Abstract

The scattering of the radiated fields of a line source by a cylindrical parabolic reflector, which has impedance boundary conditions on its surface, is obtained by using the surface integrals of the modified theory of physical optics. The reflected geometrical optics and edge diffracted fields are evaluated by using the asymptotic methods. The scattered fields are plotted numerically for various parameters such as surface impedance and angles of the edges.

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References

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  1. A. K. Bhattacharyya, High-Frequency Electromagnetic Techniques: Recent Advances and Applications (Wiley, 1995).
  2. G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedance,” Sov. Phys. Dokl. 3, 752-755 (1959).
  3. R. Tiberio, G. Pelosi, and G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. 33, 867-873 (1985).
    [CrossRef]
  4. J. L. Volakis, “A uniform geometrical theory of diffraction for an imperfectly conducting half-plane,” IEEE Trans. Antennas Propag. 34, 172-180 (1986).
    [CrossRef]
  5. Y. Z. Umul, “Modified theory of the physical-optics approach to the impedance wedge problem,” Opt. Lett. 31, 401-403 (2006).
    [CrossRef] [PubMed]
  6. Y. Z. Umul, “Modified theory of physical optics solution of impedance half plane problem,” IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
    [CrossRef]
  7. Y. Z. Umul, “Edge-dislocation waves in the diffraction process by an impedance half-plane,” J. Opt. Soc. Am. A 24, 507-511 (2007).
    [CrossRef]
  8. Y. Z. Umul, “Scattering of a Gaussian beam by an impedance half-plane,” J. Opt. Soc. Am. A 24, 3159-3167 (2007).
    [CrossRef]
  9. A. Büyükaksoy and G. Uzgören, “High-frequency scattering from the impedance discontinuity on a cylindrically curved surface,” IEEE Trans. Antennas Propag. 35, 234-236 (1987).
    [CrossRef]
  10. I. Akduman and A. Büyükaksoy, “Asymptotic expressions for the surface currents induced on a cylindrically curved impedance strip,” IEEE Trans. Antennas Propag. 43, 453-463 (1995).
    [CrossRef]
  11. O. M. Bucci and G. Franceschetti, “Rim loaded reflector antennas,” IEEE Trans. Antennas Propag. 28, 297-305 (1980).
    [CrossRef]
  12. Y. Z. Umul, “Modified theory of physical optics,” Opt. Express 12, 4959-4972 (2004).
    [CrossRef] [PubMed]
  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. Y. Z. Umul, “Diffraction by a black half plane: Modified theory of physical optics approach,” Opt. Express 13, 7276-7287 (2005).
    [CrossRef] [PubMed]
  15. S. W. Lee, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. 24, 25-34 (1976).
    [CrossRef]
  16. Y. Z. Umul, “Uniform theory for the diffraction of evanescent plane waves,” J. Opt. Soc. Am. A 24, 2426-2430 (2007).
    [CrossRef]

2007 (3)

2006 (2)

Y. Z. Umul, “Modified theory of the physical-optics approach to the impedance wedge problem,” Opt. Lett. 31, 401-403 (2006).
[CrossRef] [PubMed]

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

2005 (1)

2004 (1)

1995 (2)

I. Akduman and A. Büyükaksoy, “Asymptotic expressions for the surface currents induced on a cylindrically curved impedance strip,” IEEE Trans. Antennas Propag. 43, 453-463 (1995).
[CrossRef]

A. K. Bhattacharyya, High-Frequency Electromagnetic Techniques: Recent Advances and Applications (Wiley, 1995).

1987 (1)

A. Büyükaksoy and G. Uzgören, “High-frequency scattering from the impedance discontinuity on a cylindrically curved surface,” IEEE Trans. Antennas Propag. 35, 234-236 (1987).
[CrossRef]

1986 (1)

J. L. Volakis, “A uniform geometrical theory of diffraction for an imperfectly conducting half-plane,” IEEE Trans. Antennas Propag. 34, 172-180 (1986).
[CrossRef]

1985 (1)

R. Tiberio, G. Pelosi, and G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. 33, 867-873 (1985).
[CrossRef]

1980 (1)

O. M. Bucci and G. Franceschetti, “Rim loaded reflector antennas,” IEEE Trans. Antennas Propag. 28, 297-305 (1980).
[CrossRef]

1976 (1)

S. W. Lee, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. 24, 25-34 (1976).
[CrossRef]

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]

1959 (1)

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedance,” Sov. Phys. Dokl. 3, 752-755 (1959).

Akduman, I.

I. Akduman and A. Büyükaksoy, “Asymptotic expressions for the surface currents induced on a cylindrically curved impedance strip,” IEEE Trans. Antennas Propag. 43, 453-463 (1995).
[CrossRef]

Bhattacharyya, A. K.

A. K. Bhattacharyya, High-Frequency Electromagnetic Techniques: Recent Advances and Applications (Wiley, 1995).

Bucci, O. M.

O. M. Bucci and G. Franceschetti, “Rim loaded reflector antennas,” IEEE Trans. Antennas Propag. 28, 297-305 (1980).
[CrossRef]

Büyükaksoy, A.

