Abstract

A method based on Mueller matrix polarimetry is developed and demonstrated for determining the fibril angle and relative phase retardation of single, intact pulp fibers. The method permits quantitative and nondestructive determination of these parameters from measurements at one wavelength without any fiber alignment. The Mueller matrix of a pulp fiber and its relationship with the fibril angle and phase retardation are described. A nonmodulation method for determining the Mueller matrix is then proposed that is based on a set of intensity data registered by a single detector. Measurements were carried out with single pulp fibers as samples to test the theoretical prediction. The test measurements and results are described and presented.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. D. Preston, The Physical Biology of Plant Cell Walls (Chapman & Hall, London, 1974).
  2. D. H. Page, “A method for determining the fibrillar angle in wood tracheids,” J. Microsc. 90, 137–143 (1969).
    [CrossRef]
  3. R. E. Prud’homme, J. Noah, “Determination of fibril angle distribution in wood fibers: a comparison between the x-ray diffraction and the polarized microscope methods,” Wood Fiber 6, 282–289 (1975).
  4. C. M. Crosby, C. D. Zeeuw, R. Marton, “Fibrillar angle variation in red pine determined by Senarmont compensation,” Wood Sci. Technol. 6, 185–195 (1972).
    [CrossRef]
  5. R. C. Tang, “The microfibrillar orientation in cell-wall layers of Virginia pine tracheids,” Wood Sci. 5, 181–186 (1973).
  6. R. E. Mark, P. P. Gillis, “The relationship between fiber modulus and S2 angle,” Tappi 56, 164–167 (1973).
  7. F. El-Hosseiny, D. H. Page, “The mechanical properties of single wood pulp fibres: theories of strength,” Fibre Sci. Technol. 8, 21–30 (1975).
    [CrossRef]
  8. D. H. Page, F. El-Hosseiny, K. Winkler, A. P. S. Lancaster, “Elastic modulus of single wood pulp fibers,” Tappi 60, 114–117 (1977).
  9. D. H. Page, F. El-Hosseiny, “The mechanical properties of single wood pulp fibres. Part VI. Fibril angle and the shape of the stress–strain curve,” J. Pulp Pap. Sci. 9, 99–100 (1983).
  10. C. Ye, M. O. Sundström, K. Remes, “Microscopic transmission ellipsometry: measurement of the fibril angle and the relative retardation of single, intact wood pulp fibers,” Appl. Opt. 33, 6626–6637 (1994).
    [CrossRef] [PubMed]
  11. C. Ye, O. Sundström, “Determination of S2 -fibril-angle and fiber-wall thickness by microscopic transmission ellipsometry,” Tappi 80, 181–190 (1997).
  12. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  13. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarized light,” Opt. Lett. 10, 309–311 (1985).
    [CrossRef] [PubMed]
  14. R. M. A. Azzam, “Mueller matrix measurement using the four-detector photopolarimeter,” Opt. Lett. 11, 270–272 (1986).
    [CrossRef]
  15. R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A. 5, 681–689 (1988).
    [CrossRef]
  16. S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A. 9, 1615–1622 (1992).
    [CrossRef]
  17. D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28, 120–125 (1989).
    [CrossRef]
  18. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  19. D. A. Ramsey, K. C. Ludema, “The influence of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
    [CrossRef]
  20. W. M. McClain, W.-H. Jeng, B. Pati, Y. Shi, D. Tian, “Measurement of the Mueller scattering matrix by use of optical beats from a Zeeman laser,” Appl. Opt. 33, 1230–1241 (1994).
    [CrossRef] [PubMed]
  21. P. Y. Gerligand, B. Le. Jeune, J. Cariou, J. Lotrian, “Analysis of the spatial distribution of magneto-optic properties of γ-Fe2O3 ferrofluid using different polarimetric criteria,” J. Phys. D 28, 965–977 (1995).
    [CrossRef]
  22. D. H. Goldstein, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
    [CrossRef]
  23. J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-tuned mirror,” Opt. Eng. 34, 1593–1598 (1995).
    [CrossRef]
  24. R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
    [CrossRef] [PubMed]
  25. B. W. Bell, “Mueller matrix: an experimental and analytical tool for magneto-optics,” Opt. Eng. 28, 114–119 (1989).
  26. R. Anderson, “Measurement of Mueller matrices,” Appl. Opt. 31, 11–13 (1992).
    [CrossRef] [PubMed]
  27. P. S. Theocaria, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
    [CrossRef]
  28. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, New York, 1988).
  29. D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Harcourt Brace Jovanovich, San Diego, Calif., 1990).
  30. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1992).

