Abstract

This study demonstrates a new method for simultaneously measuring both the angle of the principal axis and the phase retardation of the linear birefringence in optical materials. We used a circular common-path interferometer (polariscope) as the basic structure modulated by an electro-optic (EO) modulator. An algorithm was developed to simultaneously measure the principal axis and the phase retardation of a λ/4 or λ/8 plate as a sample. In the case of a λ/4 plate, the average absolute error of the principal axis is approximately 3.77°, and that of the phase retardation is approximately 1.03° (1.09%). The retardation error is within the 5% uncertainty range of a commercial wave plate. Fortunately, the nonlinear error caused by the reflection phase retardation of the beam splitter dose not appear in the new system. Therefore the error could be attributed to misalignment and defects in the EO modulator or the other optical components. As for the repeatability of this new common-path heterodyne interferometer, the average deviation for the principal axis is 0.186° and the phase retardation is 0.356°. For the stability, the average deviation for the principal axis is 0.405° and the phase retardation is 0.635°. The resolution of this new system is estimated to be ∼0.5°, and the principal axis and phase retardation could be measured up to π and 2π, respectively, without ambiguity.

© 2004 Optical Society of America

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References

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  1. H. B. Serreze, R. B. Goldner, “A phase-sensitive technique for measuring small birefringence changes,” Rev. Sci. Instrum. 45, 1613–1614 (1974).
    [CrossRef]
  2. Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polymer Commun. 24, 240–244 (1983).
  3. C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
    [CrossRef]
  4. B. D. Cameron, G. L. Cote, “Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach,” IEEE Trans. Biomed. Eng. 44, 1221–1227 (1997).
    [CrossRef] [PubMed]
  5. A. Márquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
    [CrossRef]
  6. Y. L. Lo, P. F. Hsu, “Birefringence measurements by an electro-optic modulator using a new heterodyne scheme,” Opt. Eng. 41, 2764–2767 (2002).
    [CrossRef]
  7. B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
    [CrossRef]
  8. S. Ohkubo, N. Umeda, “Near-field scanning optical microscope based on fast birefringence measurements,” Sensors Mater. 13, 433–443 (2001).
  9. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  10. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Standards, Washington, D.C., 1963).
  11. K. B. Rochford, A. H. Rose, C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World, 223–227 (1997).
  12. Y. Xie, Y. Z. Wu, “Zeeman laser interferometer errors for high-precision measurements,” Appl. Opt. 31, 881–884 (1992).
    [CrossRef] [PubMed]
  13. A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
    [CrossRef]
  14. W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
    [CrossRef]
  15. Y. Bitou, “Polarization mixing error reduction in a two-beam interferometer,” Opt. Rev. 9, 227–229 (2002).
    [CrossRef]
  16. H. Z. Hu, “Polarization heterodyne interferometry using a simple rotating analyzer. 1: theory and analysis,” Appl. Opt. 22, 2052–2056 (1983).
    [CrossRef]
  17. J. Y. Lin, D. C. Su, “A new type of optical heterodyne polarimeter,” Meas. Sci. Technol. 14, 55–58 (2003).
    [CrossRef]
  18. C. K. Lee, T. W. Wu , “Differential laser interferometer for nanometer displacement measurements, AIAA J. 33, 1675–1680 (1995).

2003 (1)

J. Y. Lin, D. C. Su, “A new type of optical heterodyne polarimeter,” Meas. Sci. Technol. 14, 55–58 (2003).
[CrossRef]

2002 (2)

Y. Bitou, “Polarization mixing error reduction in a two-beam interferometer,” Opt. Rev. 9, 227–229 (2002).
[CrossRef]

Y. L. Lo, P. F. Hsu, “Birefringence measurements by an electro-optic modulator using a new heterodyne scheme,” Opt. Eng. 41, 2764–2767 (2002).
[CrossRef]

2001 (2)

S. Ohkubo, N. Umeda, “Near-field scanning optical microscope based on fast birefringence measurements,” Sensors Mater. 13, 433–443 (2001).

A. Márquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

1999 (1)

B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

1997 (2)

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
[CrossRef]

B. D. Cameron, G. L. Cote, “Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach,” IEEE Trans. Biomed. Eng. 44, 1221–1227 (1997).
[CrossRef] [PubMed]

1995 (1)

C. K. Lee, T. W. Wu , “Differential laser interferometer for nanometer displacement measurements, AIAA J. 33, 1675–1680 (1995).

1992 (2)

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Y. Xie, Y. Z. Wu, “Zeeman laser interferometer errors for high-precision measurements,” Appl. Opt. 31, 881–884 (1992).
[CrossRef] [PubMed]

1990 (1)

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

1983 (2)

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polymer Commun. 24, 240–244 (1983).