I. Akduman and A. Büyükaksoy, “Asymptotic expressions for the surface currents induced on a cylindrically curved impedance strip,” IEEE Trans. Antennas Propag. 43, 453-463 (1995).
[CrossRef]

A. Büyükaksoy and G. Uzgören, “High-frequency scattering from the impedance discontinuity on a cylindrically curved surface,” IEEE Trans. Antennas Propag. 35, 234-236 (1987).
[CrossRef]

Franceschetti, G.

O. M. Bucci and G. Franceschetti, “Rim loaded reflector antennas,” IEEE Trans. Antennas Propag. 28, 297-305 (1980).
[CrossRef]

Kouyoumjian, R. G.

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]

Lee, S. W.

S. W. Lee, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. 24, 25-34 (1976).
[CrossRef]

Maliuzhinets, G. D.

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedance,” Sov. Phys. Dokl. 3, 752-755 (1959).

Manara, G.

R. Tiberio, G. Pelosi, and G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. 33, 867-873 (1985).
[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]

Pelosi, G.

R. Tiberio, G. Pelosi, and G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. 33, 867-873 (1985).
[CrossRef]

Tiberio, R.

R. Tiberio, G. Pelosi, and G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. 33, 867-873 (1985).
[CrossRef]

Umul, Y. Z.

Uzgören, G.

A. Büyükaksoy and G. Uzgören, “High-frequency scattering from the impedance discontinuity on a cylindrically curved surface,” IEEE Trans. Antennas Propag. 35, 234-236 (1987).
[CrossRef]

Volakis, J. L.

J. L. Volakis, “A uniform geometrical theory of diffraction for an imperfectly conducting half-plane,” IEEE Trans. Antennas Propag. 34, 172-180 (1986).
[CrossRef]

IEEE Trans. Antennas Propag. (7)

R. Tiberio, G. Pelosi, and G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. 33, 867-873 (1985).
[CrossRef]

J. L. Volakis, “A uniform geometrical theory of diffraction for an imperfectly conducting half-plane,” IEEE Trans. Antennas Propag. 34, 172-180 (1986).
[CrossRef]

A. Büyükaksoy and G. Uzgören, “High-frequency scattering from the impedance discontinuity on a cylindrically curved surface,” IEEE Trans. Antennas Propag. 35, 234-236 (1987).
[CrossRef]

I. Akduman and A. Büyükaksoy, “Asymptotic expressions for the surface currents induced on a cylindrically curved impedance strip,” IEEE Trans. Antennas Propag. 43, 453-463 (1995).
[CrossRef]

O. M. Bucci and G. Franceschetti, “Rim loaded reflector antennas,” IEEE Trans. Antennas Propag. 28, 297-305 (1980).
[CrossRef]

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

S. W. Lee, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. 24, 25-34 (1976).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (2)

Opt. Lett. (1)

Proc. IEEE (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]

Sov. Phys. Dokl. (1)

G. D. Maliuzhinets, “Excitation, reflection and emission of surface waves from a wedge with given face impedance,” Sov. Phys. Dokl. 3, 752-755 (1959).

Other (1)

A. K. Bhattacharyya, High-Frequency Electromagnetic Techniques: Recent Advances and Applications (Wiley, 1995).

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Figures (13)

Fig. 1
Fig. 1

Geometry of the parabolic reflector.

Fig. 2
Fig. 2

Stationary phase geometry of the reflected and transmitted rays.

Fig. 3
Fig. 3

Reflection geometry at the stationary phase point.

Fig. 4
Fig. 4

Edge diffraction geometry of the reflector.

Fig. 5
Fig. 5

Transition regions.

Fig. 6
Fig. 6

Geometrical places of the poles.

Fig. 7
Fig. 7

Reflected scattered fields for a conducting parabola.

Fig. 8
Fig. 8

Total scattered fields by the conducting reflector.

Fig. 9
Fig. 9

Total fields for various parabola widths.

Fig. 10
Fig. 10

Reflected field for various surface impedances.

Fig. 11
Fig. 11

Levels of the diffracted fields for various surface impedances.

Fig. 12
Fig. 12

Reflected scattered field for a half-parabolic reflector.

Fig. 13
Fig. 13

Geometry of the reflection point.

