Abstract

We propose image processing algorithms for measuring two-dimensional distributions of linear birefringence using a pair of variable retarders. Several algorithms that use between two and five recorded frames allow us to optimize measurements for speed, sensitivity, and accuracy. We show images of asters, which consist of radial arrays of microtubule polymers recorded with a polarized light microscope equipped with a universal compensator. Our experimental results confirm our theoretical expectations. The lowest noise level of 0.036 nm was obtained when we used the five-frame technique and four-frame algorithm without extinction setting. The two-frame technique allows us to increase the speed of measurement with acceptable image quality.

© 2003 Optical Society of America

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References

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  1. S. Inoué, Video Microscopy (Plenum, New York, 1986).
  2. S. Inoué, R. Oldenbourg, “Microscopes,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, pp. 17.1–17.52.
  3. M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurement of 2D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
    [CrossRef]
  4. Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  5. J. L. Pezzanitti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
    [CrossRef]
  6. A. L. Bajor, “Automated polarimeter-macroscope for optical mapping of birefringence, azimuths, and transmission in large area wafer. Part I. Theory of the measurement,” Rev. Sci. Instrum. 66, 2977–2990 (1995).
    [CrossRef]
  7. A. M. Glazer, J. G. Lewis, W. Kaminsky, “An automatical optical imaging system for birefringent media,” Proc. R. Soc. London Ser. A 452, 2751–2765 (1996).
    [CrossRef]
  8. Y. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: measurement principle,” Appl. Opt. 38, 2225–2231 (1999).
    [CrossRef]
  9. Y. Zhu, T. Takada, Y. Murooka, “Two-dimensional optical measurement techniques based on optical birefringence effects,” Opt. Eng. 41, 3183–3192 (2002).
    [CrossRef]
  10. M. I. Shribak, “Autocollimating detectors of birefringence,” in International Conference on Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 805–813 (1996).
    [CrossRef]
  11. M. I. Shribak, Y. Otani, T. Yoshizawa, “Return-path polarimeter for two-dimensional birefringence distribution measurement,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 144–149 (1999).
    [CrossRef]
  12. W. H. Yeh, J. Carrier, M. Mansuripur, “Polarization microscopy of magnetic domains for magneto-optical disks,” Appl. Opt. 38, 3749–3758 (1999).
    [CrossRef]
  13. G. Mei, R. Oldenbourg, “Fast imaging polarimetry with precision universal compensator,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 29–39 (1994).
    [CrossRef]
  14. R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. (Oxford) 180, 140–147 (1995).
    [CrossRef]
  15. R. Oldenbourg, G. Mei, “Polarized light microscopy,” U.S. patent5,521,705 (12May1994). Patent is licensed to Cambridge Research and Instrumentation, Inc., Woburn, Mass.; http://www.cri-inc.com .
  16. T. Yamaguchi, S. Yoshida, A. Kinbara, “Continuous ellipsometric determination of the optical constants and thickness of a silver film during deposition,” Jpn. J. Appl. Phys. 8, 559–567 (1969).
    [CrossRef]
  17. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [CrossRef]
  18. A. M. Glazer, J. Cosier, “Method and apparatus for indicating optical anisotropy,” UK patent application2,310,925 (7February1997).
  19. M. Shribak, S. Inoué, R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high NA lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
    [CrossRef]
  20. R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
    [CrossRef]
  21. The NIH Image software is available at http://rsb.info.nih.gov/NIH-image .
  22. B. J. Schnackenberg, R. E. Palazzo, “Reconstitution of centrosome microtubule nucleation in Spisula,” Methods Cell Biol. 67, 149–165 (2001).
    [CrossRef] [PubMed]

2002 (2)

M. Shribak, S. Inoué, R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high NA lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Y. Zhu, T. Takada, Y. Murooka, “Two-dimensional optical measurement techniques based on optical birefringence effects,” Opt. Eng. 41, 3183–3192 (2002).
[CrossRef]

2001 (1)

B. J. Schnackenberg, R. E. Palazzo, “Reconstitution of centrosome microtubule nucleation in Spisula,” Methods Cell Biol. 67, 149–165 (2001).
[CrossRef] [PubMed]

1999 (2)

1998 (1)

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef]

1996 (1)

A. M. Glazer, J. G. Lewis, W. Kaminsky, “An automatical optical imaging system for birefringent media,” Proc. R. Soc. London Ser. A 452, 2751–2765 (1996).
[CrossRef]

