Abstract

It is demonstrated in the present investigation that the scattering from a layered (i.e., coated) sphere can be calculated numerically by application of the boundary-element method both inside and outside the sphere. A homogeneous sphere is treated first to prove that the computer algorithm is correctly applied, and then the same procedure is followed to deal with a coated sphere. A three-layered sphere will admit of similar treatment, but that is not demonstrated in this study. The scattering intensities from a homogeneous sphere, obtained numerically, are shown to agree well with the analytical Mie solution. It is expected that the calculation algorithm of the analytical Mie-type solution can be improved with the help of the present numerical method.

© 2001 Optical Society of America

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References

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  1. A. L. Arden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. O. B. Toon, T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef] [PubMed]
  5. M. K. Choi, J. R. Brock, “Light scattering and absorption by a radially inhomogeneous sphere: application of a hybrid numerical method,” J. Opt. Soc. Am. B 14, 620–626 (1997).
    [CrossRef]
  6. M. K. Choi, L. A. Liebman, J. R. Brock, “Finite element solution of the Maxwell equations for absorption and scattering of electromagnetic radiation by a coated dielectric particle,” Chem. Eng. Commun. 151, 5–17 (1996).
    [CrossRef]
  7. D. W. Mackowski, R. A. Altenkrich, M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
    [CrossRef] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).
  9. R. T. Ling, “Application of computational fluid dynamics methods to a numerical study of electromagnetic wave scattering phenomena,” J. Appl. Phys. 64, 3785–3792 (1985).
    [CrossRef]
  10. B. Carnahan, H. A. Luther, J. A. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

1997

1996

M. K. Choi, L. A. Liebman, J. R. Brock, “Finite element solution of the Maxwell equations for absorption and scattering of electromagnetic radiation by a coated dielectric particle,” Chem. Eng. Commun. 151, 5–17 (1996).
[CrossRef]

1990

1985

R. T. Ling, “Application of computational fluid dynamics methods to a numerical study of electromagnetic wave scattering phenomena,” J. Appl. Phys. 64, 3785–3792 (1985).
[CrossRef]

1981

1951

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

Ackerman, T. P.

Altenkrich, R. A.

Arden, A. L.

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

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

Brock, J. R.

M. K. Choi, J. R. Brock, “Light scattering and absorption by a radially inhomogeneous sphere: application of a hybrid numerical method,” J. Opt. Soc. Am. B 14, 620–626 (1997).
[CrossRef]

M. K. Choi, L. A. Liebman, J. R. Brock, “Finite element solution of the Maxwell equations for absorption and scattering of electromagnetic radiation by a coated dielectric particle,” Chem. Eng. Commun. 151, 5–17 (1996).
[CrossRef]

Carnahan, B.

B. Carnahan, H. A. Luther, J. A. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Choi, M. K.

M. K. Choi, J. R. Brock, “Light scattering and absorption by a radially inhomogeneous sphere: application of a hybrid numerical method,” J. Opt. Soc. Am. B 14, 620–626 (1997).
[CrossRef]

M. K. Choi, L. A. Liebman, J. R. Brock, “Finite element solution of the Maxwell equations for absorption and scattering of electromagnetic radiation by a coated dielectric particle,” Chem. Eng. Commun. 151, 5–17 (1996).
[CrossRef]

Huffman, D. R.

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

Kerker, M.

A. L. Arden, 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).

Liebman, L. A.

M. K. Choi, L. A. Liebman, J. R. Brock, “Finite element solution of the Maxwell equations for absorption and scattering of electromagnetic radiation by a coated dielectric particle,” Chem. Eng. Commun. 151, 5–17 (1996).
[CrossRef]

Ling, R. T.

R. T. Ling, “Application of computational fluid dynamics methods to a numerical study of electromagnetic wave scattering phenomena,” J. Appl. Phys. 64, 3785–3792 (1985).
[CrossRef]

Luther, H. A.

B. Carnahan, H. A. Luther, J. A. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Mackowski, D. W.

Menguc, M. P.

Toon, O. B.

Wilkes, J. A.

B. Carnahan, H. A. Luther, J. A. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

Appl. Opt.

Chem. Eng. Commun.

M. K. Choi, L. A. Liebman, J. R. Brock, “Finite element solution of the Maxwell equations for absorption and scattering of electromagnetic radiation by a coated dielectric particle,” Chem. Eng. Commun. 151, 5–17 (1996).
[CrossRef]

J. Appl. Phys.

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

R. T. Ling, “Application of computational fluid dynamics methods to a numerical study of electromagnetic wave scattering phenomena,” J. Appl. Phys. 64, 3785–3792 (1985).
[CrossRef]

J. Opt. Soc. Am. B

Other

B. Carnahan, H. A. Luther, J. A. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

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).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

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Figures (6)

Fig. 1
Fig. 1

Simple schematic of a homogeneous sphere: I, medium; II, particle.

Fig. 2
Fig. 2

Simple schematic of a coated sphere: I, medium; II, coating, III, core.

Fig. 3
Fig. 3

Test result of the computer code for calculation of scattering intensities for a water droplet (radius, 10 μm; m=1.179+i0.071, λ=10.6 μm). Agreement between the Mie analytical solution and the numerical results is excellent. The number of surface elements used for the numerical calculations was 700.

