Abstract

The theoretical analysis of the scattering of a beam by a spherical object centered on the propagation axis is treated exactly. For both conducting and dielectric bodies, those cases of sphere radii smaller than, equal to, and larger than that of the beam are analyzed numerically and the power radiated in the far zone is obtained. The difference between the scattering of a beam wave and that of a plane wave is discussed.

© 1975 Optical Society of America

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References

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  1. G. Mie, Ann. Physik 25, 377 (1908).
    [Crossref]
  2. P. Debye, Ann. Physik 30, 57 (1909).
    [Crossref]
  3. L. V. Lorenz, Viderskab. Selskab. Skrifter. 6, 1 (1890).
  4. N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, IEEE Trans.,  AP-16, 724 (1968).
    [Crossref]
  5. J. A. Stratton, Electromagnetic Theory, (McGraw–Hill, New York, 1941), pp. 355, 393–397, 414–419, 563–565.
  6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 717.

1968 (1)

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, IEEE Trans.,  AP-16, 724 (1968).
[Crossref]

1909 (1)

P. Debye, Ann. Physik 30, 57 (1909).
[Crossref]

1908 (1)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

1890 (1)

L. V. Lorenz, Viderskab. Selskab. Skrifter. 6, 1 (1890).

Debye, P.

P. Debye, Ann. Physik 30, 57 (1909).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 717.

Lorenz, L. V.

L. V. Lorenz, Viderskab. Selskab. Skrifter. 6, 1 (1890).

Mie, G.

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

Morita, N.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, IEEE Trans.,  AP-16, 724 (1968).
[Crossref]

Nakanishi, Y.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, IEEE Trans.,  AP-16, 724 (1968).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 717.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory, (McGraw–Hill, New York, 1941), pp. 355, 393–397, 414–419, 563–565.

Tanaka, T.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, IEEE Trans.,  AP-16, 724 (1968).
[Crossref]

Yamasaki, T.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, IEEE Trans.,  AP-16, 724 (1968).
[Crossref]

Ann. Physik (2)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

P. Debye, Ann. Physik 30, 57 (1909).
[Crossref]

IEEE Trans. (1)

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nakanishi, IEEE Trans.,  AP-16, 724 (1968).
[Crossref]

Viderskab. Selskab. Skrifter. (1)

L. V. Lorenz, Viderskab. Selskab. Skrifter. 6, 1 (1890).

Other (2)

J. A. Stratton, Electromagnetic Theory, (McGraw–Hill, New York, 1941), pp. 355, 393–397, 414–419, 563–565.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 717.

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Figures (14)

FIG. 1
FIG. 1

Physical configuration of the problem.

FIG. 2
FIG. 2

Sketch of the incident electric field.

FIG. 3
FIG. 3

Scattering patterns for conducting sphere with a = 0.20930λ, w0 = 4.18603λ. E θ r 2 in the x-z plane with a = 0.20930λ and w0 = 4.18603λ. Solid line indicates beam wave incident; coincident broken line indicates plane wave incident.

FIG. 4
FIG. 4

Scattering patterns for conducting sphere with a = 0.20930λ, w0 = 4.18603λ. E ϕ r 2 in the y-z plane with a = 0.20930λ and w0 = 4.18603λ. Solid line indicates beam wave incident; coincident broken line indicates plane wave incident.

FIG. 5
FIG. 5

Scattering patterns for conducting sphere with a = 1.46512λ, w0 = 4.18603λ. E θ r 2 in the x-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 6
FIG. 6

Scattering patterns for conducting sphere with a = 1.46512λ, w0 = 4.18603λ. E ϕ r 2 in the y-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 7
FIG. 7

Scattering patterns for conducting sphere with a = 1.46512λ, w0 = 2.0λ. E θ r 2 in x-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 8
FIG. 8

Scattering patterns for conducting sphere with a = 1.46512λ, w0 = 2.0λ. E ϕ r 2 in y-z plane. Solid line indicated beam wave incident; broken line indicates plane wave incident.

