Abstract

An improved recurrence algorithm to calculate the scattering field of a multilayered sphere is developed. The internal and external electromagnetic fields are expressed as a superposition of inward and outward waves. The alternative yet equivalent expansions of fields are proposed by use of the first kind of Bessel function and the first kind of Hankel function instead of the first and the second kinds of Bessel function. The final recursive expressions are similar in form to those of Mie theory for a homogeneous sphere and are proved to be more concise and convenient than earlier forms. The new algorithm avoids the numerical difficulties, which give rise to significant errors encountered in practice by previous methods, especially for large, highly absorbing thin shells. Various calculations and tests show that this algorithm is efficient, numerically stable, and accurate for a large range of size parameters and refractive indices.

© 2003 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. P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1998

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[CrossRef]

1997

1996

1995

1994

1991

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1990

1987

1985

1984

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

1981

1980

1976

1965

R. W. Fenn, H. Oser, “Scattering properties of concentric soot-water spheres for visible and infrared light,” Appl. Opt. 4, 1504–1509 (1965).
[CrossRef]

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, “Aerosol studies by light scattering. IV. Preparation and particle size distribution of aerosol consisting of concentric spheres,” J. Colloid Sci. 20, 501–521 (1965).
[CrossRef] [PubMed]

1951

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Ackerman, T. P.

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Altenkirch, R. A.

Artaxo, P.

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[CrossRef]

Bhandari, R.

Bohren, C. F.

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

Cheng, R. J.

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Chylek, P.

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Espenscheid, W. F.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, “Aerosol studies by light scattering. IV. Preparation and particle size distribution of aerosol consisting of concentric spheres,” J. Colloid Sci. 20, 501–521 (1965).
[CrossRef] [PubMed]

Fenn, R. W.

Fuller, K. A.

Gouesbet, G.

Grehan, G.

Guo, L. X.

Hobbs, P. V.

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[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).

Johnson, B. R.

Kai, L.

Kattawar, G. W.

Kaufman, Y. J.

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[CrossRef]

Kerker, M.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, “Aerosol studies by light scattering. IV. Preparation and particle size distribution of aerosol consisting of concentric spheres,” J. Colloid Sci. 20, 501–521 (1965).
[CrossRef] [PubMed]

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

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

Liousse, C.

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[CrossRef]

Mackowski, D. W.

Martins, J. V.

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[CrossRef]

Massoli, P.

Matijevic, E.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, “Aerosol studies by light scattering. IV. Preparation and particle size distribution of aerosol consisting of concentric spheres,” J. Colloid Sci. 20, 501–521 (1965).
[CrossRef] [PubMed]

Menguc, M. P.

Oser, H.

Ramaswamy, V.

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Reid, J. S.

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[CrossRef]

Ren, K. F.

Toon, O. B.

Wang, Y. P.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Willis, E.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, “Aerosol studies by light scattering. IV. Preparation and particle size distribution of aerosol consisting of concentric spheres,” J. Colloid Sci. 20, 501–521 (1965).
[CrossRef] [PubMed]

Wiscombe, W. J.

Wu, Z. S.

Appl. Opt.

J. Appl. Phys.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Atmos. Sci.

P. Chylek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

J. Colloid Sci.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, “Aerosol studies by light scattering. IV. Preparation and particle size distribution of aerosol consisting of concentric spheres,” J. Colloid Sci. 20, 501–521 (1965).
[CrossRef] [PubMed]

J. Geophys. Res.

J. V. Martins, P. Artaxo, C. Liousse, J. S. Reid, P. V. Hobbs, Y. J. Kaufman, “Effects of black carbon content, particle size, and mixing on the light absorption by aerosols from biomass burning in Brazil,” J. Geophys. Res. 103, 32041–32050 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Radio Sci.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Other

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

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

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Figures (8)

Fig. 1
Fig. 1

Geometry of the scattering problem by a multilayered sphere.

Fig. 2
Fig. 2

Extinction and scattering efficiencies (Q ext, Q sca) and single-scattering albedo as a function of the size parameter of the outmost layer for a soot-coated water sphere. A comparison of the results obtained from Wu et al.,11 Bhandari,9 and Toon and Ackerman8 is shown. The refractive indices of water and soot are m 1 = 1.33 + i0.00, m 2 = 1.59 + i0.66, respectively. The volume fraction of soot is 0.01.

