (2004).
[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, R.-B.

R.-B. Wu and C. H. Chen, “Variational reaction formulation of scattering problem for anisotropic dielectric cylinders,” IEEE Trans. Antennas Propag. 34, 640-645 (1986).
[CrossRef]

Wu, X. B.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic scattering by an inhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag. 53, 3982-3989 (2005).
[CrossRef]

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996-2003 (1997).
[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]

Wu, Z.-S.

Yasumoto, K.

X. B. Wu and K. Yasumoto, “Three-dimensional scattering by an infinite homogeneous anisotropic circular cylinder: an analytical solution,” J. Appl. Phys. 82, 1996-2003 (1997).
[CrossRef]

Yeh, C.

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]

Appl. Sci. Res. (1)

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Figures (6)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Incident wave.

Fig. 3
Fig. 3

Far-field pattern of | f ( v ) | 2 ( σ λ ) for an anisotropic elliptic cylinder under plane-wave incidence with different truncation orders ( θ i = 90 ° , φ i = 0 ° ), ϕ = 180°, ϵ x x = 2 ϵ y y = 2 ϵ z z = 4 ϵ 0 , ϵ x y = ϵ y x = 0 , μ x x = 2 μ y y = 2 μ z z = 4 μ 0 , μ x y = μ y x = 0 .

Fig. 4
Fig. 4

Far-field pattern of | f ( v ) | 2 and | g ( v ) | 2 for an anisotropic elliptic cylinder degenerated to a circular geometry under plane-wave incidence with ( θ i = 60 ° , φ i = 90 ° ), ϕ = 180°, u 0 = 6.0 , d λ = 1.239 E 3 , ϵ x x = 4.87526 ϵ 0 , μ ̿ = I ̿ μ 0 , ϵ y y = ϵ z z = 5.29 ϵ 0 , N = 6 .

Fig. 5
Fig. 5

Far-field pattern of | f ( v ) | 2 ( σ λ ) for an anisotropic elliptic cylinder under plane-wave incidence with ( θ i = 90 ° , φ i = 0 ° ), ϕ = 180°, ϵ x x = 2 ϵ y y = 2 ϵ z z = 4 ϵ 0 , ϵ x y = ϵ y x = 0 , μ x x = 2 μ y y = 2 μ z z = 4 μ 0 , μ x y = μ y x = 0 , N = 9 .

Fig. 6
Fig. 6

Far-field pattern of | f ( v ) | 2 and | g ( v ) | 2 for an anisotropic elliptic cylinder under plane-wave incidence with ( θ i = 75 ° or 90° and φ i = 0 ° or 90°), d λ = 0.4 , u 0 = 0.2 , ϵ x x = 2 ϵ y y = 2 ϵ z z = 4 ϵ 0 , ϵ x y = ϵ y x = 0 , μ x x = 2 μ y y = 2 μ z z = 4 μ 0 , μ x y = μ y x = 0 , N = 9 .

Equations (95)

