Abstract

A full-field technique for simultaneous measurement of the magnitude of birefringence and its orientation is presented. This is achieved using a monolithic birefringence sensitive interferometer where the interference fringes carry the information of both the birefringence phase and the orientation of the fast axis of an optically transmissive anisotropic material placed at the output of the interferometer. The interferometer consists of a suitably polarization-masked cube beam splitter, orientated as in the Gates interferometer, which serves to generate a pair of orthogonally polarized and collinearly propagating light beams. Experimental results are obtained through an algorithm incorporating eight polarization phase-shifted interferograms.

© 2011 Optical Society of America

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  1. R. S. Stein, “Measurement of birefringence in biaxially oriented films,” J. Polym. Sci. 24, 383–386 (1957).
    [CrossRef]
  2. G. R. McIntyre and A. Neureuther, “Phase shift mask interferometric birefringent monitor,” J. Vac. Sci. Technol. B 24, 2808–2814 (2006).
    [CrossRef]
  3. J.-F. Lin, T.-T. Liao, Y.-L. Lo, and S.-Y. Lee, “The optical linear birefringence measurement using Zeeman laser,” Opt. Commun. 274, 153–158 (2007).
    [CrossRef]
  4. A. K. Asundi, “Digital photoelasticity” in MATLAB for Photomechanics—A Primer (2002), pp. 37–77.
  5. A. K. Bhowmik, “Multiple-reflection effects in photoelastic stress analysis,” Appl. Opt. 40, 2687–2691 (2001).
    [CrossRef]
  6. H. Aben, J. Anton, and A. Errapart, “Modern photoelasticity for residual stress measurement in glass,” in Experimental Analysis of Nano and Engineering Materials and Structures, Proceedings of the 13th International Conference on Experimental Mechanics, E.E.Gdoutos, ed. (Springer, 2007).
    [CrossRef] [PubMed]
  7. A. S. Redner, “Photoelastic measurement by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
    [CrossRef]
  8. S. Corum and A. Brachmann, “Characterization of Ti:sapphire laser rods for installation in a polarized light source,” J. Young Investig. 8(1) (2003); http://www.jyi.org/volumes/volume8/issue1/articles/corum.html.
  9. A. Hamza, T. Z. N. Sokkar, and M. A. Kabeel, “Interferometric determination of refractive indices and birefringence of fibres with irregular transverse sections,” J. Phys. D L19, L19–L20(1986).
    [CrossRef]
  10. B. Zhao, Z. Cao, R. Fang, and A. Asundi, “Diffraction image in an optical microscope: application to detection of birefringence,” Opt. Eng. 41, 751–759 (2002).
    [CrossRef]
  11. J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
    [CrossRef] [PubMed]
  12. K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
    [CrossRef] [PubMed]
  13. W. J. Bock, W. Urbánczyk, and M. Fontaine, “Characterization of highly birefringent optical fibers using interferometric techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
    [CrossRef]
  14. F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibres,” J. Opt. A Pure Appl. Opt. 1, 197–200 (1999).
    [CrossRef]
  15. K. Bhattacharya, A. Basuray, and A. K. Chakraborty, “Photoelastic testing using a birefringence sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
    [CrossRef]
  16. N. Ghosh and K. Bhattacharya, “Cube beam-splitter interferometer for phase shifting interferometry,” J. Opt. 38, 191–198 (2009).
    [CrossRef]
  17. N. Ghosh, A. K. Chakraborty, and K. Bhattacharya, “A single element birefringence-sensitive interferometer,” J. Opt. 37, 147–152 (2008).
  18. J. A. Ferrari and E. M. Frins, “Single element interferometer,” Opt. Commun. 279, 235–239 (2007).
    [CrossRef]
  19. Y. Otani, T. Shimada, T. Oshizawa, and N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  20. J. W. Gates, “Reverse shearing interferometry,” Nature 176, 359–360 (1955).
    [CrossRef]
  21. N. Ghosh, Y. Otani, and K. Bhattacharya, “Generation of collinearly propagating orthogonally polarized beams,” Optik; available online http://dx.doi.org/10.1016/j.ijleo.2010.07.021.

