Abstract

We present fundamental studies examining the design of a phase/Doppler laser light-scattering system applicable to on-line measurements of small-diameter (<15 μm) fibers during fiberglass manufacturing. We first discuss off-line diameter measurement techniques currently used in the fiberglass industry and outline the limitations and problems associated with these methods. For the phase/Doppler design study we have developed a theoretical computer model for the response of the measurement system to cylindrical fibers, which is based on electromagnetic scattering theory. The model, valid for arbitrary fiber diameters and hardware configurations, generates simulated detector output as a function of time for a finite absorbing, cylindrical fiber oriented perpendicular to the two incident laser beams. Results of experimental measurements are presented, confirming predictions of the theoretical model. Parametric studies have also been conducted using the computer model to identify experimental arrangements that provide linear phase–diameter relationships for small-diameter fibers, within the measurement constraints imposed by the fiberglass production environment. The effect of variations in optical properties of the glass as well as fiber orientation effects are discussed. Through this research we have identified phase/Doppler arrangements that we expect to have future applications in the fiberglass industry for on-line diameter monitoring and process control.

© 1998 Optical Society of America

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  1. S. A. Schaub, A. A. Naqwi, “Light scattering based sensor for on-line monitoring of fiber diameter distribution during fiberglass manufacturing,” SBIR Final Report, Report #DOE/ER/82229–1, 6March1997 (U.S. Department of Energy, Washington, D.C.).
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    [CrossRef] [PubMed]
  3. A. Naqwi, F. Durst, “Analysis of laser light-scattering interferometric devices for in-line diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
    [PubMed]
  4. M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, 1986).
  5. R. W. Sellens, “Alignment errors in phase Doppler receiver optics,” Part. Part. Syst. Char. 7, 116–120 (1990).
    [CrossRef]
  6. A. Naqwi, F. Durst, “Contributions to the optical design of the phase/Doppler system,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 521–530.
  7. S. V. Sankar, B. J. Weber, D. Y. Damemoto, W. D. Bachalo, “Sizing fine particles with the phase Doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
    [CrossRef] [PubMed]
  8. A Naqwi, L Jenson, “Interferometric cylinder sizing and velocimetry device,” U.S. Patent5,432,605 (11July1995).
  9. A Naqwi, L Jenson, “Interferometric device for determining sizes and properties of cylinder objects based on phase shift measurements,” U.S. Patent5,453,837 (26September1995).
  10. A. Naqwi, M. Ziema, X. Liu, S. Hohmann, F. Durst, “Droplet and particle sizing using the dual cylindrical wave and the planar phase Doppler optical systems combined with a transputer based signal processor,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1992), pp. 15.3.1–15.3.6.
  11. A Naqwi, L. M. Jenson, “Device for interferometric measurements with compensation for tilt and position of measured cylindrical objects,” U.S. Patent5,513,004 (30April1996).
  12. A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.
  13. H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 35, 5180–5190 (1996).
    [CrossRef] [PubMed]
  14. S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and particle resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
    [CrossRef] [PubMed]
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  19. D. R. Alexander, J. P. Barton, S. A. Schaub, M. Emanuel, “Experimental and Theoretical Analysis of the Interaction of CO2 Laser Radiation with Fluid Cylinders and Adjacent Spheres,” in Proceedings of the 1987 CRDEC Conference on Obscuration and Aerosol Research (U.S. Army Chemical Research, Development and Engineering Center, Aberdeen Proving Grounds, Md., 1987).
  20. H. Wilhelmsson, “On the reflection of electromagnetic waves from a dielectric cylinder,” Trans. Chalmers University of Technol. 35, 3–16 (1955).
  21. E. Zimmermann, R. Dändliker, N. Souli, B. Dratinger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
    [CrossRef]
  22. G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams,” Part. Part. Syst. Char. 11, 299–308 (1994).
    [CrossRef]
  23. G. Gouesbet, G. Gréhan, “On the interaction between a Gaussian beam and an infinite cylinder, using non sigma-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
    [CrossRef]
  24. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” Part. Part. Syst. Char. 26, 225–239 (1995).
  25. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
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    [CrossRef]
  29. R. T. Wang, H. C. van de Hulst, “Application of the exact solution for scattering by an infinite cylinder to the estimation of scattering by a finite cylinder,” Appl. Opt. 34, 2811–2821 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  32. R. W. B. Ardill, K. J. M. Moriarty, “Accurate Bessel functions Jn(z), Yn(z), Hn(1)(z) and Hn(2)(z) of integer order and complex argument,” Computer Phys. Comm. 17, 321–336 (1979).
    [CrossRef]
  33. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd Ed. (Cambridge U. Press, New York, 1992).
  34. J. M. Corpus, P. K. Gupta, “Diameter dependence of the refractive index of melt-drawn glass fibers,” J. Am. Ceram. Soc. 76, 1390–1392 (1993).
    [CrossRef]
  35. W. A. Weyl, E Chostner-Marboe, The Constitution of Glasses (Interscience, New York, 1962), pp. 1351, 1517.
  36. A. Naqwi, R. Menon, L. M. Fingerson, “An adaptive phase/doppler system and its applications including particle sizing in submicron and nanometer ranges,” Exp. Fluids 20, 328–334 (1996).
    [CrossRef]
  37. J Evenstad, A Naqwi, R Menon, “A device for phase shift measurement in an advanced phase Doppler velocimeter,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1996), paper 2.1.

