Abstract

An exact theory is presented for calculating electromagnetic scattering by a model heterogeneous particle. The model is a multilayer sphere composed of alternating layers of different dielectric constants in the limit where the number of layers approaches infinity and the layer thickness approaches zero. The scattering analysis for this limiting case is based on a formulation of scattering theory known as the variable-phase method (VPM). A derivation of the VPM formulation is presented. Analytic formulas are derived for the scattering coefficients of the heterogeneous sphere that are generalizations of similar Mie theory formulas for scattering by a homogeneous sphere. The optical properties of the layered heterogeneous mixture are exactly described in this theory by the two effective-media parameters. The solution in the long-wavelength limit is used to derive a new formula for the effective dielectric constant for heterogeneous mixtures.

© 1999 Optical Society of America

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References

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  1. M. L. Dourneuf, V. K. Lan, “The variable-phase method in multichannel electron-atom or electron-ion scattering,” J. Phys. B 10, L35–L42 (1977).
    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9.
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1981), Pt. 1.
  5. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), 563–573 and 392–395.
  6. P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);erratum 134, AB1 (1964).
    [CrossRef]
  7. C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res., Sect. B 7, 113–130 (1958).
    [CrossRef]
  8. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  9. B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27, 4861–4873 (1988).
    [CrossRef] [PubMed]
  10. B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286–3296 (1996).
    [CrossRef] [PubMed]
  11. W. F. Brown, “Dielectrics,” in Encyclopedia of physics, Vol. 17, S. Flügge, ed. (Springer-Verlag, Heidelberg, 1956), Chap. 1, Sec. 53.
  12. D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
    [CrossRef]
  13. R. Landauer, in Proceedings of the First Conference on the Electrical and Transport Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., AIP Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 2–43.
  14. G. B. Smith, “Dielectric constants for mixed media,” J. Phys. D 10, L39–L42 (1977).
    [CrossRef]
  15. G. A. Niklasson, C. G. Granqvist, O. Hunderi, “Effective medium models for the optical properties of inhomogeneous materials,” Appl. Opt. 20, 26–30 (1981).
    [CrossRef] [PubMed]
  16. G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
    [CrossRef]
  17. D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
    [CrossRef]

1996

1988

1982

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

1981

1980

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[CrossRef]

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[CrossRef]

1977

G. B. Smith, “Dielectric constants for mixed media,” J. Phys. D 10, L39–L42 (1977).
[CrossRef]

M. L. Dourneuf, V. K. Lan, “The variable-phase method in multichannel electron-atom or electron-ion scattering,” J. Phys. B 10, L35–L42 (1977).
[CrossRef]

1962

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);erratum 134, AB1 (1964).
[CrossRef]

1958

C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res., Sect. B 7, 113–130 (1958).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

Bergman, D. J.

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1981), Pt. 1.

Brown, W. F.

W. F. Brown, “Dielectrics,” in Encyclopedia of physics, Vol. 17, S. Flügge, ed. (Springer-Verlag, Heidelberg, 1956), Chap. 1, Sec. 53.

Dourneuf, M. L.

M. L. Dourneuf, V. K. Lan, “The variable-phase method in multichannel electron-atom or electron-ion scattering,” J. Phys. B 10, L35–L42 (1977).
[CrossRef]

Granqvist, C. G.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1981), Pt. 1.

Hunderi, O.

Johnson, B. R.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.

Lan, V. K.

M. L. Dourneuf, V. K. Lan, “The variable-phase method in multichannel electron-atom or electron-ion scattering,” J. Phys. B 10, L35–L42 (1977).
[CrossRef]

Landauer, R.

R. Landauer, in Proceedings of the First Conference on the Electrical and Transport Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., AIP Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 2–43.

Milton, G. W.

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[CrossRef]

Niklasson, G. A.

Smith, G. B.

G. B. Smith, “Dielectric constants for mixed media,” J. Phys. D 10, L39–L42 (1977).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), 563–573 and 392–395.

Tai, C. T.

C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res., Sect. B 7, 113–130 (1958).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9.

Wyatt, P. J.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);erratum 134, AB1 (1964).
[CrossRef]

Am. J. Phys.

