Abstract

We apply functional analysis to the scattered electromagnetic field of a particle with spherical symmetry to obtain a pair of integral transforms for converting the Mie-scattering amplitudes S(θ) and S(θ) into the Mie coefficients an and bn. In the case of a homogeneous sphere, a simple mathematical construction is derived that uniquely inverts the Mie coefficients to find the refractive index and the radius of the particle. A more general method for construction of the refractive-index profile of an arbitrary sphere is discussed that follows from the treatment of Newton and Sabatier.

© 2000 Optical Society of America

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References

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  1. G. Mie, “Beitrage zur Optic trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
    [CrossRef]
  2. Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Rev. 40, 229–232 (1987).
    [CrossRef]
  3. P. J. Wyatt, “Some chemical, physical, and optical properties of fly ash particles,” Appl. Opt. 19, 975–983 (1980).
    [CrossRef] [PubMed]
  4. C. F. Bohren, E. D. Hirleman, eds., feature on Optical Particle Sizing, Appl. Opt. 30, 4685–4987 (1991).
    [CrossRef]
  5. M. R. Jones, B. P. Curry, M. Quinn Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
    [CrossRef] [PubMed]
  6. Z. Ulanowski, Z. Wang, P. H. Kaye, I. K. Ludlow, “Application of neural networks to the inverse light scattering problem for spheres,” Appl. Opt. 37, 4027–4033 (1998).
    [CrossRef]
  7. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1989).
  8. C. F. Bohren, D. R. Huffman. Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  9. L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
    [CrossRef] [PubMed]
  10. B. R. Johnson, “Exact theory of electromagnetic scattering by a hetrogeneous multilayered sphere in the infinite-layer limit: effective media approach,” J. Opt. Soc. Am. A 16, 845–852 (1999).
    [CrossRef]
  11. R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1982).
  12. R. G. Newton, “Construction of potentials from the phase shifts at fixed energy,” J. Math. Phys. 3, 75–82 (1962).
    [CrossRef]
  13. R. G. Newton, “Connection between complex angular momenta and the inverse scattering problem at fixed energy,” J. Math. Phys. 8, 1566–1570 (1967).
    [CrossRef]
  14. P. C. Sabatier, “Asymptotic properties of potentials in the inverse-scattering problem at fixed energy,” J. Math. Phys. 7, 1515–1531 (1966).
    [CrossRef]
  15. P. C. Sabatier, “Analytic properties of a class of potentials and the corresponding Jost function,” J. Math. Phys. 7, 2079–2091 (1966).
    [CrossRef]
  16. P. C. Sabatier, “General method for the inverse scattering problem of fixed energy,” J. Math. Phys. 8, 905–918 (1967).
    [CrossRef]

1999

1998

1994

1991

1987

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Rev. 40, 229–232 (1987).
[CrossRef]

1980

1967

P. C. Sabatier, “General method for the inverse scattering problem of fixed energy,” J. Math. Phys. 8, 905–918 (1967).
[CrossRef]

R. G. Newton, “Connection between complex angular momenta and the inverse scattering problem at fixed energy,” J. Math. Phys. 8, 1566–1570 (1967).
[CrossRef]

1966

P. C. Sabatier, “Asymptotic properties of potentials in the inverse-scattering problem at fixed energy,” J. Math. Phys. 7, 1515–1531 (1966).
[CrossRef]

P. C. Sabatier, “Analytic properties of a class of potentials and the corresponding Jost function,” J. Math. Phys. 7, 2079–2091 (1966).
[CrossRef]

1962

R. G. Newton, “Construction of potentials from the phase shifts at fixed energy,” J. Math. Phys. 3, 75–82 (1962).
[CrossRef]

1908

G. Mie, “Beitrage zur Optic trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Bohren, C. F.

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

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1989).

Curry, B. P.

Huffman, D. R.

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

Johnson, B. R.

Jones, M. R.

Kai, L.

Kaye, P. H.

Leong, K. H.

Ludlow, I. K.

Z. Ulanowski, Z. Wang, P. H. Kaye, I. K. Ludlow, “Application of neural networks to the inverse light scattering problem for spheres,” Appl. Opt. 37, 4027–4033 (1998).
[CrossRef]

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Rev. 40, 229–232 (1987).
[CrossRef]

Massoli, P.

Mie, G.