1997 (3)

1996 (2)

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 35, 5180–5190 (1996).
[CrossRef] [PubMed]

A. Naqwi, R. Menon, L. M. Fingerson, “An adaptive phase/doppler system and its applications including particle sizing in submicron and nanometer ranges,” Exp. Fluids 20, 328–334 (1996).
[CrossRef]

1995 (3)

1994 (3)

1993 (2)

A. Naqwi, F. Durst, “Analysis of laser light-scattering interferometric devices for in-line diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
[PubMed]

J. M. Corpus, P. K. Gupta, “Diameter dependence of the refractive index of melt-drawn glass fibers,” J. Am. Ceram. Soc. 76, 1390–1392 (1993).
[CrossRef]

1991 (2)

1990 (2)

R. W. Sellens, “Alignment errors in phase Doppler receiver optics,” Part. Part. Syst. Char. 7, 116–120 (1990).
[CrossRef]

C. F. Du Toit, “The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments,” IEEE Trans. Antennas Propagation 38, 1341–1349 (1990).
[CrossRef]

1979 (2)

R. W. B. Ardill, K. J. M. Moriarty, “Accurate Bessel functions Jn(z), Yn(z), Hn(1)(z) and Hn(2)(z) of integer order and complex argument,” Computer Phys. Comm. 17, 321–336 (1979).
[CrossRef]

W. D. Ross, “Computation of Bessel functions in light scattering,” Appl. Opt. 11, 1919–1923 (1979).
[CrossRef]

1955 (1)

H. Wilhelmsson, “On the reflection of electromagnetic waves from a dielectric cylinder,” Trans. Chalmers University of Technol. 35, 3–16 (1955).

1881 (1)

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Alexander, D. R.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and particle resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
[CrossRef] [PubMed]

D. R. Alexander, J. P. Barton, S. A. Schaub, M. Emanuel, “Experimental and Theoretical Analysis of the Interaction of CO2 Laser Radiation with Fluid Cylinders and Adjacent Spheres,” in Proceedings of the 1987 CRDEC Conference on Obscuration and Aerosol Research (U.S. Army Chemical Research, Development and Engineering Center, Aberdeen Proving Grounds, Md., 1987).

Ardill, R. W. B.

R. W. B. Ardill, K. J. M. Moriarty, “Accurate Bessel functions Jn(z), Yn(z), Hn(1)(z) and Hn(2)(z) of integer order and complex argument,” Computer Phys. Comm. 17, 321–336 (1979).
[CrossRef]

Bachalo, W. D.

Barton, J. P.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and particle resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
[CrossRef] [PubMed]

D. R. Alexander, J. P. Barton, S. A. Schaub, M. Emanuel, “Experimental and Theoretical Analysis of the Interaction of CO2 Laser Radiation with Fluid Cylinders and Adjacent Spheres,” in Proceedings of the 1987 CRDEC Conference on Obscuration and Aerosol Research (U.S. Army Chemical Research, Development and Engineering Center, Aberdeen Proving Grounds, Md., 1987).

