Abstract

The scattering of a Gaussian beam wave by a spherical object is treated exactly in terms of the vector wave functions without any restriction on the size or the position of the scatterer. Expressions obtained for the powers absorbed and scattered are given as linear combinations of the well-known Mie coefficients and can be readily applied to numerical computation. The corresponding problem for the scattering of a beam produced by a laser operating in the TEM01* mode is also solved.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [Crossref]
  2. N. Morita, T. Tanaka, T. Yamasaki, and Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. 16, 724–727 (1968).
    [Crossref]
  3. W. C. Tsai and R. J. Pogorzelski, “Eigenfunction solution of the scattering of beam radiation field by spherical objects,” J. Opt. Soc. Am. 65, 1457–1463 (1975).
    [Crossref]
  4. L. W. Casperson, C. Yeh, and W. F. Yeung, “Single particle scattering with focused laser beams,” Appl. Opt. 16, 1104–1107 (1977).
    [PubMed]
  5. W. G. Tam, “Off-beam axis scattering by spherical particles,” Appl. Opt. 16, 2016–2018 (1977).
    [Crossref] [PubMed]
  6. B. G. Schuster and R. Knollenberg, “Detection and sizing of small particles in an open cavity gas laser,” Appl. Opt. 11, 1515–1520 (1972).
    [Crossref] [PubMed]
  7. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [Crossref]
  8. A. G. van Nie, “Rigorous calculation on the electromagnetic field of wave beams,” Philips Res. Rep. 19, 378–394 (1964).
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941) pp. 28–32, 369, 392–399, 414–418, 563–565.
  10. In Ref. 3 the factor ejhz on the left-hand side of Eq. (7) was missing.
  11. I. S. Gradshteyn and I. M. Gyzhik, Table of Integrals, Series and Products (Academic, New York, 1965) p. 979.
  12. W. G. Tam and R. Corriveau, “Scattering of electromagnetic beams by spherical objects,” , February1978 (see Appendices).
  13. J. V. Dave, “Subroutine for computing the parameters of the electromagnetic radiation scattered by a sphere,” . (IBM Scientific Center, Palo Alto, California, 1968).
  14. F. OberhettingerTables of Bessel Transforms (Springer-Verlag, New York, 1972) p. 46.

1977 (2)

1975 (1)

1972 (2)

1968 (1)

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. 16, 724–727 (1968).
[Crossref]

1964 (1)

A. G. van Nie, “Rigorous calculation on the electromagnetic field of wave beams,” Philips Res. Rep. 19, 378–394 (1964).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Carter, W. H.

Casperson, L. W.

Corriveau, R.

W. G. Tam and R. Corriveau, “Scattering of electromagnetic beams by spherical objects,” , February1978 (see Appendices).

Dave, J. V.

J. V. Dave, “Subroutine for computing the parameters of the electromagnetic radiation scattered by a sphere,” . (IBM Scientific Center, Palo Alto, California, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Gyzhik, Table of Integrals, Series and Products (Academic, New York, 1965) p. 979.

Gyzhik, I. M.

I. S. Gradshteyn and I. M. Gyzhik, Table of Integrals, Series and Products (Academic, New York, 1965) p. 979.

Knollenberg, R.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Morita, N.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. 16, 724–727 (1968).
[Crossref]

Nahanishi, Y.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. 16, 724–727 (1968).
[Crossref]

Oberhettinger, F.

F. OberhettingerTables of Bessel Transforms (Springer-Verlag, New York, 1972) p. 46.

Pogorzelski, R. J.

Schuster, B. G.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941) pp. 28–32, 369, 392–399, 414–418, 563–565.

Tam, W. G.

W. G. Tam, “Off-beam axis scattering by spherical particles,” Appl. Opt. 16, 2016–2018 (1977).
[Crossref] [PubMed]

W. G. Tam and R. Corriveau, “Scattering of electromagnetic beams by spherical objects,” , February1978 (see Appendices).

Tanaka, T.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. 16, 724–727 (1968).
[Crossref]

Tsai, W. C.

van Nie, A. G.

A. G. van Nie, “Rigorous calculation on the electromagnetic field of wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Yamasaki, T.

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. 16, 724–727 (1968).
[Crossref]

Yeh, C.

Yeung, W. F.