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x = d cosh u cos v ,
y = d sinh u sin v ,
z = z ,
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 ̿ = ϵ ̿ t + ϵ z z z ̂ z ̂ , μ ̿ = μ ̿ t + μ z z z ̂ z ̂ ,
E ( r ) = E ( x , y ) e j k z i z , H ( r ) = H ( x , y ) e j k z i z ,
E ( x , y ) = E t + z ̂ E z , H ( x , y ) = H t + z ̂ H z
E t = 1 j ω ( ϵ ̿ ) 1 [ k z i ω det ( μ ̿ t ) μ ̿ t T t E z + p ̿ t H z ] ,
H t = 1 j ω ( μ ̿ ) 1 [ p ̿ t E z k z i ω det ( ϵ ̿ t ) ϵ ̿ t T t H z ] ,
ϵ ̿ = ϵ ̿ t ( k z i ω ) 2 μ ̿ t T det ( μ ̿ t ) , μ ̿ = μ ̿ t ( k z i ω ) 2 ϵ ̿ t T det ( ϵ ̿ t ) ,
p ̿ = z ̂ × .
t ϵ ̿ T t H z k z i ω det ( μ ̿ t ) t ϵ ̿ T p ̿ μ ̿ t T t E z + ω 2 μ z z det ( ϵ ̿ ) H z = 0 ,
t μ ̿ T t E z + k z i ω det ( ϵ ̿ t ) t μ ̿ T p ̿ ϵ ̿ t T t H z + ω 2 ϵ z z det ( μ ̿ ) E z = 0 .
D 1 ( t ) H z t α ̿ t E z = 0 ,
D 2 ( t ) E z + t β ̿ t H z = 0 ,
D 1 ( t ) = t ϵ ̿ T t + γ 1 2 , D 2 ( t ) = t μ ̿ T t + γ 2 2 ;
γ 1 = ω μ z z det ( ϵ ̿ ) , γ 2 = ω ϵ z z det ( μ ̿ ) ;
α ̿ = k z i ω det ( μ ̿ t ) ϵ ̿ T p ̿ μ ̿ t T , β ̿ = k z i ω det ( ϵ ̿ t ) μ ̿ T p ̿ ϵ ̿ t T .
H z = d ξ [ h a ( ξ ) e j κ + ( ξ ) ρ + h b ( ξ ) e j κ ( ξ ) ρ ] ,
E z = d ξ [ e a ( ξ ) e j κ + ( ξ ) ρ + e b ( ξ ) e j κ ( ξ ) ρ ] ,
H z = d ξ [ h a ( ξ ) e j κ + ( ξ ) ρ χ H ( ξ ) e b ( ξ ) e j κ ( ξ ) ρ ] = d ξ [ h 1 ( ξ ) M C m + + h 2 ( ξ ) M S m + χ H ( ξ ) { e 1 ( ξ ) M C m + e 2 ( ξ ) M S m } ] ,
E z = d ξ [ χ E ( ξ ) h a ( ξ ) e j κ + ( ξ ) ρ + e b ( ξ ) e j κ ( ξ ) ρ ] = d ξ [ χ E ( ξ ) { h 1 ( ξ ) M C m + + h 2 ( ξ ) M S m + } + e 1 ( ξ ) M C m + e 2 ( ξ ) M S m ] ,
where M C m ± = m = 0 j m M c m ( 1 ) ( q ± ( ξ ) , u ) c e m ( q ± ( ξ ) , v ) c e m ( q ± ( ξ ) , ξ ) ,
M S m ± = m = 1 j m M s m ( 1 ) ( q ± ( ξ ) , u ) s e m ( q ± ( ξ ) , v ) s e m ( q ± ( ξ ) , ξ ) ,
H z i = H 0 cos ϕ sin θ i e j k z i z e j k t i ρ cos ( φ φ i ) = 2 H 0 cos ϕ sin θ i e j k z i z [ m = 0 j m M c m ( 1 ) ( q 0 , u ) c e m ( q 0 , v ) c e m ( q 0 , φ i ) + m = 1 j m M s m ( 1 ) ( q 0 , u ) s e m ( q 0 , v ) s e m ( q 0 , φ i ) ] ,
E z i = η H 0 sin ϕ sin θ i e j k z i z e j k t i ρ cos ( φ φ i ) = 2 η H 0 sin ϕ sin θ i e j k z i z [ m = 0 j m M c m ( 1 ) ( q 0 , u ) c e m ( q 0 , v ) c e m ( q 0 , φ i ) + m = 1 j m M s m ( 1 ) ( q 0 , u ) s e m ( q 0 , v ) s e m ( q 0 , φ i ) ] ,
H z s = e j k z i z ( m = 0 M c m ( 4 ) ( q 0 , u ) c e m ( q 0 , v ) e m h + m = 1 M s m ( 4 ) ( q 0 , u ) s e m ( q 0 , v ) o m h ) ,