2009 (1)

N. Ghosh and K. Bhattacharya, “Cube beam-splitter interferometer for phase shifting interferometry,” J. Opt. 38, 191–198 (2009).
[CrossRef]

2008 (1)

N. Ghosh, A. K. Chakraborty, and K. Bhattacharya, “A single element birefringence-sensitive interferometer,” J. Opt. 37, 147–152 (2008).

2007 (2)

J. A. Ferrari and E. M. Frins, “Single element interferometer,” Opt. Commun. 279, 235–239 (2007).
[CrossRef]

J.-F. Lin, T.-T. Liao, Y.-L. Lo, and S.-Y. Lee, “The optical linear birefringence measurement using Zeeman laser,” Opt. Commun. 274, 153–158 (2007).
[CrossRef]

2006 (1)

G. R. McIntyre and A. Neureuther, “Phase shift mask interferometric birefringent monitor,” J. Vac. Sci. Technol. B 24, 2808–2814 (2006).
[CrossRef]

2003 (1)

S. Corum and A. Brachmann, “Characterization of Ti:sapphire laser rods for installation in a polarized light source,” J. Young Investig. 8(1) (2003); http://www.jyi.org/volumes/volume8/issue1/articles/corum.html.

2002 (1)

B. Zhao, Z. Cao, R. Fang, and A. Asundi, “Diffraction image in an optical microscope: application to detection of birefringence,” Opt. Eng. 41, 751–759 (2002).
[CrossRef]

2001 (1)

1999 (1)

F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibres,” J. Opt. A Pure Appl. Opt. 1, 197–200 (1999).
[CrossRef]

1997 (2)

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

W. J. Bock, W. Urbánczyk, and M. Fontaine, “Characterization of highly birefringent optical fibers using interferometric techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

1996 (1)

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

1994 (2)

K. Bhattacharya, A. Basuray, and A. K. Chakraborty, “Photoelastic testing using a birefringence sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Y. Otani, T. Shimada, T. Oshizawa, and N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

1986 (1)

A. Hamza, T. Z. N. Sokkar, and M. A. Kabeel, “Interferometric determination of refractive indices and birefringence of fibres with irregular transverse sections,” J. Phys. D L19, L19–L20(1986).
[CrossRef]

1985 (1)

A. S. Redner, “Photoelastic measurement by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

1957 (1)

R. S. Stein, “Measurement of birefringence in biaxially oriented films,” J. Polym. Sci. 24, 383–386 (1957).
[CrossRef]

1955 (1)

J. W. Gates, “Reverse shearing interferometry,” Nature 176, 359–360 (1955).
[CrossRef]

Aben, H.

H. Aben, J. Anton, and A. Errapart, “Modern photoelasticity for residual stress measurement in glass,” in Experimental Analysis of Nano and Engineering Materials and Structures, Proceedings of the 13th International Conference on Experimental Mechanics, E.E.Gdoutos, ed. (Springer, 2007).
[CrossRef] [PubMed]

Anton, J.

H. Aben, J. Anton, and A. Errapart, “Modern photoelasticity for residual stress measurement in glass,” in Experimental Analysis of Nano and Engineering Materials and Structures, Proceedings of the 13th International Conference on Experimental Mechanics, E.E.Gdoutos, ed. (Springer, 2007).
[CrossRef] [PubMed]

Arokoski, J. P. A.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Asundi, A.

B. Zhao, Z. Cao, R. Fang, and A. Asundi, “Diffraction image in an optical microscope: application to detection of birefringence,” Opt. Eng. 41, 751–759 (2002).
[CrossRef]

Asundi, A. K.

A. K. Asundi, “Digital photoelasticity” in MATLAB for Photomechanics—A Primer (2002), pp. 37–77.

Basuray, A.