G. Mie, “Beitrage zur Optic trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Newton, R. G.

R. G. Newton, “Connection between complex angular momenta and the inverse scattering problem at fixed energy,” J. Math. Phys. 8, 1566–1570 (1967).
[CrossRef]

R. G. Newton, “Construction of potentials from the phase shifts at fixed energy,” J. Math. Phys. 3, 75–82 (1962).
[CrossRef]

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1982).

Quinn Brewster, M.

Sabatier, P. C.

P. C. Sabatier, “General method for the inverse scattering problem of fixed energy,” J. Math. Phys. 8, 905–918 (1967).
[CrossRef]

P. C. Sabatier, “Analytic properties of a class of potentials and the corresponding Jost function,” J. Math. Phys. 7, 2079–2091 (1966).
[CrossRef]

P. C. Sabatier, “Asymptotic properties of potentials in the inverse-scattering problem at fixed energy,” J. Math. Phys. 7, 1515–1531 (1966).
[CrossRef]

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1989).

Ulanowski, Z.

Z. Ulanowski, Z. Wang, P. H. Kaye, I. K. Ludlow, “Application of neural networks to the inverse light scattering problem for spheres,” Appl. Opt. 37, 4027–4033 (1998).
[CrossRef]

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Rev. 40, 229–232 (1987).
[CrossRef]

Waites, W. M.

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Rev. 40, 229–232 (1987).
[CrossRef]

Wang, Z.

Wyatt, P. J.

Ann. Phys. (Leipzig)

G. Mie, “Beitrage zur Optic trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Appl. Opt.

FEMS Microbiol. Rev.

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Rev. 40, 229–232 (1987).
[CrossRef]

J. Math. Phys.

R. G. Newton, “Construction of potentials from the phase shifts at fixed energy,” J. Math. Phys. 3, 75–82 (1962).
[CrossRef]

R. G. Newton, “Connection between complex angular momenta and the inverse scattering problem at fixed energy,” J. Math. Phys. 8, 1566–1570 (1967).
[CrossRef]

P. C. Sabatier, “Asymptotic properties of potentials in the inverse-scattering problem at fixed energy,” J. Math. Phys. 7, 1515–1531 (1966).
[CrossRef]

P. C. Sabatier, “Analytic properties of a class of potentials and the corresponding Jost function,” J. Math. Phys. 7, 2079–2091 (1966).
[CrossRef]

P. C. Sabatier, “General method for the inverse scattering problem of fixed energy,” J. Math. Phys. 8, 905–918 (1967).
[CrossRef]

J. Opt. Soc. Am. A

Other

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1982).

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1989).

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

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Figures (2)

Fig. 1
Fig. 1

Inversion plot for α=15,β=20. The fractional deviation Δ=(z/α)-1 is incremented in steps of 10-4 between -4.0×10-4 and 4.0×10-4.

Fig. 2
Fig. 2

Discontinuous nature of the relative dielectric constant.

Equations (99)

