Abstract

Analytic equations are developed for the single-scattering properties of a spherical particle embedded in an absorbing medium, which include absorption, scattering, extinction efficiencies, the scattering phase function, and the asymmetry factor. We derive absorption and scattering efficiencies by using the near field at the surface of the particle, which avoids difficulty in obtaining the extinction based on the optical theorem when the far field is used. Computational results demonstrate that an absorbing medium significantly affects the scattering of light by a sphere.

© 2001 Optical Society of America

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References

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  1. G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).
    [CrossRef]
  2. W. C. Mundy, J. A. Roux, A. M. Smith, “Mie scattering by spheres in an absorbing medium,” J. Opt. Soc. Am. 64, 1593–1597 (1974).
    [CrossRef]
  3. P. Chylek, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67, 561–563 (1977).
    [CrossRef]
  4. C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
    [CrossRef]
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  6. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  7. W. Wiscombe, “Mie scattering calculation,” NCAR Technical Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

1979

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

1977

1974

1908

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

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

Chylek, P.

Gilra, D. P.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

Huffman, D. R.

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

Kerker, M.

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

Mie, G.

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).
[CrossRef]

Mundy, W. C.

Roux, J. A.

Smith, A. M.

Wiscombe, W.

W. Wiscombe, “Mie scattering calculation,” NCAR Technical Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

Ann. Phys. (Leipzig)

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).
[CrossRef]

J. Colloid Interface Sci.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

J. Opt. Soc. Am.

Other

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

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

W. Wiscombe, “Mie scattering calculation,” NCAR Technical Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

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Figures (9)

Fig. 1
Fig. 1

Geometry for the scattering of a linearly polarized plane wave by a spherical particle of radius a. The origin of the coordinate system is at the particle center. The positive z axis is along the direction of propagation of the incident wave with the electric vector polarized in the direction of the x axis. The direction of the scattered light is defined by polar angles θ and ϕ.

Fig. 2
Fig. 2

Extinction, scattering, and absorption efficiencies and asymmetry factor as functions of size parameters for a spherical particle embedded in a medium. A refractive index of 1.0 is used for the particle. The real refractive index of the medium is 1.34, and the imaginary refractive index of the medium is 0.0, 0.001, 0.01, and 0.05.

Fig. 3
Fig. 3

As in Fig. 2, but the refractive index of the particle is 1.34 + 0.01i and the refractive index of the medium is 1.0 + im i , where m i = 0.0, 0.001, 0.01, and 0.05.

Fig. 4
Fig. 4

As in Fig. 2, but the refractive index of the particle is 1.4 + 0.05i and the refractive index of the medium is 1.2 + im i , where m i = 0.0, 0.001, 0.01, and 0.05.

Fig. 5
Fig. 5

Scattering phase function as a function of the scattering angle for a sphere with a refractive index of 1.0 embedded in a medium with a real refractive index of 1.34 and an imaginary refractive index of 0.0, 0.001, 0.01, and 0.05. The results are presented for size parameters of 5, 25, and 100.

Fig. 6
Fig. 6

As in Fig. 5, but the refractive index of the particle is 1.34 + 0.01i and the refractive index of the medium is 1.0 + im i , where m i = 0.0, 0.001, 0.01, and 0.05.

Fig. 7
Fig. 7

As in Fig. 5, but the refractive index of the particle is 1.4 + 0.05i and the refractive index of the medium is 1.2 + im i , where m i = 0.0, 0.001, 0.01, and 0.05.

Fig. 8
Fig. 8

Electric energy density |E| 2 within an air bubble embedded in a medium with a refractive index of 1.34 + m i , where m i = 0.0, 0.001, 0.01, and 0.05. |E| 2 is in relative units with E 0 = 1. The fields are shown on a plane that is through the center of the sphere and parallel to the incident light. The positive z axis is along the direction of propagation of the incident wave with the electric vector parallel to the plane. The size parameter is 25.

Fig. 9
Fig. 9

As in Fig. 8 but with the incident electric vector perpendicular to the plane.

Equations (36)

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Ei=n=1 EnM01n1-iNe1n1,
Hi=-kωμn=1 EnMe1n1+iN01n1,
Et=n=1 EncnM01n1-idnNe1n1,
Ht=-ktωμtn=1 EndnMe1n1+icnN01n1,
Es=n=1 EnianNe1n3-bnM01n3,
Hs=kωμn=1 EnibnN01n3+anMe1n3,
M01n=cos ϕπncos θznρeˆθ-sin ϕτncos θznρeˆϕ,
Me1n=-sin ϕπncos θznρeˆθ-cos ϕτncos θznρeˆϕ,
N01n=sin ϕnn+1sin θπncos θznρ/ρer+sin ϕτncos θρznρ/ρeˆθ+cos ϕπncos θρznρ/ρeˆϕ,
Ne1n=cos ϕnn+1sin θπncos θznρ/ρeˆr+cos ϕτncos θρznρ/ρeˆθ-sin ϕπncos θ×ρznρ/ρeˆϕ,
an=mtψnαψnβ-mψnαψnβmtξnαψnβ-mξnαψnβ,
bn=mtψnαψnβ-mψnαψnβmtξnαψnβ-mξnαψnβ,
cn=mtξnαψnα-mtξnαψnαmtξnαψnβ-mξnαψnβ,
dn=mtξnαψnα-mtξnαψnαmtξnαψnβ-mξnαψnβ,
S=½ReE×H*,
Wa=-12Re  E×H*·nˆds,
E×H*·nˆ=Ei+Es×Hi+Hs*·nˆ=Et×Ht*·nˆ
Wa=-12Re  Et×Ht*·nˆds=12Re02π0πEtϕHtθ*-EtθHtϕ*a2 sin θdθdϕ,
0ππnτn+πnτnsin θdθ=0,
0ππnπn+τnτnsin θdθ=δnn2n2n+122n+1,
Wa=π|E0|2ωμtn=12n+1ImAn,
An=|cn|2ψnβψn*β-|dn|2ψnβψn*βkt.
Ws=12Re  Es×Hs*·nˆds=12Re02π0πEsθHsϕ*-EsϕHsθ*a2 sin θdθdϕ.
Ws=π|E0|2ωμn=12n+1ImBn,
Bn=|an|2ξnαξn*α-|bn|2ξnαξn*αk.
We=π|E0|2ωn=12n+1ImAnμt+Bnμ.
f=2πa2η2 I01+η-1eη,
Qa=Wa/f,
Qs=Ws/f,
Qe=We/f.
2/α2n=12n+1|an|2+|bn|2
S1=n=12n+1nn+1anπncos θ+bnτncos θ,
S2=n=12n+1nn+1anτncos θ+bnπncos θ.
Pcos θ=|S1|2+|S2|2n=12n+1|an|2+|bn|2.
g=12-11 Pcos θcos θd cos θ.
g=2 n=1nn+2n+1Reanan+1*+bnbn+1*+2n+1nn+1Reanbn*n=12n+1|an|2+|bn|2.

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