Abstract

The problem of diffraction of homogeneous and inhomogeneous plane waves at the discontinuity formed by perfectly conducting and impedance half-planes is examined by the method of modified theory of physical optics (MTPO). The MTPO integral of the reflected scattered waves by the perfectly conducting half-plane is reconstructed in order to include the effect of the diffracted wave coming from the edge of the impedance half-plane. The integrals are evaluated by a uniform asymptotic method. The results are plotted numerically and compared with the literature.

© 2007 Optical Society of America

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  1. G. Uzgoren, A. Buyukaksoy, and A. H. Serbest, "Diffraction coefficient related to a discontinuity formed by impedance and resistive half planes," Proc. IEEE 136, 19-23 (1989).
  2. A. Buyukaksoy, G. Uzgoren, and A. H. Serbest, "Diffraction of an obliquely incident plane wave by the discontinuity of a two part thin dielectric plane," Int. J. Eng. Sci. 27, 701-710 (1989).
    [CrossRef]
  3. R. Sendag and A. H. Serbest, "Scattering of plane electromagnetic waves at the junction formed by a PEC half-plane and a half-plane with anisotropic conductivity," in Proceedings of MELECON 98 9th Mediterranean Electro-technical Conference (IEEE, 1998) Vol. 1, pp. 278-282.
    [CrossRef]
  4. T. B. A. Senior, "Skew incidence on a material junction," Radio Sci. 26, 305-311 (1991).
    [CrossRef]
  5. R. Tiberio and G. Pelossi, "High-frequency scattering from the edges of impedance discontinuities on a flat plate," IEEE Trans. Antennas Propag. 31, 590-596 (1983).
    [CrossRef]
  6. R. G. Rojas, "Wiener-Hopf analysis of the EM diffraction by an impedance discontinuity in a planar surface and by an impedance half-plane," IEEE Trans. Antennas Propag. 36, 71-83 (1988).
    [CrossRef]
  7. H. C. Ly, R. G. Rojas, and P. H. Pathak, "EM plane wave diffraction by a planar junction of two thin material half-planes--oblique incidence," IEEE Trans. Antennas Propag. 41, 429-441 (1993).
    [CrossRef]
  8. A. Vallecchi, "Diffraction by a planar junction between a perfectly conducting half plane and a resistive sheet illuminated by a dipole close to its edge," in Proceedings of IEEE Antennas and Propagation Society International Symposium (IEEE, 2001) Vol. 2, pp. 208-211.
  9. Y. Z. Umul, "Modified theory of physical optics," Opt. Express 12, 4959-4972 (2004).
    [CrossRef] [PubMed]
  10. Y. Z. Umul, "Modified theory of physical optics solution of impedance half plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
    [CrossRef]
  11. T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).
    [CrossRef]
  12. S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
    [CrossRef]
  13. R. G. Kouyoumjian and P. B. Pathak, "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface," Proc. IEEE 62, 1448-1461 (1974).
    [CrossRef]
  14. L. Ronchi, V. Russo, G. T. D. Francia, and C. Zaccagnini, "Scattering of evanescent waves by cylindrical structures," Opt. Acta 8, 281-299 (1961).
    [CrossRef]
  15. H. L. Bertoni, A. C. Green, and L. B. Felsen, "Shadowing an inhomogeneous plane wave by an edge," J. Opt. Soc. Am. 68, 983-989 (1978).
    [CrossRef]
  16. G. A. Deschamps, S. W. Lee, E. Gowan III, and T. Fontana, "Diffraction of an evanescent plane wave by a half plane," Wave Motion 1, 25-35 (1979).
    [CrossRef]
  17. S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and V. H. Saward, "Edge diffraction of complex rays," Wave Motion 33, 41-49 (2001).
    [CrossRef]
  18. R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an edge," Radio Sci. 31, 1387-1397 (1996).
    [CrossRef]
  19. G. Manara, P. Nepa, R. G. Kouyoumjian, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium," IEEE Trans. Antennas Propag. 46, 1753-1755 (1998).
    [CrossRef]

2006 (1)

Y. Z. Umul, "Modified theory of physical optics solution of impedance half plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

