Abstract

A nondestructive method based on spectroscopic ellipsometry has been developed and demonstrated for the real-time measurement of a single pulp fiber's microfibril angle and phase retardation, with the latter proportional to the cell wall thickness. The method uses an optical arrangement insensitive to the sample's orientation in combination with a proper spectral analysis of the sample's image. The optical arrangement and the measurement principle of the method are described. To test the new method, equipment functioning as a spectroscopic imaging ellipsometer was constructed according to the arrangement, and measurements were carried out in which single pulp fibers and ordinary wave plates were measured. The test measurements and results are described and presented.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. D. Preston, The Physical Biology of Plant Cell Walls (Chapman and Hall, 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 and J. Noah, "Determination of fibril angle distribution in wood fibers: a comparison between the x-ray diffraction and the polarized microscope methods," Wood Fiber Sci. 6, 282-289 (1975).
  4. C. M. Crosby, C. D. Zeeuw, and 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 and P. P. Gillis, "The relationship between fiber modulus and S2 angle," Tappi J. 56, 164-167 (1973).
  7. F. El-Hosseiny and 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, and A. P. S. Lancaster, "Elastic modulus of single wood pulp fibers," Tappi J. 60, 114-117 (1977).
  9. D. H. Page and 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, and 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 and M. O. Sundström, "Determination of S2-fibril-angle and fiber-wall thickness by microscopic transmission ellipsometry," Tappi J. 80, 181-190 (1997).
  12. C. Ye, "Photopolarimetric measurement of single, intact pulp fibers by Mueller matrix imaging polarimetry," Appl. Opt. 38, 1975-1985 (1999).
    [CrossRef]
  13. P. S. Theocaria and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).
  14. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  15. C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
    [CrossRef]
  16. Spectral Imaging Ltd., http://www.specim.fi/, 17 October 2006.
  17. D. H. Page, F. El-Hosseiny, M. L. Bidmade, and R. Binet, "Birefringence and the chemical composition of wood pulp fibers," in Applied Polymer Symposium (Wiley, 1976), Vol. 28, pp. 923-929.
  18. U. Sahlberg, L. Salmen, and A. Oscarsson, "The fibrillar orientation in the S2-layer of wood fibres as determined by x-ray diffraction analysis," Wood Sci. Technol. 31, 77-86 (1997).
  19. R. Marton, P. Ruston, J. S. Sacco , and K. Sumiya, "Dimensions and ultrastructure in growing fibres," Tappi J. 55, 1499-1504 (1972).

2001

C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
[CrossRef]

1999

1997

U. Sahlberg, L. Salmen, and A. Oscarsson, "The fibrillar orientation in the S2-layer of wood fibres as determined by x-ray diffraction analysis," Wood Sci. Technol. 31, 77-86 (1997).

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

1994

1983

D. H. Page and 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).

1977

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

1975

F. El-Hosseiny and 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 and J. Noah, "Determination of fibril angle distribution in wood fibers: a comparison between the x-ray diffraction and the polarized microscope methods," Wood Fiber Sci. 6, 282-289 (1975).

1973

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

R. E. Mark and P. P. Gillis, "The relationship between fiber modulus and S2 angle," Tappi J. 56, 164-167 (1973).

1972

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

R. Marton, P. Ruston, J. S. Sacco , and K. Sumiya, "Dimensions and ultrastructure in growing fibres," Tappi J. 55, 1499-1504 (1972).

1969

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

Bidmade, M. L.

D. H. Page, F. El-Hosseiny, M. L. Bidmade, and R. Binet, "Birefringence and the chemical composition of wood pulp fibers," in Applied Polymer Symposium (Wiley, 1976), Vol. 28, pp. 923-929.

Binet, R.

D. H. Page, F. El-Hosseiny, M. L. Bidmade, and R. Binet, "Birefringence and the chemical composition of wood pulp fibers," in Applied Polymer Symposium (Wiley, 1976), Vol. 28, pp. 923-929.

Crosby, C. M.

C. M. Crosby, C. D. Zeeuw, and 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 and 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, and A. P. S. Lancaster, "Elastic modulus of single wood pulp fibers," Tappi J. 60, 114-117 (1977).

