Abstract

Existing laser-diffraction instruments that use photodiode detectors have a limited resolution for particle sizing. We attempt the implementation of a complementary metal-oxide semiconductor pixel sensor for particle-size measurement by laser diffraction. The sensor has unique features: high resolution, no blooming, and a wide dynamic range (i.e., direct measurement of the scattering pattern). The calibration of the sensor is based on each pixel. The signal-processing and the inversion schemes for obtaining the particle-size distribution are described. The results indicate an improved size resolution and an increased flexibility of application.

© 2000 Optical Society of America

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References

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  1. D. Kouzelis, S. M. Candel, E. Esposito, S. Zikikout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” Part. Charact. 4, 151–156 (1987).
    [CrossRef]
  2. C. Heffels, “On-line particle size and shape characterization by narrow angle light scattering,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1995).
  3. M. L. Strickland, “Apparatus and method for determining the size distribution of particles by light scattering,” U.S. patent5,576,827 (19November1996).
  4. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1969).
  5. J. H. Chin, C. M. Sliepcevih, M. Tribus, “Determination of particle size distributions in polydispersed systems,” J. Phys. Chem. 59, 845–848 (1955).
    [CrossRef]
  6. K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Atmos. Oceanic Phys. 2, 559–561 (1966).
  7. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  8. A. Boxman, H. G. Merkus, P. J. T. Verherijen, B. Scarlett, “Deconvolution of light-scattering patterns by observing intensity fluctuations,” Appl. Opt. 30, 4818–4823 (1991).
    [CrossRef] [PubMed]
  9. J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).
  10. M. Heuer, K. Leschonski, “Results obtained with a new instrument for the measurement of particle size distributions from diffraction patterns,” Part. Charact. 2, 7–13 (1985).
    [CrossRef]
  11. M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
    [CrossRef]
  12. R. Santer, M. Herman, “Particle size distribution from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22, 2294–2301 (1983).
    [CrossRef] [PubMed]
  13. W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).
  14. E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
    [CrossRef]
  15. S. Hayashi, “A laser small-angle scattering instrument for the determination of size and concentration distributions in sprays,” in Liquid Particle Size Measurement Techniques, American Society for Testing and Materials STP Series Volume 1083 (American Society for Testing and Materials, Philadelphia, Pa., 1990), Vol. 2, pp. 77–92.

1991

1987

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikikout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” Part. Charact. 4, 151–156 (1987).
[CrossRef]

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

1985

M. Heuer, K. Leschonski, “Results obtained with a new instrument for the measurement of particle size distributions from diffraction patterns,” Part. Charact. 2, 7–13 (1985).
[CrossRef]

1983

1977

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

1970

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

1966

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Atmos. Oceanic Phys. 2, 559–561 (1966).

1955

J. H. Chin, C. M. Sliepcevih, M. Tribus, “Determination of particle size distributions in polydispersed systems,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Abbot, D.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Beer, J. M.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Boxman, A.

Candel, S. M.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikikout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” Part. Charact. 4, 151–156 (1987).
[CrossRef]

Chahine, M. T.

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

Chin, J. H.

J. H. Chin, C. M. Sliepcevih, M. Tribus, “Determination of particle size distributions in polydispersed systems,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Esposito, E.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikikout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” Part. Charact. 4, 151–156 (1987).
[CrossRef]

Hayashi, S.

S. Hayashi, “A laser small-angle scattering instrument for the determination of size and concentration distributions in sprays,” in Liquid Particle Size Measurement Techniques, American Society for Testing and Materials STP Series Volume 1083 (American Society for Testing and Materials, Philadelphia, Pa., 1990), Vol. 2, pp. 77–92.

Heffels, C.

C. Heffels, “On-line particle size and shape characterization by narrow angle light scattering,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1995).

Herman, M.

Heuer, M.

M. Heuer, K. Leschonski, “Results obtained with a new instrument for the measurement of particle size distributions from diffraction patterns,” Part. Charact. 2, 7–13 (1985).
[CrossRef]

Hirleman, E. D.

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

Kolmakov, I. B.

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Atmos. Oceanic Phys. 2, 559–561 (1966).

Kouzelis, D.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikikout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” Part. Charact. 4, 151–156 (1987).
[CrossRef]

Leschonski, K.

M. Heuer, K. Leschonski, “Results obtained with a new instrument for the measurement of particle size distributions from diffraction patterns,” Part. Charact. 2, 7–13 (1985).
[CrossRef]

McCreath, G. C.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Menke, W.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).

Merkus, H. G.

Santer, R.

Scarlett, B.

Shifrin, K. S.

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Atmos. Oceanic Phys. 2, 559–561 (1966).

Sliepcevih, C. M.

J. H. Chin, C. M. Sliepcevih, M. Tribus, “Determination of particle size distributions in polydispersed systems,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Strickland, M. L.

M. L. Strickland, “Apparatus and method for determining the size distribution of particles by light scattering,” U.S. patent5,576,827 (19November1996).

Swithenbank, J.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Taylor, D. S.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Tribus, M.

J. H. Chin, C. M. Sliepcevih, M. Tribus, “Determination of particle size distributions in polydispersed systems,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1969).

Verherijen, P. J. T.

Zikikout, S.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikikout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” Part. Charact. 4, 151–156 (1987).
[CrossRef]

Appl. Opt.

Atmos. Oceanic Phys.

