Abstract

We consider the scattering of light by single wood fibers both theoretically and experimentally. We describe the size and the shape distributions and the internal structure and chemical composition of the wood fibers. We have modeled the random shape of the hollow, cylindrical wood fiber by using multivariate lognormal statistics. We have computed wood-fiber absorption and scattering cross sections, asymmetry parameters, and scattering phase matrices in the ray-optics approximation. Finally, we have provided experimental results from angular scattering measurements for wood fibers and present what we believe is the first comparison between these measurements and ray-optics computations for Gaussian random wood-fiber models. In spite of the complicated internal structure of the wood fiber, our model together with the ray-optics treatment explains the scattering measurements surprisingly well.

© 2001 Optical Society of America

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References

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  1. J. S. Jack, R. G. Bentley, R. L. Barron, “Optical pulp consistency sensors,” Pulp Pap. Can. 91(2), 59–64 (1990).
  2. I. Karaila, “Modeling of fiber concentration in pulp suspensions,” Ph.D dissertation (Tampere University of Technology, Tampere, Finland, 1998), Publication 232.
  3. K. Saarinen, “Consistency measurement based adaptive control system for thermo mechanical refiners,” Licentiate Thesis (University of Jyväskylä, Jyväskylä, Finland, 1993).
  4. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  5. M. Leskelä, “A model for the optical properties of paper,” Pap. Puu (Pap. Timber) 75, 683–688 (1993).
  6. J. Carlsson, W. Persson, P. Hellentin, L. Malmqvist, “The propagation of light in paper: modeling and Monte Carlo simulations,” in Proceedings of the 1995 International Paper Physics Conference (CPPA), 11–14 September 1997, Niagara-on-the-Lake, Canada (TAPPI, Atlanta, 1995), pp. 83–86.
  7. Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
    [CrossRef]
  8. W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. 18, 495–522 (1905).
    [CrossRef]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  10. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  11. M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomlyoriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
    [CrossRef] [PubMed]
  12. J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid. Interface Sci. 29, 565–571 (1969).
    [CrossRef]
  13. M. A. G. Abushagur, N. George, “Polarization and wavelength effects on the scattering from dielectric cylinders,” Appl. Opt. 24, 4141–4145 (1985).
    [CrossRef] [PubMed]
  14. S. W. Bickel, W. Gilliar, B. Bell, “Light scattering from fibers: a closer look with a new twist,” Appl. Opt. 19, 3671–3675 (1980).
    [CrossRef] [PubMed]
  15. H. K. Bustard, R. W. Smith, “Investigation into the scattering of light by human hair,” Appl. Opt. 30, 3485–3491 (1991).
    [CrossRef] [PubMed]
  16. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  17. Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders; a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
    [CrossRef] [PubMed]
  18. K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
    [CrossRef]
  19. K. Muinonen, “Light scattering by stochastically shaped particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000).
    [CrossRef]
  20. K. Green, L. Lamberg, K. Lumme, “Stochastic modeling of paper structure and Monte-Carlo simulations of light scattering,” Appl. Opt. 39, 4669–4683 (2000).
    [CrossRef]
  21. K. Muinonen, K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
    [CrossRef] [PubMed]
  22. J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
    [CrossRef] [PubMed]
  23. T. Mukai, K. Mukai, K. Weiss, R. H. Zerull, “Scattering of radiation by a large particle with a random surface,” Moon Planets 26, 197–208 (1982).
    [CrossRef]
  24. R. Schiffer, K. O. Thielheim, “A scattering model for the zodiacal light particles,” Astron. Astrophys. 116, 1–9 (1982).
  25. M. Bottlinger, H. Umhauer, “Modeling of light scattering by irregularly shaped particles using a ray-tracing method,” Appl. Opt. 30, 4732–4738 (1991).
    [CrossRef] [PubMed]
  26. E. A. Hovenac, “Calculation of far-field scattering from nonspherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4747 (1991).
    [CrossRef] [PubMed]
  27. T. Koljonen, A. Heikkurinen, “Delamination of stiff fibers,” in Proceedings of the International Mechanical Pulping Conference, Ottawa (TAPPI, Atlanta, 1995).
  28. E. Sjöström, Wood Chemistry—Fundamentals and Applications (Academic, New York, 1993).
  29. J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge University, Cambridge, UK, 1963).
  30. K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
    [CrossRef]
  31. L. Lamberg, Department of Mathematics, University of Helsinki, FIN-000014 Helsinki, Finland (personal communication, 1999).
  32. K. Muinonen, J. Lagerros, “Inversion of shape statistic for a small solar system body,” Astron. Astrophys. 333, 753–761 (1998).
  33. C. Sasse, “Development of an experimental system for optical characterization of large arbitraly shaped particles,” Rev. Sci. Instrum. 64, 864–869 (1993).
    [CrossRef]
  34. C. Sasse, J. I. Peltoniemi, “Angular scattering measurements and calculations of rough spherically shaped carbon particles,” in Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, J. C. Stover, ed., Proc. SPIE2541, 131–139 (1995).
    [CrossRef]