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

N. Morita, T. Tanaka, T. Yamasaki, and Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. 16, 724–727 (1968).
[Crossref]

J. Opt. Soc. Am. (2)

Philips Res. Rep. (1)

A. G. van Nie, “Rigorous calculation on the electromagnetic field of wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Other (6)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941) pp. 28–32, 369, 392–399, 414–418, 563–565.

In Ref. 3 the factor ejhz on the left-hand side of Eq. (7) was missing.

I. S. Gradshteyn and I. M. Gyzhik, Table of Integrals, Series and Products (Academic, New York, 1965) p. 979.

W. G. Tam and R. Corriveau, “Scattering of electromagnetic beams by spherical objects,” , February1978 (see Appendices).

J. V. Dave, “Subroutine for computing the parameters of the electromagnetic radiation scattered by a sphere,” . (IBM Scientific Center, Palo Alto, California, 1968).

F. OberhettingerTables of Bessel Transforms (Springer-Verlag, New York, 1972) p. 46.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

FIG. 1
FIG. 1

Geometrical configuration.

Equations (63)

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

I ( z = - z 0 ) e - 2 r 2 / w 0 2 ,
E ( z = - z 0 ) = u x e - r 2 / w 0 2 .
Π = ( j / ω ) e - j ω t sin ϕ ( w 0 4 / 4 ) u z × 0 e - λ 2 w 0 2 / 4 J 1 ( λ r ) e j h ( z + z 0 ) λ 2 d λ
E = - μ t × Π .
H = × × Π .
E = w 0 4 4 0 e - λ 2 w 0 2 / 4 m o 1 λ e j h ( z + z 0 ) λ 2 d λ .
e j h z m o 1 λ = l = 1 2 l + 1 l ( l + 1 ) ( l - 1 ) ! ( l + 1 ) ! j l - 1 × λ ( j d P l 1 ( cos α ) d α m o l 1 + P l 1 ( cos α ) sin α n e l 1 ) .
e j n ϕ J n ( λ r ) = m = - J n ( λ r 1 ) J n + m ( λ r ) e j ( m + n ) ϕ e - j m ϕ 1
Π = - u z w 0 4 8 ω e - j ω t 0 e - λ 2 w 0 2 / 4 e j h ( z + z 1 + z 0 ) × s = - J s ( λ r 1 ) e j s ( ϕ - ϕ 1 ) J s + 1 ( λ r ) e j ϕ + J s - 1 ( λ r ) e - j ϕ λ 2 d λ .
m s λ = m o s λ - j m e s λ ,             n s λ = n o s λ - j n e s λ .
E = j w 0 4 8 e - j ω t 0 e - λ 2 ω 0 2 / 4 e j h ( z + z 1 + z 0 ) × s = - J s ( λ r 1 ) e - j s ϕ 1 [ m s + 1 , λ + m s - 1 , λ ] λ 2 d λ .
e j h z m s λ = l = 1 2 l + 1 l ( l + 1 ) ( l - s ) ! ( l + s ) ! j l - s + 1 × λ ( d P l s ( cos α ) d α m l s + s sin α P l s ( cos α ) n l s ) .
E = l = 1 s = - D A ( l , s ) m l s + D B ( l , s ) n l s ,