1997 (1)

C. Ye, O. Sundström, “Determination of S2 -fibril-angle and fiber-wall thickness by microscopic transmission ellipsometry,” Tappi 80, 181–190 (1997).

1995 (3)

P. Y. Gerligand, B. Le. Jeune, J. Cariou, J. Lotrian, “Analysis of the spatial distribution of magneto-optic properties of γ-Fe2O3 ferrofluid using different polarimetric criteria,” J. Phys. D 28, 965–977 (1995).
[CrossRef]

D. H. Goldstein, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-tuned mirror,” Opt. Eng. 34, 1593–1598 (1995).
[CrossRef]

1994 (3)

1992 (3)

R. Anderson, “Measurement of Mueller matrices,” Appl. Opt. 31, 11–13 (1992).
[CrossRef] [PubMed]

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
[CrossRef] [PubMed]

S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A. 9, 1615–1622 (1992).
[CrossRef]

1989 (2)

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28, 120–125 (1989).
[CrossRef]

B. W. Bell, “Mueller matrix: an experimental and analytical tool for magneto-optics,” Opt. Eng. 28, 114–119 (1989).

1988 (1)

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A. 5, 681–689 (1988).
[CrossRef]

1986 (1)

1985 (1)

1983 (1)

D. H. Page, F. El-Hosseiny, “The mechanical properties of single wood pulp fibres. Part VI. Fibril angle and the shape of the stress–strain curve,” J. Pulp Pap. Sci. 9, 99–100 (1983).

1980 (1)

1978 (1)

1977 (1)

D. H. Page, F. El-Hosseiny, K. Winkler, A. P. S. Lancaster, “Elastic modulus of single wood pulp fibers,” Tappi 60, 114–117 (1977).

1975 (2)

F. El-Hosseiny, D. H. Page, “The mechanical properties of single wood pulp fibres: theories of strength,” Fibre Sci. Technol. 8, 21–30 (1975).
[CrossRef]

R. E. Prud’homme, J. Noah, “Determination of fibril angle distribution in wood fibers: a comparison between the x-ray diffraction and the polarized microscope methods,” Wood Fiber 6, 282–289 (1975).

1973 (2)

R. C. Tang, “The microfibrillar orientation in cell-wall layers of Virginia pine tracheids,” Wood Sci. 5, 181–186 (1973).

R. E. Mark, P. P. Gillis, “The relationship between fiber modulus and S2 angle,” Tappi 56, 164–167 (1973).

1972 (1)

C. M. Crosby, C. D. Zeeuw, R. Marton, “Fibrillar angle variation in red pine determined by Senarmont compensation,” Wood Sci. Technol. 6, 185–195 (1972).
[CrossRef]

1969 (1)

D. H. Page, “A method for determining the fibrillar angle in wood tracheids,” J. Microsc. 90, 137–143 (1969).
[CrossRef]

Anderson, R.

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, New York, 1988).

Bell, B. W.

B. W. Bell, “Mueller matrix: an experimental and analytical tool for magneto-optics,” Opt. Eng. 28, 114–119 (1989).

Bottiger, J. R.

Cariou, J.

P. Y. Gerligand, B. Le. Jeune, J. Cariou, J. Lotrian, “Analysis of the spatial distribution of magneto-optic properties of γ-Fe2O3 ferrofluid using different polarimetric criteria,” J. Phys. D 28, 965–977 (1995).
[CrossRef]

Chenault, D. B.

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28, 120–125 (1989).
[CrossRef]

Chipman, R. A.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-tuned mirror,” Opt. Eng. 34, 1593–1598 (1995).
[CrossRef]

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28, 120–125 (1989).
[CrossRef]

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1992).

Crosby, C. M.