2004 (1)

2001 (1)

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and V. H. Saward, "Edge diffraction of complex rays," Wave Motion 33, 41-49 (2001).
[CrossRef]

1999 (1)

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

1998 (1)

G. Manara, P. Nepa, R. G. Kouyoumjian, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium," IEEE Trans. Antennas Propag. 46, 1753-1755 (1998).
[CrossRef]

1996 (1)

R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an edge," Radio Sci. 31, 1387-1397 (1996).
[CrossRef]

1993 (1)

H. C. Ly, R. G. Rojas, and P. H. Pathak, "EM plane wave diffraction by a planar junction of two thin material half-planes--oblique incidence," IEEE Trans. Antennas Propag. 41, 429-441 (1993).
[CrossRef]

1991 (1)

T. B. A. Senior, "Skew incidence on a material junction," Radio Sci. 26, 305-311 (1991).
[CrossRef]

1989 (2)

G. Uzgoren, A. Buyukaksoy, and A. H. Serbest, "Diffraction coefficient related to a discontinuity formed by impedance and resistive half planes," Proc. IEEE 136, 19-23 (1989).

A. Buyukaksoy, G. Uzgoren, and A. H. Serbest, "Diffraction of an obliquely incident plane wave by the discontinuity of a two part thin dielectric plane," Int. J. Eng. Sci. 27, 701-710 (1989).
[CrossRef]

1988 (1)

R. G. Rojas, "Wiener-Hopf analysis of the EM diffraction by an impedance discontinuity in a planar surface and by an impedance half-plane," IEEE Trans. Antennas Propag. 36, 71-83 (1988).
[CrossRef]

1983 (1)

R. Tiberio and G. Pelossi, "High-frequency scattering from the edges of impedance discontinuities on a flat plate," IEEE Trans. Antennas Propag. 31, 590-596 (1983).
[CrossRef]

1979 (1)

G. A. Deschamps, S. W. Lee, E. Gowan III, and T. Fontana, "Diffraction of an evanescent plane wave by a half plane," Wave Motion 1, 25-35 (1979).
[CrossRef]

1978 (1)

1974 (1)

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

1961 (1)

L. Ronchi, V. Russo, G. T. D. Francia, and C. Zaccagnini, "Scattering of evanescent waves by cylindrical structures," Opt. Acta 8, 281-299 (1961).
[CrossRef]

Bertoni, H. L.

Buyukaksoy, A.

A. Buyukaksoy, G. Uzgoren, and A. H. Serbest, "Diffraction of an obliquely incident plane wave by the discontinuity of a two part thin dielectric plane," Int. J. Eng. Sci. 27, 701-710 (1989).
[CrossRef]

G. Uzgoren, A. Buyukaksoy, and A. H. Serbest, "Diffraction coefficient related to a discontinuity formed by impedance and resistive half planes," Proc. IEEE 136, 19-23 (1989).

Chapman, S. J.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and V. H. Saward, "Edge diffraction of complex rays," Wave Motion 33, 41-49 (2001).
[CrossRef]

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, S. W. Lee, E. Gowan III, and T. Fontana, "Diffraction of an evanescent plane wave by a half plane," Wave Motion 1, 25-35 (1979).
[CrossRef]

Felsen, L. B.

Fontana, T.

G. A. Deschamps, S. W. Lee, E. Gowan III, and T. Fontana, "Diffraction of an evanescent plane wave by a half plane," Wave Motion 1, 25-35 (1979).
[CrossRef]

Francia, G. T. D.

L. Ronchi, V. Russo, G. T. D. Francia, and C. Zaccagnini, "Scattering of evanescent waves by cylindrical structures," Opt. Acta 8, 281-299 (1961).
[CrossRef]

Gowan, E.

G. A. Deschamps, S. W. Lee, E. Gowan III, and T. Fontana, "Diffraction of an evanescent plane wave by a half plane," Wave Motion 1, 25-35 (1979).
[CrossRef]

Green, A. C.

Kouyoumjian, R. G.