E z s = e j k z i z ( m = 0 M c m ( 4 ) ( q 0 , u ) c e m ( q 0 , v ) e m e + m = 1 M s m ( 4 ) ( q 0 , u ) s e m ( q 0 , v ) o m e ) ,
E v = 1 j ω det ( ϵ ̿ ) h ( a 2 + b 2 ) w ϵ ̿ T t ̿ H z + k z i j ω 2 det ( μ ̿ t ) det ( ϵ ̿ ) h ( a 2 + b 2 ) w ϵ ̿ T p ̿ μ ̿ t T t ̿ E z ,
H v = 1 j ω det ( μ ̿ ) h ( a 2 + b 2 ) w μ ̿ T t ̿ E z + k z i j ω 2 det ( ϵ ̿ t ) det ( μ ̿ ) h ( a 2 + b 2 ) w μ ̿ T p ̿ ϵ ̿ t T t ̿ H z ,
w = ( a , b ) , = [ u v ] T , p ̿ = [ 0 1 1 0 ] ,
t ̿ = [ a b b a ] , h = d a 2 + b 2 ,
E v i ( s ) = k η j h ( k t i ) 2 H z i ( s ) u k z i j h ( k t i ) 2 E z i ( s ) v ,
H v i ( s ) = k j h ( k t i ) 2 η E z i ( s ) u k z i j h ( k t i ) 2 H z i ( s ) v .
E z u 0 = E z i + E z s , H z u 0 = H z i + H z s ,
E v u 0 = E v i + E v s , H v u 0 = H v i + H v s ,
E z u 0 = e j k z i z E z ( u 0 ) , H z u 0 = e j k z i z H z ( u 0 ) ,
E v u 0 = e j k z i z E v ( u 0 ) , H v u 0 = e j k z i z H v ( u 0 ) ,
d ξ [ h 1 ( ξ ) H n ( e 11 ) ( ξ ) + h 2 ( ξ ) H n ( e 12 ) ( ξ ) + e 1 ( ξ ) E n ( e 11 ) ( ξ ) + e 2 ( ξ ) E n ( e 12 ) ( ξ ) ] = F n ( 1 ) ,
d ξ [ h 1 ( ξ ) H n ( e 21 ) ( ξ ) + h 2 ( ξ ) H n ( e 22 ) ( ξ ) + e 1 ( ξ ) E n ( e 21 ) ( ξ ) + e 2 ( ξ ) E n ( e 22 ) ( ξ ) ] = F n ( 2 ) ,
d ξ [ h 1 ( ξ ) H n ( h 11 ) ( ξ ) + h 2 ( ξ ) H n ( h 12 ) ( ξ ) + e 1 ( ξ ) E n ( h 11 ) ( ξ ) + e 2 ( ξ ) E n ( h 12 ) ( ξ ) ] = F n ( 3 ) ,
d ξ [ h 1 ( ξ ) H n ( h 21 ) ( ξ ) + h 2 ( ξ ) H n ( h 22 ) ( ξ ) + e 1 ( ξ ) E n ( h 21 ) ( ξ ) + e 2 ( ξ ) E n ( h 22 ) ( ξ ) ] = F n ( 4 ) ,
M c m ( 4 ) ( q 0 , u ) 2 π d k t i cosh u e j { d k t i cosh u ( 2 m + 1 ) π 4 } M s m ( 4 ) ( q 0 , u ) ,
D M c m ( 4 ) ( q 0 , u ) M c m ( 4 ) ( q 0 , u ) ( j d k t i sinh u 1 2 ) D M s m ( 4 ) ( q 0 , u ) ,
H s = π 2 ς ( u , z ) ( n ̂ f ( v ) v ̂ g ( v ) ) ,
E s = η π 2 ς ( u , z ) ( n ̂ g ( v ) + v ̂ f ( v ) ) ,
ς ( u , z ) = 2 π d k t i cosh u e j k z i z j { d k t i cosh u π 4 } , n ̂ = k z i u ̂ + k t i z ̂ k ;
f ( v ) = 2 π sin θ i ( m = 0 e j m π 2 c e m ( q 0 , v ) e m h + m = 1 e j m π 2 s e m ( q 0 , v ) o m h ) ,
g ( v ) = 2 π sin θ i ( m = 0 e j m π 2 c e m ( q 0 , v ) e m e + m = 1 e j m π 2 s e m ( q 0 , v ) o m e ) .
j k z i p ̿ E t + j ω μ ̿ t H t p ̿ t E z = 0 ,
j k z i p ̿ H t j ω ϵ ̿ t E t p ̿ t H z = 0 ,
z ̂ × E t = j ω μ z z H z ,
z ̂ × H t = j ω ϵ z z E z .
D 1 ( j κ + ) h a ( ξ ) = κ + α ̿ κ + e a ( ξ ) ,
D 1 ( j κ ) h b ( ξ ) = κ α ̿ κ e b ( ξ ) ,
D 2 ( j κ + ) e a ( ξ ) = κ + β ̿ κ + h a ( ξ ) ,