K. Bhattacharya, A. Basuray, and A. K. Chakraborty, “Photoelastic testing using a birefringence sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Bhattacharya, K.

N. Ghosh and K. Bhattacharya, “Cube beam-splitter interferometer for phase shifting interferometry,” J. Opt. 38, 191–198 (2009).
[CrossRef]

N. Ghosh, A. K. Chakraborty, and K. Bhattacharya, “A single element birefringence-sensitive interferometer,” J. Opt. 37, 147–152 (2008).

K. Bhattacharya, A. Basuray, and A. K. Chakraborty, “Photoelastic testing using a birefringence sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

N. Ghosh, Y. Otani, and K. Bhattacharya, “Generation of collinearly propagating orthogonally polarized beams,” Optik; available online http://dx.doi.org/10.1016/j.ijleo.2010.07.021.

Bhowmik, A. K.

Bock, W. J.

W. J. Bock, W. Urbánczyk, and M. Fontaine, “Characterization of highly birefringent optical fibers using interferometric techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

Brachmann, A.

S. Corum and A. Brachmann, “Characterization of Ti:sapphire laser rods for installation in a polarized light source,” J. Young Investig. 8(1) (2003); http://www.jyi.org/volumes/volume8/issue1/articles/corum.html.

Cao, Z.

B. Zhao, Z. Cao, R. Fang, and A. Asundi, “Diffraction image in an optical microscope: application to detection of birefringence,” Opt. Eng. 41, 751–759 (2002).
[CrossRef]

Chakraborty, A. K.

N. Ghosh, A. K. Chakraborty, and K. Bhattacharya, “A single element birefringence-sensitive interferometer,” J. Opt. 37, 147–152 (2008).

K. Bhattacharya, A. Basuray, and A. K. Chakraborty, “Photoelastic testing using a birefringence sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Corum, S.

S. Corum and A. Brachmann, “Characterization of Ti:sapphire laser rods for installation in a polarized light source,” J. Young Investig. 8(1) (2003); http://www.jyi.org/volumes/volume8/issue1/articles/corum.html.

Dobai, J.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

El-Diasty, F.

F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibres,” J. Opt. A Pure Appl. Opt. 1, 197–200 (1999).
[CrossRef]

Elo, M.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

Errapart, A.

H. Aben, J. Anton, and A. Errapart, “Modern photoelasticity for residual stress measurement in glass,” in Experimental Analysis of Nano and Engineering Materials and Structures, Proceedings of the 13th International Conference on Experimental Mechanics, E.E.Gdoutos, ed. (Springer, 2007).
[CrossRef] [PubMed]

Fang, R.

B. Zhao, Z. Cao, R. Fang, and A. Asundi, “Diffraction image in an optical microscope: application to detection of birefringence,” Opt. Eng. 41, 751–759 (2002).
[CrossRef]

Ferrari, J. A.

J. A. Ferrari and E. M. Frins, “Single element interferometer,” Opt. Commun. 279, 235–239 (2007).
[CrossRef]

Fontaine, M.

W. J. Bock, W. Urbánczyk, and M. Fontaine, “Characterization of highly birefringent optical fibers using interferometric techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

Frins, E. M.

J. A. Ferrari and E. M. Frins, “Single element interferometer,” Opt. Commun. 279, 235–239 (2007).
[CrossRef]

Gates, J. W.

J. W. Gates, “Reverse shearing interferometry,” Nature 176, 359–360 (1955).
[CrossRef]

Ghosh, N.

N. Ghosh and K. Bhattacharya, “Cube beam-splitter interferometer for phase shifting interferometry,” J. Opt. 38, 191–198 (2009).
[CrossRef]

N. Ghosh, A. K. Chakraborty, and K. Bhattacharya, “A single element birefringence-sensitive interferometer,” J. Opt. 37, 147–152 (2008).

N. Ghosh, Y. Otani, and K. Bhattacharya, “Generation of collinearly propagating orthogonally polarized beams,” Optik; available online http://dx.doi.org/10.1016/j.ijleo.2010.07.021.