F. El-Hosseiny and 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, F. El-Hosseiny, M. L. Bidmade, and R. Binet, "Birefringence and the chemical composition of wood pulp fibers," in Applied Polymer Symposium (Wiley, 1976), Vol. 28, pp. 923-929.

Gdoutos, E. E.

P. S. Theocaria and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Gillis, P. P.

R. E. Mark and P. P. Gillis, "The relationship between fiber modulus and S2 angle," Tappi J. 56, 164-167 (1973).

Hyvärinen, H.

C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
[CrossRef]

Kliger, D. S.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Lancaster, A. P. S.

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

Lewis, J. W.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Mark, R. E.

R. E. Mark and P. P. Gillis, "The relationship between fiber modulus and S2 angle," Tappi J. 56, 164-167 (1973).

Marton, R.

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

R. Marton, P. Ruston, J. S. Sacco , and K. Sumiya, "Dimensions and ultrastructure in growing fibres," Tappi J. 55, 1499-1504 (1972).

Moss, P.

C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
[CrossRef]

Noah, J.

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

Nyblom, I.

C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
[CrossRef]

Oscarsson, A.

U. Sahlberg, L. Salmen, and A. Oscarsson, "The fibrillar orientation in the S2-layer of wood fibres as determined by x-ray diffraction analysis," Wood Sci. Technol. 31, 77-86 (1997).

Page, D. H.

D. H. Page and 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, and A. P. S. Lancaster, "Elastic modulus of single wood pulp fibers," Tappi J. 60, 114-117 (1977).

F. El-Hosseiny and 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]

D. H. Page, F. El-Hosseiny, M. L. Bidmade, and R. Binet, "Birefringence and the chemical composition of wood pulp fibers," in Applied Polymer Symposium (Wiley, 1976), Vol. 28, pp. 923-929.

Preston, R. D.

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

Prud'homme, R. E.

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

Randall, C. E.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Räty, J.

C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
[CrossRef]

Remes, K.

Ruston, P.

R. Marton, P. Ruston, J. S. Sacco , and K. Sumiya, "Dimensions and ultrastructure in growing fibres," Tappi J. 55, 1499-1504 (1972).

Sacco, J. S.

R. Marton, P. Ruston, J. S. Sacco , and K. Sumiya, "Dimensions and ultrastructure in growing fibres," Tappi J. 55, 1499-1504 (1972).

Sahlberg, U.

U. Sahlberg, L. Salmen, and A. Oscarsson, "The fibrillar orientation in the S2-layer of wood fibres as determined by x-ray diffraction analysis," Wood Sci. Technol. 31, 77-86 (1997).

Salmen, L.

U. Sahlberg, L. Salmen, and A. Oscarsson, "The fibrillar orientation in the S2-layer of wood fibres as determined by x-ray diffraction analysis," Wood Sci. Technol. 31, 77-86 (1997).

Sumiya, K.

R. Marton, P. Ruston, J. S. Sacco , and K. Sumiya, "Dimensions and ultrastructure in growing fibres," Tappi J. 55, 1499-1504 (1972).

Sundström, M. O.

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

C. Ye, M. O. Sundström, and 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]

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 and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Winkler, K.

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

Ye, C.

C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
[CrossRef]

C. Ye, "Photopolarimetric measurement of single, intact pulp fibers by Mueller matrix imaging polarimetry," Appl. Opt. 38, 1975-1985 (1999).
[CrossRef]

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

C. Ye, M. O. Sundström, and 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, and R. Marton, "Fibrillar angle variation in red pine determined by Senarmont compensation," Wood Sci. Technol. 6, 185-195 (1972).
[CrossRef]

Appl. Opt.

Applied Polymer Symposium

D. H. Page, F. El-Hosseiny, M. L. Bidmade, and R. Binet, "Birefringence and the chemical composition of wood pulp fibers," in Applied Polymer Symposium (Wiley, 1976), Vol. 28, pp. 923-929.

Fibre Sci. Technol.