H. Z. Hu, “Polarization heterodyne interferometry using a simple rotating analyzer. 1: theory and analysis,” Appl. Opt. 22, 2052–2056 (1983).
[CrossRef]

1974 (1)

H. B. Serreze, R. B. Goldner, “A phase-sensitive technique for measuring small birefringence changes,” Rev. Sci. Instrum. 45, 1613–1614 (1974).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Standards, Washington, D.C., 1963).

Bitou, Y.

Y. Bitou, “Polarization mixing error reduction in a two-beam interferometer,” Opt. Rev. 9, 227–229 (2002).
[CrossRef]

Bobroff, N.

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Cameron, B. D.

B. D. Cameron, G. L. Cote, “Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach,” IEEE Trans. Biomed. Eng. 44, 1221–1227 (1997).
[CrossRef] [PubMed]

Chang, J. G.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
[CrossRef]

Chang, M.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
[CrossRef]

Chou, C.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
[CrossRef]

Cote, G. L.

B. D. Cameron, G. L. Cote, “Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach,” IEEE Trans. Biomed. Eng. 44, 1221–1227 (1997).
[CrossRef] [PubMed]

Davis, J. A.

A. Márquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

Feng, C. M.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
[CrossRef]

Franich, D. J.

A. Márquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

Goldner, R. B.

H. B. Serreze, R. B. Goldner, “A phase-sensitive technique for measuring small birefringence changes,” Rev. Sci. Instrum. 45, 1613–1614 (1974).
[CrossRef]

Hanabusa, H.

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polymer Commun. 24, 240–244 (1983).

Hou, W.

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Hsu, P. F.

Y. L. Lo, P. F. Hsu, “Birefringence measurements by an electro-optic modulator using a new heterodyne scheme,” Opt. Eng. 41, 2764–2767 (2002).
[CrossRef]

Hu, H. Z.

Huang, Y. C.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
[CrossRef]

Lee, C. K.

C. K. Lee, T. W. Wu , “Differential laser interferometer for nanometer displacement measurements, AIAA J. 33, 1675–1680 (1995).

Lin, J. Y.

J. Y. Lin, D. C. Su, “A new type of optical heterodyne polarimeter,” Meas. Sci. Technol. 14, 55–58 (2003).
[CrossRef]

Lo, Y. L.

Y. L. Lo, P. F. Hsu, “Birefringence measurements by an electro-optic modulator using a new heterodyne scheme,” Opt. Eng. 41, 2764–2767 (2002).
[CrossRef]

Márquez, A.

A. Márquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

Oakberg, T. C.

B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Ohkubo, S.

S. Ohkubo, N. Umeda, “Near-field scanning optical microscope based on fast birefringence measurements,” Sensors Mater. 13, 433–443 (2001).

Rochford, K. B.

K. B. Rochford, A. H. Rose, C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World, 223–227 (1997).

Rose, A. H.

K. B. Rochford, A. H. Rose, C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World, 223–227 (1997).

Rosenbluth, A. E.

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Serreze, H. B.

H. B. Serreze, R. B. Goldner, “A phase-sensitive technique for measuring small birefringence changes,” Rev. Sci. Instrum. 45, 1613–1614 (1974).
[CrossRef]

Shindo, Y.

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polymer Commun. 24, 240–244 (1983).

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Standards, Washington, D.C., 1963).

Su, D. C.

J. Y. Lin, D. C. Su, “A new type of optical heterodyne polarimeter,” Meas. Sci. Technol. 14, 55–58 (2003).
[CrossRef]

Umeda, N.

S. Ohkubo, N. Umeda, “Near-field scanning optical microscope based on fast birefringence measurements,” Sensors Mater. 13, 433–443 (2001).

Wang, B.

B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Wang, C. M.

K. B. Rochford, A. H. Rose, C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World, 223–227 (1997).

Wilkening, G.

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Wu, T. W.

C. K. Lee, T. W. Wu , “Differential laser interferometer for nanometer displacement measurements, AIAA J. 33, 1675–1680 (1995).

Wu, Y. Z.

Xie, Y.

Yamauchi, M.

A. Márquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

AIAA J. (1)

C. K. Lee, T. W. Wu , “Differential laser interferometer for nanometer displacement measurements, AIAA J. 33, 1675–1680 (1995).