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E0(r, t)=E0p exp[i(kz-ωt)]=-E0 exp(-iωt)n=1in+1(2n+1)2n(n+1){(px-ipy)×[N1n1(kr)+M1n1(kr)]+(px+ipy)×[N-1n1(kr)-M-1n1(kr)]},
Es(r, t)=E0 exp(-iωt)n=1in+1(2n+1)2n(n+1){(px-ipy)×[anN1n3(kr)+bnM1n3(kr)]+(px+ipy)×[anN-1n3(kr)-bnM-1n3(kr)]},
Ei(r, t)=E0 exp(-iωt)n=1in+1(2n+1)2n(n+1){(px-ipy)×[cnN1n1(mkr)+dnM1n1(mkr)]+(px+ipy)×[cnN-1n1(mkr)-dnM-1n1(mkr)]}.
H(r, t)=1iωμ0×E(r, t).
Mmn(kr)=1krψn(kr)ξn(kr)Cmn(θ, ϕ),
Nmn(kr)=n(n+1)(kr)2ψn(kr)ξn(kr)Pmn(θ, ϕ)+1krψn(kr)ξn(kr)Bmn(θ, ϕ),
Pmn(θ, ϕ)=aˆrPn|m|(θ)exp(imϕ),
Bmn(θ, ϕ)=aˆθθ+aˆϕsin θϕPn|m|(θ)×exp(imϕ),
Cmn(θ, ϕ)=aˆθsin θϕ-aˆϕθPn|m|(θ)×exp(imϕ),
Pmn·Bmn*dΩ=Pmn·Cmn*dΩ=Bmn·Cmn*dΩ=0,
Bmn·Bmn*dΩ=Cmn·Cmn*dΩ=n(n+1)Pmn·Pmn*dΩ=Λmnδmmδnn,
Λmn=4πn(n+1)(n+|m|)!(2n+1)(n-|m|)!.
cnψn(β)=anξn(α)-ψn(α),
cnmψn(β)=anξn(α)-ψn(α),
dnmψn(β)=bnξn(α)-ψn(α),
dnψn(β)=bnξn(α)-ψn(α),
an=ψn(β)ψn(α)-mψn(β)ψn(α)ψn(β)ξn(α)-mψn(β)ξn(α),
bn=mψn(β)ψn(α)-ψn(β)ψn(α)mψn(β)ξn(α)-ψn(β)ξn(α),
cn=imψn(β)ξn(α)-mψn(β)ξn(α),
dn=immψn(β)ξn(α)-ψn(β)ξn(α).
Es(r, t)=exp(ikr)rE0 exp(-iωt)F(θ, ϕ),
F(θ, ϕ)=i2k(px-ipy)n=1(2n+1)n(n+1)×[anB1n(θ, ϕ)-ibnC1n(θ, ϕ)]+(px+ipy)n=1(2n+1)n(n+1)[anB-1n(θ, ϕ)+ibnC-1n(θ, ϕ)]
1Λ1nF(θ, ϕ)·B1n*dΩ=i2k(px-ipy)2n+1n(n+1)an.
S(θ)=n=1(2n+1)n(n+1)[anπn(θ)+bnτn(θ)],
S(θ)=n=1(2n+1)n(n+1)[bnπn(θ)+anτn(θ)],
πn(θ)=Pn1(θ)sin θ,τn=Pn1(θ)θ.
FX(θ, ϕ)=ik[aˆθS(θ)cos ϕ-aˆϕS(θ)sin ϕ],
FY(θ, ϕ)=ik[aˆθS(θ)sin ϕ+aˆϕS(θ)cos ϕ].
an=12n(n+1)0π[S(θ)τn(θ)+S(θ)πn(θ)]sin θdθ,
bn=12n(n+1)0π[S(θ)πn(θ)+S(θ)τn(θ)]sin θdθ,
m2=anξn(α)-ψn(α)anξn(α)-ψn(α)bnξn(α)-ψn(α)bnξn(α)-ψn(α),
fn(z)=anξn(z)-ψn(z)anξn(z)-ψn(z)bnξn(z)-ψn(z)bnξn(z)-ψn(z)
fn(z)=mψn(β)D1-ψn(β)Cmψn(β)E-ψn(β)D2ψn(β)E-mψn(β)D2ψn(β)D1-mψn(β)C,
C=ψn(α)χn(z)-χn(α)ψn(z),
D1=ψn(α)χn(z)-χn(α)ψn(z),
D2=ψn(α)χn(z)-χn(α)ψn(z),
E=ψn(α)χn(z)-χn(α)ψn(z),
CE=D1D2+1.
(fn-1)m{D1Eψn2(β)+D2C[ψn(β)]2}-[(fnm2-1)×CE+(fn-m2)D1D2]ψn(β)ψn(β)=0.
(fn-1)mD1E=0,
(fn-1)mCD2=0,
(fn-1)(m2+1)D1D2+(fnm2-1)=0,
fn=1,m=1,
fn=1/m2,D1=0,D2=0,CE=1,
fn=m2,C=0,E=0,D1D2=-1.
