Abstract

A method for solving the inverse-scattering problem empirically is introduced. The method is called the strip-map technique, and, for now, we restrict its use to the identification of single homogeneous spherical particles First the technique is illustrated in a gedanken experiment with an ideal spherical microparticle Then the technique is applied to the study of four particles from which light-scattering data have been collected. Nonspherical or inhomogeneous spherical particles are immediately distinguished from particles that are homogeneous spheres, the latter being specified to a high degree of precision. The technique can be made extremely rapid: with modest computer means, individual microparticles can be characterized in less than 1 msec.

© 1985 Optical Society of America

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Corrections

Gregory M. Quist and Philip J. Wyatt, "Empirical solution to the inverse-scattering problem by the optical strip-map technique: errata," J. Opt. Soc. Am. A 3, 671-671 (1986)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-3-5-671

References

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  1. P. J. Wyatt, “Differential light scattering: a physical method for identifying living bacterial cells,” Appl. Opt. 7, 1879 (1968).
    [CrossRef] [PubMed]
  2. L. D. Faddeyev, “The inverse problem in the quantum theory of scattering,” J. Math. Phys. 4, 72 (1963).
    [CrossRef]
  3. R. G. Newton, “Construction of potentials from phase shifts at fixed energy,” J. Math. Phys. 3, 75 (1962).
    [CrossRef]
  4. D. T. Phillips, P. J. Wyatt, R. M. Berkman, “Measurement of the Lorenz-Mie scattering of a single particle: polystyrene latex,” J. Colloid Interface Sci. 34, 159 (1970).
    [CrossRef]
  5. P. J. Wyatt, D. T. Phillips, “Structure of single bacteria from light scattering,” J. Theoret. Biol. 37, 493 (1972).
    [CrossRef]
  6. P. J. Wyatt, “Cell wall thickness, size distribution, refractive index ratio, and dry weight content of living bacteria (Staphylococcus aureus),” Nature 226, 277 (1970).
    [CrossRef] [PubMed]
  7. P. J. Wyatt, “Observations on the structure of spores,” J. Appl. Bacteriol. 38, 47 (1975).
    [CrossRef] [PubMed]
  8. H. M. Blau, D. J. McClesse, D. Watson, “Scattering by individual transparent spheres,” Appl. Opt. 9, 2522 (1970).
    [CrossRef] [PubMed]
  9. P. J. Wyatt, “Some chemical, physical, and optical properties of flyash particles,” Appl. Opt. 19, 975 (1980).
    [CrossRef] [PubMed]
  10. R. Mireles, “The inverse problem of electromagnetic scattering theory. I. Uniqueness theorem for cylinders,” J. Math. Phys. 45, 179 (1966).
  11. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  12. P. J. Wyatt, “Differential light scattering techniques for microbiology,” in Methods in Microbiology, J. R. Norris, D. W. Ribbons, eds. (Academic, New York, 1973), Vol. 8, p. 183.
    [CrossRef]
  13. P. J. Wyatt, D. T. Phillips, “A new instrument for the study of individual aerosol particles,” J. Colloid Interface Sci. 39, 125 (1972).
    [CrossRef]
  14. P. J. Wyatt, ed., Atlas of Light Scattering Curves (Science Spectrum, Santa Barbara, Calif., 1975).
  15. G. J. Culler, CHI Systems Inc., Santa Barbara, Calif. (personal communication).
  16. P. Chylek, “Partial wave resonances and ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285 (1976).
    [CrossRef]
  17. P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229 (1978).
    [CrossRef]
  18. J. Shmoys, “Proposed diagnostic method for cylindrical plasmas,” J. Appl. Phys. 32, 689 (1961).
    [CrossRef]

1980 (1)

1978 (1)

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

1976 (1)

1975 (1)

P. J. Wyatt, “Observations on the structure of spores,” J. Appl. Bacteriol. 38, 47 (1975).
[CrossRef] [PubMed]

1972 (2)

P. J. Wyatt, D. T. Phillips, “A new instrument for the study of individual aerosol particles,” J. Colloid Interface Sci. 39, 125 (1972).
[CrossRef]

P. J. Wyatt, D. T. Phillips, “Structure of single bacteria from light scattering,” J. Theoret. Biol. 37, 493 (1972).
[CrossRef]

1970 (3)