FIG. 9
FIG. 9

Scattering patterns for conducting sphere with a = 3.0λ, w0 = 2.0λ. E θ r 2 in x-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 10
FIG. 10

Scattering patterns for conducting sphere with a = 3.0λ, w0 = 2.0λ. E ϕ r 2 in y-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 11
FIG. 11

Scattering patterns for conducting sphere with a = 3.0λ, w0 = 4.0λ. E θ r 2 in x-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 12
FIG. 12

Scattering patterns for conducting sphere with a = 3.0λ, w0 = 4.0λ. E ϕ r 2 in y-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 13
FIG. 13

Scattering patterns for dielectric sphere with a = 3.0λ, w0 = 2.0λ. E θ r 2 in x-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

FIG. 14
FIG. 14

Scattering patterns for dielectric sphere with a = 3.0λ, w0 = 2.0λ. E ϕ r 2 in y-z plane. Solid line indicates beam wave incident; broken line indicates plane wave incident.

Equations (45)

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Y l m ( θ , ϕ ) = P l m ( cos θ ) cos sin m ϕ .
E ( z = - z 0 ) = u x e - r 2 / w 0 2 ,
E ( z = - z 0 ) = ( u r cos ϕ - u ϕ sin ϕ ) e - r 2 / w 0 2 .
π = - u z r e - r 2 / w 0 2 sin ϕ f ( z ) ,
r e - r 2 / w 0 2 = 0 g ( λ ) J 1 ( λ r ) λ d λ .
g ( λ ) = 0 r e - r 2 / w 0 2 J 1 ( λ r ) r d r = w 0 4 4 e - λ 2 w 0 2 / 4 .
π z ( z = - z 0 ) = - sin ϕ w 0 4 4 0 e - λ 2 w 0 2 / 4 J 1 ( λ r ) λ 2 d λ .
π z = - e - j ω t sin ϕ w 0 4 4 0 e - λ 2 w 0 2 / 4 J 1 ( λ r ) × e - j ( k 2 - λ 2 ) 1 / 2 ( z + z 0 ) λ 2 d λ .
E = - u r 1 r π z ϕ + u ϕ π z r .
E = w 0 4 4 0 e - λ 2 w 0 2 / 4 ( u r 1 r J 1 ( λ r ) cos ϕ - u ϕ J 1 ( λ r ) r sin ϕ ) e - j ( k 2 - λ 2 ) 1 / 2 ( z + z 0 ) λ 2 d λ ,
E = w 0 4 4 0 e - λ 2 w 0 2 / 4 m o 1 λ e - j ( k 2 - λ 2 ) 1 / 2 ( z + z 0 ) λ 2 d λ ,
m o 1 λ = u r 1 r J 1 ( λ r ) cos ϕ - u ϕ J 1 ( λ r ) r sin ϕ .
π z = - e - j ω t sin ϕ r e - r 2 / w 0 2 ,
E = u r e - r 2 / w 0 2 cos ϕ - u ϕ e - r 2 / w 0 2 ( 1 - 2 r 2 / w 0 2 ) sin ϕ .
E = u x e - r 2 / w 0 2 .
m o 1 λ = l = 1 2 l + 1 l ( l + 1 ) ( l - 1 ) ! ( l + 1 ) ! j l - 1 λ ( d P l 1 ( cos ζ ) d ζ | ζ = α j m o l 1 + 1 sin α P l 1 ( cos α ) n e l 1 ) ,
E = w 0 4 4 0 e - λ 2 w 0 2 / 4 l = 1 2 l + 1 l 2 ( l + 1 ) 2 j l = 1 ( d P l 1 ( cos α ) d α j m o l 1 + 1 sin α P l 1 ( cos α ) n e l 1 ) e - j ( k 2 - λ 2 ) 1 / 2 z 0 λ 2 d λ ,