Hamza, A.

A. Hamza, T. Z. N. Sokkar, and M. A. Kabeel, “Interferometric determination of refractive indices and birefringence of fibres with irregular transverse sections,” J. Phys. D L19, L19–L20(1986).
[CrossRef]

Helminen, H. J.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Hyttinen, M. M.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Kabeel, M. A.

A. Hamza, T. Z. N. Sokkar, and M. A. Kabeel, “Interferometric determination of refractive indices and birefringence of fibres with irregular transverse sections,” J. Phys. D L19, L19–L20(1986).
[CrossRef]

Király, K.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

Kiviranta, I.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

Kosztáczky, B.

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Kovanen, V.

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Lapveteläinen, T.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Lee, S.-Y.

J.-F. Lin, T.-T. Liao, Y.-L. Lo, and S.-Y. Lee, “The optical linear birefringence measurement using Zeeman laser,” Opt. Commun. 274, 153–158 (2007).
[CrossRef]

Liao, T.-T.

J.-F. Lin, T.-T. Liao, Y.-L. Lo, and S.-Y. Lee, “The optical linear birefringence measurement using Zeeman laser,” Opt. Commun. 274, 153–158 (2007).
[CrossRef]

Lin, J.-F.

J.-F. Lin, T.-T. Liao, Y.-L. Lo, and S.-Y. Lee, “The optical linear birefringence measurement using Zeeman laser,” Opt. Commun. 274, 153–158 (2007).
[CrossRef]

Lo, Y.-L.

J.-F. Lin, T.-T. Liao, Y.-L. Lo, and S.-Y. Lee, “The optical linear birefringence measurement using Zeeman laser,” Opt. Commun. 274, 153–158 (2007).
[CrossRef]

McIntyre, G. R.

G. R. McIntyre and A. Neureuther, “Phase shift mask interferometric birefringent monitor,” J. Vac. Sci. Technol. B 24, 2808–2814 (2006).
[CrossRef]

Módis, L.

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Neureuther, A.

G. R. McIntyre and A. Neureuther, “Phase shift mask interferometric birefringent monitor,” J. Vac. Sci. Technol. B 24, 2808–2814 (2006).
[CrossRef]

Oshizawa, T.

Y. Otani, T. Shimada, T. Oshizawa, and N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Otani, Y.

Y. Otani, T. Shimada, T. Oshizawa, and N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

N. Ghosh, Y. Otani, and K. Bhattacharya, “Generation of collinearly propagating orthogonally polarized beams,” Optik; available online http://dx.doi.org/10.1016/j.ijleo.2010.07.021.

Redner, A. S.

A. S. Redner, “Photoelastic measurement by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

Shimada, T.

Y. Otani, T. Shimada, T. Oshizawa, and N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Sokkar, T. Z. N.

A. Hamza, T. Z. N. Sokkar, and M. A. Kabeel, “Interferometric determination of refractive indices and birefringence of fibres with irregular transverse sections,” J. Phys. D L19, L19–L20(1986).
[CrossRef]

Stein, R. S.

R. S. Stein, “Measurement of birefringence in biaxially oriented films,” J. Polym. Sci. 24, 383–386 (1957).
[CrossRef]

Takács, P.

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Umeda, N.

Y. Otani, T. Shimada, T. Oshizawa, and N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Urbánczyk, W.

W. J. Bock, W. Urbánczyk, and M. Fontaine, “Characterization of highly birefringent optical fibers using interferometric techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

Zhao, B.