P. J. Wyatt, “Cell wall thickness, size distribution, refractive index ratio, and dry weight content of living bacteria (Staphylococcus aureus),” Nature 226, 277 (1970).
[CrossRef] [PubMed]

D. T. Phillips, P. J. Wyatt, R. M. Berkman, “Measurement of the Lorenz-Mie scattering of a single particle: polystyrene latex,” J. Colloid Interface Sci. 34, 159 (1970).
[CrossRef]

H. M. Blau, D. J. McClesse, D. Watson, “Scattering by individual transparent spheres,” Appl. Opt. 9, 2522 (1970).
[CrossRef] [PubMed]

1968 (1)

1966 (1)

R. Mireles, “The inverse problem of electromagnetic scattering theory. I. Uniqueness theorem for cylinders,” J. Math. Phys. 45, 179 (1966).

1963 (1)

L. D. Faddeyev, “The inverse problem in the quantum theory of scattering,” J. Math. Phys. 4, 72 (1963).
[CrossRef]

1962 (1)

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

1961 (1)

J. Shmoys, “Proposed diagnostic method for cylindrical plasmas,” J. Appl. Phys. 32, 689 (1961).
[CrossRef]

Berkman, R. M.

D. T. Phillips, P. J. Wyatt, R. M. Berkman, “Measurement of the Lorenz-Mie scattering of a single particle: polystyrene latex,” J. Colloid Interface Sci. 34, 159 (1970).
[CrossRef]

Blau, H. M.

Chylek, P.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

P. Chylek, “Partial wave resonances and ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285 (1976).
[CrossRef]

Culler, G. J.

G. J. Culler, CHI Systems Inc., Santa Barbara, Calif. (personal communication).

Faddeyev, L. D.

L. D. Faddeyev, “The inverse problem in the quantum theory of scattering,” J. Math. Phys. 4, 72 (1963).
[CrossRef]

Kerker, M.

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

Kiehl, J. T.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

McClesse, D. J.

Mireles, R.

R. Mireles, “The inverse problem of electromagnetic scattering theory. I. Uniqueness theorem for cylinders,” J. Math. Phys. 45, 179 (1966).

Newton, R. G.

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

Phillips, D. T.

P. J. Wyatt, D. T. Phillips, “Structure of single bacteria from light scattering,” J. Theoret. Biol. 37, 493 (1972).
[CrossRef]

P. J. Wyatt, D. T. Phillips, “A new instrument for the study of individual aerosol particles,” J. Colloid Interface Sci. 39, 125 (1972).
[CrossRef]

D. T. Phillips, P. J. Wyatt, R. M. Berkman, “Measurement of the Lorenz-Mie scattering of a single particle: polystyrene latex,” J. Colloid Interface Sci. 34, 159 (1970).
[CrossRef]

Shmoys, J.

J. Shmoys, “Proposed diagnostic method for cylindrical plasmas,” J. Appl. Phys. 32, 689 (1961).
[CrossRef]

Watson, D.

Wyatt, P. J.

P. J. Wyatt, “Some chemical, physical, and optical properties of flyash particles,” Appl. Opt. 19, 975 (1980).
[CrossRef] [PubMed]

P. J. Wyatt, “Observations on the structure of spores,” J. Appl. Bacteriol. 38, 47 (1975).
[CrossRef] [PubMed]

P. J. Wyatt, D. T. Phillips, “Structure of single bacteria from light scattering,” J. Theoret. Biol. 37, 493 (1972).
[CrossRef]

P. J. Wyatt, D. T. Phillips, “A new instrument for the study of individual aerosol particles,” J. Colloid Interface Sci. 39, 125 (1972).
[CrossRef]

P. J. Wyatt, “Cell wall thickness, size distribution, refractive index ratio, and dry weight content of living bacteria (Staphylococcus aureus),” Nature 226, 277 (1970).
[CrossRef] [PubMed]

D. T. Phillips, P. J. Wyatt, R. M. Berkman, “Measurement of the Lorenz-Mie scattering of a single particle: polystyrene latex,” J. Colloid Interface Sci. 34, 159 (1970).
[CrossRef]

P. J. Wyatt, “Differential light scattering: a physical method for identifying living bacterial cells,” Appl. Opt. 7, 1879 (1968).
[CrossRef] [PubMed]