Bohren, C. F.

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

Buchhave, P.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, 1986).

Chostner-Marboe, E

W. A. Weyl, E Chostner-Marboe, The Constitution of Glasses (Interscience, New York, 1962), pp. 1351, 1517.

Corpus, J. M.

J. M. Corpus, P. K. Gupta, “Diameter dependence of the refractive index of melt-drawn glass fibers,” J. Am. Ceram. Soc. 76, 1390–1392 (1993).
[CrossRef]

Damemoto, D. Y.

Dändliker, R.

Dratinger, B.

Du Toit, C. F.

C. F. Du Toit, “The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments,” IEEE Trans. Antennas Propagation 38, 1341–1349 (1990).
[CrossRef]

Durst, F.

A. Naqwi, F. Durst, “Analysis of laser light-scattering interferometric devices for in-line diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
[PubMed]

A. Naqwi, F. Durst, “Contributions to the optical design of the phase/Doppler system,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 521–530.

A. Naqwi, M. Ziema, X. Liu, S. Hohmann, F. Durst, “Droplet and particle sizing using the dual cylindrical wave and the planar phase Doppler optical systems combined with a transputer based signal processor,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1992), pp. 15.3.1–15.3.6.

Emanuel, M.

D. R. Alexander, J. P. Barton, S. A. Schaub, M. Emanuel, “Experimental and Theoretical Analysis of the Interaction of CO2 Laser Radiation with Fluid Cylinders and Adjacent Spheres,” in Proceedings of the 1987 CRDEC Conference on Obscuration and Aerosol Research (U.S. Army Chemical Research, Development and Engineering Center, Aberdeen Proving Grounds, Md., 1987).

Evenstad, J

J Evenstad, A Naqwi, R Menon, “A device for phase shift measurement in an advanced phase Doppler velocimeter,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1996), paper 2.1.

Fingerson, L. M.

A. Naqwi, R. Menon, L. M. Fingerson, “An adaptive phase/doppler system and its applications including particle sizing in submicron and nanometer ranges,” Exp. Fluids 20, 328–334 (1996).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd Ed. (Cambridge U. Press, New York, 1992).

Gouesbet, G.

Gréhan, G.

Gupta, P. K.

J. M. Corpus, P. K. Gupta, “Diameter dependence of the refractive index of melt-drawn glass fibers,” J. Am. Ceram. Soc. 76, 1390–1392 (1993).
[CrossRef]

Hassenboehler, C.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Havir, D.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Hohmann, S.

A. Naqwi, M. Ziema, X. Liu, S. Hohmann, F. Durst, “Droplet and particle sizing using the dual cylindrical wave and the planar phase Doppler optical systems combined with a transputer based signal processor,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1992), pp. 15.3.1–15.3.6.

Huffman, D. R.

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

Jenson, L

A Naqwi, L Jenson, “Interferometric cylinder sizing and velocimetry device,” U.S. Patent5,432,605 (11July1995).

A Naqwi, L Jenson, “Interferometric device for determining sizes and properties of cylinder objects based on phase shift measurements,” U.S. Patent5,453,837 (26September1995).

Jenson, L. M.

A Naqwi, L. M. Jenson, “Device for interferometric measurements with compensation for tilt and position of measured cylindrical objects,” U.S. Patent5,513,004 (30April1996).

Kerker, M.

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

Liu, X.

A. Naqwi, M. Ziema, X. Liu, S. Hohmann, F. Durst, “Droplet and particle sizing using the dual cylindrical wave and the planar phase Doppler optical systems combined with a transputer based signal processor,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1992), pp. 15.3.1–15.3.6.

Lock, J. A.

Mahon, T.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Menon, R

J Evenstad, A Naqwi, R Menon, “A device for phase shift measurement in an advanced phase Doppler velocimeter,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1996), paper 2.1.

Menon, R.

A. Naqwi, R. Menon, L. M. Fingerson, “An adaptive phase/doppler system and its applications including particle sizing in submicron and nanometer ranges,” Exp. Fluids 20, 328–334 (1996).
[CrossRef]

Mignon, H.