ψn(α)χn(α)=ψn(z)χn(z),ψn(α)χn(α)=ψn(z)χn(z)
fn(z)=m2+fn(z)(z-α)+fn(z)(z-α)22+,
fn(α)=m(m2-1)gnψn(β)ψn(β)-ψn(β)ψn(β),
fn(α)=2(m2-1)m2gnψn(β)ψn(β)-ψn(β)ψn(β)2+mn(n+1)α3ψn(β)ψn(β)+(m2+1)gn
gn=1-n(n+1)/α2.
ψn(β)ψn(β)=±[1-n(n+1)/α2]1/2.
an=12[1-exp(i2un)],
bn=12[1-exp(i2vn)].
tan un=ψn(β)ψn(α)-mψn(β)ψn(α)ψn(β)χn(α)-mψn(β)χn(α),
tan vn=mψn(β)ψn(α)-ψn(β)ψn(α)mψn(β)χn(α)-ψn(β)χn(α).
fn(z)=sin[θn(z)+un]cos[ϕn(z)+vn]cos[ϕn(z)+un]sin[θn(z)+vn],
d2dr2ϕn(kr)+k2-n(n+1)r2ϕn(kr)=V(r)ϕn(kr)
ϕn(r)=ψn(r)-0rK(r, ρ)ψn(ρ)ρ2dρ,
ϕn(r)An sin(kr-nπ/2+un),
dϕn(r)drkAn cos(kr-nπ/2+un),
V(r)=-2rddrK(r, r)r.
K(r, ρ)=n=0cnϕn(r)ψn(ρ)
E1(r)=Fn(r)Cmn(θ, ϕ),
H1(r)=1iωμ0n(n+1)Fn(r)rPmn(θ, ϕ)+1rddr[rFn(r)]Bmn(ϑ, ϕ),
××E1(r)=m2(r)k2E1(r),
d2dr2(rFn)+m2(r)k2-n(n+1)r2(rFn)=0.
H2(r)=Gn(r)Cmn(θ, ϕ),
E2(r)=-1iω0m2(r)n(n+1)Gn(r)rPmn(θ, ϕ)+1rddr[rGn(r)]Bmn(θ, ϕ),
×1m2(r)×H2(r)=k2H2(r)
m2(r)ddr1m2(r)ddr(rGn)+m2(r)k2-n(n+1)r2(rGn)=0.
d2dr2ϕn(kr)+k2-n(n+1)r2ϕn(kr)
=V1(r)ϕn(kr)V1(r)ϕn(kr)+V2(r)ddrϕn(kr),
V1(r)=-[m2(r)-1]k2,
V2(r)=-ddrln1m2(r).
m2(r)=m2,ra1,ra,
V1(r)=-(m2-1)k2[1-θ(r-a)],
V2(r)=ddr{2 ln m[1-θ(r-a)]}=-2 ln mδ(r-a),
×E(r)=iωμ0H(r),
×H(r)=-iωμ0m2(r)E(r),
·D(r)=·[0m2(r)E(r)]=0,
·B(r)=·[μ0H(r)]=0.
×[f(r)Cmn(θ, ϕ)]=n(n+1)f(r)rPmn(θ, ϕ)+1rddr[rf(r)]Bmn(θ, ϕ),
×[f(r)Pmn(θ, ϕ)]=f(r)rCmn(θ, ϕ),
×[f(r)Bmn(θ, ϕ)]=-1rddr[rf(r)]Cmn(θ, ϕ),
·[f(r)Cmn(θ, ϕ)]=0,
·[f(r)Pmn(θ, ϕ)]=1r2ddr[r2f(r)]Ymn(θ, ϕ),
·[f(r)Bmn(θ, ϕ)]=-n(n+1)f(r)rYmn(θ, ϕ).
E1(r)=Fn(r)Cmn(θ, ϕ)
H1(r)=×E1(r)iωμ0=1iωμ0n(n+1)Fn(r)rPmn(θ, ϕ)+1rddr[rFn(r)]Bmn(θ, ϕ)
D1(r)=0m2(r)Fn(r)Cmn(θ, ϕ),
B1(r)=1iωn(n+1)Fn(r)rPmn(θ, ϕ)+1rddr[rFn(r)]Bmn(θ, ϕ),
××E1(r)=m2(r)k2E1(r).
d2dr2(rFn)+m2(r)k2-n(n+1)r2(rFn)=0
H2(r)=Gn(r)Cmn(θ, ϕ)
E2(r)=-×H2(r)iω0m2(r)=-1iω0m2(r)n(n+1)Gn(r)rPmn(θ, ϕ)+1rddr[rGn(r)]Bmn(θ, ϕ).
D2(r)=-1iωn(n+1)Gn(r)rPmn(θ, ϕ)+1rddr[rGn(r)]Bmn(θ, ϕ),
B2(r)=μ0Gn(r)Cmn(θ, ϕ),
×1m2(r)×H2(r)=k2H2(r).
m2(r)ddr1m2(r)ddr(rGn)+m2(r)k2-n(n+1)r2×(rGn)=0.

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