P. J. Wyatt, “Differential light scattering techniques for microbiology,” in Methods in Microbiology, J. R. Norris, D. W. Ribbons, eds. (Academic, New York, 1973), Vol. 8, p. 183.
[CrossRef]

Appl. Opt. (3)

J. Appl. Bacteriol. (1)

P. J. Wyatt, “Observations on the structure of spores,” J. Appl. Bacteriol. 38, 47 (1975).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

J. Shmoys, “Proposed diagnostic method for cylindrical plasmas,” J. Appl. Phys. 32, 689 (1961).
[CrossRef]

J. Colloid Interface Sci. (2)

P. J. Wyatt, D. T. Phillips, “A new instrument for the study of individual aerosol particles,” J. Colloid Interface Sci. 39, 125 (1972).
[CrossRef]

D. T. Phillips, P. J. Wyatt, R. M. Berkman, “Measurement of the Lorenz-Mie scattering of a single particle: polystyrene latex,” J. Colloid Interface Sci. 34, 159 (1970).
[CrossRef]

J. Math. Phys. (3)

L. D. Faddeyev, “The inverse problem in the quantum theory of scattering,” J. Math. Phys. 4, 72 (1963).
[CrossRef]

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

R. Mireles, “The inverse problem of electromagnetic scattering theory. I. Uniqueness theorem for cylinders,” J. Math. Phys. 45, 179 (1966).

J. Opt. Soc. Am. (1)

J. Theoret. Biol. (1)

P. J. Wyatt, D. T. Phillips, “Structure of single bacteria from light scattering,” J. Theoret. Biol. 37, 493 (1972).
[CrossRef]

Nature (1)

P. J. Wyatt, “Cell wall thickness, size distribution, refractive index ratio, and dry weight content of living bacteria (Staphylococcus aureus),” Nature 226, 277 (1970).
[CrossRef] [PubMed]

Phys. Rev. A (1)

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Other (4)

P. J. Wyatt, ed., Atlas of Light Scattering Curves (Science Spectrum, Santa Barbara, Calif., 1975).

G. J. Culler, CHI Systems Inc., Santa Barbara, Calif. (personal communication).

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

P. J. Wyatt, “Differential light scattering techniques for microbiology,” in Methods in Microbiology, J. R. Norris, D. W. Ribbons, eds. (Academic, New York, 1973), Vol. 8, p. 183.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

The geometry for scattered light from a homogeneous sphere. ϕ = 90° corresponds to the V plane, ϕ = 0° corresponds to the H plane; θ is measured from the forward-scattering direction, r is the radius vector defining the observation point and is much greater than the radius of the scattering particle (i.e., the far field).

Fig. 2
Fig. 2

A typical differential light-scattering pattern in the V plane generated from Eqs. (1). The scatterer is a homogeneous sphere of size parameter x = 6.752 and refractive index n = 1.50 illuminated with vertically polarized light. Optical observables of the variety discussed in Section 4 can be generated (with unnormalized intensities) from these data normalized to the intensity IH(90).

Fig. 3
Fig. 3

A strip map for the optical observable O1 = IV(20)/IH(90) for the restricted range 1.4 ≤ n ≤ 1.6, 6.0 ≤ x ≤ 7.0. The region depicted corresponds to those particles whose optical observable O1 is in the range OO1 ≤ 1.00.

Fig. 4
Fig. 4

A strip map for the optical observable O4 = the number of peaks in the V plane = 6 overlaid onto the strip map from Fig. 3. The region of intersection is shaded.

Fig. 5
Fig. 5

A strip map for the optical observable O2 = IV(180)/IH(90) in the range 4.50 ≤ O2 ≤ 6.00 is overlaid onto the strip maps from Fig. 4. The region of intersection is shaded.

Fig. 6
Fig. 6

A fourth strip map for the optical observable O3 = IV(90)/IH(90) in the range 1.00 ≤ O3 ≤ 1.60 is overlaid onto the three previous strip maps from Fig. 5. The region of intersection is shown: its geometrical center and extent is n = 1.50 ± 0.01 and x = 6.70 ± 0.03.

Fig. 7
Fig. 7

A computer-generated strip map corresponding to those particles whose vertical DLS patterns have five peaks in the extended region 1.4 ≤ n ≤ 1.75 and 4.0 ≤ x ≤ 9.0. Each dot represents a rectangular region of dimension Δn = 0.01 and Δx = 0.05.