1995 (3)

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. (Oxford) 180, 140–147 (1995).
[CrossRef]

J. L. Pezzanitti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

A. L. Bajor, “Automated polarimeter-macroscope for optical mapping of birefringence, azimuths, and transmission in large area wafer. Part I. Theory of the measurement,” Rev. Sci. Instrum. 66, 2977–2990 (1995).
[CrossRef]

1994 (1)

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

1980 (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

1969 (1)

T. Yamaguchi, S. Yoshida, A. Kinbara, “Continuous ellipsometric determination of the optical constants and thickness of a silver film during deposition,” Jpn. J. Appl. Phys. 8, 559–567 (1969).
[CrossRef]

Bajor, A. L.

A. L. Bajor, “Automated polarimeter-macroscope for optical mapping of birefringence, azimuths, and transmission in large area wafer. Part I. Theory of the measurement,” Rev. Sci. Instrum. 66, 2977–2990 (1995).
[CrossRef]

Carrier, J.

Chipman, R. A.

J. L. Pezzanitti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Cosier, J.

A. M. Glazer, J. Cosier, “Method and apparatus for indicating optical anisotropy,” UK patent application2,310,925 (7February1997).

Glazer, A. M.

A. M. Glazer, J. G. Lewis, W. Kaminsky, “An automatical optical imaging system for birefringent media,” Proc. R. Soc. London Ser. A 452, 2751–2765 (1996).
[CrossRef]

A. M. Glazer, J. Cosier, “Method and apparatus for indicating optical anisotropy,” UK patent application2,310,925 (7February1997).

Hauge, P. S.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Inoué, S.

M. Shribak, S. Inoué, R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high NA lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

S. Inoué, Video Microscopy (Plenum, New York, 1986).

S. Inoué, R. Oldenbourg, “Microscopes,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, pp. 17.1–17.52.

Ishikawa, T.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurement of 2D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Kaminsky, W.

A. M. Glazer, J. G. Lewis, W. Kaminsky, “An automatical optical imaging system for birefringent media,” Proc. R. Soc. London Ser. A 452, 2751–2765 (1996).
[CrossRef]

Kinbara, A.

T. Yamaguchi, S. Yoshida, A. Kinbara, “Continuous ellipsometric determination of the optical constants and thickness of a silver film during deposition,” Jpn. J. Appl. Phys. 8, 559–567 (1969).
[CrossRef]

Koyama, T.

Lewis, J. G.

A. M. Glazer, J. G. Lewis, W. Kaminsky, “An automatical optical imaging system for birefringent media,” Proc. R. Soc. London Ser. A 452, 2751–2765 (1996).
[CrossRef]

Mansuripur, M.

Mei, G.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. (Oxford) 180, 140–147 (1995).
[CrossRef]

G. Mei, R. Oldenbourg, “Fast imaging polarimetry with precision universal compensator,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 29–39 (1994).
[CrossRef]

R. Oldenbourg, G. Mei, “Polarized light microscopy,” U.S. patent5,521,705 (12May1994). Patent is licensed to Cambridge Research and Instrumentation, Inc., Woburn, Mass.; http://www.cri-inc.com .

Murooka, Y.

Y. Zhu, T. Takada, Y. Murooka, “Two-dimensional optical measurement techniques based on optical birefringence effects,” Opt. Eng. 41, 3183–3192 (2002).
[CrossRef]

Y. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: measurement principle,” Appl. Opt. 38, 2225–2231 (1999).
[CrossRef]

Noguchi, M.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurement of 2D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Ohno, M.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurement of 2D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Oldenbourg, R.

M. Shribak, S. Inoué, R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high NA lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef]

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. (Oxford) 180, 140–147 (1995).
[CrossRef]

G. Mei, R. Oldenbourg, “Fast imaging polarimetry with precision universal compensator,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 29–39 (1994).
[CrossRef]

R. Oldenbourg, G. Mei, “Polarized light microscopy,” U.S. patent5,521,705 (12May1994). Patent is licensed to Cambridge Research and Instrumentation, Inc., Woburn, Mass.; http://www.cri-inc.com .

S. Inoué, R. Oldenbourg, “Microscopes,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, pp. 17.1–17.52.

Otani, Y.