2000 (2)

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

K. Green, L. Lamberg, K. Lumme, “Stochastic modeling of paper structure and Monte-Carlo simulations of light scattering,” Appl. Opt. 39, 4669–4683 (2000).
[CrossRef]

1998 (1)

K. Muinonen, J. Lagerros, “Inversion of shape statistic for a small solar system body,” Astron. Astrophys. 333, 753–761 (1998).

1996 (2)

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomlyoriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

1993 (2)

C. Sasse, “Development of an experimental system for optical characterization of large arbitraly shaped particles,” Rev. Sci. Instrum. 64, 864–869 (1993).
[CrossRef]

M. Leskelä, “A model for the optical properties of paper,” Pap. Puu (Pap. Timber) 75, 683–688 (1993).

1991 (3)

1990 (1)

J. S. Jack, R. G. Bentley, R. L. Barron, “Optical pulp consistency sensors,” Pulp Pap. Can. 91(2), 59–64 (1990).

1989 (2)

1985 (1)

1982 (2)

T. Mukai, K. Mukai, K. Weiss, R. H. Zerull, “Scattering of radiation by a large particle with a random surface,” Moon Planets 26, 197–208 (1982).
[CrossRef]

R. Schiffer, K. O. Thielheim, “A scattering model for the zodiacal light particles,” Astron. Astrophys. 116, 1–9 (1982).

1980 (2)

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1969 (1)

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid. Interface Sci. 29, 565–571 (1969).
[CrossRef]

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

1905 (1)

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. 18, 495–522 (1905).
[CrossRef]

1881 (1)

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

Abushagur, M. A. G.

Aitchison, J.

J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge University, Cambridge, UK, 1963).

Barron, R. L.

J. S. Jack, R. G. Bentley, R. L. Barron, “Optical pulp consistency sensors,” Pulp Pap. Can. 91(2), 59–64 (1990).

Bell, B.

Bentley, R. G.

J. S. Jack, R. G. Bentley, R. L. Barron, “Optical pulp consistency sensors,” Pulp Pap. Can. 91(2), 59–64 (1990).

Bickel, S. W.

Bottlinger, M.

Brown, J. A. C.

J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge University, Cambridge, UK, 1963).

Bustard, H. K.

Carlsson, J.

J. Carlsson, W. Persson, P. Hellentin, L. Malmqvist, “The propagation of light in paper: modeling and Monte Carlo simulations,” in Proceedings of the 1995 International Paper Physics Conference (CPPA), 11–14 September 1997, Niagara-on-the-Lake, Canada (TAPPI, Atlanta, 1995), pp. 83–86.

Fast, P.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

George, N.

Gilliar, W.

Green, K.

Heikkurinen, A.

T. Koljonen, A. Heikkurinen, “Delamination of stiff fibers,” in Proceedings of the International Mechanical Pulping Conference, Ottawa (TAPPI, Atlanta, 1995).

Hellentin, P.

J. Carlsson, W. Persson, P. Hellentin, L. Malmqvist, “The propagation of light in paper: modeling and Monte Carlo simulations,” in Proceedings of the 1995 International Paper Physics Conference (CPPA), 11–14 September 1997, Niagara-on-the-Lake, Canada (TAPPI, Atlanta, 1995), pp. 83–86.

Hovenac, E. A.

Irvine, W. M.

Jack, J. S.

J. S. Jack, R. G. Bentley, R. L. Barron, “Optical pulp consistency sensors,” Pulp Pap. Can. 91(2), 59–64 (1990).

Karaila, I.

I. Karaila, “Modeling of fiber concentration in pulp suspensions,” Ph.D dissertation (Tampere University of Technology, Tampere, Finland, 1998), Publication 232.

