Abstract

A method of calculating the scattered electromagnetic fields of an infinite cylinder of arbitrary orientation illuminated with a shaped beam is presented. The method relies on the use of a theory known as the generalized Lorenz–Mie theory that provides the general framework. The three-dimensional nature of the incident shaped beam is considered. For the case of a tightly focused Gaussian beam propagating perpendicular to the cylinder axis, the scattering characteristics that are different from those for an incident plane wave are described in detail, and numerical results of the normalized differential scattering cross section are evaluated.

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References

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  1. J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189-195 (1955).
    [CrossRef]
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
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    [CrossRef]
  4. G. Gouesbet and G. Gréhan, "Interaction between a Gaussian beam and an infinite cylinder with the use of non-Σ-separable potentials," J. Opt. Soc. Am. A 11, 3261-3273 (1994).
    [CrossRef]
  5. G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
    [CrossRef]
  6. G. Gouesbet, G. Gréhan, and K. F. Ren, "Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz-Mie theory for cylinders," J. Opt. Soc. Am. A 15, 511-523 (1998).
    [CrossRef]
  7. G. Gouesbet, K. F. Ren, L. Mees, and G. Gréhan, "Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorenz-Mie theory for cylinders, with arbitrary location and orientation of the scatterer," Appl. Opt. 38, 2647-2665 (1999).
    [CrossRef]
  8. G. Gouesbet, "Validity of the cylindrical localized approximation for arbitrary shaped beams in generalized Lorenz-Mie theory for circular cylinders," J. Mod. Opt. 46, 1185-1200 (1999).
    [CrossRef]
  9. K. F. Ren, G. Gréhan, and G. Gouesbet, "Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory formulation and numerical results," J. Opt. Soc. Am. A 14, 3014-3025 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.
  15. S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).
  16. H. Y. Zhang, Y. P. Han, and G. X. Han, "Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions," J. Opt. Soc. Am. B 24, 1383-1391 (2007).
    [CrossRef]
  17. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
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    [CrossRef] [PubMed]
  19. L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  20. J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  21. C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory (International Textbook Company, 1971).

2007 (1)

1999 (3)

1998 (1)

1997 (3)

1995 (1)

G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
[CrossRef]

1994 (1)

1990 (1)

1989 (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

1988 (2)

1979 (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1961 (1)

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

1955 (1)

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189-195 (1955).
[CrossRef]

Appl. Opt. (4)

Can. J. Phys. (1)

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189-195 (1955).
[CrossRef]

J. Appl. Phys. (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

J. Mod. Opt. (1)

G. Gouesbet, "Validity of the cylindrical localized approximation for arbitrary shaped beams in generalized Lorenz-Mie theory for circular cylinders," J. Mod. Opt. 46, 1185-1200 (1999).
[CrossRef]

J. Opt. (Paris) (1)

G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (1)

Phys. Rev. A (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Q. Appl. Math. (1)

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

Other (4)

C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory (International Textbook Company, 1971).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957), Chap. 4.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (3)

Fig. 1
Fig. 1

The Cartesian coordinate system O x y z is attached to an infinite cylinder and is obtained by a rigid-body rotation of the system o x y z through Euler angles α, β, γ. O x y z is parallel to the shaped beam coordinate system O x y z , and the Cartesian coordinates of O in O x y z are ( 0 , 0 , z 0 ) .

Fig. 2
Fig. 2

α 1 ( ζ ) sin ζ (solid curve) and α 2 ( ζ ) sin ζ (dotted curve) versus ζ for an infinite cylinder ( n ̃ = 1.33 , k r 0 = 12.57 , and z 0 = 0 ) with Euler angles α = γ = 0 , β = π 2 illuminated by the Gaussian beam (TE mode) with s = 0.15 .