Fig. 5
Fig. 5

Vertically polarized intensity i as a function of the scattering angle θ for a radially multilayered sphere with a size parameter x L = 5. The refractive indices in the lth layer, are, for case 1, m 1 = 1.0834 + i0.0, m l = x L /x l , and, for case 2, m 1 = 1.0779 + i0.0, m l = 1.01(x L /x l )2. The size parameter is x l = x 1 + (x L - x 1)(l - 1)/(L - 1), where the total number of layers is L = 1500.

Fig. 3
Fig. 3

Computational domain for a multilayered sphere. The solid curve is the affordable number of layers, and region B is the computational domain obtained by Kai and Massoli.14 Region A is the computational domain of our algorithm. The refractive index m l = n l + ik l is n l = n 1 + 0.5(n L - n 1)(1 - cos tπ), and k l = 0, where t = (l - 1)/(L - 1), n 1 = 1.01n L , and n L = 1.33. The size parameter is x l = x 1 + t(x L - x 1), where l = 1, 2, …, L, x 1 = 0.001x L , and L is the total number of layers.

Fig. 4
Fig. 4

Scattering efficiency Q sca versus the dimensionless size parameter of the core x 1/x L . The refractive index m l = n l + ik l is n l = n 1 + 0.5(n L - n 1)(1 -cos tπ), and k l = 0, where t = (l - 1)/(L - 1), n 1 = 1.43, and n L = 1.33. The size parameter is x l = x 1 + t(x L - x 1), l = 1, 2, …, L, and L is the total number of layers.

Fig. 6
Fig. 6

Absorption cross section per unit volume of absorbing material, σabs/V, as a function of its volume fraction f v for a two-layer sphere model with a radius r 2 = 5 μm at the outer rim. The refractive index of the water shell is m 2 = 1.33 + i0.00, and that of the absorbing core is m 1 = 2.0 + ik, where k = 0.22, 0.44, 0.66. The case of k = 0.66 corresponds to a soot particle. The wavelength is λ = 0.5 μm.

Fig. 7
Fig. 7

As in Fig. 6, but the absorbing material in the two-layer sphere model exists as a thin shell on the outside of a water droplet.

Fig. 8
Fig. 8

Reduced differential cross section dσ/d(a 2Ω) as a function of scattering angle θ for a spherical Luneburg lens with a size parameter x L = 60. The size parameter in the lth layer is x l = lx L /500. The refractive index is m l = [2 - (x¯/x L )2]1/2, with x¯ = (x¯ l-1 + x¯ l )/2, for l = 2, 3, …, L. For the geometrical optics theory, dσ/dΩ = a 2 cos(θ).

Tables (3)

Tables Icon

Table 1 Asymptotic Behavior of the Logarithmic Derivatives of Riccati-Bessel Functions Dni(z) (i = 1, 2, 3) for a Large, Absorbing Particlea

Tables Icon

Table 2 Asymptotic Behavior of the Ratios of Riccati-Bessel Functions ψn(z)/χn(z) and ψn(z)/ζn(z) for a Large, Absorbing Particlea

Tables Icon

Table 3 Extinction, Scattering, and Backscattering Efficiencies and Single-Scattering Albedo (Qext, Qsca, Qback and ϖ)a

Equations (49)