{ D A ( l , s ) D B ( l , s ) } = j w 0 4 8 e - j ω t 0 d λ e - λ 2 w 0 2 / 4 e j h ( z 1 + z 0 ) λ 2 × [ J s - 1 ( λ r 1 ) e - j ( s - 1 ) ϕ 1 + J s + 1 ( λ r 1 ) e - j ( s + 1 ) ϕ 1 ] × { A ( l , s ; λ ) B ( l , s ; λ ) } ,
{ A ( l , s ; λ ) B ( l , s ; λ ) } = 2 l + 1 l ( l + 1 ) ( l - s ) ! ( l + s ) ! j l - s + 1 λ { d P l s ( cos α ) d α s P l s ( cos α ) sin α } .
u R × ( E + E r ) = u R × E t ,
u R × ( H + H r ) = u R × H t ,
E r = l = 1 s = - D A ( l , s ) a l s r m l s ( 3 ) + D B ( l , s ) b l s r n l s ( 3 ) ,
E t = l = 1 s = - D A ( l , s ) a l s t m l s ( 1 ) + D B ( l , s ) b l s t n l s ( 1 ) .
a l s r = a l s r ,             b l s r = b l s r ,
a l s t = a l s t ,             b l s t = b l s t .
a l s r = - j l ( ρ 1 ) [ ρ j l ( ρ ) ] - j l ( ρ ) [ ρ 1 j l ( ρ 1 ) ] j l ( ρ 1 ) [ ρ h l ( 1 ) ( ρ ) ] - h l ( 1 ) ( ρ ) [ ρ 1 j l ( ρ 1 ) ] ,
b l s r = - j l ( ρ ) [ ρ 1 j l ( ρ 1 ) ] - ( k / k ) 2 j l ( ρ 1 ) [ ρ j l ( ρ ) ] h l ( 1 ) ( ρ ) [ ρ 1 j l ( ρ 1 ) ] - ( k / k ) 2 j l ( ρ 1 ) [ ρ h l ( 1 ) ( ρ ) ] ,
E r = l = 1 s ( a l r s = - D A ( l , s ) m l s ( 3 ) + b l s r s = - D B ( l , s ) n l s ( 3 ) ) .
H = k j ω l = 1 s = - D A ( l , s ) n l s + D B ( l , s ) m l s
H r = k j ω l = 1 ( a l r s = - D A ( l , s ) n l s ( 3 ) + b l r s = - D B ( l , s ) m l s ( 3 ) ) .
W s = lim R 1 2 0 π 0 2 π R 2 ( E θ r H ϕ r * - E ϕ r H θ r * ) sin θ d θ d ϕ = 2 π k ω l = 1 s = - l ( l + 1 ) 2 l + 1 ( l + s ) ! ( l - s ) ! × { a l r D A ( l , s ) 2 + b l r D B ( l , s ) 2 } .
0 π 0 2 π ( m l s θ ( 3 ) m l s ϕ ( 3 ) - m l s ϕ ( 3 ) m l s θ ( 3 ) * ) sin θ d θ d ϕ = 0 ,
0 π 0 2 π ( n l s ϕ ( 3 ) n l s θ ( 3 ) * - n l s θ ( 3 ) n l s ϕ ( 3 ) * ) sin θ d θ d ϕ = 0 ,
0 π 0 2 π ( m l s θ ( 3 ) n l s ϕ ( 3 ) * - m l s ϕ ( 3 ) n l s θ ( 3 ) * ) sin θ d θ d ϕ = δ l , l δ s , s 4 π l 2 l + 1 ( l + 1 ) ( l + s ) ! ( l - s ) ! × h l ( 1 ) ( ρ ) 1 ρ d d ρ [ ρ h l ( 1 ) * ( ρ ) ] .
W = lim R - ½ Re 0 π 0 2 π R 2 sin θ ( E θ H ϕ r * + E θ r H ϕ * - E ϕ H θ r * - E ϕ r H θ * ) d θ d ϕ
W = 2 π k ω Re l = 1 s = - l ( l + 1 ) 2 l + 1 ( l + s ) ! ( l - s ) ! × { a l r D A ( l , s ) 2 + b l r D B ( l , s ) 2 } .
{ D A ( l , s ) D B ( l , s ) } = j w 0 4 8 e - j ω t 0 e - λ 2 w 0 2 / 4 e j h ( z 1 + z 0 ) × ( δ s , 1 + δ s , - 1 ) × { A ( l , s ; λ ) B ( l , s ; λ ) } λ 2 d λ .
W s = 2 π k ω l = 1 2 ( 2 l + 1 ) [ l ( l + 1 ) ] 2 [ d A ( l ) a l r 2 + d B ( l ) b l r 2 ]
W t = 2 π k ω l = 1 2 ( 2 l + 1 ) [ l ( l + 1 ) ] 2 [ a l r d A ( l ) 2 + b l r d B ( l ) 2 ] ,