Moriarty, K. J. M.

R. W. B. Ardill, K. J. M. Moriarty, “Accurate Bessel functions Jn(z), Yn(z), Hn(1)(z) and Hn(2)(z) of integer order and complex argument,” Computer Phys. Comm. 17, 321–336 (1979).
[CrossRef]

Naqwi, A

A Naqwi, L. M. Jenson, “Device for interferometric measurements with compensation for tilt and position of measured cylindrical objects,” U.S. Patent5,513,004 (30April1996).

A Naqwi, L Jenson, “Interferometric device for determining sizes and properties of cylinder objects based on phase shift measurements,” U.S. Patent5,453,837 (26September1995).

A Naqwi, L Jenson, “Interferometric cylinder sizing and velocimetry device,” U.S. Patent5,432,605 (11July1995).

J Evenstad, A Naqwi, R Menon, “A device for phase shift measurement in an advanced phase Doppler velocimeter,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1996), paper 2.1.

Naqwi, A.

A. Naqwi, R. Menon, L. M. Fingerson, “An adaptive phase/doppler system and its applications including particle sizing in submicron and nanometer ranges,” Exp. Fluids 20, 328–334 (1996).
[CrossRef]

A. Naqwi, F. Durst, “Analysis of laser light-scattering interferometric devices for in-line diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
[PubMed]

A. Naqwi, F. Durst, “Contributions to the optical design of the phase/Doppler system,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 521–530.

A. Naqwi, M. Ziema, X. Liu, S. Hohmann, F. Durst, “Droplet and particle sizing using the dual cylindrical wave and the planar phase Doppler optical systems combined with a transputer based signal processor,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1992), pp. 15.3.1–15.3.6.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Naqwi, A. A.

S. A. Schaub, A. A. Naqwi, “Light scattering based sensor for on-line monitoring of fiber diameter distribution during fiberglass manufacturing,” SBIR Final Report, Report #DOE/ER/82229–1, 6March1997 (U.S. Department of Energy, Washington, D.C.).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd Ed. (Cambridge U. Press, New York, 1992).

Rayleigh,

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Ross, W. D.

Saffman, M.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, 1986).

Sankar, S. V.

Schaub, S. A.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and particle resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
[CrossRef] [PubMed]

D. R. Alexander, J. P. Barton, S. A. Schaub, M. Emanuel, “Experimental and Theoretical Analysis of the Interaction of CO2 Laser Radiation with Fluid Cylinders and Adjacent Spheres,” in Proceedings of the 1987 CRDEC Conference on Obscuration and Aerosol Research (U.S. Army Chemical Research, Development and Engineering Center, Aberdeen Proving Grounds, Md., 1987).

S. A. Schaub, A. A. Naqwi, “Light scattering based sensor for on-line monitoring of fiber diameter distribution during fiberglass manufacturing,” SBIR Final Report, Report #DOE/ER/82229–1, 6March1997 (U.S. Department of Energy, Washington, D.C.).

Sellens, R. W.

R. W. Sellens, “Alignment errors in phase Doppler receiver optics,” Part. Part. Syst. Char. 7, 116–120 (1990).
[CrossRef]

Souli, N.

Tanger, H.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. J. Adrian, D. F. Durao, F. Durst, H. Mishina, J. Whitelaw, eds. (LADOAN, Lisbon, 1986).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd Ed. (Cambridge U. Press, New York, 1992).

Tropea, C.

Tsai, P.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

van de Hulst, H. C.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, 2nd Ed. (Cambridge U. Press, New York, 1992).

Wadsworth, L.

A. Naqwi, T. Mahon, D. Havir, P. Tsai, C. Hassenboehler, L. Wadsworth, “On-line sizing of meltblown and spunbond fibers using adaptive phase/Doppler velocimeter (APV) method,” in Book of Papers, INDA-TEC 95 (Association of Nonwoven Fabrics Industry, Cary, N.C., 1995), pp. 167–184.

Wang, R. T.

Weber, B. J.

Weyl, W. A.

W. A. Weyl, E Chostner-Marboe, The Constitution of Glasses (Interscience, New York, 1962), pp. 1351, 1517.

Wilhelmsson, H.