G. Manara, P. Nepa, R. G. Kouyoumjian, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium," IEEE Trans. Antennas Propag. 46, 1753-1755 (1998).
[CrossRef]

R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an edge," Radio Sci. 31, 1387-1397 (1996).
[CrossRef]

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

Lawry, J. M. H.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and V. H. Saward, "Edge diffraction of complex rays," Wave Motion 33, 41-49 (2001).
[CrossRef]

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Lee, S. W.

G. A. Deschamps, S. W. Lee, E. Gowan III, and T. Fontana, "Diffraction of an evanescent plane wave by a half plane," Wave Motion 1, 25-35 (1979).
[CrossRef]

Ly, H. C.

H. C. Ly, R. G. Rojas, and P. H. Pathak, "EM plane wave diffraction by a planar junction of two thin material half-planes--oblique incidence," IEEE Trans. Antennas Propag. 41, 429-441 (1993).
[CrossRef]

Manara, G.

G. Manara, P. Nepa, R. G. Kouyoumjian, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium," IEEE Trans. Antennas Propag. 46, 1753-1755 (1998).
[CrossRef]

R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an edge," Radio Sci. 31, 1387-1397 (1996).
[CrossRef]

Nepa, P.

G. Manara, P. Nepa, R. G. Kouyoumjian, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium," IEEE Trans. Antennas Propag. 46, 1753-1755 (1998).
[CrossRef]

R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an edge," Radio Sci. 31, 1387-1397 (1996).
[CrossRef]

Ockendon, J. R.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and V. H. Saward, "Edge diffraction of complex rays," Wave Motion 33, 41-49 (2001).
[CrossRef]

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Pathak, P. B.

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

Pathak, P. H.

H. C. Ly, R. G. Rojas, and P. H. Pathak, "EM plane wave diffraction by a planar junction of two thin material half-planes--oblique incidence," IEEE Trans. Antennas Propag. 41, 429-441 (1993).
[CrossRef]

Pelossi, G.

R. Tiberio and G. Pelossi, "High-frequency scattering from the edges of impedance discontinuities on a flat plate," IEEE Trans. Antennas Propag. 31, 590-596 (1983).
[CrossRef]

Rojas, R. G.

H. C. Ly, R. G. Rojas, and P. H. Pathak, "EM plane wave diffraction by a planar junction of two thin material half-planes--oblique incidence," IEEE Trans. Antennas Propag. 41, 429-441 (1993).
[CrossRef]

R. G. Rojas, "Wiener-Hopf analysis of the EM diffraction by an impedance discontinuity in a planar surface and by an impedance half-plane," IEEE Trans. Antennas Propag. 36, 71-83 (1988).
[CrossRef]

Ronchi, L.

L. Ronchi, V. Russo, G. T. D. Francia, and C. Zaccagnini, "Scattering of evanescent waves by cylindrical structures," Opt. Acta 8, 281-299 (1961).
[CrossRef]

Russo, V.

L. Ronchi, V. Russo, G. T. D. Francia, and C. Zaccagnini, "Scattering of evanescent waves by cylindrical structures," Opt. Acta 8, 281-299 (1961).
[CrossRef]

Saward, V. H.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and V. H. Saward, "Edge diffraction of complex rays," Wave Motion 33, 41-49 (2001).
[CrossRef]

Sendag, R.

R. Sendag and A. H. Serbest, "Scattering of plane electromagnetic waves at the junction formed by a PEC half-plane and a half-plane with anisotropic conductivity," in Proceedings of MELECON 98 9th Mediterranean Electro-technical Conference (IEEE, 1998) Vol. 1, pp. 278-282.
[CrossRef]

Senior, T. B. A.

T. B. A. Senior, "Skew incidence on a material junction," Radio Sci. 26, 305-311 (1991).
[CrossRef]

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).
[CrossRef]

Serbest, A. H.

A. Buyukaksoy, G. Uzgoren, and A. H. Serbest, "Diffraction of an obliquely incident plane wave by the discontinuity of a two part thin dielectric plane," Int. J. Eng. Sci. 27, 701-710 (1989).
[CrossRef]

G. Uzgoren, A. Buyukaksoy, and A. H. Serbest, "Diffraction coefficient related to a discontinuity formed by impedance and resistive half planes," Proc. IEEE 136, 19-23 (1989).

