Abstract

The conventional Lorenz-Mie formalism is extended to the case for a coated sphere embedded in an absorbing medium. The apparent and inherent scattering cross sections of a particle, derived from the far field and near field, respectively, are different if the host medium is absorptive. The effect of absorption within the host medium on the phase-matrix elements associated with polarization depends on the dielectric properties of the scattering particle. For the specific cases of a soot particle coated with a water layer and an ice sphere containing an air bubble, the phase-matrix elements -P 12/P 11 and P 33/P 11 are unique if the shell is thin. The radiative transfer equation for a multidisperse particle system embedded within an absorbing medium is discussed. Conventional multiple-scattering computational algorithms can be applied if scaled apparent single-scattering properties are applied.

© 2002 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).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1963).
  4. W. J. Wiscombe, “Mie scattering calculation,” NCAR Tech. Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  6. J. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” IBM Rep. 320-3237 (IBM Scientific Center, Palo Alto, Calif., 1968).
  7. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  8. O. B. Toon, T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef] [PubMed]
  9. 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]
  10. P. Chylek, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67, 561–563 (1977).
    [CrossRef]
  11. I. W. Sudiarta, and P. Chylek,“Mie scattering formalism in absorbing medium,” in the Proceedings of the Fifth Conference on Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, G. Videen, Q. Fu, P. Chylek, eds. (U.S. Army Research Laboratory, Adelphi, Md., 2000), pp. 115–118.
  12. Q. Fu, W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40, 1354–1361 (2001).
    [CrossRef]
  13. W. D. Ross, “Computation of Bessel functions in light scattering studies,” Appl. Opt. 11, 1919–1923 (1972).
    [CrossRef] [PubMed]
  14. G. W. Kattawar, D. A. Hood, “Electromagnetic scattering from a spherical polydispersion of coated spheres,” Appl. Opt. 15, 1996–1999 (1976).
    [CrossRef] [PubMed]
  15. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).
  16. R. W. Preisendorfer, C. D. Mobely, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
    [CrossRef]
  17. S.-C. Tsay, K. Stamnes, K. Jayaweera, “Radiative energy budget in the cloudy and hazy arctic,” J. Atmos. Sci. 46, 1002–1018 (1989).
    [CrossRef]
  18. S. Platnick, F. P. J. Valero, “A validation of a satellite cloud retrieval during ASTEX,” J. Atmos. Sci. 15, 2985–3001 (1995).
    [CrossRef]
  19. S. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
    [CrossRef] [PubMed]
  20. G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).
  21. A. Macke, M. I. Mishchenko, B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
    [CrossRef]
  22. L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
    [CrossRef]
  23. M. I. Mishchenko, A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116, 167–179 (2000).
    [CrossRef]

2001 (2)

Q. Fu, W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40, 1354–1361 (2001).
[CrossRef]

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

2000 (1)

M. I. Mishchenko, A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116, 167–179 (2000).
[CrossRef]

1996 (1)

A. Macke, M. I. Mishchenko, B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[CrossRef]

1995 (1)

S. Platnick, F. P. J. Valero, “A validation of a satellite cloud retrieval during ASTEX,” J. Atmos. Sci. 15, 2985–3001 (1995).
[CrossRef]

1989 (1)

S.-C. Tsay, K. Stamnes, K. Jayaweera, “Radiative energy budget in the cloudy and hazy arctic,” J. Atmos. Sci. 46, 1002–1018 (1989).
[CrossRef]

1984 (2)

R. W. Preisendorfer, C. D. Mobely, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

S. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
[CrossRef] [PubMed]

1981 (1)

1980 (1)

1977 (1)

1976 (1)

1974 (1)

1972 (1)

1908 (1)

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

Ackerman, T. P.

Bohren, C. F.

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

Brogniez, G.

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

Buriez, J.-C.

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

Cairns, B.

A. Macke, M. I. Mishchenko, B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

Chylek, P.

d’Almeida, G. A.

G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

Dave, J.

J. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” IBM Rep. 320-3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

Doutriaux-Boucher, M.

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

Fu, Q.

Gayet, J. F.

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

Hood, D. A.

Huffman, D. R.

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

Jayaweera, K.

S.-C. Tsay, K. Stamnes, K. Jayaweera, “Radiative energy budget in the cloudy and hazy arctic,” J. Atmos. Sci. 46, 1002–1018 (1989).
[CrossRef]

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1963).

Koepke, P.

G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

Labonnote, L. C.

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116, 167–179 (2000).
[CrossRef]

Macke, A.