Kerker, M.

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

Koljonen, T.

T. Koljonen, A. Heikkurinen, “Delamination of stiff fibers,” in Proceedings of the International Mechanical Pulping Conference, Ottawa (TAPPI, Atlanta, 1995).

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Lagerros, J.

K. Muinonen, J. Lagerros, “Inversion of shape statistic for a small solar system body,” Astron. Astrophys. 333, 753–761 (1998).

Lamberg, L.

K. Green, L. Lamberg, K. Lumme, “Stochastic modeling of paper structure and Monte-Carlo simulations of light scattering,” Appl. Opt. 39, 4669–4683 (2000).
[CrossRef]

L. Lamberg, Department of Mathematics, University of Helsinki, FIN-000014 Helsinki, Finland (personal communication, 1999).

Leskelä, M.

M. Leskelä, “A model for the optical properties of paper,” Pap. Puu (Pap. Timber) 75, 683–688 (1993).

Lumme, K.

Lundberg, J. L.

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid. Interface Sci. 29, 565–571 (1969).
[CrossRef]

Macke, A.

Malmqvist, L.

J. Carlsson, W. Persson, P. Hellentin, L. Malmqvist, “The propagation of light in paper: modeling and Monte Carlo simulations,” in Proceedings of the 1995 International Paper Physics Conference (CPPA), 11–14 September 1997, Niagara-on-the-Lake, Canada (TAPPI, Atlanta, 1995), pp. 83–86.

Mishchenko, M. I.

Muinonen, K.

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

K. Muinonen, J. Lagerros, “Inversion of shape statistic for a small solar system body,” Astron. Astrophys. 333, 753–761 (1998).

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

K. Muinonen, K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

K. Muinonen, “Light scattering by stochastically shaped particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000).
[CrossRef]

Mukai, K.

T. Mukai, K. Mukai, K. Weiss, R. H. Zerull, “Scattering of radiation by a large particle with a random surface,” Moon Planets 26, 197–208 (1982).
[CrossRef]

Mukai, T.

T. Mukai, K. Mukai, K. Weiss, R. H. Zerull, “Scattering of radiation by a large particle with a random surface,” Moon Planets 26, 197–208 (1982).
[CrossRef]

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Nousiainen, T.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Peltoniemi, J. I.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

K. Muinonen, K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

C. Sasse, J. I. Peltoniemi, “Angular scattering measurements and calculations of rough spherically shaped carbon particles,” in Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, J. C. Stover, ed., Proc. SPIE2541, 131–139 (1995).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Persson, W.

J. Carlsson, W. Persson, P. Hellentin, L. Malmqvist, “The propagation of light in paper: modeling and Monte Carlo simulations,” in Proceedings of the 1995 International Paper Physics Conference (CPPA), 11–14 September 1997, Niagara-on-the-Lake, Canada (TAPPI, Atlanta, 1995), pp. 83–86.

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Rayleigh, Lord

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

Saarinen, K.

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

K. Saarinen, “Consistency measurement based adaptive control system for thermo mechanical refiners,” Licentiate Thesis (University of Jyväskylä, Jyväskylä, Finland, 1993).

Sasse, C.

C. Sasse, “Development of an experimental system for optical characterization of large arbitraly shaped particles,” Rev. Sci. Instrum. 64, 864–869 (1993).
[CrossRef]

C. Sasse, J. I. Peltoniemi, “Angular scattering measurements and calculations of rough spherically shaped carbon particles,” in Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, J. C. Stover, ed., Proc. SPIE2541, 131–139 (1995).
[CrossRef]

Schiffer, R.

R. Schiffer, K. O. Thielheim, “A scattering model for the zodiacal light particles,” Astron. Astrophys. 116, 1–9 (1982).

Sjöström, E.

E. Sjöström, Wood Chemistry—Fundamentals and Applications (Academic, New York, 1993).

Smith, R. W.

Takano, Y.

Tanaka, M.

Thielheim, K. O.

R. Schiffer, K. O. Thielheim, “A scattering model for the zodiacal light particles,” Astron. Astrophys. 116, 1–9 (1982).

Travis, L. D.

Umhauer, H.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

von Ignatowsky, W.

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. 18, 495–522 (1905).
[CrossRef]

Weiss, K.