R. Sendag and A. H. Serbest, "Scattering of plane electromagnetic waves at the junction formed by a PEC half-plane and a half-plane with anisotropic conductivity," in Proceedings of MELECON 98 9th Mediterranean Electro-technical Conference (IEEE, 1998) Vol. 1, pp. 278-282.
[CrossRef]

Taute, B. J. E.

G. Manara, P. Nepa, R. G. Kouyoumjian, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium," IEEE Trans. Antennas Propag. 46, 1753-1755 (1998).
[CrossRef]

R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an edge," Radio Sci. 31, 1387-1397 (1996).
[CrossRef]

Tew, R. H.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Tiberio, R.

R. Tiberio and G. Pelossi, "High-frequency scattering from the edges of impedance discontinuities on a flat plate," IEEE Trans. Antennas Propag. 31, 590-596 (1983).
[CrossRef]

Umul, Y. Z.

Y. Z. Umul, "Modified theory of physical optics solution of impedance half plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

Y. Z. Umul, "Modified theory of physical optics," Opt. Express 12, 4959-4972 (2004).
[CrossRef] [PubMed]

Uzgoren, G.

G. Uzgoren, A. Buyukaksoy, and A. H. Serbest, "Diffraction coefficient related to a discontinuity formed by impedance and resistive half planes," Proc. IEEE 136, 19-23 (1989).

A. Buyukaksoy, G. Uzgoren, and A. H. Serbest, "Diffraction of an obliquely incident plane wave by the discontinuity of a two part thin dielectric plane," Int. J. Eng. Sci. 27, 701-710 (1989).
[CrossRef]

Vallecchi, A.

A. Vallecchi, "Diffraction by a planar junction between a perfectly conducting half plane and a resistive sheet illuminated by a dipole close to its edge," in Proceedings of IEEE Antennas and Propagation Society International Symposium (IEEE, 2001) Vol. 2, pp. 208-211.

Volakis, J. L.

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).
[CrossRef]

Zaccagnini, C.

L. Ronchi, V. Russo, G. T. D. Francia, and C. Zaccagnini, "Scattering of evanescent waves by cylindrical structures," Opt. Acta 8, 281-299 (1961).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

R. Tiberio and G. Pelossi, "High-frequency scattering from the edges of impedance discontinuities on a flat plate," IEEE Trans. Antennas Propag. 31, 590-596 (1983).
[CrossRef]

R. G. Rojas, "Wiener-Hopf analysis of the EM diffraction by an impedance discontinuity in a planar surface and by an impedance half-plane," IEEE Trans. Antennas Propag. 36, 71-83 (1988).
[CrossRef]

H. C. Ly, R. G. Rojas, and P. H. Pathak, "EM plane wave diffraction by a planar junction of two thin material half-planes--oblique incidence," IEEE Trans. Antennas Propag. 41, 429-441 (1993).
[CrossRef]

Y. Z. Umul, "Modified theory of physical optics solution of impedance half plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

G. Manara, P. Nepa, R. G. Kouyoumjian, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an impedance wedge in a lossy medium," IEEE Trans. Antennas Propag. 46, 1753-1755 (1998).
[CrossRef]

Int. J. Eng. Sci. (1)

A. Buyukaksoy, G. Uzgoren, and A. H. Serbest, "Diffraction of an obliquely incident plane wave by the discontinuity of a two part thin dielectric plane," Int. J. Eng. Sci. 27, 701-710 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

L. Ronchi, V. Russo, G. T. D. Francia, and C. Zaccagnini, "Scattering of evanescent waves by cylindrical structures," Opt. Acta 8, 281-299 (1961).
[CrossRef]

Opt. Express (1)

Proc. IEEE (2)

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

G. Uzgoren, A. Buyukaksoy, and A. H. Serbest, "Diffraction coefficient related to a discontinuity formed by impedance and resistive half planes," Proc. IEEE 136, 19-23 (1989).