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

A. Macke, M. I. Mishchenko, B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[CrossRef]

Mie, G.

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

Mishchenko, M. I.

M. I. Mishchenko, A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116, 167–179 (2000).
[CrossRef]

A. Macke, M. I. Mishchenko, B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[CrossRef]

Mobely, C. D.

R. W. Preisendorfer, C. D. Mobely, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

Mundy, W. C.

Platnick, S.

S. Platnick, F. P. J. Valero, “A validation of a satellite cloud retrieval during ASTEX,” J. Atmos. Sci. 15, 2985–3001 (1995).
[CrossRef]

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobely, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

Ross, W. D.

Roux, J. A.

Shettle, E. P.

G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

Smith, A. M.

Stamnes, K.

S.-C. Tsay, K. Stamnes, K. Jayaweera, “Radiative energy budget in the cloudy and hazy arctic,” J. Atmos. Sci. 46, 1002–1018 (1989).
[CrossRef]

Sudiarta, I. W.

I. W. Sudiarta, and P. Chylek,“Mie scattering formalism in absorbing medium,” in the Proceedings of the Fifth Conference on Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, G. Videen, Q. Fu, P. Chylek, eds. (U.S. Army Research Laboratory, Adelphi, Md., 2000), pp. 115–118.

Sun, W.

Toon, O. B.

Tsay, S.-C.

S.-C. Tsay, K. Stamnes, K. Jayaweera, “Radiative energy budget in the cloudy and hazy arctic,” J. Atmos. Sci. 46, 1002–1018 (1989).
[CrossRef]

Valero, F. P. J.

S. Platnick, F. P. J. Valero, “A validation of a satellite cloud retrieval during ASTEX,” J. Atmos. Sci. 15, 2985–3001 (1995).
[CrossRef]

van de Hulst, H. C.

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

Warren, S.

Wiscombe, W. J.

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

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

Ann. Phys. (Leipzig) (1)

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

Appl. Math. Comput. (1)

M. I. Mishchenko, A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116, 167–179 (2000).
[CrossRef]

Appl. Opt. (6)

J. Atmos. Sci. (2)

S.-C. Tsay, K. Stamnes, K. Jayaweera, “Radiative energy budget in the cloudy and hazy arctic,” J. Atmos. Sci. 46, 1002–1018 (1989).
[CrossRef]

S. Platnick, F. P. J. Valero, “A validation of a satellite cloud retrieval during ASTEX,” J. Atmos. Sci. 15, 2985–3001 (1995).
[CrossRef]

J. Geophys. Res. (2)

A. Macke, M. I. Mishchenko, B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[CrossRef]

L. C. Labonnote, G. Brogniez, J.-C. Buriez, M. Doutriaux-Boucher, J. F. Gayet, A. Macke, “Polarized light scattering by inhomogeneous hexagonal monocrystals: validation with ADEOS-POLDER measurements,” J. Geophys. Res. 106, 12139–12154 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

Limnol. Oceanogr. (1)

R. W. Preisendorfer, C. D. Mobely, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

Other (8)

I. W. Sudiarta, and P. Chylek,“Mie scattering formalism in absorbing medium,” in the Proceedings of the Fifth Conference on Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, G. Videen, Q. Fu, P. Chylek, eds. (U.S. Army Research Laboratory, Adelphi, Md., 2000), pp. 115–118.

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1963).

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

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

J. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” IBM Rep. 320-3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

G. A. d’Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

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Figures (11)

Fig. 1
Fig. 1

Geometry for scattering by a coated sphere embedded in an absorbing medium.

Fig. 2
Fig. 2

Inherent and apparent extinction efficiency and single-scattering albedo values for soot spheres embedded in an ice medium at wavelengths of 1.38, 3.75, and 11.0 µm. Also shown are the asymmetry factor values.

Fig. 3
Fig. 3

Same as Fig. 2, except that the scatterers are air bubbles in an ice medium.

Fig. 4
Fig. 4

Interception efficiency defined for particles embedded in an ice medium at three wavelengths.

Fig. 5
Fig. 5

Nonzero phase-matrix elements for homogeneous (uncoated) spheres that are embedded in an absorbing media. Results are provided for three values of the imaginary refractive index and for three size parameters. The particle refractive index is assumed to be 1.33 + i0.0, which is essentially the refractive index for water at a visible wavelength.

Fig. 6
Fig. 6

Same as Fig. 5, except for a multidisperse particle system.

Fig. 7
Fig. 7

Same as Fig. 6, except that the particle refractive index is 1.75 + i0.435, which is the refractive index for soot at a visible wavelength.