B. Zhao, Z. Cao, R. Fang, and A. Asundi, “Diffraction image in an optical microscope: application to detection of birefringence,” Opt. Eng. 41, 751–759 (2002).
[CrossRef]

Ann. Rheum. Dis. (1)

J. P. A. Arokoski, M. M. Hyttinen, T. Lapveteläinen, P. Takács, B. Kosztáczky, L. Módis, V. Kovanen, and H. J. Helminen, “Decreased direfringence of the superficial zone collagen network in the canine knee (stifle) articular cartilage after long distance running training, detected by quantitative polarised light microscopy,” Ann. Rheum. Dis. 55, 253–264 (1996).
[CrossRef] [PubMed]

Appl. Opt. (1)

Exp. Mech. (1)

A. S. Redner, “Photoelastic measurement by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

Histochem. J. (1)

K. Király, M. M. Hyttinen, T. Lapveteläinen, M. Elo, I. Kiviranta, J. Dobai, L. Módis, H. J. Helminen, and J. P. A. Arokoski, “Specimen preparation and quantification of collagen birefringence in unstained sections of articular cartilage using image analysis and polarizing light microscopy,” Histochem. J. 29, 317–327 (1997).
[CrossRef] [PubMed]

IEEE Trans. Instrum. Meas. (1)

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the single-element birefringent interferometer.

Fig. 2
Fig. 2

Surface map of a uniform quarter-wave plate with (a)  φ = 100 ° and (b)  δ = 90 ° .

Fig. 3
Fig. 3

A sample comprising three quarter-wave plate sheets in different orientations of φ pasted on a glass plate.

Fig. 4
Fig. 4

(a) and (b) show the two-dimensional and three-dimensional spatial variation of the fast axis, respectively, over the sample plane; the dark regions indicate the absence of any birefringent material.

Fig. 5
Fig. 5

(a) and (b) show the two-dimensional and three-dimensional spatial variation of the birefringence, respectively, over the sample plane. Note that the three zones containing the sample shows a uniform value of δ = 90 ° since all of them are quarter-wave plates.

Equations (29)

