Abstract

We investigate the scattering process of plane waves by a conducting half-plane between two dielectric media by introducing novel boundary conditions, in terms of soft and hard surfaces. The cases of soft and hard half-planes are studied independently. The scattered waves are examined numerically. The numerical results show that the behavior of the fields is in harmony with the theory. The transition between the two dielectric media is continuous, and the structure of the method enables one also to examine more complex geometries, such as wedges having soft and hard boundary conditions.

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References

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  1. C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).
  2. A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
    [CrossRef]
  3. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
    [CrossRef]
  4. T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).
    [CrossRef]
  5. A. Büyükaksoy and G. Uzgören, “Secondary diffraction of a plane wave by a metallic wide strip residing on the plane interface of two dielectric media,” Radio Sci. 22, 183–191(1987).
    [CrossRef]
  6. Y. Z. Umul, “Closed form series solution of the diffraction problem of plane waves by an impedance half-plane,” J. Opt. A: Pure Appl. Opt. 11, 045709 (2009).
    [CrossRef]
  7. G. D. Malyughinetz, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. 461, 107–112 (1960).
    [CrossRef]
  8. T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. Lond. A 213, 436–458 (1952).
    [CrossRef]
  9. Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” PIER M 8, 39–50 (2009).
    [CrossRef]
  10. R. G. Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, “The diffraction of an inhomogeneous plane wave by a wedge,” Radio Sci. 31, 1387–1397 (1996).
    [CrossRef]

2009 (2)

Y. Z. Umul, “Closed form series solution of the diffraction problem of plane waves by an impedance half-plane,” J. Opt. A: Pure Appl. Opt. 11, 045709 (2009).
[CrossRef]

Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” PIER M 8, 39–50 (2009).
[CrossRef]

1996 (1)

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

1987 (1)

A. Büyükaksoy and G. Uzgören, “Secondary diffraction of a plane wave by a metallic wide strip residing on the plane interface of two dielectric media,” Radio Sci. 22, 183–191(1987).
[CrossRef]

1960 (1)

G. D. Malyughinetz, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. 461, 107–112 (1960).
[CrossRef]

1957 (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
[CrossRef]

1952 (1)

T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. Lond. A 213, 436–458 (1952).
[CrossRef]

1896 (1)

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

Büyükaksoy, A.

A. Büyükaksoy and G. Uzgören, “Secondary diffraction of a plane wave by a metallic wide strip residing on the plane interface of two dielectric media,” Radio Sci. 22, 183–191(1987).
[CrossRef]

Kouyoumjian, R. G.

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

Malyughinetz, G. D.

G. D. Malyughinetz, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. 461, 107–112 (1960).
[CrossRef]

Manara, G.

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

Nepa, P.

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

Rubinowicz, A.

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
[CrossRef]

Senior, T. B. A.

T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. Lond. A 213, 436–458 (1952).
[CrossRef]

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

Sommerfeld, A.

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
[CrossRef]

Taute, B. J. E.

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

Umul,

Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” PIER M 8, 39–50 (2009).
[CrossRef]

Umul, Y. Z.

Y. Z. Umul, “Closed form series solution of the diffraction problem of plane waves by an impedance half-plane,” J. Opt. A: Pure Appl. Opt. 11, 045709 (2009).
[CrossRef]

Uzgören, G.

A. Büyükaksoy and G. Uzgören, “Secondary diffraction of a plane wave by a metallic wide strip residing on the plane interface of two dielectric media,” Radio Sci. 22, 183–191(1987).
[CrossRef]

Volakis, J. L.

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

Ann. Phys. (1)

G. D. Malyughinetz, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. 461, 107–112 (1960).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

Y. Z. Umul, “Closed form series solution of the diffraction problem of plane waves by an impedance half-plane,” J. Opt. A: Pure Appl. Opt. 11, 045709 (2009).
[CrossRef]

Math. Ann. (1)

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
[CrossRef]

Nature (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957).
[CrossRef]

PIER M (1)

Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” PIER M 8, 39–50 (2009).
[CrossRef]

Proc. R. Soc. Lond. A (1)

T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. Lond. A 213, 436–458 (1952).
[CrossRef]

Radio Sci. (2)

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

A. Büyükaksoy and G. Uzgören, “Secondary diffraction of a plane wave by a metallic wide strip residing on the plane interface of two dielectric media,” Radio Sci. 22, 183–191(1987).
[CrossRef]

Other (2)

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

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

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Figures (10)

Fig. 1
Fig. 1

Geometry of the dielectric interface.