Radio Sci. (2)

T. B. A. Senior, "Skew incidence on a material junction," Radio Sci. 26, 305-311 (1991).
[CrossRef]

R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, "The diffraction of an inhomogeneous plane wave by an edge," Radio Sci. 31, 1387-1397 (1996).
[CrossRef]

SIAM Rev. (1)

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Wave Motion (2)

G. A. Deschamps, S. W. Lee, E. Gowan III, and T. Fontana, "Diffraction of an evanescent plane wave by a half plane," Wave Motion 1, 25-35 (1979).
[CrossRef]

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and V. H. Saward, "Edge diffraction of complex rays," Wave Motion 33, 41-49 (2001).
[CrossRef]

Other (3)

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).
[CrossRef]

R. Sendag and A. H. Serbest, "Scattering of plane electromagnetic waves at the junction formed by a PEC half-plane and a half-plane with anisotropic conductivity," in Proceedings of MELECON 98 9th Mediterranean Electro-technical Conference (IEEE, 1998) Vol. 1, pp. 278-282.
[CrossRef]

A. Vallecchi, "Diffraction by a planar junction between a perfectly conducting half plane and a resistive sheet illuminated by a dipole close to its edge," in Proceedings of IEEE Antennas and Propagation Society International Symposium (IEEE, 2001) Vol. 2, pp. 208-211.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the planar junction.

Fig. 2
Fig. 2

Total diffracted field for sin θ = 2 .

Fig. 3
Fig. 3

Total diffracted field for sin θ = 2 , 4 , 10 .

Fig. 4
Fig. 4

Scattered, GO, and diffracted fields from the first half-plane.

Fig. 5
Fig. 5

Scattered, GO, and diffracted fields from the second half-plane.

Fig. 6
Fig. 6

Total scattered field.

Fig. 7
Fig. 7

Total field for different values of ϕ 02 .

Fig. 8
Fig. 8

Propagation of an inhomogeneous plane wave.

Fig. 9
Fig. 9

Variation of the total field for sin θ = 4 , 10 .

Equations (41)

