Abstract

An inverse, Monte Carlo (IMC) technique is developed to solve the electromagnetic inverse-scattering problem from generally complex distributions of dielectric particles. One can verify the technique using simulated scattering data from aerosols composed of spherical dielectrics. The IMC method is found to give accurate inversion results even when the data have a signal-to-noise ratio to as low as 3:1.

© 1996 Optical Society of America

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References

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  1. J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
    [CrossRef] [PubMed]
  2. E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
    [CrossRef]
  3. K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).
  4. J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
    [CrossRef]
  5. F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
    [CrossRef]
  6. O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
    [CrossRef]
  7. H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991); “Simultaneous determination of size distribution and refractive index of colloidal particles from static light-scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
    [CrossRef] [PubMed]
  8. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983) Chaps. 3 and 4.
  9. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 353–354.

1991 (2)

1987 (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
[CrossRef]

1980 (2)

F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
[CrossRef]

1966 (1)

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

1955 (1)

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Agrawal, Y. C.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983) Chaps. 3 and 4.

Chin, J. H.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Glatter, O.

Hagan, F.

F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

Hirleman, E. D.

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983) Chaps. 3 and 4.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 353–354.

Riley, J. B.

Schnablegger, H.

Shifrin, K. S.

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

Sliepcevich, C. M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Tribus, M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Appl. Opt. (2)

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

J. Appl. Crystallogr. (1)

O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
[CrossRef]

J. Comput. Phys. (1)

F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

J. Part. Charact. (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
[CrossRef]

J. Phys. Chem. (1)

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Other (2)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983) Chaps. 3 and 4.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 353–354.

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Figures (12)

Fig. 1
Fig. 1

Inversion results for distribution m1r. The distribution was solved using 16 mesh points over radius a and ten mesh points over refractive index n. The IMC was used to determine the single mode.

Fig. 2
Fig. 2

Inversion results for distribution m1r using the refractive index found from the coarse, two-dimensional inversion.

Fig. 3
Fig. 3

Inversion results for distribution m1r using data in the diffraction angular region. Note the flat distribution over the refractive index.

Fig. 4
Fig. 4

Inversion results for distribution m1r in the diffraction angular region using a refractive index of 1.46. The solid curve represents the exact distribution.

Fig. 5
Fig. 5

Inversion results for distribution m1r in the backscattered angular region.

Fig. 6
Fig. 6

Surface plot of the inversion results for bimodal distribution m2r in the full angular region.

Fig. 7
Fig. 7

Results of inversion for multispectral light scattering for measurements taken at (a) 168 deg and (b) 168 and 170 deg. Plot (a) clearly shows the aliasing effect that occurs for incomplete angular data. The addition of another angular set resolves this problem.

Fig. 8
Fig. 8

Inversion results for distribution m1r when the grid has been overspecified.

Fig. 9
Fig. 9

Results of inversion for distribution m1r when random noise is included. The signal-to-noise ratios were (a) 8.5, (b) 4.5, (c) 3.5, (d) 2.0. The inversion was accurate for signal-to-noise ratios to approximately 3:1.

Fig. 10
Fig. 10

Surface plot of the inversion results for distribution m2r when random (shot) noise is present. The signal-to-noise ratio for this data set was 5.0.

Fig. 11
Fig. 11

Inversion results showing the distribution for the core radius for a distribution of coated spheres. The solid curve represents the exact distribution.

Fig. 12
Fig. 12

Inversion results showing the coat thickness distribution for an ensemble of coated spheres. The solid curve represents the exact distribution.

Equations (9)

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i ( λ, θ ) = m = 1 M i m ( λ, θ , ζ ˜ m ) ,
i ( λ , θ ) = N 0 g ( ζ ˜ ) i ( λ , θ, ζ ˜ ) d ζ ˜ ,
i μ = N 0 j i ˜ μ j f j .
i ˜ μ j i ˜ ( λ, θ, a j , m ) = 1 δ a a j δ a / 2 a j + δ a / 2 i ( λ, θ, a , m ) d a .
s μ ¯ = s μ s μ , i μ ¯ = i μ i μ = 1 P i ˜ μ j f j 1 P i ˜ 0 j f j .
R 2 = μ ( s μ ¯ i μ ¯ s μ ¯ ) 2 .
N 0 = 1 n data μ n data ( s μ i μ ) .
χ 2 = μ n data ( s μ ¯ i μ ¯ ) 2 μ ,
g ( a ) = 1 2 π a σ g exp [ ( ln a ln a m ) 2 2 σ g 2 ] ,

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