Fig. 2
Fig. 2

Geometry of the half-plane between two dielectric media.

Fig. 3
Fig. 3

Scattered fields in region I for the angle of incidence at 60 ° .

Fig. 4
Fig. 4

Scattered waves in region II for the angle of incidence at 60 ° .

Fig. 5
Fig. 5

Total scattered and diffracted waves for the angle of incidence at 60 ° .

Fig. 6
Fig. 6

Scattered fields by a hard half-plane for the angle of incidence at 60 ° .

Fig. 7
Fig. 7

Scattered field in the second region for the angle of incidence at 60 ° .

Fig. 8
Fig. 8

Total scattered and diffracted waves for the angle of incidence at 60 ° .

Fig. 9
Fig. 9

Complex contour of C.

Fig. 10
Fig. 10

Complex contour of C 1 C 2 .

Equations (55)

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2 u 1 , 2 + k 1 , 2 2 u 1 , 2 = 0 ,
u 1 | S = u 2 | S ,
u 1 n | S = ε 2 ε 1 u 2 n | S ,
u 1 = e j k 1 2 ξ 2 x ( A ξ e j ξ y + B ξ e j ξ y ) ,
u 2 = e j k 2 2 ζ 2 x ( A ζ e j ζ y + B ζ e j ζ y ) ,
cos ϕ 1 = ε 1 ε 2 cos ϕ 0 ,
u 0 + B ξ = A ζ ,
u 0 B ξ = η A ζ ,
η = ε 2 sin ϕ 1 ε 1 sin ϕ 0 .
B ξ = u 0 1 η 1 + η ,
A ζ = u 0 2 1 + η ,
e j x cos θ = n = 0 χ n J n ( x ) e j n π 2 cos ( n θ ) ,
u 1 = u 0 [ n = 0 χ n J n ( k 1 ρ ) e j n π 2 cos n ( ϕ ϕ 0 ) + 1 η 1 + η n = 0 χ n J n ( k 1 ρ ) e j n π 2 cos n ( ϕ + ϕ 0 ) ] ,
u 2 = u 0 2 1 + η n = 0 χ n J n ( k 2 ρ ) e j n π 2 cos n [ ( ϕ 2 π ) ϕ 1 ] ,
u s = 4 u 0 n = 1 J n ( k ρ ) e j n π 2 sin ( n ϕ ) sin ( n ϕ 0 ) ,
u h = 2 u 0 n = 0 χ n J n ( k ρ ) e j n π 2 cos ( n ϕ ) cos ( n ϕ 0 ) ,
u 1 = 2 u 0 [ 2 α n = 1 J n ( k 1 ρ ) e j n π 2 sin ( n ϕ ) sin ( n ϕ 0 ) + β n = 0 χ n J n ( k 1 ρ ) e j n π 2 cos ( n ϕ ) cos ( n ϕ 0 ) ] ,
u 2 = 2 u 0 β { 2 n = 1 J n ( k 2 ρ ) e j n π 2 sin n ( ϕ 2 π ) sin ( n ϕ 1 ) + n = 0 χ n J n ( k 2 ρ ) e j n π 2 cos n ( ϕ 2 π ) cos ( n ϕ 1 ) } ,
u 1 = α u s 1 + β u h 1 ,
u 2 = β ( u s 2 + u h 2 ) .
α = η 1 + η ,
β = 1 1 + η ,
u i = exp [ j k 1 ( x cos ϕ 0 + y sin ϕ 0 ) ] ,
u s 1 | ϕ = 0 = 0 ,
u s 1 = α u s 1 s + β u s 1 h ,
u s 2 | ϕ = 2 π = 0 ,
u s 2 = β ( u s 2 s + u s 2 h ) ,
u s 1 = J v ( k 1 ρ ) [ A v sin ( v ϕ ) + B v cos ( v ϕ ) ] ,
u s 2 = J ξ ( k 2 ρ ) [ A ξ sin ( ξ ϕ ) + B ξ cos ( ξ ϕ ) ] ,
u s 1 = A v J v ( k 1 ρ ) sin ( v ϕ ) ,
