Abstract

We have applied an anomalous diffraction approximation in order to obtain extinction cross sections of a variety of nonspherical particles in a simple analytical form. Specifically, we have derived extinction cross sections of partial spheres, cones, and prismatic columns with triangular, rectangular, trapezium, and hexagonal bases. We have also derived an addition theorem showing how the extinction cross section of an arbitrary column with a polygonal base can be obtained. The extinction cross section of a long cylinder is obtained as the limiting case of a polygonal-based prismatic column with the number of sides approaching infinity.

© 1991 Optical Society of America

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References

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  1. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]
  2. P. C. Waterman, C. V. McCarthy, “Numerical solution of electromagnetic scattering problems,” (Mitre Corporation, Bedford, Mass., June1968).
  3. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  4. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
    [CrossRef] [PubMed]
  5. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 740.
  6. D. H. Napper, “A diffraction theory approach to the total scattering by cubes,” Kolloid Z. Z. Polym. 218, 41–46 (1967).
    [CrossRef]
  7. G. L. Stephens, “Scattering of plane waves by soft obstacles: anomalous diffraction theory for circular cylinders,” Appl. Opt. 23, 954–959 (1984).
    [CrossRef] [PubMed]
  8. P. Chýlek, “Interference structure of the Mie extinction cross section,” J. Opt. Soc. Am. A 6, 1846–1851 (1989).
    [CrossRef]
  9. A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric sphere,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [CrossRef]
  10. P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
    [CrossRef]
  11. G. J. Rosasco, H. S. Bennett, “Internal field resonance structure: implication for optical absorption and scattering by microscopic particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
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  12. J. R. Probert-Jones, “Resonance component of backscatter by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
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  14. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]

1989 (1)

1988 (3)

1984 (2)

1979 (1)

1978 (2)

1977 (1)

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric sphere,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1967 (1)

D. H. Napper, “A diffraction theory approach to the total scattering by cubes,” Kolloid Z. Z. Polym. 218, 41–46 (1967).
[CrossRef]

Asano, S.

Ashkin, A.

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric sphere,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

Bennett, H. S.

Box, M. A.

Chýlek, P.

P. Chýlek, “Interference structure of the Mie extinction cross section,” J. Opt. Soc. Am. A 6, 1846–1851 (1989).
[CrossRef]

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric sphere,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

Goedecke, G. H.

Hunter, B. A.

Kiehl, J. T.

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Ko, M. K. W.

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Lock, J. A.

Maier, B.

McCarthy, C. V.

P. C. Waterman, C. V. McCarthy, “Numerical solution of electromagnetic scattering problems,” (Mitre Corporation, Bedford, Mass., June1968).

Napper, D. H.

D. H. Napper, “A diffraction theory approach to the total scattering by cubes,” Kolloid Z. Z. Polym. 218, 41–46 (1967).
[CrossRef]

O’Brien, S. G.

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Probert-Jones, J. R.

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Rosasco, G. J.

Stephens, G. L.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 740.

Waterman, P. C.

P. C. Waterman, C. V. McCarthy, “Numerical solution of electromagnetic scattering problems,” (Mitre Corporation, Bedford, Mass., June1968).

Appl. Opt. (3)

Astrophys. J. (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Kolloid Z. Z. Polym. (1)

D. H. Napper, “A diffraction theory approach to the total scattering by cubes,” Kolloid Z. Z. Polym. 218, 41–46 (1967).
[CrossRef]

Phys. Rev. A (1)

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Phys. Rev. Lett. (1)

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric sphere,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

Other (2)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 740.

P. C. Waterman, C. V. McCarthy, “Numerical solution of electromagnetic scattering problems,” (Mitre Corporation, Bedford, Mass., June1968).

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Figures (9)

Fig. 1
Fig. 1

Particle of refractive index m casts a shadow on a projection plane P. The geometrical path of an individual ray within a particle is denoted by l.

Fig. 2
Fig. 2

System of coordinates used for scattering by a sphere.

Fig. 3
Fig. 3

Normalized extinction cross section Qext as a function of the phase parameter ρ = 2πa(m − 1)/λ for the case of a spherical particle with refractive index m = 1.2 for Mie theory, a running mean of Mie calculations over the size parameter interval Δx = 2 (〈MIE〉), and the ADA approximation.

Fig. 4
Fig. 4

Scattering by a right circular cone topped by a spherical cap. The geometrical paths of a ray within the cone and the cap are lcone and lcap, respectively.

Fig. 5
Fig. 5

(a) The addition theorem states that the real part of the forward-scattering amplitude can be written as a sum of the real parts of the forward-scattering amplitudes of nonoverlapping sections, (b) column with a triangular base with one side parallel to the direction of incoming radiation, (c) trapezoidal base as a difference of two triangles.

Fig. 6
Fig. 6

(a) Column with triangular base in arbitrary orientation, (b) column with rectangular base in arbitrary orientation, (c) column with trapezium base.

Fig. 7
Fig. 7

Column with hexagonal base: (a) in arbitrary orientation, (b) with special cases of edge-on incidence, (c) flat incidence.

