Abstract

Glauber's approximation method valid at high energy is used to study light scattering by a stratified sphere. The scattering amplitude is obtained in a simple closed form. The resulting efficiency factors and intensities of the scattered light are numerically evaluated for dielectric bubbles and wet dielectric balls and are shown to agree well with the exact values from Mie's method.

© 1987 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, New York, 1969).
  2. R. Bhandari, “Scattering Coefficients for a Multilayered Sphere: Analytic Expressions and Algorithms,” Appl. Opt. 24, 1960 (1985).
    [CrossRef] [PubMed]
  3. R. Bhandari, “Tiny Core or Thin Layer as a Perturbation in Scattering by a Single-layered Sphere,” J. Opt. Soc. Am. A 3, 319 (1986).
    [CrossRef]
  4. L. I. Schiff, “Approximation Method for Short Wavelength or High-Energy Scattering,” Phys. Rev. 104, 1481 (1956).
    [CrossRef]
  5. R. J. Glauber, “High-Energy Collision Theory,” in Lecture in. Theoretical Physics, W. E. Britten, L. C. Dunham, Eds. (Interscience, New York, 1959).
  6. J. M. Perrin, P. Chiappeta, “Light Scattering by Large Particle I. A New Theoretical Description in the Eikonal Picture,” Opt. Acta 32, 907 (1985);J. M. Perrin, P. L. Lamy, “Light Scattering by Large Particles II. A Vectorial Description in the Eikonal Picture,” Opt. Acta 33, 1001 (1986).
    [CrossRef]
  7. D. S. Saxon, “Lecture on the Scattering of Light,” UCLA Department of Meteorological Science Report9 (1955).
  8. T. W. Chen, “Generalized Eikonal Approximation,” Phys. Rev. C30, 585 (1984).
  9. C. J. Joachain, C. Quigg, “Multiple Scattering Expansions in Several Particle Dynamics,” Rev. Mod. Phys. 46, 279 (1974) and reference herein.
    [CrossRef]

1986

1985

J. M. Perrin, P. Chiappeta, “Light Scattering by Large Particle I. A New Theoretical Description in the Eikonal Picture,” Opt. Acta 32, 907 (1985);J. M. Perrin, P. L. Lamy, “Light Scattering by Large Particles II. A Vectorial Description in the Eikonal Picture,” Opt. Acta 33, 1001 (1986).
[CrossRef]

R. Bhandari, “Scattering Coefficients for a Multilayered Sphere: Analytic Expressions and Algorithms,” Appl. Opt. 24, 1960 (1985).
[CrossRef] [PubMed]

1984

T. W. Chen, “Generalized Eikonal Approximation,” Phys. Rev. C30, 585 (1984).

1974

C. J. Joachain, C. Quigg, “Multiple Scattering Expansions in Several Particle Dynamics,” Rev. Mod. Phys. 46, 279 (1974) and reference herein.
[CrossRef]

1956

L. I. Schiff, “Approximation Method for Short Wavelength or High-Energy Scattering,” Phys. Rev. 104, 1481 (1956).
[CrossRef]

1955

D. S. Saxon, “Lecture on the Scattering of Light,” UCLA Department of Meteorological Science Report9 (1955).

Bhandari, R.

Chen, T. W.

T. W. Chen, “Generalized Eikonal Approximation,” Phys. Rev. C30, 585 (1984).

Chiappeta, P.

J. M. Perrin, P. Chiappeta, “Light Scattering by Large Particle I. A New Theoretical Description in the Eikonal Picture,” Opt. Acta 32, 907 (1985);J. M. Perrin, P. L. Lamy, “Light Scattering by Large Particles II. A Vectorial Description in the Eikonal Picture,” Opt. Acta 33, 1001 (1986).
[CrossRef]

Glauber, R. J.

R. J. Glauber, “High-Energy Collision Theory,” in Lecture in. Theoretical Physics, W. E. Britten, L. C. Dunham, Eds. (Interscience, New York, 1959).

Joachain, C. J.

C. J. Joachain, C. Quigg, “Multiple Scattering Expansions in Several Particle Dynamics,” Rev. Mod. Phys. 46, 279 (1974) and reference herein.
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, New York, 1969).

Perrin, J. M.

J. M. Perrin, P. Chiappeta, “Light Scattering by Large Particle I. A New Theoretical Description in the Eikonal Picture,” Opt. Acta 32, 907 (1985);J. M. Perrin, P. L. Lamy, “Light Scattering by Large Particles II. A Vectorial Description in the Eikonal Picture,” Opt. Acta 33, 1001 (1986).
[CrossRef]

Quigg, C.

C. J. Joachain, C. Quigg, “Multiple Scattering Expansions in Several Particle Dynamics,” Rev. Mod. Phys. 46, 279 (1974) and reference herein.
[CrossRef]

Saxon, D. S.

D. S. Saxon, “Lecture on the Scattering of Light,” UCLA Department of Meteorological Science Report9 (1955).

Schiff, L. I.

L. I. Schiff, “Approximation Method for Short Wavelength or High-Energy Scattering,” Phys. Rev. 104, 1481 (1956).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Acta

J. M. Perrin, P. Chiappeta, “Light Scattering by Large Particle I. A New Theoretical Description in the Eikonal Picture,” Opt. Acta 32, 907 (1985);J. M. Perrin, P. L. Lamy, “Light Scattering by Large Particles II. A Vectorial Description in the Eikonal Picture,” Opt. Acta 33, 1001 (1986).
[CrossRef]

Phys. Rev.