C. M. Crosby, C. D. Zeeuw, R. Marton, “Fibrillar angle variation in red pine determined by Senarmont compensation,” Wood Sci. Technol. 6, 185–195 (1972).
[CrossRef]

El-Hosseiny, F.

D. H. Page, F. El-Hosseiny, “The mechanical properties of single wood pulp fibres. Part VI. Fibril angle and the shape of the stress–strain curve,” J. Pulp Pap. Sci. 9, 99–100 (1983).

D. H. Page, F. El-Hosseiny, K. Winkler, A. P. S. Lancaster, “Elastic modulus of single wood pulp fibers,” Tappi 60, 114–117 (1977).

F. El-Hosseiny, D. H. Page, “The mechanical properties of single wood pulp fibres: theories of strength,” Fibre Sci. Technol. 8, 21–30 (1975).
[CrossRef]

Elminyawi, I. M.

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A. 5, 681–689 (1988).
[CrossRef]

El-Saba, A. M.

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A. 5, 681–689 (1988).
[CrossRef]

Fry, E. S.

Gdoutos, E. E.

P. S. Theocaria, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Gerligand, P. Y.

P. Y. Gerligand, B. Le. Jeune, J. Cariou, J. Lotrian, “Analysis of the spatial distribution of magneto-optic properties of γ-Fe2O3 ferrofluid using different polarimetric criteria,” J. Phys. D 28, 965–977 (1995).
[CrossRef]

Gillis, P. P.

R. E. Mark, P. P. Gillis, “The relationship between fiber modulus and S2 angle,” Tappi 56, 164–167 (1973).

Goldstein, D. H.

D. H. Goldstein, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
[CrossRef] [PubMed]

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28, 120–125 (1989).
[CrossRef]

Jeng, W.-H.

Jeune, B. Le.

P. Y. Gerligand, B. Le. Jeune, J. Cariou, J. Lotrian, “Analysis of the spatial distribution of magneto-optic properties of γ-Fe2O3 ferrofluid using different polarimetric criteria,” J. Phys. D 28, 965–977 (1995).
[CrossRef]

Kliger, D. S.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Harcourt Brace Jovanovich, San Diego, Calif., 1990).

Krishnan, S.

S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A. 9, 1615–1622 (1992).
[CrossRef]

Lancaster, A. P. S.

D. H. Page, F. El-Hosseiny, K. Winkler, A. P. S. Lancaster, “Elastic modulus of single wood pulp fibers,” Tappi 60, 114–117 (1977).

Lewis, J. W.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Harcourt Brace Jovanovich, San Diego, Calif., 1990).

Lotrian, J.

P. Y. Gerligand, B. Le. Jeune, J. Cariou, J. Lotrian, “Analysis of the spatial distribution of magneto-optic properties of γ-Fe2O3 ferrofluid using different polarimetric criteria,” J. Phys. D 28, 965–977 (1995).
[CrossRef]

Ludema, K. C.

D. A. Ramsey, K. C. Ludema, “The influence of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Mark, R. E.

R. E. Mark, P. P. Gillis, “The relationship between fiber modulus and S2 angle,” Tappi 56, 164–167 (1973).

Marton, R.

C. M. Crosby, C. D. Zeeuw, R. Marton, “Fibrillar angle variation in red pine determined by Senarmont compensation,” Wood Sci. Technol. 6, 185–195 (1972).
[CrossRef]

McClain, W. M.

Noah, J.

R. E. Prud’homme, J. Noah, “Determination of fibril angle distribution in wood fibers: a comparison between the x-ray diffraction and the polarized microscope methods,” Wood Fiber 6, 282–289 (1975).

Page, D. H.

D. H. Page, F. El-Hosseiny, “The mechanical properties of single wood pulp fibres. Part VI. Fibril angle and the shape of the stress–strain curve,” J. Pulp Pap. Sci. 9, 99–100 (1983).

D. H. Page, F. El-Hosseiny, K. Winkler, A. P. S. Lancaster, “Elastic modulus of single wood pulp fibers,” Tappi 60, 114–117 (1977).