Fig. 8
Fig. 8

Comparison of inherent and apparent extinction efficiency and single-scattering albedo for a soot sphere coated with water. The refractive indices for the host medium were chosen to be 1.0 + i0.001 and 1.0 + i0.01.

Fig. 9
Fig. 9

Same as Fig. 8, except that the scattering particles are hollow ice spheres.

Fig. 10
Fig. 10

Phase-matrix elements for a particle of water-coated soot. The host refractive index was chosen as 1.0 + i0.01.

Fig. 11
Fig. 11

Same as Fig. 10, except for a hollow ice sphere.

Tables (1)

Tables Icon

Table 1 Complex Refractive Index for Soot and Ice at Three Wavelengthsa

Equations (62)

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Eix, y, z=êxE0 expikm0z,
Hix, y, z=-ik ×Eix, y, z=êym0E0 expikm0z,
Eiθϕ, θ, r=cos ϕm0krn=1 Enπncos θψnm0kr-iτncos θψnm0kr,
Eiϕϕ, θ, r=- sin ϕm0krn=1 Enτncos θψnm0kr-iπncos θψnm0kr,
Hiθϕ, θ, r= sin ϕkrn=1 Enπncos θψnm0kr-iτncos θψnm0kr,
Hiϕϕ, θ, r=cos ϕkrn=1 Enτncos θψnm0kr-iπncos θψn m0kr,
Esθϕ, θ, r =cos ϕm0krn=1 Enianτncos θξn m0kr-bnπncos θξnm0kr,
Esϕϕ, θ, r=- sin ϕm0krn=1 Enianπncos θξn× m0kr-bnτncos θξnm0kr,
Hsθ ϕ, θ, r=-sin ϕkrn=1 Enanπncos θξnm0kr-ibnτncos θξn m0kr,
Hsϕϕ, θ, r=- cos ϕkrn=1 Enanτncos θξnm0kr-ibnπncos θξn m0kr,
En=E0in2n+1/n n+1.
an= D˜nm0/m2+n/m0kR2ψnm0kR2-ψn-1m0kR2D˜nm0/m2+n/m0kR2ξnm0kR2-ξn-1m0kR2,
bn= G˜nm2/m0+n/ m0kR2ψnm0kR2-ψn-1m0kR2G˜nm2/m0+n/ m0kR2ξnm0kR2-ξn-1m0kR2,
D˜n=Dnm2kR2-Anχn m2kR2/ψnm2kR21-Anχnm2kR2/ψnm2kR2,
G˜n= Dnm2kR2-Bnχn m2kR2/ψnm2kR21-Bnχnm2kR2/ψnm2kR2,
An= m2Dnm1kR1-m1Dnm2kR1m2Dnm1kR1χnm2kR1/ψnm2kR1-m1χn m2kR1/ψnm2kR1,
Bn= m1Dnm1kR1-m2Dnm2kR1m1Dnm1kR1χnm2kR1/ψnm2kR1-m2χn m2kR1/ψnm2kR1,
an=m0Dnm0kR/m1+n/ m0kRψnm0kR-ψn-1m0kRm0Dnm0kR/m1+n/ m0kRξnm0kR-ξn-1m0kR,
bn=m1Dnm1kR/m0+n/ m0kRψnm0kR-ψn-1m0kRm1Dnm1kR/m0+n/ m0kRξnm0kR-ξn-1m0kR,
Fz= c8πReEi×Hi* =F0 exp-2m0,ikz,
σi= 1F002ππ/2π F R2 cos θR22sin θdθdϕ=πR2222m0ikR2-1exp2m0ikR2+12m0ikR22.
σs= 1F002π0πc8πReEsϕ, θ, R2×Hs* ϕ, θ, R2rˆ R22sin θdθdϕ= 1F002π0πc8πReEsθϕ, θ, R2Hsϕ* ϕ, θ, R2-Esϕϕ, θ, R2Hsθ* ϕ, θ, R2R22sin θdθdϕ= 2πm0rk2Im1m0n=12n+1|an|2 ξn m0kR2ξn* m0kR2-|bn|2 ξnm0kR2ξn* m0kR2,