H. Wilhelmsson, “On the reflection of electromagnetic waves from a dielectric cylinder,” Trans. Chalmers University of Technol. 35, 3–16 (1955).

Xu, T. H.

Ziema, M.

A. Naqwi, M. Ziema, X. Liu, S. Hohmann, F. Durst, “Droplet and particle sizing using the dual cylindrical wave and the planar phase Doppler optical systems combined with a transputer based signal processor,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1992), pp. 15.3.1–15.3.6.

Zimmermann, E.

Appl. Opt. (8)

Computer Phys. Comm. (1)

R. W. B. Ardill, K. J. M. Moriarty, “Accurate Bessel functions Jn(z), Yn(z), Hn(1)(z) and Hn(2)(z) of integer order and complex argument,” Computer Phys. Comm. 17, 321–336 (1979).
[CrossRef]

Exp. Fluids (1)

A. Naqwi, R. Menon, L. M. Fingerson, “An adaptive phase/doppler system and its applications including particle sizing in submicron and nanometer ranges,” Exp. Fluids 20, 328–334 (1996).
[CrossRef]

IEEE Trans. Antennas Propagation (1)

C. F. Du Toit, “The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments,” IEEE Trans. Antennas Propagation 38, 1341–1349 (1990).
[CrossRef]

J. Am. Ceram. Soc. (1)

J. M. Corpus, P. K. Gupta, “Diameter dependence of the refractive index of melt-drawn glass fibers,” J. Am. Ceram. Soc. 76, 1390–1392 (1993).
[CrossRef]

J. Opt. Soc. Am. A (4)

Part. Part. Syst. Char. (3)

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams,” Part. Part. Syst. Char. 11, 299–308 (1994).
[CrossRef]

R. W. Sellens, “Alignment errors in phase Doppler receiver optics,” Part. Part. Syst. Char. 7, 116–120 (1990).
[CrossRef]

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” Part. Part. Syst. Char. 26, 225–239 (1995).

Philos. Mag. (1)

Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Trans. Chalmers University of Technol. (1)

H. Wilhelmsson, “On the reflection of electromagnetic waves from a dielectric cylinder,” Trans. Chalmers University of Technol. 35, 3–16 (1955).

Other (16)

S. A. Schaub, A. A. Naqwi, “Light scattering based sensor for on-line monitoring of fiber diameter distribution during fiberglass manufacturing,” SBIR Final Report, Report #DOE/ER/82229–1, 6March1997 (U.S. Department of Energy, Washington, D.C.).

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

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

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Figures (15)

Fig. 1
Fig. 1

Schematic of the phase/Doppler arrangement used for theoretical modeling.

Fig. 2
Fig. 2

Phase/diameter response curves for the arrangements examined by Mignon et al.13 λ = 488.0 nm, θ b = 2.72 deg, FLTran = 310 mm; real refractive index, 1.50. Top, θ1 = 25.178 deg, θ2 = 34.822 deg, receiver focal length x d = 160 mm. Bottom, θ1 = 28.067 deg, θ2 = 31.933 deg, receiver focal length x d = 400 mm.

Fig. 3
Fig. 3

Schematic showing the desired regions for component location. Δθ i is the instrument angle.

Fig. 4
Fig. 4

Correlation coefficient as a function of instrument angle for initial parametric study. Phase factor restricted to 16 deg/μm < F < 22 deg/μm.

Fig. 5
Fig. 5

Phase response for λ = 0.4579 μm, FLTran = x d = 200 mm, aperture diameter y d,max = 50 mm, beam separation is 8.2 mm, θ1 = 161 deg, θ2 = 192 deg, n̅ = 1.52 + 10-6 i, z polarization.

Fig. 6
Fig. 6

Phase response for λ = 0.4579 μm, FLTran = x d = 200 mm, aperture diameter y d,max = 50 mm, beam separation is 5.5 mm, θ1 = 51 deg, θ2 = 276 deg, n̅ = 1.52 + 10-6 i, y polarization.

Fig. 7
Fig. 7

Schematic showing the various order geometric rays.

Fig. 8
Fig. 8

Intensity as a function of scattering angle for parallel and perpendicular polarization and various order rays. Real index, 1.52.