F. El-Hosseiny, D. H. Page, “The mechanical properties of single wood pulp fibres: theories of strength,” Fibre Sci. Technol. 8, 21–30 (1975).
[CrossRef]

D. H. Page, “A method for determining the fibrillar angle in wood tracheids,” J. Microsc. 90, 137–143 (1969).
[CrossRef]

Pati, B.

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-tuned mirror,” Opt. Eng. 34, 1593–1598 (1995).
[CrossRef]

Preston, R. D.

R. D. Preston, The Physical Biology of Plant Cell Walls (Chapman & Hall, London, 1974).

Prud’homme, R. E.

R. E. Prud’homme, J. Noah, “Determination of fibril angle distribution in wood fibers: a comparison between the x-ray diffraction and the polarized microscope methods,” Wood Fiber 6, 282–289 (1975).

Ramsey, D. A.

D. A. Ramsey, K. C. Ludema, “The influence of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Randall, C. E.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Harcourt Brace Jovanovich, San Diego, Calif., 1990).

Remes, K.

Shi, Y.

Sundström, M. O.

Sundström, O.

C. Ye, O. Sundström, “Determination of S2 -fibril-angle and fiber-wall thickness by microscopic transmission ellipsometry,” Tappi 80, 181–190 (1997).

Tang, R. C.

R. C. Tang, “The microfibrillar orientation in cell-wall layers of Virginia pine tracheids,” Wood Sci. 5, 181–186 (1973).

Theocaria, P. S.

P. S. Theocaria, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Thompson, R. C.

Tian, D.

Winkler, K.

D. H. Page, F. El-Hosseiny, K. Winkler, A. P. S. Lancaster, “Elastic modulus of single wood pulp fibers,” Tappi 60, 114–117 (1977).

Ye, C.

C. Ye, O. Sundström, “Determination of S2 -fibril-angle and fiber-wall thickness by microscopic transmission ellipsometry,” Tappi 80, 181–190 (1997).

C. Ye, M. O. Sundström, K. Remes, “Microscopic transmission ellipsometry: measurement of the fibril angle and the relative retardation of single, intact wood pulp fibers,” Appl. Opt. 33, 6626–6637 (1994).
[CrossRef] [PubMed]

Zeeuw, C. D.

C. M. Crosby, C. D. Zeeuw, R. Marton, “Fibrillar angle variation in red pine determined by Senarmont compensation,” Wood Sci. Technol. 6, 185–195 (1972).
[CrossRef]

Appl. Opt. (5)

Fibre Sci. Technol. (1)

F. El-Hosseiny, D. H. Page, “The mechanical properties of single wood pulp fibres: theories of strength,” Fibre Sci. Technol. 8, 21–30 (1975).
[CrossRef]

J. Microsc. (1)

D. H. Page, “A method for determining the fibrillar angle in wood tracheids,” J. Microsc. 90, 137–143 (1969).
[CrossRef]

J. Opt. Soc. Am. A. (2)

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A. 5, 681–689 (1988).
[CrossRef]

S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A. 9, 1615–1622 (1992).
[CrossRef]

J. Phys. D (1)

P. Y. Gerligand, B. Le. Jeune, J. Cariou, J. Lotrian, “Analysis of the spatial distribution of magneto-optic properties of γ-Fe2O3 ferrofluid using different polarimetric criteria,” J. Phys. D 28, 965–977 (1995).
[CrossRef]

J. Pulp Pap. Sci. (1)

D. H. Page, F. El-Hosseiny, “The mechanical properties of single wood pulp fibres. Part VI. Fibril angle and the shape of the stress–strain curve,” J. Pulp Pap. Sci. 9, 99–100 (1983).

Opt. Eng. (4)

D. H. Goldstein, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-tuned mirror,” Opt. Eng. 34, 1593–1598 (1995).
[CrossRef]

B. W. Bell, “Mueller matrix: an experimental and analytical tool for magneto-optics,” Opt. Eng. 28, 114–119 (1989).

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28, 120–125 (1989).
[CrossRef]

Opt. Lett. (3)

Rev. Sci. Instrum. (1)

D. A. Ramsey, K. C. Ludema, “The influence of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Tappi (3)

C. Ye, O. Sundström, “Determination of S2 -fibril-angle and fiber-wall thickness by microscopic transmission ellipsometry,” Tappi 80, 181–190 (1997).