T. Mukai, K. Mukai, K. Weiss, R. H. Zerull, “Scattering of radiation by a large particle with a random surface,” Moon Planets 26, 197–208 (1982).
[CrossRef]

Zerull, R. H.

T. Mukai, K. Mukai, K. Weiss, R. H. Zerull, “Scattering of radiation by a large particle with a random surface,” Moon Planets 26, 197–208 (1982).
[CrossRef]

Ann. Phys. (1)

W. von Ignatowsky, “Reflexion elektromagnetischer Wellen an einem Draht,” Ann. Phys. 18, 495–522 (1905).
[CrossRef]

Appl. Opt. (10)

Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders; a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
[CrossRef] [PubMed]

S. W. Bickel, W. Gilliar, B. Bell, “Light scattering from fibers: a closer look with a new twist,” Appl. Opt. 19, 3671–3675 (1980).
[CrossRef] [PubMed]

M. A. G. Abushagur, N. George, “Polarization and wavelength effects on the scattering from dielectric cylinders,” Appl. Opt. 24, 4141–4145 (1985).
[CrossRef] [PubMed]

K. Muinonen, K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

H. K. Bustard, R. W. Smith, “Investigation into the scattering of light by human hair,” Appl. Opt. 30, 3485–3491 (1991).
[CrossRef] [PubMed]

M. Bottlinger, H. Umhauer, “Modeling of light scattering by irregularly shaped particles using a ray-tracing method,” Appl. Opt. 30, 4732–4738 (1991).
[CrossRef] [PubMed]

E. A. Hovenac, “Calculation of far-field scattering from nonspherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4747 (1991).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomlyoriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

K. Green, L. Lamberg, K. Lumme, “Stochastic modeling of paper structure and Monte-Carlo simulations of light scattering,” Appl. Opt. 39, 4669–4683 (2000).
[CrossRef]

Astron. Astrophys. (2)

R. Schiffer, K. O. Thielheim, “A scattering model for the zodiacal light particles,” Astron. Astrophys. 116, 1–9 (1982).

K. Muinonen, J. Lagerros, “Inversion of shape statistic for a small solar system body,” Astron. Astrophys. 333, 753–761 (1998).

Astrophys. J. (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

J. Colloid. Interface Sci. (1)

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid. Interface Sci. 29, 565–571 (1969).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Moon Planets (1)

T. Mukai, K. Mukai, K. Weiss, R. H. Zerull, “Scattering of radiation by a large particle with a random surface,” Moon Planets 26, 197–208 (1982).
[CrossRef]

Pap. Puu (Pap. Timber) (1)

M. Leskelä, “A model for the optical properties of paper,” Pap. Puu (Pap. Timber) 75, 683–688 (1993).

Philos. Mag. (1)

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

Pulp Pap. Can. (1)

J. S. Jack, R. G. Bentley, R. L. Barron, “Optical pulp consistency sensors,” Pulp Pap. Can. 91(2), 59–64 (1990).

Rev. Sci. Instrum. (1)

C. Sasse, “Development of an experimental system for optical characterization of large arbitraly shaped particles,” Rev. Sci. Instrum. 64, 864–869 (1993).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (11)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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

I. Karaila, “Modeling of fiber concentration in pulp suspensions,” Ph.D dissertation (Tampere University of Technology, Tampere, Finland, 1998), Publication 232.

K. Saarinen, “Consistency measurement based adaptive control system for thermo mechanical refiners,” Licentiate Thesis (University of Jyväskylä, Jyväskylä, Finland, 1993).

J. Carlsson, W. Persson, P. Hellentin, L. Malmqvist, “The propagation of light in paper: modeling and Monte Carlo simulations,” in Proceedings of the 1995 International Paper Physics Conference (CPPA), 11–14 September 1997, Niagara-on-the-Lake, Canada (TAPPI, Atlanta, 1995), pp. 83–86.

C. Sasse, J. I. Peltoniemi, “Angular scattering measurements and calculations of rough spherically shaped carbon particles,” in Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, J. C. Stover, ed., Proc. SPIE2541, 131–139 (1995).
[CrossRef]

L. Lamberg, Department of Mathematics, University of Helsinki, FIN-000014 Helsinki, Finland (personal communication, 1999).

K. Muinonen, “Light scattering by stochastically shaped particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000).
[CrossRef]

T. Koljonen, A. Heikkurinen, “Delamination of stiff fibers,” in Proceedings of the International Mechanical Pulping Conference, Ottawa (TAPPI, Atlanta, 1995).