cos α = h / k = 1 k ( k 2 - λ 2 ) 1 / 2 = ( 1 - ( λ / k ) 2 ) 1 / 2 .
d P l 1 ( cos α ) d α = l ( l + 1 ) 2 l + 1 ( ( 2 l + 1 ) P l ( cos α ) + cos α sin 2 α [ P l + 1 ( cos α ) - P l - 1 ( cos α ) ] ) ,
E = 2 A 2 l = 1 2 l + 1 l ( l + 1 ) j l 0 e - A y P l ( ( 1 - y ) 1 / 2 ) e - j B ( 1 - y ) 1 / 2 y d y m o l 1 + 2 A 2 l = 1 1 l ( l + 1 ) j l 0 ( 1 - y ) 1 / 2 e - A y [ P l + 1 ( ( 1 - y ) 1 / 2 ) - P l - 1 ( ( 1 - y ) 1 / 2 ) ] e - j B ( 1 - y ) 1 / 2 d y m o l 1 - 2 A 2 l = 1 1 l ( l + 1 ) j l - 1 0 e - A y [ P l + 1 ( ( 1 - y ) 1 / 2 - P l - 1 ( ( 1 - y ) 1 / 2 ) ] e - j B ( 1 - y ) 1 / 2 d y n e l 1 ,
E = l = 1 2 l + 1 l ( l + 1 ) j l I 1 m o l 1 + l = 1 1 l ( l + 1 ) j l I 2 m o l 1 - l = 1 1 l ( l + 1 ) j l - 1 I 3 n e l 1 ,
I 1 = 2 A 2 0 e - A y - j B ( 1 - y ) 1 / 2 P l ( ( 1 - y ) 1 / 2 ) y d y ,
I 2 = 2 A 2 0 ( 1 - y ) 1 / 2 e - A y - j B ( 1 - y ) 1 / 2 × [ P l + 1 ( ( 1 - y ) 1 / 2 - P l - 1 ( ( 1 - y ) 1 / 2 ) ] d y ,
I 3 = 2 A 2 0 e - A y - j B ( 1 - y ) 1 / 2 × [ P l + 1 ( ( 1 - y ) 1 / 2 ) - P l - 1 ( ( 1 - y ) 1 / 2 ) ] d y .
E = l = 1 2 l + 1 l ( l + 1 ) j l ( m o l 1 - j n e l 1 ) ,
m o l 1 ( 3 ) = u θ 1 sin θ h l ( 1 ) ( ρ ) P l 1 cos ϕ - u ϕ h l ( 1 ) ( ρ ) P l 1 θ sin ϕ
n e l 1 ( 3 ) = u r l ( l + 1 ) h l ( 1 ) ( ρ ) ρ P l 1 cos ϕ + u θ [ ρ h l ( 1 ) ( ρ ) ] ρ P l 1 θ cos ϕ - u ϕ [ ρ h l ( 1 ) ( ρ ) ] ρ 1 sin θ P l 1 sin ϕ ,
P l 1 = - 1 sin θ l ( l + 1 ) 2 l + 1 ( P l + 1 - P l - 1 ) ,
P l 1 θ = l ( l + 1 ) 2 l + 1 [ ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ] ,
[ ( ρ h l ( 1 ) ( ρ ) ] ρ = - l h l ( 1 ) ( ρ ) ρ + h l - 1 ( 1 ) ( ρ ) ,
m o l 1 ( 3 ) = - u θ 1 sin 2 θ h l ( 1 ) ( ρ ) l ( l + 1 ) 2 l + 1 ( P l + 1 - P l - 1 ) cos ϕ - u ϕ h l ( 1 ) ( ρ ) l ( l + 1 ) 2 l + 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) sin ϕ
n e l 1 ( 3 ) = - u r l ( l + 1 ) h l ( 1 ) ( ρ ) ρ 1 sin θ l ( l + 1 ) 2 l + 1 ( P l + 1 - P l - 1 ) cos ϕ + u θ ( l h l ( 1 ) ( ρ ) ρ + h l - 1 ( 1 ) ( ρ ) ) l ( l + 1 ) 2 l + 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) cos ϕ + u ϕ ( - l h l ( 1 ) ( ρ ) ρ + h l - 1 ( 1 ) ( ρ ) ) 1 sin 2 θ l ( l + 1 ) 2 l + 1 ( P l + 1 - P l - 1 ) sin ϕ .