D. H. Page, F. El-Hosseiny, K. Winkler, A. P. S. Lancaster, “Elastic modulus of single wood pulp fibers,” Tappi 60, 114–117 (1977).

R. E. Mark, P. P. Gillis, “The relationship between fiber modulus and S2 angle,” Tappi 56, 164–167 (1973).

Wood Fiber (1)

R. E. Prud’homme, J. Noah, “Determination of fibril angle distribution in wood fibers: a comparison between the x-ray diffraction and the polarized microscope methods,” Wood Fiber 6, 282–289 (1975).

Wood Sci. (1)

R. C. Tang, “The microfibrillar orientation in cell-wall layers of Virginia pine tracheids,” Wood Sci. 5, 181–186 (1973).

Wood Sci. Technol. (1)

C. M. Crosby, C. D. Zeeuw, R. Marton, “Fibrillar angle variation in red pine determined by Senarmont compensation,” Wood Sci. Technol. 6, 185–195 (1972).
[CrossRef]

Other (5)

R. D. Preston, The Physical Biology of Plant Cell Walls (Chapman & Hall, London, 1974).

P. S. Theocaria, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, (North-Holland, New York, 1988).

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Harcourt Brace Jovanovich, San Diego, Calif., 1990).

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1992).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic representation of the layer structure of a single wood fiber: φ, fibril angle (see text).

Fig. 2
Fig. 2

Single wood fiber (represented by the S2 layer) illuminated by polarized light. The two fiber walls are described as two linear retarders of the same relative retardation Δ and the same orientation angle φ (fibril angle) pointing in opposite directions.

Fig. 3
Fig. 3

Schematic diagram of the experimental setup; P, A, orientation angles of the polarizer and the analyzer. The two quarter-wave retarders are oriented at 45°.

Fig. 4
Fig. 4

Measured results of a λ/2 and λ/4 wave plates for (a) retardation Δ and (b) orientation angle φ. The wave plates were aligned at 0°, 22.5°, 45°, and 90° for Mueller matrix measurement, and Δ and φ were determined from the corresponding Mueller matrices obtained.

Fig. 5
Fig. 5

(a) Polarizing micrograph (λ = 550 nm) of unbleached pine kraft pulp. (b) Fiber segment to be measured (520×; P, A, azimuths of the polarizer and the analyzer). The Mueller matrix was measured at the five marked points in the center region.

Fig. 6
Fig. 6

Measured (a) retardation Δ and (b) orientation angle φ of the fiber in Fig. 3 obtained by the Mueller matrix method compared with the previous polarimetry method. For better evaluation of the measurement results, the fiber was oriented at θ ≈ -45°, θ ≈ 0°, θ ≈ -70°, and θ ≈ -90° for the measurement.

Fig. 7
Fig. 7

Results for angles δθ1, δθ2, and δθ3 formed when the fiber in Fig. 3 was turned from the position θ ≈ -45° to θ ≈ 0°, θ ≈ -70°, and θ ≈ -90°, respectively.

Fig. 8
Fig. 8

(a) Polarizing micrograph (λ = 550 nm) of unbleached pine kraft pulp. (b) Fiber segment to be measured (520×; P, A, azimuths of the polarizer and the analyzer). The Mueller matrix was measured at the three marked points in the center region.

Fig. 9
Fig. 9

Measured (a) retardation Δ and (b) orientation angle φ of the fiber in Fig. 8 obtained by the Mueller matrix method compared with those of the previous polarimetry method. For better evaluation of the measurement results, the fiber was oriented at θ ≈ -45°, θ ≈ -42.4°, and θ ≈ 0° for the measurement.

Fig. 10
Fig. 10

Results for angles δθ1 and δθ2 formed when the fiber in Fig. 8 was turned from the position θ ≈ -45° to θ ≈ -42.4° and to θ ≈ 0°, respectively.

Fig. 11
Fig. 11

Results of measurements performed on 10 single fibers of a pulp sample by the Mueller matrix method (a) for the retardation Δ and (b) for the fibril angle φ compared with those obtained by the previous method. The error bars represent the upper and lower bounds of 10% on the data series of the previous method for Δ and 4.0° for φ.