E. Sjöström, Wood Chemistry—Fundamentals and Applications (Academic, New York, 1993).

J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge University, Cambridge, UK, 1963).

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Figures (11)

Fig. 1
Fig. 1

Comparison between the probability density function p F ) [Eq. (4), solid curve] and its approximation [Eq. (5), dotted curve]. Shape parameters are as in the best-fit column in Table 2.

Fig. 2
Fig. 2

Cross sections of two layered Gaussian random cylinders. Both cylinders were generated with the same set of shape parameters, the values of which can be found in the best-fit column in Table 2.

Fig. 3
Fig. 3

Coordinate system of the particle (xyz′) and the incident ray (xyz). In xyz′ the direction of the incident radiation is defined by Ω i = (0, 0) T . The scattering angles are given in xyz by Ω = (θ, ϕ) T .

Fig. 4
Fig. 4

Degree of linear polarization of a layered circular cylinder with three different ratios of a L /ã F : 0.4, 0.6, and 0.8. The corresponding critical angles are marked by Φ0.4, Φ0.6, and Φ0.8.

Fig. 5
Fig. 5

Measurement setup.

Fig. 6
Fig. 6

Laser beam scattered by wood fiber. A paper sheet has been placed at the front of the detector ring. The silhouette of the sample holder and the toothpick can be seen.

Fig. 7
Fig. 7

Measurement points in the (ϕ′, Δ) coordinate system.

Fig. 8
Fig. 8

Measured scattering phase matrix elements of 10 wood fibers in the xy′ plane of the particle coordinate system: (a) normalized phase function; (b) degree of linear polarization.

Fig. 9
Fig. 9

Comparison of the projected-area-weighted mean-scattering matrix elements of the entire set of measurements and simulated matrix elements in the xy′ plane of the particles coordinate system: solid curve, measurements; dotted curve, best fit simulation results for (a) the normalized phase function [Eq. (21)], and (b) the degree of linear polarization; (a) dashed curve, simulation results for the normalized phase function [Eq. (21)] of the infinite hollow circular cylinder with the same mean radius of the lumen a L and the mean cell thickness a C as in the best-fit solution; (b) dashed curve, simulation results in which the mean cell thickness a C has been increased by 1.0 µm.

Fig. 10
Fig. 10

Comparison of the measured and simulated quantile ranges W p (ϕ′) in the (ϕ′, Δ) coordinate system [Eq. (34)]: solid curve, quantile range for the projected-area-weighted mean-scattering phase function of the entire set of measurements; dotted curve, quantile ranges for the best-fit simulation results; dash-dotted curve, quantile ranges for the simulation results where l z,L has been increased by 1; upper curves, quantile ranges W 0.07; lower curves, quantile ranges W 0.31.

Fig. 11
Fig. 11

Comparison of the measured and simulated phase functions as functions of elevation angle Δ at (a) ϕ′ = 30°, (b) ϕ′ = 60°, and (c) ϕ′ = 90°: solid curve, projected-area-weighted mean-scattering phase function of the entire set of measurements; dotted curve, best-fit simulation results.

Tables (2)

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Table 1 Diameters of Measured Wood Fibers and Their Standard Deviations

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Table 2 Shape Parameters

Equations (37)