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

E i = e z E i e j k ρ cos ( ϕ ϕ 0 )
E r s 1 = e z k E i e j π 4 2 π 0 Γ ( β , ϕ 0 , θ ) f ( β ) e j k x cos ϕ 0 e j k R k R d x ,
E r s 2 = e z k E i e j π 4 2 π 0 g ( β ) e j k x cos ϕ 0 e j k R k R d x
Γ ( β , ϕ 0 , θ ) = sin β + ϕ 0 2 sin θ sin β + ϕ 0 2 + sin θ .
0 K ( α ) e j k ψ ( α ) d α e j π 4 2 π k K ( α s ) e j k ψ ( α s ) ψ ( α s ) U ( k [ ψ ( α s ) ψ ( 0 ) ] ) + K ( 0 ) e j k ψ ( 0 ) j k ψ ( 0 )
E r s 1 = E i { Γ ( ϕ 0 , ϕ 0 , θ ) f ( ϕ 0 ) sin ϕ 0 e j k ρ cos ( ϕ + ϕ 0 ) U ( 2 k ρ cos ϕ + ϕ 0 2 ) + e j π 4 2 π Γ ( π ϕ , ϕ 0 , θ ) f ( π ϕ ) cos ϕ + cos ϕ 0 e j k ρ k ρ } ,
E r s 2 = E i { g ( ϕ 0 ) sin ϕ 0 e j k ρ cos ( ϕ + ϕ 0 ) U ( 2 k ρ cos ϕ + ϕ 0 2 ) e j π 4 2 π g ( π ϕ ) cos ϕ + cos ϕ 0 e j k ρ k ρ }
n × E t y = 0 = 0 ,
n × ( n × E t ) y = 0 = Z n × H t y = 0
n × ( E i + E r 2 ) y = 0 = 0 ,
n × ( E d 1 + E d 2 ) y = 0 = 0 .
g ( ϕ 0 ) = sin ϕ 0 .
g ( π ) = Γ ( π , ϕ 0 , θ ) f ( π ) ,
n × [ n × ( E i + E r 1 ) ] y = 0 = Z n × ( H i + H r 1 ) y = 0 ,
n × ( E d 1 + E d 2 ) y = 0 0 , for k ρ 1 ,
f ( ϕ 0 ) = sin ϕ 0 ,
g ( 0 ) = Γ ( 0 , ϕ 0 , θ ) f ( 0 )
f ( β ) = ( sin β + ϕ 0 2 sin β ϕ 0 2 ) sin β ϕ 0 2 sin θ sin β ϕ 0 2 + sin θ ,
g ( β ) = sin β ϕ 0 2 sin β + ϕ 0 2 ,
E r s 1 = e z k E i e j π 4 2 π 0 Γ ( β , ϕ 0 , θ ) Γ ( β , ϕ 0 , θ ) ( sin β + ϕ 0 2 sin β ϕ 0 2 ) e j k x cos ϕ 0 e j k R k R d x ,
E r s 2 = e z k E i e j π 4 2 π 0 ( sin β + ϕ 0 2 sin β ϕ 0 2 ) e j k x cos ϕ 0 e j k R k R d x
E st = E i + E r 1 GO + E r 2 GO + E d t ,
E r 1 GO = E i sin ϕ 0 sin θ sin ϕ 0 + sin θ U ( 2 k ρ cos ϕ + ϕ 0 2 ) e j k ρ cos ( ϕ + ϕ 0 ) ,
E r 2 GO = E i U ( 2 k ρ cos ϕ + ϕ 0 2 ) e j k ρ cos ( ϕ + ϕ 0 )
E d 1 = e j π 4 2 π Γ ( π ϕ , ϕ 0 , θ ) Γ ( π ϕ , ϕ 0 , θ ) sin ϕ 2 sin ϕ 0 2 cos ϕ + cos ϕ 0 e j k ρ k ρ ,
E d 2 = e j π 4 2 π sin ϕ 2 sin ϕ 0 2 cos ϕ + cos ϕ 0 e j k ρ k ρ ,
T ( ξ ) = F [ ξ ] F ̂ ( ξ ) ,
F [ ξ ] = e j π 4 π ξ e j t 2 d t ,
F ̂ ( ξ ) = e j π 4 2 π e j ξ 2 ξ .
E d t = e j π 4 2 π [ 1 Γ ( π ϕ , ϕ 0 , θ ) Γ ( π ϕ , ϕ 0 , θ ) ] sin ϕ 2 sin ϕ 0 2 cos ϕ + cos ϕ 0 T ( 2 k ρ i cos ϕ + ϕ 0 2 ) e j k ρ k ρ
z = x y + 2 y e j π 4 ,
e j π 4 π α e j t 2 d t = sgn ( x y ) [ e j π 4 π z e j t 2 d t U ( y x ) ] ,
T ( z ) = F [ α ] F ̂ ( α )
ξ 1 = 2 k ρ cos ϕ + ϕ 01 2 cosh ϕ 02 2 ,
ξ 2 = 2 k ρ sin ϕ + ϕ 01 2 sinh ϕ 02 2 ,
T ( ξ ) = F [ ξ 1 ξ 2 + 2 ξ 2 e j π 4 ] F ̂ ( ξ 1 ξ 2 + 2 ξ 2 e j π 4 ) ,
E r 1 GO = E i sin ϕ 0 sin θ sin ϕ 0 + sin θ U ( ξ 1 ξ 2 ) e j k ρ cos ( ϕ + ϕ 0 ) ,
E r 2 GO = E i U ( ξ 2 ξ i ) e j k ρ cos ( ϕ + ϕ 0 ) ,
E d t = e j π 4 8 π η K ( η , cos ϕ ) K ( η , cos ϕ 0 ) sin ϕ 2 sin ϕ 0 2 cos ϕ + cos ϕ 0 T ( 2 k ρ cos ϕ + ϕ 0 2 ) e j k ρ k ρ ,
K ( η , cos χ ) = 4 η sin χ 2 [ ψ x ( 3 π 2 χ θ ) ψ π ( π 2 χ + θ ) ] 2 [ ψ π ( π 2 ) ] 4 [ 1 + 2 cos ( π 2 χ + θ 2 ) ] [ 1 + 2 cos ( π 2 χ θ 2 ) ] ,
ψ π ( ν ) = exp ( 1 8 π 0 ν π sin t 2 2 sin t 2 + 2 t cos t d t ) .

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