Fig. 8
Fig. 8

A DLS curve in the V plane from a homogeneous spherical flyash particle illuminated by 632.8-nm linearly polarized light.

Fig. 9
Fig. 9

A DLS curve in the V plane of a spherically concentric structured bacterial spore, Bacillus sphaericus, illuminated by 514.5-nm linearly polarized light.

Fig. 10
Fig. 10

A DLS curve in the V plane of a rod-shaped bacterium, E. coli, illuminated by 514.5-nm linearly polarized light.

Fig. 11
Fig. 11

A DLS curve in the V plane of a homogeneous latex sphere illuminated by 514.5-nm linearly polarized light. Note that the light-scattering intensity at V = 20° is unavailable.

Equations (14)

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I θ = | λ 2 4 π r 2 | S 2 2 cos 2 ϕ , I ϕ = | λ 2 4 π r 2 | S 1 2 sin 2 ϕ ,
S 1 = | n = 1 2 n + 1 n ( n + 1 ) [ e B n π n ( cos θ ) + m B n τ n ( cos θ ) ] | , S 2 = | n = 1 2 n + 1 n ( n + 1 ) [ e B n π n ( cos θ ) + m B n τ n ( cos θ ) ] | , π n ( cos θ ) = [ P n ( 1 ) ( cos θ ) ] sin θ , τ n ( cos θ ) = d d θ [ P n ( 1 ) ( cos θ ) ] = n ( n + 1 ) P n ( cos θ ) π n ( cos θ ) .
O 1 = I V ( 20 ° ) / I H ( 90 ° ) = 0.72 , O 2 = I V ( 180 ° ) / I H ( 90 ° ) = 5.35 , O 3 = I V ( 90 ° ) / I H ( 90 ° ) = 1.17 , O 4 = number of peaks in V plan = 6 .
O 1 = I V ( 20 ° ) / I V ( 90 ° ) ( when available ) , O 2 = I V ( 55 ° ) / I V ( 90 ° ) , O 3 = I V ( 125 ° ) / I V ( 90 ° ) , O 4 = I V ( 160 ° ) / I V ( 90 ° ) , O 5 = number of peaks in V plane
O 1 = 64.40 ± 2.40 , O 2 = 4.61 ± 0.18 , O 3 = 0.94 ± 0.05 , O 4 = 3.78 ± 0.15 , O 5 = 5 .
n = 1.534 ± 0.01 , x = 5.93 ± 0.05 .
I V ( 20 ° ) = 116.5 ± 7.0 , I V ( 55 ° ) = 18.4 ± 1.0 , I V ( 90 ° ) = 6.3 ± 1.0 , I V ( 125 ° ) = 5.3 ± 1.0 , I V ( 160 ° ) = 9.2 ± 1.0 .
15.00 O 1 23.30 , measured value 18.49 , 2.56 O 2 3.66 , measured value 2.92 , 0.59 O 3 1.19 , measured value 0.84 , 1.19 O 4 1.92 , measured value 1.46 , O 5 = 5 .
x = 4.72 ± 0.03 , n = 1.515 ± 0.010 .
V ( 20 ° ) = 20.5 ± 6.0 , V ( 55 ° ) = 7.0 ± 4.0 , V ( 90 ° ) = 10.0 ± 3.0 , V ( 125 ° ) = 4.0 ± 2.0 , V ( 160 ° ) = 12.5 ± 4.0 , number of peaks in V plane = 5 = O 5 .
O 1 = V ( 20 ° ) / V ( 90 ° ) = 2 ± 0.8 , O 2 = V ( 55 ° ) / V ( 90 ° ) = 0.7 ± 0.5 , O 3 = V ( 125 ° ) / V ( 90 ° ) = 0.4 ± 0.2 , O 4 = V ( 160 ° ) / V ( 90 ° ) = 1.3 ± 0.6 .
V ( 55 ° ) = 184 ± 10.75 , V ( 90 ° ) = 7.0 ± 3.25 , V ( 125 ° ) = 11.0 ± 3.75 , V ( 160 ° ) = 9.0 ± 4.0 , peaks in V plan = 5 = O 5 .
O 2 = 26 ± 12 , O 3 = 1.6 ± 0.9 , O 4 = 1.3 ± 0.8 .
x = 5.8 ± 0.2 , n = 1.61 ± 0.03 ,

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