Appl. Opt. (2)

IEEE Trans. Biomed. Eng. (1)

B. D. Cameron, G. L. Cote, “Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach,” IEEE Trans. Biomed. Eng. 44, 1221–1227 (1997).
[CrossRef] [PubMed]

Meas. Sci. Technol. (1)

J. Y. Lin, D. C. Su, “A new type of optical heterodyne polarimeter,” Meas. Sci. Technol. 14, 55–58 (2003).
[CrossRef]

Opt. Commun. (2)

A. Márquez, M. Yamauchi, J. A. Davis, D. J. Franich, “Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer,” Opt. Commun. 190, 129–133 (2001).
[CrossRef]

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314–321 (1997).
[CrossRef]

Opt. Eng. (1)

Y. L. Lo, P. F. Hsu, “Birefringence measurements by an electro-optic modulator using a new heterodyne scheme,” Opt. Eng. 41, 2764–2767 (2002).
[CrossRef]

Opt. Rev. (1)

Y. Bitou, “Polarization mixing error reduction in a two-beam interferometer,” Opt. Rev. 9, 227–229 (2002).
[CrossRef]

Polymer Commun. (1)

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polymer Commun. 24, 240–244 (1983).

Precis. Eng. (2)

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Rev. Sci. Instrum. (2)

H. B. Serreze, R. B. Goldner, “A phase-sensitive technique for measuring small birefringence changes,” Rev. Sci. Instrum. 45, 1613–1614 (1974).
[CrossRef]

B. Wang, T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Sensors Mater. (1)

S. Ohkubo, N. Umeda, “Near-field scanning optical microscope based on fast birefringence measurements,” Sensors Mater. 13, 433–443 (2001).

Other (3)

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Standards, Washington, D.C., 1963).

K. B. Rochford, A. H. Rose, C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World, 223–227 (1997).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the measuring system.

Fig. 2
Fig. 2

Algorithm of the signal process.

Fig. 3
Fig. 3

Simulation results for the principal axis and the phase retardation.

Fig. 4
Fig. 4

Experimental results with a λ/4 wave plate as a sample.

Fig. 5
Fig. 5

Repeatability measurements of the new system: (a) with full scale on the Y axis; (b) with an enlarged scale on the Y axis.

Fig. 6
Fig. 6

Stability measurements of the new system: (a) with full scale on the Y axis; (b) with an enlarged scale on the Y axis.

Fig. 7
Fig. 7

Experimental results with a λ/8 wave plate as a sample.

Fig. 8
Fig. 8

Nonlinear error caused by misalignment of the EO modulator. (a) Error in principal angle α, (b) error in phase retardation β.

Equations (19)

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S=cosβ2+i cos2αsinβ2i sin2αsinβ2i sin2αsinβ2cosβ2-i cos2αsinβ2,
E1=P10°λ4-45°cosβ2+i cos2αsinβ2i sin2αsinβ2i sin2αsinβ2cosβ2-i cos2αsinβ2×λ445°cosΓ2-i sinΓ2-isinΓ2cosΓ2EO-45°P0(90°01He-Ne,
E2=P245°λ4-45°cosβ2+i cos2αsinβ2i sin2αsinβ2i sin2αsinβ2cosβ2-i cos2αsinβ2×λ445°cosΓ2-isinΓ2-isinΓ2cosΓ2EO-45°P090°01He-Ne.
I1=E1E1*=121-cosϕc sinωctcosβ-sinϕc sinωctsin2αsinβ,
I2=E2E2*=184-4 cosϕc sinωctcos2αsinβ-2 sinϕc sinωctsin4α+2 sinϕc sinωctsin4αcosβ
I1=-J1ϕcsinωctsin2αsinβ,
I2=-J2ϕccos2ωctcos2αsinβ,
I1=-12 J1ϕcsin2αsinβ,
I2=-12 J2ϕccos2αsinβ.
Itan2α=I1/I2=tan2α.
I5=-12 J2ϕcsin2αcosβ.
Itanβ=I1/I5=tanβ.
INo-Sample=sin2Γ=121-cos2Γ,
INo-Sample=12 J1ϕcsinϕdc.
EOε=R45°-εexpiΓ200exp-iΓ2R-45°+ε=cosΓ2+i sin2εsinΓ2-i cos2εsinΓ2-i cos2εsinΓ2cosΓ2-i sin2εsinΓ2.
α=arctanIε1Iε2,
Iε1=-12 J1ϕcsin2αsinβcos2ε, Iε2=-12 J2ϕc12cos2αsinβ+12cos2αsinβcos4ε+1+cosβ-sin4ε+cosβcos4αsin4ε,β=arctanIε1Iε5, Iε5=14 J2ϕc12sin4εsinβsin4α-cos4ε+1cosβsin2α.
BS=exp-iϕBS/200expiϕBS/2,
I2,BS=184-4 cosΓcos2α-ϕBSsinβ-sinΓ2 sin4α-ϕBS-sin4α-β-ϕBS-sinβ-ϕBS-sin4α+β-ϕBS+2 sinϕBS+sinβ+ϕBS

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