Fig. 4
Fig. 4

Test result of the computer code made for a coated sphere for calculation of scattering intensities for a water droplet (radius, 10 μm; m=1.179+i0.071, λ=10.6 μm). Agreement between the Mie analytical solution and the numerical results is acceptable. The number of outer surface elements used for the numerical calculations was 600.

Fig. 5
Fig. 5

L2 norms of the numerical error versus the number of surface elements employed for the computer code for a homogeneous sphere. The radius of the sphere treated here is 10 μm.

Fig. 6
Fig. 6

L2 norms of the numerical error versus the number of particle surface elements employed for the computer code for a coated sphere. The radius of the sphere treated here is 10 μm.

Equations (62)

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2E+k2E=0,
2H+k2H=0,
n×(EK-EJ)=0,
n×(HK-HJ)=0,
2uK+(kK)2uK=0,
2vK+(kK)2vK=0,
iωK(ρuK)ρ=ri=iωJ(ρuJ)ρ=ri,
ρ (ρuK)ρ=ri=ρ (ρuJ)ρ=ri,
iωμK(ρvK)ρ=ri=iωμJ(ρvJ)ρ=ri,
ρ (ρvK)ρ=ri=ρ (ρvJ)ρ=ri.
-2πui,1I=S1-uIg1Iρ+g1IuIρdS-4πui,1(inc),
-2πvi,1I=S1-vIg1Iρ+g1IvIρdS-4πvi,1inc,
g1I=exp(ikI|x-xi,1|)|x-xi,1|.
ui,1(inc)=cos ϕi(kI)2r1 W(r1, θi),
vi,1(inc)=sin ϕi(kI)2r1 W(r1, θi),
W(ρ, θ)=exp(ikIρ cos θ)sin θ-cotθ2exp(ikIρ)2-tanθ2exp(-ikIρ)2.
-2πui,1II=S1-uIIg1IIρ+g1IIuIIρdS,
-2πvi,1II=S1-vIIg1IIρ+g1IIvIIρdS,
g1II=exp(ikII|x-xi,1|)|x-xi,1|.
u(ρ, θ, ϕ)=cos ϕ sin θw(ρ, θ),
v(ρ, θ, ϕ)=sin ϕ sin θy(ρ, θ).
-2πwiI=-wIg1Iρ+g1IwIρr12sin2 θ cos ϕsin θicos ϕidθdϕ-4π W(r1, θi)(kI)2r1sin θi,
-2πyiI=-yIg1Iρ+g1Iyρr12sin2 θ sin ϕsin θisin ϕidθdϕ-4π W(r1, θi)(kI)2r1sin θi.
θ=0πwIϕ=02πcos ϕ-g1Iρdϕ r12sin2 θsin θicos ϕidθ.
θ=0πwIr12sin2 θsin θicos ϕiϕ=02πcos ϕ-g1Iρdϕdθ
=(B11IwI)i.
ϕ=02πθ=0πg1Icos ϕ wIρr12sin2 θsin θicos ϕidθdϕ
=θ=0πwIρr12sin2 θsin θicos ϕiϕ=02πcos ϕ g1Iρdθ=C11IwIρi.
(-2πI-B11I)wI-C11IwIρ=-4π W(kI)2r1sin θi
m12(-2πI-B11I)-1-m12r1C11IwII-C11IwIIρ
=-4π W(kI)2r1sin θi.
-2πwiII=-wIIg1IIρ+g1IIwIIρ×r12sin2 θ cos ϕsin θicos ϕidθdϕ,
(2πI+B11II)wII+C11IIwIIρ=0.
Kx=f,
K11=m12(-2πI-B11I)-1-m12r1C11I,
K12=-1r1 C11I,
K21=2πI+B11II,K22=(1/r1)C11II,
x1=wII,x2=r1wIIρr1,
f1=-4πW(kI)2r1sin θi,f2=0.
-4πuks=S1-uIg1Iρ+g1IuIρdS,
-4πvks=S1-vIg1Iρ+g1IvIρdS.
Eks=×[(ruks)×xk]+iωμ×(rvksxk),
EsθE0exp(ikr)-ikrcos ϕS2(cos θ),
Esθ-E0exp(ikr)-ikrsin ϕS1(cos θ).
-2πui,1I=S1-uIg1Iρ+g1IuIρdS-4πui(inc),
-2πui,1II=S1-uIIg1IIρ+g1IIuIIρdS+S2-uIIg1IIρ+g1IIuIIρdS,
-2πui,2II=S1-uIIg2IIρ+g2IIuIIρdS+S2-uIIg2IIρ+g2IIuIIρdS,
-2πui,2III=S2-uIIIg2IIIρ+g2IIIuIIIρdS.
Kx=f,
K11=m12(-2πI-B11I)-1-m12r1C11I,
K12=-1r1C11I,K13=K14=0,
K21=-2πI+B11II,K22=1r1C11II,
K23=-m22B12II-1-m22r2C12II,K24=-1r2C12II,
K31=B21II,K32=1r1C21II,
K33=-m22(2πI+B22I)-1-m22r2C22II,
K34=-1r2C22II,
K41=K42=0,K43=-2πI+B22III,
K44=1r2C22III,
x1=wII,x2=r1wIIρr1,x3=wIII,
x4=r2wIIIρr2,
f1=-4πW(kI)2r1sin θi,f2=f3=f4=0.
L2=-11(error)2d(-cos θ).

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