D 2 ( j κ ) e b ( ξ ) = κ β ̿ κ h b ( ξ ) .
D 1 ( j κ ) D 2 ( j κ ) + ( κ α ̿ κ ) ( κ β ̿ κ ) = 0 .
κ ± 2 ( ξ ) = B + ± B 2 4 γ 1 2 γ 2 2 ( κ ̂ α ̿ κ ̂ ) ( κ ̂ β ̿ κ ̂ ) 2 [ ( κ ̂ ϵ ̿ κ ̂ ) ( κ ̂ μ ̿ κ ̂ ) + ( κ ̂ α ̿ κ ̂ ) ( κ ̂ β ̿ κ ̂ ) ] ,
κ + 2 ω 2 μ z z det ( ϵ ̿ t ) κ ̂ ϵ ̿ t κ ̂ ; κ 2 ω 2 ϵ z z det ( μ ̿ t ) κ ̂ μ ̿ t κ ̂ .
χ H ( ξ ) = κ α ̿ κ D 1 ( j κ ) = h b e b , χ E ( ξ ) = κ + β ̿ κ + D 2 ( j κ + ) = e a h a .
j ( k t i ) 2 E t i ( s ) = k z i t E z i ( s ) k η z ̂ × t H z i ( s ) ,
j ( k t i ) 2 H t i ( s ) = k z i t H z i ( s ) + k η z ̂ × t E z i ( s ) ,
H n ( e 11 ) ( ξ ) = m = 0 j m c e m ( q + ( ξ ) , ξ ) { D M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 1 C m + C n f 2 C 3 C m + C n ] + M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 2 D C m + C n f 2 C 4 D C m + C n + η sin 2 θ i D M c n ( 4 ) ( q 0 , u ) M c n ( 4 ) ( q 0 , u ) C m + C n ] } ,
H n ( e 12 ) ( ξ ) = m = 1 j m s e m ( q + ( ξ ) , ξ ) { D M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 1 S m + C n f 2 C 3 S m + C n ] + M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 2 D S m + C n f 2 C 4 D S m + C n + χ E cos θ i π sin 2 θ i m = 1 ( S m + S m D S m C n ) ] } ,
E n ( e 11 ) ( ξ ) = m = 0 j m c e m ( q ( ξ ) , ξ ) { D M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 1 C m C n + f 4 C 3 C m C n ] + M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 2 D C m C n + f 4 C 4 D C m C n η χ H sin 2 θ i D M c n ( 4 ) ( q 0 , u ) M c n ( 4 ) ( q 0 , u ) C m C n ] } ,
E n ( e 12 ) ( ξ ) = m = 1 j m s e m ( q ( ξ ) , ξ ) { D M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 1 S m C n + f 4 C 3 S m C n ] + M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 2 D S m C n + f 4 C 4 D S m C n + cos θ i π sin 2 θ i m = 1 ( S m S m D S m C n ) ] } ;
H n ( e 21 ) ( ξ ) = m = 0 j m c e m ( q + ( ξ ) , ξ ) { D M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 1 C m + S n f 2 C 3 C m + S n ] + M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 2 D C m + S n f 2 C 4 D C m + S n + χ E cos θ i π sin 2 θ i m = 0 ( C m + C m D C m S n ) ] } ,
H n ( e 22 ) ( ξ ) = m = 1 j m s e m ( q + ( ξ ) , ξ ) { D M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 1 S m + S n f 2 C 3 S m + S n ] + M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 1 C 2 D S m + S n f 2 C 4 D S m + S n + η sin 2 θ i D M s n ( 4 ) ( q 0 , u ) M s n ( 4 ) ( q 0 , u ) S m + S n ] } ,