Tables (1)

Tables Icon

Table 1 Set of 16 Light Intensities Ik (k = 1, 2, … , 16) for Calculating All 16 Mueller Matrix Elements tij (i; j = 1, 2, 3, 4) and the Corresponding Configuration Parameters of the Imaging Polarimeter for the Intensity Measurements

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

Ts=RφT0ΔR-2φT0ΔRφ,
Rφ=10000cos 2φsin 2φ00-sin 2φcos 2φ00001, T0Δ=1000010000cos Δsin Δ00-sin Δcos Δ,
Ts=10000m22m23m240-m23m33m340m24-m34m44,
m22=1-2 sin2 4φ sin4Δ2, m23=-m32=2 sin 4φ sin2Δ22 cos2 2φ sin2Δ2-1, m24=m42=-4 sin 2φ cos2 2φ sin2Δ2sin Δ, m33=1+8 sin2Δ2cos2 2φsin2Δ2cos2 2φ-1, m34=-m43=2 cos 2φ sin Δ1-2 cos2 2φ sin2Δ2, m44=1-2 cos2 2φ sin2 Δ.
Ts=10000t22t23t240t32t33t340t42t43t44=R-θTsRθ.
t22=m22 cos2 2θ+m33 sin2 2θ, t23=m23+m22-m33sin 2θ cos 2θ, t24=m24 cos 2θ-m34 sin 2θ, t32=m32+m22-m33sin 2θ cos 2θ, t33=m22 sin2 2θ+m33 cos2 2θ, t34=m24 sin 2θ+m34 cos 2θ, t42=m42 cos 2θ-m43 sin 2θ, t43=m42 sin 2θ+m43 cos 2θ, t44=m44.
m22=12t22+t33+t22-t33cos 4θ+t23+t32sin 4θ, m23=-m32=t23-t322, m33=12t22+t33-t22-t33cos 4θ-t23+t32sin 4θ, m24=m42=t24 cos 2θ+t34 sin 2θ=t42 cos 2θ+t43 sin 2θ, m34=-m43=t34 cos 2θ-t24 sin 2θ=t43×cos 2θ-t42 sin 2θ,  m44t44.
tan 2θ=t34+t43t24+t42=t24-t42t43-t34
tan 4θ=t23+t32t22-t33.
S=I021cos 2Asin 2A0cos 2Acos2 2Asin 2A cos 2A0sin 2Asin 2A cos 2Asin2 2A00000×10000cos Δ10-sin Δ100100sin Δ10cos Δ1t11t12t13t14t21t22t23t24t31t32t33t34t41t42t43t44×10000cos Δ20-sin Δ200100sin Δ20cos Δ21cos 2Psin 2P0,
I=I02t11+t12 cos Δ1 cos 2P+t13 sin 2P+t14 sin Δ1 cos 2P+cos Δ2×cos 2At21+t22 cos Δ1 cos 2P+t23×sin 2P+t24 sin Δ1 cos 2P+sin 2A×t31+t32 cos Δ1 cos 2P+t33×sin 2P+t34 sin Δ1 cos 2P-sin Δ2×cos 2At41+t42 cos Δ1 cos 2P+t43 sin 2P+t44 sin Δ1 cos 2P.
Mλ/2,0=1.0000-0.0125-0.0419-0.0158-0.03810.95570.07420.04090.0164-0.0594-0.9314-0.0304-0.05540.0382-0.0306-0.8511, Δ=182.00°, φ=0.12°  Mλ/4,0=1.0000-0.00560.0027-0.0020-0.05660.93520.05340.1645-0.0156-0.01400.04980.98210.02980.0060-0.9361-0.0354, Δ=86.89°, φ=-1.37°.
Ts=1.0000-0.00730.01700.0086-0.00330.9889-0.01200.00770.01560.01200.0788-0.85430.01690.00640.85630.0291.
Ts=1.0000-0.0014-0.0004-0.00300.03280.9862-0.0410-0.0177-0.00390.04100.58320.7587-0.0001-0.0187-0.79670.5883.

Metrics