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Atmos. Oceanic Phys. 2, 559–561 (1966).

J. Atmos. Sci.

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

J. Phys. Chem.

J. H. Chin, C. M. Sliepcevih, M. Tribus, “Determination of particle size distributions in polydispersed systems,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Part. Charact.

M. Heuer, K. Leschonski, “Results obtained with a new instrument for the measurement of particle size distributions from diffraction patterns,” Part. Charact. 2, 7–13 (1985).
[CrossRef]

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

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikikout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” Part. Charact. 4, 151–156 (1987).
[CrossRef]

Prog. Astronaut. Aeronaut.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Other

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

C. Heffels, “On-line particle size and shape characterization by narrow angle light scattering,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1995).

M. L. Strickland, “Apparatus and method for determining the size distribution of particles by light scattering,” U.S. patent5,576,827 (19November1996).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1969).

S. Hayashi, “A laser small-angle scattering instrument for the determination of size and concentration distributions in sprays,” in Liquid Particle Size Measurement Techniques, American Society for Testing and Materials STP Series Volume 1083 (American Society for Testing and Materials, Philadelphia, Pa., 1990), Vol. 2, pp. 77–92.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).

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Figures (17)

Fig. 1
Fig. 1

Experimental setup for measuring particle-size distributions: 1, He–Ne laser; 2, beam expander; 3, sample cell; 4, Fourier lens; 5, CMOS detector; 6, interface; 7, computer.

Fig. 2
Fig. 2

Correlation between the pixel value (the readout-signal value) and the average pixel value for two pixels (A and B) at different light intensities. The squares represent pixel A, and the circles, pixel B.

Fig. 3
Fig. 3

Correlated calibration curve between the real intensity and the average pixel value for the CMOS sensor.

Fig. 4
Fig. 4

Simulated sensor signal and the accordingly scaled signals of the left-hand side of Eq. (6).

Fig. 5
Fig. 5

Effects of regularization and the scale parameter on the solution error in the Phillips–Twomey [nonnegativity least-squares (nnls)] method and the Chahine inversion (ch): The main particle-size distribution has a 1.5 g.s.d. The number of particle-size classes is 20, and the condition number is the ith number of log-space divisions for the smoothing factor (0.001–16) and the number of iterations (31–1000), for a total in this case of 20.

Fig. 6
Fig. 6

Effects of regularization and the scale parameter on the solution error in the Phillips–Twomey [nonnegativity least-squares (nnls)] method and the Chahine inversion (ch): The main particle-size distribution has a 1.5 g.s.d. The number of particle-size classes is 50. (See Fig. 5 for a definition of the condition number.)

Fig. 7
Fig. 7

Effects of regularization and the scale parameter on the solution error in the Phillips–Twomey [nonnegativity least-squares (nnls)] method and the Chahine inversion (ch): The main particle-size distribution has a 1.2 g.s.d. The number of particle-size classes is 20. (See Fig. 5 for a definition of the condition number.)

Fig. 8
Fig. 8

Effects of regularization and the scale parameter on the solution error in the Phillips–Twomey [nonnegativity least-squares (nnls)] method and the Chahine inversion (ch): The main particle-size distribution has a 1.2 g.s.d. The number of particle-size classes is 50. (See Fig. 5 for a definition of the condition number.)

Fig. 9
Fig. 9

Comparison of the measured (filled circles) and the theoretical (solid curve) scattered-light intensities for a 1-mm pinhole, as measured with a 300-mm lens.

Fig. 10
Fig. 10

Scattering image recorded by the detector for 103-µm polymer microspheres.

Fig. 11
Fig. 11

Recovered particle-size distribution for the 103-µm polymer microspheres. The results obtained with the CMOS sizer are represented by the solid curve with the filled circles, and those obtained with the Mastersizer are represented by the dashed curve with the open circles.

Fig. 12
Fig. 12

Recovered particle-size distribution for a mixed sample with 20- and 30-µm polymer microspheres. The open squares represent the Mastersizer, and the filled circles, the CMOS sizer.

Fig. 13
Fig. 13

Averaged sample-signal and background-signal profiles for potato starch particles.

Fig. 14
Fig. 14

Standard deviation of the signals at different angle positions for whole-sweep measurements for potato starch particles.

Fig. 15
Fig. 15

Pixel number for averaging one scattering image obtained from potato starch particles.

Fig. 16
Fig. 16

Comparison between the measured, scaled intensity profile (filled circles) and the calculated intensity profile (solid curve) by means of inversion.

Fig. 17
Fig. 17

Measured results obtained with the CMOS method (squares) compared with results obtained with a commercial instrument (Mastersizer; circles) for potato starch particles.

Tables (1)

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Table 1 Definitions for Symbols and Abbreviations

Equations (10)

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Iθ=I0k2d2 α4J1αθαθ2,
Iθ=CI00 α4J1αθαθ2qαdα.
I=Kq,
q=KTK+γH-1KTI,
qmi+1=Cm1Cmi-1Cmiqm0,  Cmi=j=1M Km,jrjj=1M Km,j,  rm=Im/Imi,  Imi=j=1M Km,jqji,
Iθiθia=C 0J12αθiαθi2-a qαα3dα,
δ=i=1NIi,real-Ii,noise21/2i=1N Ii,real21/2,
=i=1Mqi,real-qi,regu21/2i=1M qi,real21/2,
Obs=1-Isam,cIback,c,
I=Isam-1-ObsIback,

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