Fig. 9
Fig. 9

Phase shift as a function of diameter for real index 1.52 and variable imaginary part k, for λ = 0.4579 μm, FLTran = x d = 200 mm, aperture diameter 50 mm, beam separation 8.2 mm, θ1 = 161 deg, θ2 = 192 deg, z polarization.

Fig. 10
Fig. 10

Phase shift as a function of diameter for real index 1.52 and variable imaginary part k, for λ = 0.4579 μm, FLTran = x d = 200 mm, aperture diameter 50 mm, beam separation 5.5 mm, θ1 = 51 deg, θ2 = 276 deg, y polarization.

Fig. 11
Fig. 11

Relative error in measured mean and standard deviation for simulated log-normal fiber diameter distributions with nominal mean 4.00 μm and standard deviation 2.26 μm. n̅ = 1.52 + 10-6 i; (a), (b) 31-deg backscatter angle of Fig. 5; (c), (d) 225-deg sidescatter angle of Fig. 6.

Fig. 12
Fig. 12

Phase response for λ = 0.5145 μm, FLTran = x d = 310 mm, aperture diameter 10 mm, beam separation 5 mm, θ1 = 144 deg, θ2 = 156 deg, z polarization, n̅ = 1.52 + 10-6 i.

Fig. 13
Fig. 13

Photograph showing the hardware components used in the laboratory experimental measurements.

Fig. 14
Fig. 14

Phase as a function of diameter for both theory and experiment: λ = 514.5 nm; beam separation, 17 mm; θ1 = 159 deg; θ2 = 195 deg; aperture, 35 mm; x d = 310 mm; FLTran, 300 mm; n̅ = 1.52 + 10-6 i, z polarization.

Fig. 15
Fig. 15

Phase as a function of diameter for both theory and experiment. λ = 514.5 nm; beam separation, 17 mm; θ1 = 69 deg; θ2 = 281 deg; aperture, 35 mm; x d = 310 mm; FLTran, 500 mm; n̅ = 1.52 + 10-6 i, y polarization.

Tables (1)

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Table 1 Experimental Parameters Used in Systematic Theoretical Study

Equations (25)

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δ = λ / 2   sin θ b ,
x 0 = x d cos   θ d - y d sin   θ d ,
y 0 = y d cos   θ d + x d sin   θ d .
x 1 = x 0 cos   θ b - y 0 sin   θ b ,
y 1 = y 0 cos   θ b + x 0 sin   θ b ,
x 2 = x 0 cos   θ b + y 0 sin   θ b ,
y 2 = y 0 cos   θ b - x 0 sin   θ b .
E 0 = E d = E 1 + exp - i ϕ E 2 / 2 ,
H 0 = H d = H 1 + exp - i ϕ H 2 / 2 .
S r Y - Pol = Re + E θ H z * ,
S r Z - Pol = Re - E z H θ * .
P n ϕ ,   x d =   S r ϕ ,   x d ,   y d d y d .
P n = A n + B n sin ϕ - n ,
V n = P max - P min P max + P min = A n + B n - A n - B n A n + B n + A n - B n = B n A n .
τ 12 = | 2 - 1 | .
tan   n = sin   ϕ 1 - sin   ϕ 3 - P ˜ n sin   ϕ 2 - sin   ϕ 3 P ˜ n cos   ϕ 3 - cos   ϕ 2 - cos   ϕ 3 - cos   ϕ 1 ,
P ˜ n = P n ϕ 1 - P n ϕ 3 P n ϕ 2 - P n ϕ 3 .
B n = P n ϕ 2 - P n ϕ 3 sin ϕ 2 - n - sin ϕ 3 - n ,
A n = P n ϕ 3 - B n sin ϕ 3 - n .
τ 12 = FD ,
F = 1 N i = 1 N τ 12 , i D i .
R 2 = i = 1 N D i - D avg τ 12 , i - τ avg i = 1 N D i - D avg 2 1 / 2 i = 1 N τ 12 , i - τ avg 2 1 / 2 2
P D = 1 σ 2 π exp - log D - μ 2 2 σ 2 ,
τ 12 = aD 2 + bD + c ,
D meas = b 2 - 4 a c - τ 12 1 / 2 - b 2 a .

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