Equations (59)

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E = e z E o e j k ρ k ρ ,
E 0 = e j ( π 4 ) w μ 0 I 0 2 2 π ,
H i = e ϕ E o Z 0 e j k ρ k ρ ,
sin θ = Z 0 Z
ρ = 2 f 1 + cos ϕ = f cos 2 ϕ 2 ,
n = cos ϕ 2 e ρ + sin ϕ 2 e ϕ .
E s = e z E 0 [ e j k ρ k ρ + k e j ( π 4 ) 2 π ϕ 0 ϕ 0 Γ ( α , β , θ ) f ( α , β ) e j k ρ k ρ e j k R k R ρ cos ( ϕ 2 ) d ϕ ] ,
Γ ( α , β , θ ) = sin β + α 2 sin θ sin β + α 2 + sin θ ,
f ( α , β ) = sin β + α 2 sin β α 2 ,
g ( ϕ ) = ρ cos γ + ρ [ 1 cos ( α + β ) ] ,
γ + ϕ ϕ = α + β ,
d g ( ϕ ) d ϕ = ρ sin γ d γ d ϕ + ρ cos α sin α [ 1 cos ( α + β ) ] + ρ sin ( α + β ) ( d α d ϕ + d β d ϕ ) ,
d ρ d ϕ = ρ cos α sin α .
d g ( ϕ ) d ϕ = ρ cos α sin α [ 1 cos ( α + β ) ] ρ sin ( α + β ) ,
ρ sin γ = ρ sin ( α + β ) ,
d γ d ϕ = d α d ϕ + d β d ϕ + 1 ,
d g ( ϕ ) d ϕ = ρ sin α ( cos α cos β ) ,
ϕ 2 = π 2 α ,
β s r = α s ,
β s t = α s ,
d 2 g ( ϕ ) d ϕ 2 = d d ϕ ( ρ sin α ) ( cos α cos β ) + ρ sin α ( sin α d α d ϕ + sin β d β d ϕ ) .
d α d ϕ = 1 2 ,
d β d ϕ = 1 2 + ρ cos α sin ( α + β ) ρ cos γ sin α R sin α ,
g ( ϕ ) l + l 0 + 1 2 l 0 2 l ( ϕ ϕ s ) 2 ,
f ( ϕ ) e j ( π 4 ) E 0 2 π sin α s sin θ sin α s + sin θ l 0 l .
E r GO = E 0 sin α s sin θ sin α s + sin θ e j k l 0 k l 0 e j k l ,
sin α s = 1 + cos ϕ s 2 ,
cos ϕ s = 4 f 2 ρ 2 sin 2 ϕ 4 f 2 + ρ 2 sin 2 ϕ .
g ( ϕ ) ρ + 1 2 ρ ρ s R s ( ϕ ϕ s ) 2 ,
f ( ϕ ) e j ( π 4 ) E 0 2 π ρ s R s .
E t GO = E 0 e j k ρ k ρ [ U ( ϕ ϕ 0 ) U ( ϕ 2 π + ϕ 0 ) ] ,
β e = sin 1 ρ sin ( ϕ ϕ e ) l e ϕ e 2 ,
l e = ρ 2 + ρ e 2 2 ρ ρ e cos ( ϕ ϕ e ) .
ρ e = f cos 2 ( ϕ e 2 ) .
E d = E 0 e j k ( ρ e + l e ) k ρ e l e [ D ( ϕ 0 ) D ( ϕ 0 ) ] ,
D ( ϕ e ) = e j ( π 4 ) 2 2 π Γ ( α e , β e , θ ) ( 1 sin β e α e 2 1 sin β e + α e 2 ) .
sin β e α e 2 = 0 ,
sin β e + α e 2 = 0 ,
E r s = e z E 0 k e j ( π 4 ) 2 π ϕ 0 ϕ 0 Γ ( α , β , θ ) sin β + α 2 e j k ρ k ρ e j k R k R ρ cos ( ϕ 2 ) d ϕ ,
E r s = e z E 0 k e j ( π 4 ) 2 π ϕ 0 ϕ 0 sin α e j k ρ k ρ e j k R k R ρ cos ( ϕ 2 ) d ϕ .
t = sin ϕ 2 e ρ + cos ϕ 2 e ϕ .
n = sin v t + cos v n ,
α = π 2 ϕ 2 .
u = π 2 β + α 2 ,
v = β α 2 ,
I = a b f ( x ) e j Ω g ( x ) d x ,
I = a b f ( x ) g ( x ) g ( x ) e j Ω g ( x ) d x
I = 1 j Ω [ f ( a ) g ( a ) e j Ω g ( a ) f ( b ) g ( b ) e j Ω g ( b ) ] + 1 j Ω a b f ( x ) g ( x ) f ( x ) g ( x ) [ g ( x ) ] 2 e j Ω g ( x ) d x .
I 1 j Ω [ f ( a ) g ( a ) e j Ω g ( a ) f ( b ) g ( b ) e j Ω g ( b ) ] .
ξ = k ( l d l GO ) ,
ξ = 2 k l i l s l d + l GO cos σ 2 .
u d n u = e j ( π 4 ) 2 2 π 1 cos σ 2 e j k l d k l d .
u d n u = e j ( π 4 ) 2 π e j ξ 2 ξ ,
u d = sgn ( ξ ) F [ ξ ] ,
F [ x ] = e j ( π 4 ) π x e j t 2 d t .
T = F [ ξ ] F ̂ [ ξ ] .
u d n u = e j k l GO 2 l i l s l d ( l d + l GO ) F ̂ [ ξ ]
2 l i l s l d ( l d + l GO ) e j k l GO .
u d = e j k l GO 2 l i l s l d ( l d + l GO ) sgn ( ξ ) F [ ξ ] ,

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