σa=- 1F002π0πc8πReEiϕ, θ, R2+ Esϕ, θ, R2× Hi* ϕ, θ, R2+ Hs* ϕ, θ, R2rˆR22sin θdθdϕ=- 1F002π0πc8πReEiθϕ, θ, R2+Esθϕ, θ, R2 Hiϕ* ϕ, θ, R2+Hsϕ* ϕ, θ, R2- Eiϕϕ, θ, R+Esϕϕ, θ, R2 Hiθ* ϕ, θ, R2+Hsθ* ϕ, θ, R2 R22sin θdθdϕ =2πm0rk2Im1m0n=12n+1ψnm0kR2ψn*× m0kR2-ψn m0kR2ψn* m0kR2+anξn m0kR2ψn* m0kR2-bnξn m0kR2ψn m0kR2+an*ψn m0kR2ξn* m1kR-bn*ψnm0kR2ξn* m0kR2-|an|2 ξn m0kR2ξn* m0kR2+|bn|2 ξnm0kR2ξn* m0kR2.
σe=σa+σs.
Et= n=1 EnfnMo1n1-ignNe1n1+νnMo1n2-iwnNe1n2,
Et= n=1 EncnMo1n1-idnNe1n1.
Qe=σe/ πR22,
Qs=σs/ πR22.
Qi=σi/ πR22,
Qe,inh=Qe/Qi,
Qs,inh=Qs/Qi,
ω0,inh=Qs,inh/Qe,inh=Qs/Qe,
Esθϕ, θ, r||m0kr|=cos ϕ expim0kr-im0kr E0S2,
Esϕϕ, θ, r||m0kr| =-sin ϕexp im0kr-im0kr E0S1,
S1= n=12n+1nn+1anπncos θ+bnτncos θ,
S2= n=12n+1nn+1anτncos θ+bnπncos θ.
Esϕ, θ, rEsϕ, θ, r= expim0kr-im0krS200S1Ei0Ei0,
Fsϕ, θ, r= cm0, r8π|Es,|2+|Es,|2= cm0, r8π |E0|2exp-2m0,ikr|m0|2k2r2|S1|2+|S2|2=exp-2m0,ik r-R2F0× exp-2m0, ikR2|m0|2 k2r2|S1|2+|S2|2.
σ˜s= exp2m0ik r-R2F0× 02π0π Fsϕ, θ, rr2 sin θdθdϕ= exp-2m0ikR2|m0|2 k202π0π|S1|2+|S2|2sin θdθdϕ = 2π exp-2m0ikR2|m0|2 k2× n=12n+1|an|2+|bn|2.
Isϕ, θ, rQsϕ, θ, rUsϕ, θ, rVsϕ, θ, r= exp-2m0ik r-R2σ˜sr2× P114π1P12/P1100P12/P1110000P33/P11-P33/P1100P43/P11P33/P11Ii0Qi0Ui0Vi0,
Iiϕ, θ, rQiϕ, θ, rUiϕ, θ, rViϕ, θ, r=exp-2m0ikr cos θIi0Qi0Ui0Vi0.
P11= |S1|2+|S2|2n=12n+1|an|2+|bn|2,
P12/P11= |S2|2-|S1|2|S2|2+|S1|2,
P33/P11=2ReS1S2*|S2|2+|S1|2,
P43/P11= 2ImS1S2*|S2|2+|S1|2.
g= 120π P11cos θcos θ sin θdθ= 2n=2Ren-1n+1an-1an*+bn-1bn*/n+ 2n-1/ n-1nRean-1bn-1*n=12n+1|an|2+|bn|2,
σ˜e=σa+σ˜s,
Q˜e=σ˜e/ πR22,
Q˜s=σ˜s/ πR22.
Qe,app=Q˜e/Qi,
Qs,app=Q˜s/Qi,
ω0,app=Qs,app/Qe,app=Q˜s/Q˜e.
Isϕ, θ, r= exp-2m0ik r-Rσ˜sr2P114π Ii0,
σ˜s,scaled=exp2m0kRσ˜s,
σ˜e,scaled=σa+σ˜s,scaled.
Isϕ, θ, r= exp-2m0ikrσ˜s,scaledr2P114π Ii0.
β˜e, ps= RminRmax σ˜e,scaleds, RN s, RdR,
β˜s, ps= RminRmax σ˜s,scaleds, RN s, RdR,
βe,hosts=2m0isk.
ΩˆI Ωˆ, s=- βe, ps+βe, hostsI Ωˆ, s+J Ωˆ, s,
JΩˆ, s= βˆs,ps4π4π I Ωˆ, sP11s, Ωˆ, ΩˆdΩˆ.
P11s, Ωˆ, Ωˆ= RminRmax σ˜s, scaleds, RP11s, R, Ωˆ, ΩˆNs, RdRRminRmax σ˜s, scaleds, RN s, RdR.

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