T. W. Chen, “Generalized Eikonal Approximation,” Phys. Rev. C30, 585 (1984).

L. I. Schiff, “Approximation Method for Short Wavelength or High-Energy Scattering,” Phys. Rev. 104, 1481 (1956).
[CrossRef]

Rev. Mod. Phys.

C. J. Joachain, C. Quigg, “Multiple Scattering Expansions in Several Particle Dynamics,” Rev. Mod. Phys. 46, 279 (1974) and reference herein.
[CrossRef]

UCLA Department of Meteorological Science Report

D. S. Saxon, “Lecture on the Scattering of Light,” UCLA Department of Meteorological Science Report9 (1955).

Other

R. J. Glauber, “High-Energy Collision Theory,” in Lecture in. Theoretical Physics, W. E. Britten, L. C. Dunham, Eds. (Interscience, New York, 1959).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, New York, 1969).

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Figures (6)

Fig. 1
Fig. 1

Extinction efficiency factor (Qext) vs the Mie size for dielectric bubbles and wet dielectric balls with α = 0.8. (A), (B), and (C) are for the bubbles with n equal to 1.1, 1.33, and 1.33 + 0.1i, respectively. (D), (E), and (F) are for the wet balls with n fixed at 1.1, 1.5, and 1.5 + 0.1i, respectively, and nwater = 1.33. The solid curves from the Mie method are compared with the slashed curves from the HEA method. In (E) and (F) the dotted curves are the HEA results with n2 − 1 instead of 2 (n − 1).

Fig. 2
Fig. 2

Qext vs the real part of the index of refraction at x = 50 and α = 0.8. The lower two curves are for dielectric bubbles without absorption [curve (A)] and with absorption [Im(n) = 0.1 in curve (B)]. The upper three curves are for wet dielectric balls with Im(nball) = 0 and nwater = 1.33 in (C) with Im(nball) = 0 and nwater = 1.33 + i0.05 in (D) and with Im(nball) = 0.05 and nwater = 1.33 +i0.05 in (E). The solid curves are exact values, and the slashed curves are HEA results.

Fig. 3
Fig. 3

Qext vs the imaginary part of the refractive index for dielectric bubbles and wet dielectric balls. Curves (A) are for a water bubble with Re(n) = 1.33 at x = 24 and α = 0.833; curves (B) are for a dielectric bubble with Re(n) = 1.4 at x = 50 and a = 0.8; curves (C) are for a dielectric bubble with Re(n) = 2 at x = 24 and α = 0.833; curves (D) are for a wet dielectric ball with Re(n) = 1.5 and nwater = 1.33 at x = 50 and α = 0.8; curves (E) are for a wet dielectric ball with Re(n) = 1.2 and nwater = 1.33 at x = 50 and α = 0.8; and curves (F) are for a dielectric ball with Re(n) = 2 and nwater = 1.33 + i0.05 at x = 24 and α = 0.833.

Fig.4
Fig.4

Relative intensity vs scattering angle θ for dielectric bubbles at x = 12 and α = 0.833. Curves (A), (B), and (C) are for dielectrics with a refractive index of 1.1 + i0.1, 1.33 + i0.1, and 1.5 + i0.1, respectively.

Fig. 5
Fig. 5

Relative intensity vs scattering angle for wet dielectric balls at x = 24 and α = 0.833. Water with an index of 1.33 + i0.1 surrounds a core with an index of 1.1 + i0.1 in curves (A), 1.5 + i0.1 in curves (B), and 2.0 + i0.1 in curves (C).

Fig. 6
Fig. 6

Relative intensity vs scattering angle for water bubbles with n = 1.33 + i0.1. Curves (A) are for x = 80 and α = 0.25, curves (B) are for x = 60 and α = 0.33, and curves (C) are for x = 40 and α = 0.5.

Equations (12)

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E ( r ) r ~ = E 0 + exp ( ikr ) ikr [ S ( r ̂ S ) r ̂ ] ,
S = k 4 π i exp ( i k r ) V ( r ) E ( r ) d 3 r ,
V ( r ) = k 2 [ ( r ) 1 ] .
S ( θ ) = E 0 k 2 0 J 0 ( q b ) { 1 exp [ i χ ( b ) ] } bdb ,
q = 2 k sin ( θ / 2 ) ,
χ = 1 2 k V ( b , z ) d z .
V ( r ) = k 2 l = 1 N ( n l 2 1 ) θ ( r a l 1 ) θ ( a l r ) ,
S = E 0 k 2 { l = 1 N a l 1 a l bdb J 0 ( q b ) g l ( b ) } ,
g l ( b ) = 1 exp { i p l a l 2 b 2 + i j = l + 1 N p j ( a j 2 b 2 a j 1 2 b 2 ) } , p l = k ( n l 2 1 ) 2 k ( n l 1 ) , since n l 1 .
S = E 0 k 2 0 a 1 bdb J 0 ( q b ) { 1 exp ( i p 1 a 1 2 b 2 ) } .
S = E 0 k 2 { 0 a 1 bdb J 0 ( q b ) [ 1 exp [ i p 1 a 1 2 b 2 ) + i p 2 ( a 2 2 b 2 ) a 1 2 b 2 ) ] ] + a 1 a 2 bdb J 0 ( q b ) [ 1 exp ( i p 2 a 2 2 b 2 ) ] } .
S = E 0 x 2 { 0 α ydy J 0 ( 2 x y sin θ 2 ) [ 1 exp [ i p 1 α 2 y 2 ) + i ρ 2 ( 1 y 2 α 2 y 2 ) ] ] + α 1 ydy J 0 ( 2 x y sin θ 2 ) [ 1 exp ( i ρ 2 1 y 2 ) ] }

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