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

J. Microsc.

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

J. Pulp Pap. Sci.

D. H. Page and 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).

Nord. Pulp Pap. Res. J.

C. Ye, J. Räty, I. Nyblom, H. Hyvärinen, and P. Moss, "Estimation of lignin content in single, intact pulp fibers by UV photometry and VIS Mueller matrix polarimetry," Nord. Pulp Pap. Res. J. 2, 143-148 (2001).
[CrossRef]

Tappi J.

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

R. E. Mark and P. P. Gillis, "The relationship between fiber modulus and S2 angle," Tappi J. 56, 164-167 (1973).

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

R. Marton, P. Ruston, J. S. Sacco , and K. Sumiya, "Dimensions and ultrastructure in growing fibres," Tappi J. 55, 1499-1504 (1972).

Wood Fiber Sci.

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

Wood Sci.

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

Wood Sci. Technol.

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

U. Sahlberg, L. Salmen, and A. Oscarsson, "The fibrillar orientation in the S2-layer of wood fibres as determined by x-ray diffraction analysis," Wood Sci. Technol. 31, 77-86 (1997).

Other

Spectral Imaging Ltd., http://www.specim.fi/, 17 October 2006.

P. S. Theocaria and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

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

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 (16)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Single pulp fiber (represented by the S 2 layer) illuminated by polarized light with its cell walls equivalent to two identical linear retarders of the retardation Δ oriented with the axes pointing in opposite directions around the fiber axis.

Fig. 3
Fig. 3

Schematic of an optical arrangement used for real-time measurement of pulp fibers, comprising two achromatic quarter-wave retarders arranged between a pair of polarizers: pl 1 , pl 2 , linear polarizers ( P 1 , P 2 , azimuths); qr 1 , qr 2 , achromatic quarter wave retarders ( φ 1 , φ 2 , orientation angles; Δ 1 ( λ ) , Δ 2 ( λ ) , retardations).

Fig. 4
Fig. 4

Calculated error factors (a) |R c | and (b) |R c | of the optical arrangement in Fig. 3 containing a pulp fiber of retardation Δ and fibril angle φ for measurement as a function of Δ at a few selected angles of φ when the quarter-wave retarders of the arrangement have retardation error, and they are crossed and parallel to each other, respectively (see the text).

Fig. 5
Fig. 5

Schematic of the experimental setup (a modified microscope): P 1 , P 2 , azimuths of the polarizers; φ 1 , φ 2 , orientation angles of the quarter-wave retarders.

Fig. 6
Fig. 6

(a) Micrograph of a fiber (pine kraft pulp) measured by the experimental setup, I, selected fiber segment for measurement and I 0 , selected segment of the background image formed by the empty experimental setup. (b) Spectral image dispersed from the scanned image part in (a); I [ Δ ( λ ) , φ ] , I 0 ( λ ) , spectral distributions dispersed from the image segments I and I 0 .

Fig. 7
Fig. 7

Measured spectral transmission function T [ Δ ( λ ) , φ ] of the fiber shown in Fig. 6(a) and its fit curve in the range of 400 to 710   nm . T [ Δ ( λ ) , φ ] was determined from the spectral image segments I [ Δ ( λ ) , φ ] and I 0 ( λ ) in Fig. 6(b) with T [ Δ ( λ ) , φ ] = I [ Δ ( λ ) , φ ] / I 0 ( λ ) (see the text).

Fig. 8
Fig. 8

(a) Measured retardation Δ at λ = 550   nm , and (b) microfibril angle φ of the fiber in Fig. 6(a) by the real-time method as a function of the fiber orientation angle θ. For a better test of the method, the fiber was repeatedly measured when it was rotated to different orientations.

Fig. 9
Fig. 9

(a) Micrograph of a fiber (birch kraft pulp) measured by the experimental setup, I. Selected fiber segment for measurement and I 0 , selected segment of the background image formed by the empty experimental setup. (b) Spectral image dispersed from the scanned image part in (a), I [ Δ ( λ ) , φ ] , I 0 ( λ ) , spectral distributions dispersed from the segments I and I 0 .