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[CrossRef]

Appl. Sci. Res., Sect. B

C. T. Tai, “The electromagnetic theory of the spherical Luneberg lens,” Appl. Sci. Res., Sect. B 7, 113–130 (1958).
[CrossRef]

J. Phys. B

M. L. Dourneuf, V. K. Lan, “The variable-phase method in multichannel electron-atom or electron-ion scattering,” J. Phys. B 10, L35–L42 (1977).
[CrossRef]

J. Phys. D

G. B. Smith, “Dielectric constants for mixed media,” J. Phys. D 10, L39–L42 (1977).
[CrossRef]

Phys. Rev.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962);erratum 134, AB1 (1964).
[CrossRef]

Phys. Rev. Lett.

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[CrossRef]

Other

R. Landauer, in Proceedings of the First Conference on the Electrical and Transport Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., AIP Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 2–43.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1981), Pt. 1.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), 563–573 and 392–395.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

W. F. Brown, “Dielectrics,” in Encyclopedia of physics, Vol. 17, S. Flügge, ed. (Springer-Verlag, Heidelberg, 1956), Chap. 1, Sec. 53.

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Figures (3)

Fig. 1
Fig. 1

Plot of the difference functions Dx(N)=|Q¯x(N)-Qx| versus N for x=ext, sca. Q¯x(N) is the extinction/scattering efficiency calculated for the model N-layer sphere (see the text for a description of the sphere), and Qx is the same efficiency calculated by the present generalized Mie theory. The results show that Q¯x(N) converges to Qx in the limit N.

Fig. 2
Fig. 2

(a) Differential cross section calculated by the present generalized Mie theory for scattering unpolarized light by the model mixed-media multilayer sphere in the infinite-layer limit, (b) absolute difference between the differential cross section shown in (a) and the differential cross sections calculated for the model multilayer sphere for N=200, 2000, and 7000 layers.

Fig. 3
Fig. 3

Effective dielectric constants of a two-component heterogeneous mixture, where 1=1 and 2=9, versus the volume fraction of 2. The four curves are calculated by the two Maxwell–Garnett approximations MG1 and MG2, the Bruggeman approximation, and the approximation given by Eq. (45).

Equations (71)