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

M. I. Shribak, Y. Otani, T. Yoshizawa, “Return-path polarimeter for two-dimensional birefringence distribution measurement,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 144–149 (1999).
[CrossRef]

Palazzo, R. E.

B. J. Schnackenberg, R. E. Palazzo, “Reconstitution of centrosome microtubule nucleation in Spisula,” Methods Cell Biol. 67, 149–165 (2001).
[CrossRef] [PubMed]

Pezzanitti, J. L.

J. L. Pezzanitti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Salmon, E. D.

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef]

Schnackenberg, B. J.

B. J. Schnackenberg, R. E. Palazzo, “Reconstitution of centrosome microtubule nucleation in Spisula,” Methods Cell Biol. 67, 149–165 (2001).
[CrossRef] [PubMed]

Shimada, T.

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

Shribak, M.

M. Shribak, S. Inoué, R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high NA lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Shribak, M. I.

M. I. Shribak, Y. Otani, T. Yoshizawa, “Return-path polarimeter for two-dimensional birefringence distribution measurement,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 144–149 (1999).
[CrossRef]

M. I. Shribak, “Autocollimating detectors of birefringence,” in International Conference on Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 805–813 (1996).
[CrossRef]

Tachihara, S.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurement of 2D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Takada, T.

Y. Zhu, T. Takada, Y. Murooka, “Two-dimensional optical measurement techniques based on optical birefringence effects,” Opt. Eng. 41, 3183–3192 (2002).
[CrossRef]

Y. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: measurement principle,” Appl. Opt. 38, 2225–2231 (1999).
[CrossRef]

Tran, P. T.

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef]

Umeda, N.

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

Yamaguchi, T.

T. Yamaguchi, S. Yoshida, A. Kinbara, “Continuous ellipsometric determination of the optical constants and thickness of a silver film during deposition,” Jpn. J. Appl. Phys. 8, 559–567 (1969).
[CrossRef]

Yeh, W. H.

Yoshida, S.

T. Yamaguchi, S. Yoshida, A. Kinbara, “Continuous ellipsometric determination of the optical constants and thickness of a silver film during deposition,” Jpn. J. Appl. Phys. 8, 559–567 (1969).
[CrossRef]

Yoshizawa, T.

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

M. I. Shribak, Y. Otani, T. Yoshizawa, “Return-path polarimeter for two-dimensional birefringence distribution measurement,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 144–149 (1999).
[CrossRef]

Zhu, Y.

Y. Zhu, T. Takada, Y. Murooka, “Two-dimensional optical measurement techniques based on optical birefringence effects,” Opt. Eng. 41, 3183–3192 (2002).
[CrossRef]

Y. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: measurement principle,” Appl. Opt. 38, 2225–2231 (1999).
[CrossRef]

Appl. Opt. (2)

Biophys. J. (1)

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef]

J. Microsc. (Oxford) (1)

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. (Oxford) 180, 140–147 (1995).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Yamaguchi, S. Yoshida, A. Kinbara, “Continuous ellipsometric determination of the optical constants and thickness of a silver film during deposition,” Jpn. J. Appl. Phys. 8, 559–567 (1969).
[CrossRef]

Methods Cell Biol. (1)

B. J. Schnackenberg, R. E. Palazzo, “Reconstitution of centrosome microtubule nucleation in Spisula,” Methods Cell Biol. 67, 149–165 (2001).
[CrossRef] [PubMed]

Opt. Eng. (4)

Y. Zhu, T. Takada, Y. Murooka, “Two-dimensional optical measurement techniques based on optical birefringence effects,” Opt. Eng. 41, 3183–3192 (2002).
[CrossRef]

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

J. L. Pezzanitti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

M. Shribak, S. Inoué, R. Oldenbourg, “Polarization aberrations caused by differential transmission and phase shift in high NA lenses: theory, measurement and rectification,” Opt. Eng. 41, 943–954 (2002).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

A. M. Glazer, J. G. Lewis, W. Kaminsky, “An automatical optical imaging system for birefringent media,” Proc. R. Soc. London Ser. A 452, 2751–2765 (1996).
[CrossRef]

Rev. Sci. Instrum. (1)

A. L. Bajor, “Automated polarimeter-macroscope for optical mapping of birefringence, azimuths, and transmission in large area wafer. Part I. Theory of the measurement,” Rev. Sci. Instrum. 66, 2977–2990 (1995).
[CrossRef]

Surf. Sci. (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Other (9)

A. M. Glazer, J. Cosier, “Method and apparatus for indicating optical anisotropy,” UK patent application2,310,925 (7February1997).