Fig. 8
Fig. 8

Comparison of normalized extinction cross section of an infinitely long circular cylinder using Mie theory, a running average of Mie calculations over the size parameter interval Δx = 2 (〈MIE〉), and the ADA approximation.

Fig. 9
Fig. 9

ADA-normalized extinction cross sections Qext of a sphere, a spherical cap, and a triangular column as a function of the phase parameter ρ = 2πlmax(m − 1)/λ, where lmax is the largest possible ray path within the considered particle with given orientation. With this choice of phase parameter as the independent variable, the oscillations of the Qext curves (due to interference effects) have the same period of oscillation. The spherical angle θ0 = 67.5° (cf. Fig. 4).

Equations (29)

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ψ = k l ( m - 1 ) .
S ( 0 ) = k 2 2 π P { 1 - exp [ - i ψ ( ξ , η ) ] } d P .
σ ext = 4 π k 2 Re S ( 0 ) .
Q ext = 4 π k 2 P Re S ( 0 ) .
Re S ( 0 ) = k 2 2 π P ( 1 - cos ψ ) d P ,
Re S ( 0 ) = k 2 π P sin 2 ( ψ 2 ) d P .
ψ = ρ cos θ ,
Re S ( 0 ) = x 2 π ϕ max θ 1 θ 2 sin 2 ( ψ 2 ) sin θ cos θ d θ = ϕ max x 2 2 [ cos 2 θ 2 - cos θ sin ( ρ cos θ ) ρ - cos ( ρ cos θ ) ρ 2 ] θ 2 θ 1 .
P = ϕ max a 2 2 ( sin 2 θ 1 - sin 2 θ 2 ) ,
Q ext = 4 ( 1 2 - sin ρ ρ + 1 - cos ρ ρ 2 ) ,
ψ = ρ csc θ 0 sin ( θ 0 - θ ) ,
Re S ( 0 ) = 2 x 2 0 θ 0 sin 2 [ ρ 2 csc θ 0 sin ( θ 0 - θ ) ] sin θ cos θ d θ ,
Re S ( 0 ) = 2 k 2 0 r sin 2 { k ( m - 1 ) 2 ( a 2 - ω ¯ 2 ) 1 / 2 - H } ω ¯ d ω ¯ = 8 ( m - 1 ) 2 0 ρ cap / 2 [ v + k H ( m - 1 ) 2 ] sin 2 v d v = k 2 ( a - H ) 2 [ 1 2 - sin ρ cap ρ cap + ( 1 - cos ρ cap ) ρ cap 2 ] + k H ρ cap ( m - 1 ) ( 1 - sin ρ cap ρ cap ) .
Re S ( 0 ) = 2 x 2 0 1 v sin 2 [ ρ H 2 ( 1 - v ) ] d v = x 2 [ 1 2 - ( 1 - cos ρ H ) ρ H 2 ] ,
Re S ( 0 ) = j = 1 n Re S j ( 0 ) .
Re S ( 0 ) = k 2 2 π 0 L d l 0 H ( 1 - cos ψ ) d h = k 2 2 π H L ( 1 - sin ρ t ρ t ) .
Q ext = 2 ( 1 - sin ρ t ρ t ) .
Re S ( 0 ) = k 2 2 π P ( 1 - sin ρ 1 - sin ρ 2 ρ 1 - ρ 2 ) ,
Re S ( 0 ) = k 2 2 π P ( 1 - cos ρ ) ,
Re S ( 0 ) = k 2 π P sin 2 ( ρ 2 ) ,
Q ext = 2 ( 1 - 1 P K P i sin ρ i ρ i - 1 P M P i sin ρ 1 i - sin ρ 2 i ρ 1 i - ρ 2 i - 1 P N P i cos ρ i ) ,
P = K P i + M P i + N P i
Q ext = 2 ( 1 - 2 a sin α a sin α + b cos α sin ρ ρ - b cos α - a sin α b cos α + a sin α cos ρ ) .
Q ext = 2 ( 1 - 2 sin α cos α + sin α sin ρ ρ - cos α - sin α sin α + cos α cos ρ ) .
Q ext = 2 { 1 - [ a sin α b cos β + ( 1 - a sin α + c sin γ b cos β ) ρ 1 ρ 1 - ρ 2 ] × sin ρ 1 ρ 1 - [ c sin γ b cos β - ( 1 - a sin α + c sin γ b cos β ) ρ 2 ρ 1 - ρ 2 ] × sin ρ 2 ρ 2 } ,
ρ 1 = k a ( m - 1 ) cos α ( 1 + tan α tan β ) , ρ 2 = k c ( m - 1 ) cos γ ( 1 - tan γ tan β ) .
Q ext = 2 ( 1 - 2 sin ρ - sin ρ / 2 ρ ) ,
Q ext = 2 ( 1 - sin ρ 2 ρ - cos ρ 2 ) ,
Q ext = 2 [ 1 - 1 P lim N ( N P i cos ρ i ) ] = 2 { 1 - 1 2 a - a a cos [ 2 k ( m - 1 ) ( a 2 - y 2 ) 1 / 2 ] d y } = 2 - 1 1 sin 2 [ ρ 2 ( 1 - v 2 ) 1 / 2 ] d v = π H 1 ( ρ ) ,

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