E n ( e 21 ) ( ξ ) = m = 0 j m c e m ( q ( ξ ) , ξ ) { D M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 1 C m S n + f 4 C 3 C m S n ] + M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 2 D C m S n + f 4 C 4 D C m S n + cos θ i π sin 2 θ i m = 0 ( C m C m D C m S n ) ] } ,
E n ( e 22 ) ( ξ ) = m = 1 j m s e m ( q ( ξ ) , ξ ) { D M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 1 S m S n + f 4 C 3 S m S n ] + M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 3 C 2 D S m S n + f 4 C 4 D S m S n η χ H sin 2 θ i D M s n ( 4 ) ( q 0 , u ) M s n ( 4 ) ( q 0 , u ) S m S n ] } ;
H n ( h 11 ) ( ξ ) = m = 0 j m c e m ( q + ( ξ ) , ξ ) { D M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 5 C m + C n + f 6 C 7 C m + C n ] + M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 6 D C m + C n + f 6 C 8 D C m + C n χ E η sin 2 θ i D M c n ( 4 ) ( q 0 , u ) M c n ( 4 ) ( q 0 , u ) C m + C n ] } ,
H n ( h 12 ) ( ξ ) = m = 1 j m s e m ( q + ( ξ ) , ξ ) { D M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 5 S m + C n + f 6 C 7 S m + C n ] + M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 6 D S m + C n + f 6 C 8 D S m + C n + cos θ i π sin 2 θ i m = 1 ( S m + S m D S m C n ) ] } ,
E n ( h 11 ) ( ξ ) = m = 0 j m c e m ( q ( ξ ) , ξ ) { D M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 5 C m C n f 8 C 7 C m C n ] + M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 6 D C m C n f 8 C 8 D C m C n 1 η sin 2 θ i D M c n ( 4 ) ( q 0 , u ) M c n ( 4 ) ( q 0 , u ) C m C n ] } ,
E n ( h 12 ) ( ξ ) = m = 1 j m s e m ( q ( ξ ) , ξ ) { D M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 5 S m C n f 8 C 7 S m C n ] + M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 6 D S m C n f 8 C 8 D S m C n χ H cos θ i π sin 2 θ i m = 1 ( S m S m D S m C n ) ] } ;
H n ( h 21 ) ( ξ ) = m = 0 j m c e m ( q + ( ξ ) , ξ ) { D M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 5 C m + S n + f 6 C 7 C m + S n ] + M c m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 6 D C m + S n + f 6 C 8 D C m + S n + cos θ i π sin 2 θ i m = 0 ( C m + C m D C m S n ) ] } ,
H n ( h 22 ) ( ξ ) = m = 1 j m s e m ( q + ( ξ ) , ξ ) { D M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 5 S m + S n + f 6 C 7 S m + S n ] + M s m ( 1 ) ( q + ( ξ ) , u 0 ) [ f 5 C 6 D S m + S n + f 6 C 8 D S m + S n χ E η sin 2 θ i D M s n ( 4 ) ( q 0 , u ) M s n ( 4 ) ( q 0 , u ) S m + S n ] } ,