u s 2 = A ξ J ξ ( k 2 ρ ) sin [ ξ ( ϕ 2 π ) ] cos ( 2 π ξ ) ,
u s 1 = α n = 1 A n J n ( k 1 ρ ) sin ( n ϕ ) + β n = 0 B n J ϑ n ( k 1 ρ ) sin ( ϑ n ϕ ) ,
u s 2 = β { n = 1 C n J n ( k 2 ρ ) sin [ n ( ϕ 2 π ) ] cos ( 2 π n ) + n = 1 D n J ϑ n ( k 2 ρ ) sin [ ϑ n ( ϕ 2 π ) ] cos ( 2 π ϑ n ) } ,
u s 2 = β { n = 1 C n J n ( k 2 ρ ) sin [ n ( ϕ 2 π ) ] n = 1 D n J ϑ n ( k 2 ρ ) sin [ ϑ n ( ϕ 2 π ) ] } .
u s 1 = 4 α n = 1 J n ( k 1 ρ ) e j n π 2 sin ( n ϕ ) sin ( n ϕ 0 ) + 4 β n = 0 J ϑ n ( k 1 ρ ) e j ϑ n π 2 sin ( ϑ n ϕ ) sin ( ϑ n ϕ 0 ) ,
u s 2 = 4 β { n = 1 J n ( k 2 ρ ) e j n π 2 sin [ n ( ϕ 2 π ) ] sin ( n ϕ 1 ) n = 1 J ϑ n ( k 2 ρ ) e j ϑ n π 2 sin [ ϑ n ( ϕ 2 π ) ] sin ( ϑ n ϕ 1 ) } ,
u GO 1 = e j k 1 ρ cos ( ϕ ϕ 0 ) + e j k 1 ρ cos ( ϕ + ϕ 0 ) [ RU ( t + ) U ( t + ) ] ,
u GO 2 = T e j k 2 ρ cos ( ϕ ϕ 0 ) U ( t ) ,
t = 2 k 1 , 2 ρ cos ϕ ϕ 0 , 1 2 .
u d 1 = u s 1 u GO 1 ,
u d 2 = u s 2 u GO 2 ,
u s 1 ϕ | ϕ = 0 = 0 ,
u s 2 ϕ | ϕ = 2 π = 0 ,
u s 1 = 2 β n = 0 χ n J n ( k 1 ρ ) e j n π 2 cos ( n ϕ ) cos ( n ϕ 0 ) + 4 α n = 0 J ϑ n ( k 1 ρ ) e j ϑ n π 2 cos ( ϑ n ϕ ) cos ( ϑ n ϕ 0 ) ,
u s 2 = β { 2 n = 0 χ n J n ( k 2 ρ ) e j n π 2 cos [ n ( ϕ 2 π ) ] cos ( n ϕ 1 ) 4 n = 1 J ϑ n ( k 2 ρ ) e j ϑ n π 2 cos [ ϑ n ( ϕ 2 π ) ] cos ( ϑ n ϕ 1 ) } ,
S = 4 n = 1 J n ( k 1 ρ ) e j n π 2 sin ( n ϕ ) sin ( n ϕ 0 ) ,
S = S + + S ,
S = n = 1 J n ( k 1 ρ ) exp ( j n π 2 ) { exp [ j n ( ϕ ϕ 0 ) ] exp [ j n ( ϕ + ϕ 0 ) ] } .
J v ( x ) = 1 2 π C exp [ j ( x cos ξ + v ξ ) ] d ξ
S = I 1 + I 2 ,
I 1 = 1 2 π C 1 exp ( j k 1 ρ cos ξ ) { n = 1 exp [ j n ( ξ + ϕ ϕ 0 ) ] n = 1 exp [ j n ( ξ + ϕ + ϕ 0 ) ] } d ξ ,
I 2 = 1 2 π C 2 exp ( j k 1 ρ cos ξ ) { n = 1 exp [ j n ( ξ + ϕ ϕ 0 ) ] n = 1 exp [ j n ( ξ + ϕ + ϕ 0 ) ] } d ξ ,
I 1 = j π C 1 exp ( j k 1 ρ cos ξ ) [ exp ( j ξ + ϕ ϕ 0 2 ) sin ( ξ + ϕ ϕ 0 2 ) exp ( j ξ + ϕ + ϕ 0 2 ) sin ( ξ + ϕ + ϕ 0 2 ) ] d ξ ,
I 2 = j π C 2 exp ( j k 1 ρ cos ξ ) [ exp ( j ξ + ϕ ϕ 0 2 ) sin ( ξ + ϕ ϕ 0 2 ) exp ( j ξ + ϕ + ϕ 0 2 ) sin ( ξ + ϕ + ϕ 0 2 ) ] d ξ ,
S = 2 j π C 1 C 2 exp ( j k 1 ρ cos ξ ) [ cot g ( ξ + ϕ + ϕ 0 2 ) cot g ( ξ + ϕ ϕ 0 2 ) ] d ξ .

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