Abstract

The problem of plane-wave scattering by an infinitely long perfectly conducting circular cylinder that is partially buried in a perfectly conducting ground plane is studied by the method of images, for which the incident electric-field vector is assumed to be in a plane perpendicular to the axis of the cylinder (TE polarization). The incident field, the reflected field from the ground plane in the absence of the cylinder, and the scattered field from the cylinder and its image are expressed in terms of cylindrical vector wave functions. By imposing the boundary conditions on the surface of the cylinder, we obtain a set of two coupled infinite systems of equations for the even–and odd–mode expansion coefficients of the scattered field. We solve these equations numerically by truncating the infinite summations and using a subsequent Gaussian elimination procedure. The scattered power patterns in the far–field region are obtained, and their variations with the angle of incidence, height (or depth) above (or below) the ground plane, and the electrical radius are studied. Comparisons with the corresponding TM–case results are made.

© 1991 Optical Society of America

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References

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  1. T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried into a ground plane I: TM case,” J. Opt. Soc. Am. 6 A, 1270–1280 (1989).
    [CrossRef]
  2. T. C. Rao, R. Barakat, “Plane-wave scattering by a finite array of conducting circular cylinders partially buried into a conducting ground plane,” submitted to Philos. Trans. R. Soc. London.
  3. J. van Bladel, Electromagnetic Fields (Hemisphere, New York, 1985).
  4. Lord Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.

1989 (1)

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried into a ground plane I: TM case,” J. Opt. Soc. Am. 6 A, 1270–1280 (1989).
[CrossRef]

Barakat, R.

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried into a ground plane I: TM case,” J. Opt. Soc. Am. 6 A, 1270–1280 (1989).
[CrossRef]

T. C. Rao, R. Barakat, “Plane-wave scattering by a finite array of conducting circular cylinders partially buried into a conducting ground plane,” submitted to Philos. Trans. R. Soc. London.

Rao, T. C.

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried into a ground plane I: TM case,” J. Opt. Soc. Am. 6 A, 1270–1280 (1989).
[CrossRef]

T. C. Rao, R. Barakat, “Plane-wave scattering by a finite array of conducting circular cylinders partially buried into a conducting ground plane,” submitted to Philos. Trans. R. Soc. London.

Rayleigh, Lord

Lord Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.

van Bladel, J.

J. van Bladel, Electromagnetic Fields (Hemisphere, New York, 1985).

J. Opt. Soc. Am. (1)

T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried into a ground plane I: TM case,” J. Opt. Soc. Am. 6 A, 1270–1280 (1989).
[CrossRef]

Other (3)

T. C. Rao, R. Barakat, “Plane-wave scattering by a finite array of conducting circular cylinders partially buried into a conducting ground plane,” submitted to Philos. Trans. R. Soc. London.

J. van Bladel, Electromagnetic Fields (Hemisphere, New York, 1985).

Lord Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.

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Figures (10)

Fig. 1
Fig. 1

Scattered power (in decibels) as a function of ϕ for a normally incident plane wave (γ = π/2, α = −π/2) for = 1: solid curve, h = a; dashed curve, h = 0.75α; dotted–dashed curve, h = 0.5α.

Fig. 2
Fig. 2

Scattered power (in decibels) as a function of ϕ for a normally incident plane wave (γ = π/2, α = −π/2) for = 1 for a partially protruding cylinder: solid curve, h = −0.5a; dashed curve, h = −0.625a; dotted–dashed curve, h = −0.75a.

Fig. 3
Fig. 3

Scattered power (in decibels) as a function of ϕ for a normally incident plane wave (γ = π/2, α = −π/2) for ka = 1: solid curve, h = a; dashed curve, h = 0.75a; dotted–dashed curve, h = 0.5a.

Fig. 4
Fig. 4

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/2) for ka = 4: solid curve, h = a; dashed curve, h = 0.75a; dotted–dashed curve, h = 0.5a.

Fig. 5
Fig. 5

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, a = −π/5) for ka = 1: solid curve, h = a; dashed curve, h = 0.75a; dotted–dashed curve, h = 0.5a.

Fig. 6
Fig. 6

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for ka = 2: solid curve, h = a; dashed curve, h = 0.75a; dotted–dashed curve, h = 0.5a.

Fig. 7
Fig. 7

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for ka = 4: solid curve, h = a; dashed curve, h = 0.75a; dotted-dashed curve, h = 0.5a.

Fig. 8
Fig. 8

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for h = 0 (Rayleigh case) for ka = 1: solid curve, TM mode; dashed curve, TE mode.

Fig. 9
Fig. 9

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for h = 0 (Rayleigh case) for ka = 2: solid curve, TM mode; dashed curve, TE mode.

Fig. 10
Fig. 10

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for h = 0 (Rayleigh case) for ka = 4: solid curve, TM mode; dashed curve, TE mode.