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W ( δ , ϕ ) = ( cos ( δ / 2 ) + i sin ( δ / 2 ) cos ( 2 φ ) i sin ( δ / 2 ) sin ( 2 φ ) i sin ( δ / 2 ) sin ( 2 φ ) cos ( δ / 2 ) i sin ( δ / 2 ) cos ( 2 φ ) ) .
H ( π , θ ) = ( i cos ( 2 θ ) i sin ( 2 θ ) i sin ( 2 θ ) i cos ( 2 θ ) ) ,
P ( γ ) = ( cos 2 γ cos γ sin γ cos γ sin γ sin 2 γ ) .
E x = P ( γ ) W ( δ , φ ) H ( π , θ ) ( 1 0 ) ,
E y = P ( γ ) W ( δ , φ ) H ( π , θ ) ( 0 1 ) .
E x = P ( γ ) ( E x 11 E x 21 ) ,
E x 11 = i cos ( δ / 2 ) cos ( 2 θ ) sin ( δ / 2 ) cos ( 2 θ 2 φ ) ,
E x 21 = i cos ( δ / 2 ) sin ( 2 θ ) + sin ( δ / 2 ) sin ( 2 θ 2 φ ) .
a x = [ i cos ( δ / 2 ) cos ( 2 θ ) sin ( δ / 2 ) cos ( 2 θ 2 φ ) ] cos γ + [ i cos ( δ / 2 ) sin ( 2 θ ) + sin ( δ / 2 ) sin ( 2 θ 2 φ ) ] sin γ = sin ( δ / 2 ) cos ( 2 θ 2 φ + γ ) + i cos ( δ / 2 ) cos ( 2 θ γ ) ,
Δ x = tan 1 [ cos ( δ / 2 ) cos ( 2 θ γ ) sin ( δ / 2 ) cos ( 2 θ 2 φ + γ ) ] .
a y = ( E y 11 cos γ + E y 21 sin γ ) = sin ( δ / 2 ) sin ( 2 θ 2 φ + γ ) + i cos ( δ / 2 ) sin ( 2 θ γ ) = r y e i Δ y ,
Δ y = tan [ cos ( δ / 2 ) sin ( 2 θ γ ) sin ( δ / 2 ) sin ( 2 θ 2 φ + γ ) ] .
a x = r x exp ( i Δ x ) ,
a y = r y exp { i ( Δ y + ) } = r y exp ( i Δ y ) ,
a y = [ sin ( δ / 2 ) sin ( 2 θ 2 φ + γ ) + i cos ( δ / 2 ) sin ( 2 θ γ ) ] [ cos ( ε ) + i sin ( ε ) ] = [ cos ( δ / 2 ) sin ( 2 θ γ ) sin ε sin ( δ / 2 ) sin ( 2 θ 2 φ + γ ) cos ε ] + i [ cos ( δ / 2 ) sin ( 2 θ γ ) cos ε sin ( δ / 2 ) sin ( 2 θ 2 φ + γ ) sin ε ] ,
Δ y = tan 1 [ [ cos ( δ / 2 ) sin ( 2 θ γ ) cos ε sin ( δ / 2 ) sin ( 2 θ 2 φ + γ ) sin ε ] [ cos ( δ / 2 ) sin ( 2 θ γ ) sin ε sin ( δ / 2 ) sin ( 2 θ 2 φ + γ ) cos ε ] ] .
I = | a x + a y | 2 = | r x exp ( i Δ x ) + r y exp ( i Δ y / ) | 2 = r x 2 + r y 2 + 2 r x r y cos Δ ,
I = 1 + 1 2 sin ( 2 γ + 4 θ 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] 1 2 sin ( 2 γ 4 θ ) cos ( ε ) [ 1 + cos ( δ ) ] sin ( 2 γ 2 φ ) sin ( ε ) sin δ .
I no_sample = 1 sin ( 2 γ 4 θ ) cos ( ε ) .
I 1 ( θ = 0 , γ = 0 ) = 1 1 2 sin ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] + sin ( 2 φ ) sin ( ε ) sin ( δ ) ,
I 2 ( θ = 0 , γ = π / 4 ) = 1 + 1 2 cos ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] 1 2 cos ( ε ) [ 1 + cos ( δ ) ] cos ( 2 φ ) sin ( ε ) sin ( δ ) ,
I 3 ( θ = 0 , γ = π / 2 ) = 1 + 1 2 sin ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] ,
I 4 ( θ = 0 , γ = 3 π / 4 ) = 1 + 1 2 cos ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] + 1 2 cos ( ε ) [ 1 + cos ( δ ) ] + cos ( 2 φ ) sin ( ε ) sin ( δ ) ,
I 5 ( θ = π / 4 , γ = 0 ) = 1 + 1 2 sin ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] + sin ( 2 φ ) sin ( ε ) sin ( δ ) ,
I 6 ( θ = π / 4 , γ = π / 4 ) = 1 1 2 cos ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] + 1 2 cos ( ε ) [ 1 + cos ( δ ) ] cos ( 2 φ ) sin ( ε ) sin ( δ ) ,
I 7 ( θ = π / 4 , γ = π / 2 ) = 1 1 2 sin ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] sin ( 2 φ ) sin ( ε ) sin ( δ ) ,
I 8 ( θ = π / 4 , γ = 3 π / 4 ) = 1 + 1 2 cos ( 4 φ ) cos ( ε ) [ 1 cos ( δ ) ] 1 2 cos ( ε ) [ 1 + cos ( δ ) ] + cos ( 2 φ ) sin ( ε ) sin ( δ ) .
tan ( 2 φ ) = 2 ( I 7 I 1 ) ( I 4 I 2 ) + ( I 8 I 6 ) φ = 1 2 tan 1 [ 2 ( I 7 I 1 ) ( I 4 I 2 ) + ( I 8 I 6 ) ] .
δ = 2 tan 1 { ( I 1 + I 7 ) ( I 3 + I 5 ) [ 1 + tan 2 ( 2 φ ) ] ( I 1 + I 7 ) ( I 3 + I 5 ) [ 1 tan 2 ( 2 φ ) ] 2 ( I 4 + I 6 ) ( I 2 + I 8 ) tan ( 2 φ ) } 1 / 2 .

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