Fig. 10
Fig. 10

(a) Micrograph of a fiber (pine kraft pulp) measured by the experimental setup, I. Selected fiber segment for measurement and I 0 , selected segment of the background image formed by the empty experimental setup. (b) Spectral image dispersed from the scanned image part in (a), I [ Δ ( λ ) , φ ] , I 0 ( λ ) , spectral distributions dispersed from the segments I and I 0 .

Fig. 11
Fig. 11

Measured spectral transmission function T [ Δ ( λ ) , φ ] of the fiber shown in Fig. 9(a) and its fit curve in the range of 400 to 710   nm . T [ Δ ( λ ) , φ ] was determined from the spectral image segments I [ Δ ( λ ) , φ ] and I 0 ( λ ) in Fig. 9(b) with T [ Δ ( λ ) , φ ] = I [ Δ ( λ ) , φ ] / I 0 ( λ ) (see the text).

Fig. 12
Fig. 12

Measured spectral transmission function T [ Δ ( λ ) , φ ] of the fiber shown in Fig. 10(a) and its fit curve in the range of 400 to 710   nm . T [ Δ ( λ ) , φ ] was determined from the spectral image segments I [ Δ ( λ ) , φ ] and I 0 ( λ ) in Fig. 10(b) with T [ Δ ( λ ) , φ ] = I [ Δ ( λ ) , φ ] / I 0 ( λ ) (see the text).

Fig. 13
Fig. 13

(a) Measured retardation Δ at λ = 550   nm and (b) microfibril angle φ of the fiber in Fig. 9(a) by the real-time method as a function of the fiber orientation angle θ. For a better test of the method, the fiber was repeatedly measured when it was rotated to different orientations.

Fig. 14
Fig. 14

(a) Measured retardation Δ at λ = 550   nm and (b) microfibril angle φ of the fiber in Fig. 10(a) by the real-time method as a function of the fiber orientation angle θ. For a better test of the method, the fiber was repeatedly measured when it was rotated to different orientations.

Fig. 15
Fig. 15

(a) Measured half-retardation Δ ( 550   nm ) and (b) imaginary microfibril angle φ of a mica λ / 2 plate as a function of its orientation angle θ, which was used as a special fiber sample with the microfibril angle φ = 0 ° and the retardation of its cell wall equal to half the nominal retardation of the λ / 2 plate for further testing the real-time method. The results for half-retardation Δ are compared with those generated when φ = 0 ° was set.

Fig. 16
Fig. 16

(a) Measured half-retardation Δ ( 560   nm ) and (b) imaginary microfibril angle φ of a plastic λ / 4 plate as a function of its orientation angle θ, which was used as a special fiber sample with the microfibril angle φ = 0 ° and the retardation of its cell wall equal to one half the nominal retardation of the λ / 4 plate for further testing the real-time method. The results for half-retardation Δ are compared with those generated when φ = 0 ° was set.

Equations (25)