Equations (15)

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E inc = E 0 [ ( sin α ) a x ( cos α ) a y ] exp ( i k r cos Θ ) ,
E inc = E 0 i k sin γ m = 0 ( 2 δ om ) i m × [ cos m α M em ( 1 ) ( r , γ ) + sin m α M om ( 1 ) ( r , γ ) ] .
H inc = 1 i k ( 0 μ 0 ) 1 / 2 × E inc = ( 0 μ 0 ) 1 / 2 E 0 k sin γ m = 0 ( 2 δ om ) i m × [ cos m α N em ( 1 ) ( r , γ ) + sin m α N om ( 1 ) ( r , γ ) ] .
E ref = E 0 i k sin γ m = 0 ( 2 δ om ) i m × [ cos m α M em ( 1 ) ( r , γ ) sin m α M om ( 1 ) ( r , γ ) ] ,
H ref = ( 0 μ 0 ) 1 / 2 E 0 k sin γ m = 0 ( 2 δ om ) i m [ cos m α N em ( 1 ) ( r , γ ) sin m α N om ( 1 ) ( r , γ ) ] ,
E scat = ( E 0 i k sin γ ) m = 0 ( 2 δ om ) i m [ b em cos m α M em ( 3 ) ( r , γ ) + b om sin m α M om ( 3 ) ( r , γ ) ] ,
E scat = ( E 0 i k sin γ ) m = 0 ( 2 δ om ) i m [ b em cos m α M em ( 3 ) ( r , γ ) b om sin m α M om ( 3 ) ( r , γ ) ] ,
b em = b em , b om = b om ,
E ϕ inc ( a , ϕ , z ) + E ϕ ref ( a , ϕ , z ) + E ϕ scat ( a , ϕ , z ) + [ a ϕ E scat ( r , γ ) ] ρ = a = 0 .
b em = [ 1 + exp ( i 2 λ h sin α ) ] J m ( λ a ) H m ( 1 ) ( λ a ) 1 2 ( 1 + δ om ) cos m α J m ( λ a ) H m ( 1 ) ( λ a ) × [ m = 0 m ( 2 δ o m ) i m m A m m ( e , m e , m ) cos m α b e m + m = 0 ( 2 δ o m ) i m m A m + m ( e , m e , m ) cos m α b e m + ( 1 ) m ( 1 δ om ) m = m ( 2 δ o m ) i m m × A m m ( e , m e , m ) cos m α b e m m = 0 m ( 2 δ o m ) i m m A m m ( o , m e , m ) sin m α b o m m = 0 ( 2 δ o m ) i m m A m + m ( o , m e , m ) sin m α b o m ( 1 ) m ( 1 δ om ) m = m ( 2 δ o m ) i m m × A m m ( o , m e , m ) sin m α b o m ] ,
b om = [ 1 exp ( i 2 λ h sin α ) ] J m ( λ a ) H m ( 1 ) ( λ a ) + 1 2 sin m α J m ( λ a ) H m ( 1 ) ( λ a ) × [ m = 0 m ( 2 δ o m ) i m m A m m ( o , m o , m ) sin m α b o m + m = 0 ( 2 δ o m ) i m m A m + m ( o , m o , m ) sin m α b o m + ( 1 ) m + 1 ( 1 δ om ) m = m ( 2 δ o m ) i m m × A m m ( o , m o , m ) sin m α b o m m = 0 m ( 2 δ o m ) i m m A m m ( e , m o , m ) cos m α b e m m = 0 ( 2 δ o m ) i m m A m + m ( e , m o , m ) cos m α b e m ( 1 ) m + 1 ( 1 δ om ) m = m ( 2 δ o m ) i m m × A m m ( e , m o , m ) cos m α b e m ] ,
E ρ scat ( 2 E 0 i k ρ sin γ ) ( 2 π k ρ sin γ ) 1 / 2 × exp ( i k z cos γ + i k ρ sin γ i π 4 ) m = 0 ( 2 δ om ) × m ( b em cos m α sin m ϕ b om sin m α cos m ϕ ) ,
E ϕ scat 2 E 0 ( 2 π k ρ sin γ ) 1 / 2 × exp ( i k z cos γ + i k ρ sin γ i π 4 ) m = 0 ( 2 δ om ) × ( b em cos m α cos m ϕ + b om sin m α sin m ϕ ) .
b em = 2 J m ( λ a ) H m ( 1 ) ( λ a ) b om = 0 .
E ϕ scat = E 0 i k sin γ m = 0 ( 2 δ om ) i m 2 J m ( λ a ) H m ( 1 ) ( λ a ) × cos m α cos m ϕ H m ( 1 ) ( k ρ sin γ ) = E 0 i k sin γ m = 0 ( 2 δ om ) i m J m ( λ a ) H m ( 1 ) ( λ a ) H m ( 1 ) ( k ρ sin γ ) × { exp [ i m ( α ϕ ) ] + exp [ + i m ( α + ϕ ) ] } .

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