The NIH Image software is available at http://rsb.info.nih.gov/NIH-image .

M. I. Shribak, “Autocollimating detectors of birefringence,” in International Conference on Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 805–813 (1996).
[CrossRef]

M. I. Shribak, Y. Otani, T. Yoshizawa, “Return-path polarimeter for two-dimensional birefringence distribution measurement,” in Polarization: Measurement, Analysis, and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 144–149 (1999).
[CrossRef]

S. Inoué, Video Microscopy (Plenum, New York, 1986).

S. Inoué, R. Oldenbourg, “Microscopes,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, pp. 17.1–17.52.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurement of 2D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

G. Mei, R. Oldenbourg, “Fast imaging polarimetry with precision universal compensator,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 29–39 (1994).
[CrossRef]

R. Oldenbourg, G. Mei, “Polarized light microscopy,” U.S. patent5,521,705 (12May1994). Patent is licensed to Cambridge Research and Instrumentation, Inc., Woburn, Mass.; http://www.cri-inc.com .

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Figures (5)

Fig. 1
Fig. 1

Schematics of the polarized light microscope with a liquid-crystal universal compensator in the illumination path (first configuration) and in the imaging path (second configuration). α and β are the retardance of the liquid-crystal plates LCA and LCB, respectively; λ/4 is a quarter-wave plate; and P and A are the linear polarizer and analyzer.

Fig. 2
Fig. 2

Settings of the probe beam on the Poincaré sphere. ∑0 is the setting with right-circular polarization; ∑1, ∑2, ∑3, and ∑4 are settings with elliptical polarizations. Diagrams show orientation and shape of vibration ellipsis with components Ex and Ey.

Fig. 3
Fig. 3

Five intensity images of aster (top) and background (bottom) at polarization settings ∑0, ∑1, ∑2, ∑3, and ∑4 of the universal compensator. The images were contrast enhanced for better visibility.

Fig. 4
Fig. 4

Reconstituted aster consisting of microtubules that were polymerized off a spherical organizing center called centrosome. Top left image: retardance magnitude image calculated with the five-frame algorithm with background correction. Bright microtubule arrays and bundles radiate from the dark centrosome. White corresponds to 1.2-nm birefringence retardation; black corresponds to zero birefringence. The right portion of the image is contrast enhanced to improve the visibility of fine fibers and the background noise. Bottom left image: orientation of the birefringence axis recorded with the PolScope. Black corresponds to horizontal orientation; increasing brightness corresponds to increasing orientation angles measured from the horizontal direction. Inset: enlarged region showing a microtubule bundle. The magnitude image is overlaid by lines indicating the orientation of the measured birefringence axis in each pixel. In pixels representing the microtubule bundle, the birefringence axis (high-refractive-index axis) is parallel to the bundle.

Fig. 5
Fig. 5

Magnitude images of the aster that were calculated with (a) the two-frame, (b) the original four-frame, (c) the five-frame algorithms. In all cases a background correction was applied. White corresponds to 1.2- or 0.6-nm birefringence retardation, as shown in the areas, and black corresponds to zero birefringence. Areas 1, 2, and 3 in (c) were used for measuring the noise level. Here white corresponds to 0.3-nm retardance.

Tables (1)

Tables Icon

Table 1 Retardance Noise Levels Generated by the Different Algorithms

Equations (28)