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××E-k2(r)E=0,
M(r, θ, ϕ)=×[Ψ(r, θ, ϕ)r],
N(r, θ, ϕ)=1k(r)××[Φ(r, θ, ϕ)r],
2Ψ+k2(r)Ψ=0,
2Φ-1ε(r)d(r)drΦr+k2(r)Φ=0.
Ψn,m=1krSn(r)Pnm(cos θ)exp(imϕ),
Φn,m=1krTn(r)Pnm(cos θ)exp(imϕ),
d2Sn(r)dr2+k2(r)-n(n+1)r2Sn(r)=0,
d2Tn(r)dr2-1(r)d(r)drdTn(r)dr
+k2(r)-n(n+1)r2Tn(r)=0.
Sn(r)=Anψn(kr)+Bnζn(kr),
Tn(r)=Cnψn(kr)+Dnζn(kr).
an=Bn/An,
bn=Dn/Cn,
dan(r)dr=ik[(r)-1][ψn(kr)+ζn(kr)an(r)]2,
dbn(r)dr=ik[(r)-1][ψn(kr)+ζn(kr)bn(r)]2+n(n+1)(kr)2[1--1(r)]×[ψn(kr)+ζn(kr)bn(r)]2.
ri=R(i/N)1/3,i=2, 4,, N,
ri=R[(i+f1-f2)/N]1/3,i=1, 3,, N-1.
da¯n(r)dr=ik[(r)-1][ψn(kr)+ζn(kr)a¯n(r)]2,
db¯n(r)dr=ik[(r)-1][ψn(kr)+ζn(kr)b¯n(r)]2+n(n+1)(kr)2[1--1(r)]×[ψn(kr)+ζn(kr)b¯n(r)]2,
(r)=f11+f22,
-1(r)=f1/1+f2/2.
an(r)=-ψn(mkr)ψn(kr)-mψn(kr)ψn(mkr)ψn(mkr)ζn(kr)-mζn(kr)ψn(mkr),
bn(r)=-mψn(mkr)ψn(kr)-ψn(kr)ψn(mkr)mψn(mkr)ζn(kr)-ζn(kr)ψn(mkr).
m¯=(r)1/2,
μ=(r)-1(r).
a¯n(r)=-ψn(m¯kr)ψn(kr)-m¯ψn(kr)ψn(m¯kr)ψn(m¯kr)ζn(kr)-m¯ζn(kr)ψn(m¯kr).
d2Tn(μ, z)dz2+1-μn(n+1)z2Tn(μ, z)=0,
γn(μ, z)=(πz/2)1/2Jν(z),
ν=[μn(n+1)+1/4]1/2.
b¯n(r)=-m¯γn(μ, m¯kr)ψn(kr)-ψn(kr)γn(μ, m¯kr)m¯γn(μ, m¯kr)ζn(kr)-ζn(kr)γn(μ, m¯kr).
a¯n(r)=-ψn(kr)ζn(kr)Un(kr)-m¯Un(m¯kr)Gn(kr)-m¯Un(m¯kr),
b¯n(r)=-ψn(kr)ζn(kr)m¯Un(kr)-Vn(μ,m¯kr)m¯Gn(kr)-Vn(μ, m¯kr),
Gn(z)=ζn(z)/ζn(z),
Un(z)=ψn(z)/ψn(z),
Vn(μ, z)=γn(μ, z)/γn(μ, z).
Qext=-(2/α2)n=1(2n+1)Re(an+bn),
Qsca=(2/α2)n=1(2n+1)(|an|2+|bn|2),
Dext(N)=0.922/N1.05,
Dsca(N)=0.591/N1.04.
a1=(i/45)(m2-1)α5,
b1=(2i/3)[(m2-1)/(m2+2)]α3.
a¯1(R)=(i/45)(m¯2-1)α5,
b¯1(R)=i32m¯2-(1.5μ1/2+0.5)m¯2+(1.5μ1/2+0.5)α3,
2i3m2-1m2+2α3=i32m¯2-(1.5μ1/2+0.5)m¯2+(1.5μ1/2+0.5)α3.
eff=21.5μ1/2+0.5.
eff-heff+2h=f11-h1+2h+f22-h2+2h.
Sn(r)=An(r)ψn(kr)+Bn(r)ζn(kr),
Sn(r)=k[An(r)ψn(kr)+Bn(r)ζn(kr)],
An(r)ψn(kr)+Bn(r)ζn(kr)=0.
An(r)ψn(kr)+Bn(r)ζn(kr)=-k[(r)-1].
ψn(z)ζn(z)-ψn(z)ζn(z)=i
An(r)=-ik[(r)-1]ζn(kr)Sn(r),
Bn(r)=ik[(r)-1]ψn(kr)Sn(r).
an(r)=Bn(r)/An(r).
d2Fn(μ, r)dr2-1(r)d(r)drdFn(μ, r)dr
+k2(r)-μn(n+1)r2Fn(μ, r)=0.
Fn(μ, r)=Cn(μ, r)ψn(kr)+Dn(μ, r)ζn(kr),
Fn(μ, r)=k(r)[Cn(μ, r)ψn(kr)+Dn(μ, r)ζn(kr)].
Cψ+Dζ=(1--1)F.
Cψ+Dζ=-(1-μ-1)gF,
g=n(n+1)/r2.
C=-i[(1--1)ζF+k-1(1-μ-1)gζF],
D=-i[(1--1)ψF+k-1(1-μ-1)gψF].
b(μ, r)=C(μ, r)/D(μ, r).
bn(μ, r)=ik{[(r)-1][ψn(kr)+ζn(kr)bn(μ, r)]2+[n(n+1)/(kr)2][1-μ-1(r)]×[ψn(kr)+ζn(kr)bn(μ, r)]}.
γν-1(μ, z)+γν+1(μ, z)=(2ν/z)γν(μ, z),
zγν(μ, z)=(1/2+ν)γν(μ, z)-zγν+1(μ, z).
Rν(μ, z)=γν(μ, z)/γν+1(μ, z).
Rν-1(μ, z)=(2ν/z)-1/Rν(μ, z).
Vn(μ, z)=γν(μ, z)/γν(μ, z)=(1/2+ν)/z-Rν-1(μ, z),

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