d A ( l ) = w 0 4 8 0 e - λ 2 w 0 2 / 4 e j h ( z 1 + z 0 ) d P l 1 ( cos α ) d α λ 2 d λ ,
d B ( l ) = w 0 4 8 0 e - λ 2 w 0 2 / 4 e j h ( z 1 + z 0 ) P l 1 ( cos α ) sin α λ 2 d λ .
W s = π k ω l = 1 [ ( 2 l + 1 ) a l r 2 + b l r 2 ]
W t = π k ω Re l = 1 ( 2 l + 1 ) ( a l r + b l r )
I ( z = - z 0 ) r 2 e - 2 r 2 / w 0 2 ,
E ( z = - z 0 ) = u x r e - r 2 / w 0 2 .
E ( 1 ) = × × Π ( 1 ) ,             H ( 1 ) = ɛ t × ( 1 ) ,
E ( 2 ) = - μ t × Π ( 2 ) ,             H ( 2 ) = × × Π ( 2 ) .
Π ( 1 ) = ψ u z ,             Π ( 2 ) = χ u z .
ψ = e - j ω t ( r 2 / 2 ) cos 2 ϕ             e - r 2 / w 0 2 f ( z ) ,
χ = ( 1 / 2 ω ) e - j ω t r 2 sin 2 ϕ e - r 2 / w 0 2 g ( z ) .
d f d z | z = - z 0 = 1 ,             g ( z = - z 0 ) = 1.
q s i ( λ ) = 0 r 2 e - r 2 / w 0 2 J s i ( λ ) r d r ,
ψ = i = 1 3 e - j ω t + j s i ϕ δ i 8 0 q s i ( λ ) h J s i ( λ r ) e j h ( z + z 0 ) λ d λ ,
χ = - i = 1 3 e - j ω t + j s i ϕ ( - 1 ) i δ i 8 ω × 0 q s i ( λ ) J s i ( λ r ) e j h ( z + z 0 ) λ d λ ,
E ( 1 ) = e - j ω t i = 1 3 e j s i ϕ δ i 8 0 q s i ( λ ) e j h ( z + z 0 ) × ( j J s i ( λ r ) r u r - s i r J s i ( λ r ) u ϕ + λ 2 h J s i ( λ r ) u z ) λ d λ ,
E ( 2 ) = e - j ω t i = 1 3 e j s i ϕ ( - 1 ) i δ i j 8 0 q s i ( λ ) e j h ( z + z 0 ) × ( - j s i r J s i ( λ r ) u r + J s j ( λ r ) r u ϕ ) λ d λ .
E = E ( 1 ) + E ( 2 ) = e - j ω t i = 1 3 δ i j 8 0 q s i ( λ ) e j h ( z + z 0 ) × ( k h n s i + ( - 1 ) i j n s i λ ) λ d λ .
E = e - j ω t 8 i = 1 3 s = - 0 ξ ( s , i , λ ) e j h z × ( j k h n s + s i , λ - ( - 1 ) i m s + s i , λ ) λ d λ ,
ξ ( s , i , λ ) = δ i λ q s i ( λ ) e j h ( z 1 + z 0 ) J s ( λ r 1 ) e - j s ϕ 1 ,
e j h z n s λ = l = 1 2 l + 1 l ( l + 1 ) ( l - s ) ! ( l + s ) ! j l - s + 1 λ × [ d P l s ( cos α ) d α m l s + s sin α P l s ( cos α ) n l s ] ,
E = l = 1 s = - [ J ( l , s ) m l s + K ( l , s ) n l s ] ,
J ( l , s ) = e - j ω t 8 i = 1 3 0 ξ ( s - s i , i , λ ) × [ j k h B ( l , s ; λ ) - ( - 1 ) i A ( l , s ; λ ) ] d λ ,
E r = l = 1 s = - [ a l r J ( l , s ) m l s ( 3 ) + b l r K ( l , s ) n l s ( 3 ) ] ,
H r = k j ω l = 1 s = - [ a l r J ( l , s n l s ( 3 ) + b l r K ( l , s ) n l s ( 3 ) ] .
W s = 2 π k ω l = 1 s = - l ( l + 1 ) 2 l + 1 ( l + s ) ! ( l - s ) ! × { a l r J ( l , s ) 2 + b l r K ( l , s ) 2 } ,
W t = 2 π k ω Re l = 1 s = - l ( l + 1 ) 2 l + 1 ( l + s ) ! ( l - s ) ! × { a l r J ( l , s ) 2 + b l r K ( l , s ) 2 } .
0 d λ e - ( λ 2 w 0 2 / 4 ) J s ( λ r 1 ) λ m