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

T = [ 1 0 0 0 0 t 22 t 23 t 24 0 t 32 t 33 t 34 0 t 42 t 43 t 44 ] ,
t 22 = m 22 cos 2 2 θ + m 33 sin 2 2 θ ,
t 23 = m 23 + ( m 22 m 33 ) sin 2 θ cos 2 θ .
t 24 = m 24 cos 2 θ m 34 sin 2 θ ,
t 32 = m 32 + ( m 22 m 33 ) sin 2 θ cos 2 θ ,
t 33 = m 22 sin 2 2 θ + m 33 cos 2 2 θ ,
t 34 = m 24 sin 2 θ + m 34 cos 2 θ ,
t 42 = m 42 cos 2 θ m 43 sin 2 θ ,
t 43 = m 42 sin 2 θ + m 43 cos 2 θ ,
t 44 = 1 2 cos 2 2 φ sin 2 Δ .
m 22 = 1 2 sin 2 4 φ sin 4 Δ 2 ,
m 23 = m 32 = 2 sin 4 φ sin 2 Δ 2 ( 2 cos 2 2 φ sin 2 Δ 2 1 ) ,
m 24 = m 42 = 4 sin 2 φ cos 2 2 φ sin 2 Δ 2 sin Δ ,
m 33 = 1 + 8 sin 2 Δ 2 cos 2 2 φ ( sin 2 Δ 2 cos 2 2 φ 1 ) .
m 34 = m 43 = 2 cos 2 φ sin Δ ( 1 2 cos 2 2 φ sin 2 Δ 2 ) ,
I = I 0 2 [ 1 1 0 0 ] 1 [ 1 0 0 0 0 cos Δ 2 ( λ ) 0 ± sin Δ 2 ( λ ) 0 0 1 0 0 sin Δ 2 ( λ ) 0 cos Δ 2 ( λ ) ] [ 1 0 0 0 0 t 22 t 23 t 24 0 t 32 t 33 t 34 0 t 42 t 43 t 44 ] [ 1 0 0 0 0 cos Δ 1 ( λ ) 0 sin Δ 1 ( λ ) 0 0 1 0 0 sin Δ 1 ( λ ) 0 cos Δ 1 ( λ ) ] [ 1 1 0 0 ]
= { I 0 2 { 1 + cos Δ 2 ( λ ) [ t 22 cos Δ 1 ( λ ) + t 24 sin Δ 1 ( λ ) ] + sin Δ 2 ( λ ) [ t 42 cos Δ 1 ( λ ) + t 44 sin Δ 1 ( λ ) ] } ,   for   φ 1 = 45°   and   φ 2 = 45° I 0 2 { 1 + cos Δ 2 ( λ ) [ t 22 cos Δ 1 ( λ ) + t 24 sin Δ 1 ( λ ) ] sin Δ 2 ( λ ) [ t 42 cos Δ 1 ( λ ) + t 44 sin Δ 1 ( λ ) ] } , for   φ 1 = 45°   and   φ 2 = 45° ,
[ 1 1 0 0 ] 1 = [ 1 1 0 0 ] .
I = { I 0 2 [ 1 + t 22 cos Δ 1 2 ( λ ) + ( t 24 + t 42 ) sin Δ 1 ( λ ) cos Δ 1 ( λ ) + t 44 sin Δ 1 2 ( λ ) ] ,   for   φ 1 = 45 °   and   φ 2 = 45 ° , (5a) I 0 2 [ 1 + t 22 cos Δ 1 2 ( λ ) + ( t 24 t 42 ) sin Δ 1 ( λ ) cos Δ 1 ( λ ) t 44 sin Δ 1 2 ( λ ) ] ,   for   φ 1 = 45 °   and   φ 2 = 45 ° . (5b)
I = { I 0 2 ( 1 + t 44 ) = I 0 [ 1 cos 2 2 φ sin 2 Δ ( λ ) ] ,   for   φ 1 = 45 °   and   φ 2 = 45 ° , (6a) I 0 2 ( 1 t 44 ) = I 0 cos 2 2 φ sin 2 Δ ( λ ) ,   for   φ 1 = 45 °   and   φ 2 = 45 ° . (6b)
T = { 1 cos 2 2 φ sin 2 Δ ( λ ) ,  for φ 1 = 45 °   and   φ 2 = 45 ° , (7a) cos 2 2 φ sin 2 Δ ( λ ) ,  for φ 1 = 45 °   and   φ 2 = 45 ° . (7b)
δ I c = I 0 2 [ ( t 22 t 44 ) cos Δ 1 ( λ ) + ( t 24 + t 42 ) sin Δ 1 ( λ ) ] × cos Δ 1 ( λ ) I 0 R c cos 2 θ δ Δ 1 ( λ ) ,
δ I p = I 0 2 [ ( t 22 + t 44 ) cos Δ 1 ( λ ) + ( t 24 t 42 ) sin Δ 1 ( λ ) ] × cos Δ 1 ( λ ) I 0 R p sin 2 θ δ Δ 1 ( λ ) ,
R c = 2 sin 2 φ cos 2 2 φ ( 1 cos Δ ) sin Δ ,
R p = 2 cos 2 φ sin Δ ( 1 2 cos 2 2 φ    sin 2 Δ 2 ) .

Metrics