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

Iα, β, x, y=12 τx, yImaxx, y1+sin α cos β cos Δx, y-sin α sin β cos 2ϕx, y×sin Δx, y+cos α sin 2ϕx, ysin Δx, y+Iminx, y.
E=cosα2exp-i β2-i sinα2expi β2.
tan 2γ=tan α sin β, sin 2=-sin α cos β,
tan =ba.
=45°-χ/2,
ba=1-π180° χ,
I0x, y=12 τx, yImax1-cos Δx, y+Iminx, y, I1x, y=12 τx, yImax1-cos χ cos Δx, y+sin χ sin 2ϕx, ysin Δx, y+Iminx, y, I2x, y=12 τx, yImax1-cos χ cos Δx, y-sin χ sin 2ϕx, ysin Δx, y+Iminx, y, I3x, y=12 τx, yImax1-cos χ cos Δx, y-sin χ cos 2ϕx, ysin Δx, y+Iminx, y, I4x, y=12 τx, yImax1-cos χ cos Δx, y+sin χ cos 2ϕx, ysin Δx, y+Iminx, y.
I1-I2=τImax sin χ sin 2ϕ sin Δ, I1+I2-2I3=τImax sin χ cos 2ϕ sin Δ, I1+I2-2I0=τImax1-cos χcos Δ.
0°Δ90° when I1+I2-2I00, 90°<Δ<180° when I1+I2-2I0<0, 0°ϕ90° when I1-I20, 90°<ϕ<180° when I1-I2<0.
AI1-I2I1+I2-2I0tanχ2=sin 2ϕ tan Δ, BI1+I2-2I3I1+I2-2I0tanχ2=cos 2ϕ tan Δ.
Δ=arctanA2+B21/2 when I1+I2-2I00, Δ=180°-arctanA2+B21/2 when I1+I2-2I0<0, ϕ=12arctanAB.
Mo·Mb, or Mb·Mo,
Mo=cosΔo2+i sinΔo2cos 2ϕoi sinΔo2sin 2ϕoi sinΔo2sin 2ϕocosΔo2-i sinΔo2cos 2ϕo1+i2 Δo cos 2ϕoi2 Δo sin 2ϕoi2 Δo sin 2ϕo1-i2 Δo cos 2ϕo, Mb=cosΔb2+i sinΔb2cos 2ϕbi sinΔb2sin 2ϕbi sinΔb2sin 2ϕbcosΔb2-i sinΔb2cos 2ϕb1+i2 Δb cos 2ϕbi2 Δb sin 2ϕbi2 Δb sin 2ϕb1-i2 Δb cos 2ϕb.
Mo·Mb=Mb·Mo=1+i2Δo cos 2ϕo+Δb cos 2ϕbi2Δo sin 2ϕo+Δb sin 2ϕbi2Δo sin 2ϕo+Δb sin 2ϕb1-i2Δo cos 2ϕo+Δb cos 2ϕb.
Me=1+i2 Δe cos 2ϕei2 Δe sin 2ϕei2 Δe sin 2ϕe1-i2 Δe cos 2ϕe.
Δe sin 2ϕe=Δo sin 2ϕo+Δb sin 2ϕb, Δe cos 2ϕe=Δo cos 2ϕo+Δb cos 2ϕb.
Δo sin 2ϕo=Δe sin 2ϕe-Δb sin 2ϕb, Δo cos 2ϕo=Δe cos 2ϕe-Δb cos 2ϕb.
AI1-I3I1+I2=12sin χ1-cos χ cos Δsin2ϕ+45°sin Δ, BI2-I3I1+I2=12sin χ1-cos χ cos Δcos2ϕ+45°sin Δ.
Δ=2 arctan2A2+B21/21+1-2A2+B21/2tanχ2, ϕ=12arctanAB-22.5°.
AI1-I2I1+I2-2I0tanχ2=sin 2ϕ tan Δ, BI4-I3I4+I3-2I0tanχ2=cos 2ϕ tan Δ.
Δ=arctanA2+B21/2 when I1+I2-2I00, Δ=180°-arctanA2+B21/2when I1+I2-2I0<0,ϕ=12arctanAB.
AI1-I2I1+I2=sin χ1-cos χ cos Δsin 2ϕ sin Δ, BI4-I3I4+I3=sin χ1-cos χ cos Δcos 2ϕ sin Δ.
Δ=2 arctanA2+B21/21+1-A2+B21/2tanχ2, ϕ=12arctanAB.
I1x, y=12 Imax1-cos χ cos Δx, y+sin χ sin 2ϕx, ysin Δx, y+Iminx, y, I3x, y=12 Imax1-cos χ cos Δx, y-sin χ cos 2ϕx, ysin Δx, y+Iminx, y, Ibg1x, y=Imaxsin2χ2+Iminx, y.
AI1-Ibg1Ibg1tanχ2sin 2ϕ sin Δ, BI3-Ibg1Ibg1tanχ2cos 2ϕ sin Δ.
AI1-Ibg1Ibg1-Ibg0tanχ2sin 2ϕ sin Δ, BI3-Ibg3Ibg3-Ibg0tanχ2cos 2ϕ sin Δ.
Δ=arcsinA2+B21/2λ2πA2+B21/2, ϕ=12arctanAB.
Δ=arcsinA2+B21/2.

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