E θ r = - l = 1 a l r I 1 P l + 1 - P l - 1 sin 2 θ cos ϕ h l ( 1 ) ( ρ ) j l - l = 1 a l r I 2 1 2 l + 1 P l + 1 - P l - 1 sin 2 θ cos ϕ h l ( 1 ) ( ρ ) j l + l = 1 b l r I 3 1 2 l + 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) cos ϕ ( - l h l ( 1 ) ( ρ ) ρ + h l - 1 ( 1 ) ( ρ ) ) j l + 1 ,
E ϕ r = - l = 1 a l r I 1 [ ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ] sin ϕ h l ( 1 ) ( ρ ) j l - l = 1 a l r I 2 1 2 l + 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) sin ϕ h l ( 1 ) ( ρ ) j l + l = 1 b l r I 3 1 2 l + 1 P l + 1 - P l - 1 sin 2 θ sin ϕ ( - l h l ( 1 ) ( ρ ) ρ + h l - 1 ( 1 ) ( ρ ) ) j l + 1 ,
h l ( 1 ) ( ρ ) e j ρ ρ ( - j ) l + 1 ,
lim ρ ρ E θ r = [ - l = 1 a l r I 1 P l + 1 - P l - 1 sin 2 θ cos ϕ ( - j ) - l = 1 a l r I 2 1 2 l + 1 P l + 1 - P l - 1 sin 2 θ cos ϕ ( - j ) - l = 1 b l r I 3 1 2 l + 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) cos ϕ ( - j ) ] e j ρ
lim ρ ρ E ϕ r = [ - l = 1 a l r I 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) sin ϕ ( - j ) - l = 1 a l r I 2 1 2 l + 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) sin ϕ ( - j ) - l = 1 b l r I 3 1 2 l + 1 P l + 1 - P l - 1 sin 2 θ sin ϕ ( - j ) ] e j ρ .
H = - k j ω μ ( l + 1 I 1 2 l + 1 l ( l + 1 ) j l n o l 1 + l = 1 I 2 1 l ( l + 1 ) j l n o l 1 - I = 1 I 3 1 l ( l + 1 ) j l - 1 m e l 1 ) .
lim ρ ω μ k ρ H θ r = [ l = 1 a l r I 1 ( - j ) ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) sin ϕ + l = 1 a l r I 2 ( - j ) ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) sin ϕ + l = 1 b l r I 3 ( - j ) 1 2 l + 1 P l + 1 - P l - 1 sin 2 θ sin ϕ ] e j ρ ,
lim ρ ω μ k ρ H ϕ r = [ - l = 1 a l r I i ( - j ) P l + 1 - P l - 1 sin 2 θ cos ϕ - l = 1 a l r I 2 ( - j ) 1 2 l + 1 P l + 1 - P l - 1 sin 2 θ cos ϕ - l = 1 b l r I 3 ( - j ) 1 2 l + 1 ( ( 2 l + 1 ) P l + cos θ sin 2 θ ( P l + 1 - P l - 1 ) ) cos ϕ ] e j ρ .
lim ρ ( ρ E θ r ) ( ρ H ϕ r * ω μ / k ) = lim ρ ( ρ E θ r ) ( ρ E θ r ) *
lim ρ ( ρ E ϕ r ) ( ρ H θ r * ω μ / k ) = lim ρ ( ρ E ϕ r ) ( ρ E ϕ r ) * ,
P = 1 2 Re lim ρ ρ 2 ( E r × H r * ) = 1 2 Re lim ρ u r ( ρ E θ r ρ H ϕ r * - ρ E ϕ r ρ H θ r * ) = k ρ 2 2 ω μ u r ( E θ r 2 + E ϕ r 2 ) .
P = k ρ 2 2 ω μ u r E θ r 2 .
P = k ρ 2 2 ω μ u r E ϕ r 2 .