Fig. 3
Fig. 3

Normalized differential scattering cross section k 2 σ ( ϕ ) 4 for an infinite cylinder ( n ̃ = 1.33 , k r 0 = 12.57 , and z 0 = 0 ) (solid curve) and that for another one ( n ̃ = 1.33 , k r 0 = 18.85 , and z 0 = 0 ) (dotted curve), all with Euler angles α = γ = 0 , β = π 2 , illuminated by the Gaussian beam (TE mode) with s = 0.15 .

Equations (36)

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E i = E 0 m = 0 π [ I m , TE ( ζ ) m m λ + I m , TM ( ζ ) n m λ ] exp ( i h z ) sin ζ d ζ ,
H i = i E 0 k ω μ m = 0 π [ I m , TE ( ζ ) n m λ + I m , TM ( ζ ) m m λ ] exp ( i h z ) sin ζ d ζ ,
I m , TE = ( 1 ) m 1 i m + 1 k sin 2 ζ L m , TE ,
I m , TM = ( 1 ) m 1 i m + 1 k sin 2 ζ L m , TM ,
L m , TE = n = m ( n m ) ! ( n + m ) ! 2 n + 1 2 n ( n + 1 ) g n [ m 2 P n m ( cos β ) sin β P n m ( cos ζ ) + d P n m ( cos β ) d β d P n m ( cos ζ ) d ζ sin ζ ] ,
L m , TM = m n = m ( n m ) ! ( n + m ) ! 2 n + 1 2 n ( n + 1 ) g n [ P n m ( cos β ) sin β d P n m ( cos ζ ) d ζ sin ζ + d P n m ( cos β ) d β P n m ( cos ζ ) ] ,
g n = 1 1 + 2 i s z 0 w 0 exp ( i k z 0 ) exp [ s 2 ( n + 1 2 ) 2 1 + 2 i s z 0 w 0 ] ,
E s = E 0 m = 0 π [ α m ( ζ ) m m λ ( 3 ) + β m ( ζ ) n m λ ( 3 ) ] exp ( i h z ) sin ζ d ζ ,
H s = i E 0 k ω μ m = 0 π [ α m ( ζ ) n m λ ( 3 ) + β m ( ζ ) m m λ ( 3 ) ] exp ( i h z ) sin ζ d ζ ,
E w = E 0 m = 0 π [ χ m ( ζ ) m m λ ( 1 ) + τ m ( ζ ) n m λ ( 1 ) ] exp ( i h z ) sin ζ d ζ ,
H w = i E 0 k ω μ m = 0 π [ χ m ( ζ ) n m λ ( 1 ) + τ m ( ζ ) m m λ ( 1 ) ] exp ( i h z ) sin ζ d ζ ,
E ϕ i + E ϕ s = E ϕ w E z i + E z s = E z w H ϕ i + H ϕ s = H ϕ w H z i + H z s = H z w } .
ξ d d ξ J m ( ξ ) I m , TE + h m k J m ( ξ ) I m , TM + ξ d d ξ H m ( 1 ) ( ξ ) α m ( ζ ) + h m k H m ( 1 ) ( ξ ) β m ( ζ ) = η d d η J m ( η ) χ m ( ζ ) + h m k J m ( η ) τ m ( ζ ) ,
J m ( ξ ) I m , TM + H m ( 1 ) ( ξ ) β m ( ζ ) = λ 2 k k λ 2 J m ( η ) τ m ( ζ ) ,
ξ d d ξ J m ( ξ ) I m , TM + h m k J m ( ξ ) I m , TE + ξ d d ξ H m ( 1 ) ( ξ ) β m ( ζ ) + h m k H m ( 1 ) ( ξ ) α m ( ζ ) = k k [ η d d η J m ( η ) τ m ( ζ ) + h m k J m ( η ) χ m ( ζ ) ] ,