E n ( h 21 ) ( ξ ) = m = 0 j m c e m ( q ( ξ ) , ξ ) { D M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 5 C m S n f 8 C 7 C m S n ] + M c m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 6 D C m S n f 8 C 8 D C m S n χ H cos θ i π sin 2 θ i m = 0 ( C m C m D C m S n ) ] } ,
E n ( h 22 ) ( ξ ) = m = 1 j m s e m ( q ( ξ ) , ξ ) { D M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 5 S m S n f 8 C 7 S m S n ] + M s m ( 1 ) ( q ( ξ ) , u 0 ) [ f 7 C 6 D S m S n f 8 C 8 D S m S n 1 η sin 2 θ i D M s n ( 4 ) ( q 0 , u ) M s n ( 4 ) ( q 0 , u ) S m S n ] } ;
F n ( 1 ) = 2 j n H 0 η π cos ϕ c e n ( q 0 , φ i ) sin θ i [ D M c n ( 1 ) ( q 0 , u 0 ) D M c n ( 4 ) ( q 0 , u 0 ) M c n ( 4 ) ( q 0 , u 0 ) M c n ( 1 ) ( q 0 , u 0 ) ] ,
F n ( 2 ) = 2 j n H 0 η π cos ϕ s e n ( q 0 , φ i ) sin θ i [ D M s n ( 1 ) ( q 0 , u 0 ) D M s n ( 4 ) ( q 0 , u 0 ) M s n ( 4 ) ( q 0 , u 0 ) M s n ( 1 ) ( q 0 , u 0 ) ] ,
F n ( 3 ) = 2 j n H 0 π sin ϕ c e n ( q 0 , φ i ) sin θ i [ D M c n ( 1 ) ( q 0 , u 0 ) D M c n ( 4 ) ( q 0 , u 0 ) M c n ( 4 ) ( q 0 , u 0 ) M c n ( 1 ) ( q 0 , u 0 ) ] ,
F n ( 4 ) = 2 j n H 0 π sin ϕ s e n ( q 0 , φ i ) sin θ i [ D M s n ( 1 ) ( q 0 , u 0 ) D M s n ( 4 ) ( q 0 , u 0 ) M s n ( 4 ) ( q 0 , u 0 ) M s n ( 1 ) ( q 0 , u 0 ) ] ;
f 1 = χ E cos θ i det ( μ ̿ t ) det ( ϵ ̿ ) , f 2 = η det ( ϵ ̿ ) , f 3 = cos θ i det ( μ ̿ t ) det ( ϵ ̿ ) , f 4 = η χ H det ( ϵ ̿ ) ,
f 5 = χ E η det ( μ ̿ ) , f 6 = cos θ i det ( ϵ ̿ t ) det ( μ ̿ ) , f 7 = 1 η det ( μ ̿ ) , f 8 = χ H cos θ i det ( ϵ ̿ t ) det ( μ ̿ ) ;
C k ( C S ) m ± C n = 0 2 π c k ( c e s e ) m ( q ± ( ξ ) , v ) c e n ( q 0 , v ) d v , k = 1 , 3 , 5 , 7 ,
C l ( D C D S ) m ± C n = 0 2 π c l ( D c e D s e ) m ( q ± ( ξ ) , v ) c e n ( q 0 , v ) d v , l = 2 , 4 , 6 , 8 ,
S m ± S n = 0 2 π s e m ( q ± ( ξ ) , v ) s e n ( q 0 , v ) d v ,
C m ± C n = 0 2 π c e m ( q ± ( ξ ) , v ) c e n ( q 0 , v ) d v ,
C i ( S C ) m ± S n = 0 2 π c i ( s e c e ) m ( q ± ( ξ ) , v ) s e n ( q 0 , v ) d v , i = 1 , 3 , 5 , 7 ,
C p ( D S D C ) m ± S n = 0 2 π c p ( D s e D c e ) m ( q ± ( ξ ) , v ) s e n ( q 0 , v ) d v , p = 2 , 4 , 6 , 8 ,
D S m C n = 0 2 π D s e m ( q 0 , v ) c e n ( q 0 , v ) d v ,
D C m S n = 0 2 π D c e m ( q 0 , v ) s e n ( q 0 , v ) d v ;
( c 1 , c 2 ) = 1 a 2 + b 2 w ϵ ̿ T p ̿ μ ̿ t T t ̿ , ( c 3 , c 4 ) = 1 a 2 + b 2 w ϵ ̿ T t ̿ ,
( c 5 , c 6 ) = 1 a 2 + b 2 w μ ̿ T t ̿ , ( c 7 , c 8 ) = 1 a 2 + b 2 w μ ̿ T p ̿ ϵ ̿ t T t ̿ .

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