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ρFφ, z=ρLφ, z+ρCφ, z,
l1ρ, a, σ=12π βρexp-12s2β2,
s=12 β2+lnρa,  β=ln1+σ21/2,
pρF=-12 βL2+lnρFaL dsn1s, 0, βLl1ρF-aL ×exps-12 βL2, aC, σC,
l1ρF, aL+aC, σL2aL2+σC2aC21/2aL+aC
Csγ, ζ=m=0k=0 cmk cosmφcoskKz, K=πLz,m=0k=0 cmk=1,
Σsγ, ζ=β2Csγ, ζ,
Σργ, ζ=a2expΣsγ, ζ-1, Cργ, ζ=expΣsγ, ζ-1expβ2-1=1σ2a2 Σργ, ζ, σ2=expβ2-1.
ρφ, z=a1+σ21/2expsφ, z, sφ, z=m=0k=0 smk expimφ+kKz,
varResmk=181+δm0+δk0+5δm0δk0cmkβ2, varImsmk=181+δm0+δk0-3δm0δk0cmkβ2.
s-m,-k*=smk,  m, k=0, 1,, ,
Csγ, ζ=exp-2lφ2sin212 γ-12lz2 ζ2,
cmk=2-δm0exp-1lφ2Im1lφ2×2-δk0lzLzexp-12 k2π2lz2Lz2, m, k=0, 1,, .
Γφ=2 arcsin12 lφ.
ãF=aL+aC, σ˜F=σL2aL2+σC2aC21/2aL+ac, Σ˜ρ,Fγ, ζ=Σρ,Lγ, ζ+Σρ,Cγ, ζ, Σ˜s,Fγ, ζ=ln1ãF2 Σ˜ρ,Fγ, ζ+1, l˜φ,F2=-Σ˜s,F0, 0Σ˜s,F,γγ0, 0, l˜z,F2=-Σ˜s,F0, 0Σ˜s,F,ζζ0, 0.
χφ,L2=-Σs,L,γγ0, 0=βL2lφ,L2, χz,L2=-Σs,L,ξξ0, 0=βL2lz,L2.
χ˜φ,F2=-Σ˜s,F,γγ0, 0=aL2χφ,L2 expβL2+aC2χφ,C2 expβC2ãF2 expβ˜F2, χ˜z,F2=-Σ˜s,F,ζζ0, 0=aL2χz,L2 expβL2+aC2χz,C2 expβC2ãF2 expβ˜F2,
χφ,C2=-Σs,C,γγ0, 0=βC2lφ,C2, χz,C2=-Σs,C,ξξ0, 0=βC2lz,C2,
V˜z=πãF2 expβ˜F2-πaL2 expβL2,
Ãz=2πãF1+12 χ˜φ,F2+χ˜z,F2ãF2 exp3β˜F2×-183χ˜φ,F4+3χ˜z,F4ãF4 exp10β˜F2+2χ˜φ,F2χ˜z,F2ãF2 exp3β˜F2+0χ˜6.
ĪsΩ, Ωi=σsΩi4πr2P¯Ω, Ωi·ĪiΩi, 4πdΩ4π P11Ω, Ωi=1,
σsΩi=σsDΩi+σsGΩi, P¯Ω, Ωi=1σsΩiσsDΩiP¯DΩ, Ωi+σsGΩiP¯GΩ, Ωi,
σeΩi=σaΩi+σsΩi=2AΩi, σsDΩi=AΩi,
σaΩi=σaDΩi+σaGΩi=σaGΩi, σaDΩi=0.
gΩi=4πdΩ4πcos θP11Ω, Ωi, ϖGΩi=σsGΩiAΩi,
ϖ=σsΩiσeΩi=121+ϖG.
P¯DΩ, Ωi  k24π2ZAΩi uΩ, Ωi2×1+cos θ21¯, uΩ, Ωi=-ZZdx y1x,Ωiy2x,Ωidy exp-ikx sin θ cos ϕ+y sin θ sin ϕ,
uΩ, Ωi=2 sinkc sin θ sin ϕk sin θ sin ϕ2 sinkZ sin θ cos ϕk sin θ cos φ.
kRF|Mo|  1, kRL|MC|  1, 2kRF|MC-Mo|  1, 2kRL|ML-MC|  1,
χφ,L2=v1lφ,L2,  χ˜φ,F2=v1v2lφ,L2+v3lφ,C2, χz,L2=v1lz,L2,  χ˜z,F2=v1v2lz,L2+v3lz,C2,
v1=ln1+ãF2aL2 σ˜F2-ãFaL-12 σC2, v2=aL2ãF21+ãF2aL2 σ˜F2-ãFaL-12 σC21+σ˜F2, v3=1-aLãF21+σC2 ln1+σC21+σ˜F2,
Fqp=-qpdxfx=p;  qp=F-1p.
qpϕ=F-1p,  Fqpϕ=-90°qpϕdΔS11Δ, ϕ-90°90°dΔS11Δ, ϕ,
Wpϕ=q1-pϕ-qpϕ.
ξ  -60°, -40°, -20°, -10°, -5°, 0°, 5°, 10°, 20°, 40°, 60°.
d2σd2=VarρFφ=φ0+VarρFφ=φ0+180°ãF2σ˜F2+ãF2σ˜F2  σ˜F=2 σd,
aLσL=ãFσ˜F2-aCσC21/2=2.5 μm.

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