J m ( ξ ) I m , TE + H m ( 1 ) ( ξ ) α m ( ζ ) = λ 2 λ 2 J m ( η ) χ m ( ζ ) ,
α m ( ζ ) = ( 1 ) m 1 i m + 1 k sin 2 ζ α m ( ζ ) ,
β m ( ζ ) = ( 1 ) m 1 i m + 1 k sin 2 ζ β m ( ζ ) ,
α m ( ζ ) = 1 W m ( ζ ) V m ( ζ ) D m 2 ( ζ ) × { [ A m ( ζ ) V m ( ζ ) + C m ( ζ ) D m ( ζ ) ] L m , TE [ C m ( ζ ) V m ( ζ ) + B m ( ζ ) D m ( ζ ) ] L m , TM } ,
β m ( ζ ) = 1 W m ( ζ ) V m ( ζ ) D m 2 ( ζ ) × { [ B m ( ζ ) W m ( ζ ) + C m ( ζ ) D m ( ζ ) ] L m , TM [ C m ( ζ ) W m ( ζ ) + A m ( ζ ) D m ( ζ ) ] L m , TE } ,
A m ( ζ ) = ξ [ η d d ξ J m ( ξ ) J m ( η ) ξ J m ( ξ ) d d η J m ( η ) ] ,
B m ( ζ ) = ξ [ η d d ξ J m ( ξ ) J m ( η ) n ̃ 2 ξ J m ( ξ ) d d η J m ( η ) ] ,
C m ( ζ ) = m cos ζ η J m ( ξ ) J m ( η ) ( ξ 2 η 2 1 ) ,
D m ( ζ ) = m cos ζ η H m ( 1 ) ( ξ ) J m ( η ) ( ξ 2 η 2 1 ) ,
W m ( ζ ) = ξ [ ξ H m ( 1 ) ( ξ ) d d η J m ( η ) η d d ξ H m ( 1 ) ( ξ ) J m ( η ) ] ,
V m ( ζ ) = ξ [ n ̃ 2 ξ H m ( 1 ) ( ξ ) d d η J m ( η ) η d d ξ H m ( 1 ) ( ξ ) J m ( η ) ] .
E s = E 0 exp ( i π 4 ) m = ( i ) m exp ( i m ϕ ) [ a m e ̂ ϕ + b m e ̂ r + c m e ̂ z ] ,
a m = ( i ) m 2 π k r 0 π α m ( ζ ) 1 sin ζ exp [ i k ( r sin ζ + z cos ζ ) ] d ζ ,
b m = ( i ) m + 1 2 π k r 0 π β m ( ζ ) cos ζ sin ζ exp [ i k ( r sin ζ + z cos ζ ) ] d ζ ,
c m = ( i ) m + 1 2 π k r 0 π β m ( ζ ) sin ζ exp [ i k ( r sin ζ + z cos ζ ) ] d ζ .
e ̂ y exp [ i k ( x sin ζ + z cos ζ ) ] = m = ( i ) m + 1 k sin ζ m m λ exp ( i h z ) .
E s = E 0 exp ( i π 4 ) 2 π k r sin ζ exp [ i k ( r sin ζ + z cos ζ ) ] × m = ( 1 ) m + 1 exp ( i m ϕ ) [ a m e ̂ ϕ + i b m ( cos ζ e ̂ r sin ζ e ̂ z ) ] ,
a m = A m ( ζ ) V m ( ζ ) + C m ( ζ ) D m ( ζ ) W m ( ζ ) V m ( ζ ) D m 2 ( ζ ) ,
b m = C m ( ζ ) W m ( ζ ) + A m ( ζ ) D m ( ζ ) W m ( ζ ) V m ( ζ ) D m 2 ( ζ ) .
E s = 2 i E 0 1 k r exp ( i k r ) m = ( 1 ) m exp ( i m ϕ ) α m ( π 2 ) e ̂ ϕ .
σ ( ϕ ) = r 2 E s E 0 2 = 4 k 2 m = ( 1 ) m exp ( i m ϕ ) α m ( π 2 ) 2 .

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