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Ein=n=1 EncnlMo1n1-idnlNe1n1
Eout=n=1 EnianlNe1n3-bnlMo1n3,
El=n=1 EncnlMo1n1-idnlNe1n1+ianlNe1n3-bnlMo1n3,
Hl=-klωμn=1 EndnlMe1n1+icnlNo1n1-ibnlNo1n3-anlMe1n3,
E1=n=1 Encn1Mo1n1-idn1Ne1n1,
H1=-k1ωμn=1 Endn1Me1n1+icn1No1n1.
Ei=n=1 EnMo1n1-iNe1n1,
Hi=-kωμn=1 EnMe1n1+iNo1n1,
Es=n=1 EnianNe1n3-bnMo1n3,
Hs=-kωμn=1 En-ibnNo1n3-anMe1n3.
Dn1z=ψnz/ψnz,
Dn3z=ζnz/ζnz,
Rnz=ψnz/ζnz.
An1=0, Hnam1x1=Dn1m1x1,
Hnamlxl=RnmlxlDn1mlxl-AnlDn3mlxlRnmlxl-Anl,
Anl+1=Rml+1xlml+1Hnamlxl-m1Dn1ml+1xlml+1Hnamlxl-mlDn3ml+1xl.
Bn1=0, Hnbm1x1=Dn1m1x1,
Hnbmlxl=RnmlxlDn1mlxl-BnlDn3mlxlRnmlxl-Bnl,
Bnl+1=Rml+1xlmlHnbmlxl-ml+1Dn1ml+1xlmlHnbmlxl-ml+1Dn3ml+1xl.
an=AnL+1=HnamLxL/mL+n/xLψnxL-ψn-1xLHnamLxL/mL+n/xLζnxL-ζn-1xL,
bn=BnL+1=mLHnbmLxL+n/xLψnxL-ψn-1xLmLHnbmLxL+n/xLζnxL-ζn-1xL
Bnz=n/zBn-1z-Bn-1z,
Bnz+n/zBnz=Bn-1z,
ψ-1z=cos z, ψ0z=sin z, ζ-1z=cos z+i sin z, ζ0z=sin z-i cos z.
Dniz=n/z-Dn-1iz-1-n/z, i=1, 3.
ψnzχnz=ψn-1zχn-1zDn2z+n/zDn1z+n/z,
ψnzζnz=ψn-1zζn-1zDn3z+n/zDn1z+n/z,
ψ0zχ0z=i 1-exp2aiexp-2b1+exp2aiexp-2b,
ψ0zζ0z=1-exp-2aiexp2b2,
Qnl=Rnz1Rnz2=ψnmlxl-1ζnmlxl-1ψnmlxlζnmlxl.
Hnamlxl=G2Dn1mlxl-QnlG1Dn3mlxlG2-QnlG1.
Hnbmlxl=G˜2Dn1mlxl-QnlG˜1Dn3mlxlG˜2-QnlG˜1.
G1=mlHnaml-1xl-1-ml-1Dn1mlxl-1,
G2=mlHnaml-1xl-1-ml-1Dn3mlxl-1,
G˜1=ml-1Hnbml-1xl-1-mlDn1mlxl-1,
G˜2=ml-1Hnbml-1xl-1-mlDn3mlxl-1.
Nstop=xL+4xL1/3+10.02xL8xL+4.05xL1/3+28xL4200xL+4xL1/3+24200xL20,000;
D03z=i,
ψ0zζ0z=121-cos 2a+i sin 2aexp-2b,
Qnl=Qn-1lDn3z1+n/z1Dn1z1+n/z1Dn3z2+n/z2Dn1z2+n/z2,
Q0l=R0z1R0z2=exp-i2a1-exp-2b1exp-i2a2-exp-2b2×exp-2b2-b1,
Qnl=Qn-1lxl-1xl2z2Dn1z2+nz1Dn1z1+nn-z2Dn-13z2n-z1Dn-13z1.
meff2r=mmed21+3fvrβ1-fvrβ,
mr=2-r/a21/2,
dσ/dΩ=a2 cosθ.
El+1-El×eˆr=0, Hl+1-Hl×eˆr=0,
dnl+1m1ψnml+1xl-anl+1mlζnml+1xl-dnlml+1ψnmlxl+anlml+1ζnmlxl=0, cnl+1mlψnml+1xl-bnl+1mlζnml+1xl-cnlml+1ψnmlxl+bnlml+1ζnmlxl=0, cnl+1ψnml+1xl-bnl+1ζnml+1xl-cnlψnmlxl+bnlζnmlxl=0, dnl+1ψnml+1xl-anl-1ζnml+1xl-dnlψnmlxl+anlζnmlxl=0,
EL+1=Ei+Es, HL+1=Hi+Hs.
mLψnxL-anmLζnxL-dnLψnmLxL+anLζnmLxL=0, mLψnxL-bnmLζnxL-cnLψnmLxL+bnLζnmLxL=0, ψnxL-bnζnxL-cnLψnmLxL+bnLζnmLxL=0, ψnxL-anζnxL-dnLψnmLxL+anLζnmLxL=0

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