Abstract

An imaging dynamic photopolarimeter in an optical configuration P(α)SA(n) (polarizer-sample-analyzer) having harmonically rotating polarizer (P) and analyzer (A), is presented for studies of inhomogeneous birefringent media. The method is capable of simple and precise measuring of birefringence and optical rotation across a two-dimensional field. Although the basic principle upon which the method operates has been proposed earlier, the focus here is on its extension, applicable to a whole field, and immediate measurement of three optical parameters, which have direct physical meaning. Earlier versions of the instrument either operate with a single beam or measure components of Jones or Mueller matrices. The operation of the given method is consistent with a general idea of dynamic photopolarimetry by discrete Fourier analysis and entails digital procession of intensity images, obtained at nine equispaced positions of the polarizer and the analyzer during synchronous rotation of them. A technique of data acquisition is described, and a detailed error analysis is provided. Theoretical considerations are supplemented with testing measurements of a quartz phase plate.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. A. Patterson, Z. F. Wang, “Towards full-field automatic photoelastic analysis of complex components,” Strain 27(2), 49–56 (1991).
    [CrossRef]
  2. A. V. S. S. R. Sarma, S. A. Pillai, G. Subramanian, T. K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Exp. Mech. 31, 24–29 (1992).
    [CrossRef]
  3. T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
    [CrossRef]
  4. Yu. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  5. M. H. Chiu, C. D. Chen, D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1930 (1996).
    [CrossRef]
  6. A. D. Nurse, “Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping,” Appl. Opt. 36, 5781–5786 (1997).
    [CrossRef] [PubMed]
  7. J. A. Quiroga, A. Gonzalez-Cano, “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
    [CrossRef]
  8. J. F. deBoer, T. E. Milner, M. J. C. van Gemert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
    [CrossRef]
  9. T. W. Ng, “Derivation of retardation phase in computer-aided photoelasticity by using carrier fringe phase shifting,” Appl. Opt. 36, 8259–8263 (1997).
    [CrossRef]
  10. R. A. Tomlinson, E. A. Patterson, “Determination of characteristic parameters for integrated photoelasticity using phase stepping,” in Proceedings of the SEM Annual Meeting on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.
  11. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  12. J. L. Pezanniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
    [CrossRef]
  13. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  14. R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
    [CrossRef]
  15. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1997).
  16. R. A. Williams, A. H. Rose, C. M. Wang, “Rotating-polarizer polarimeter for accurate retardance measurement,” Appl. Opt. 36, 6466–6472 (1997).
    [CrossRef]
  17. J. L. Pezanniti, R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng. 34, 1569–1573 (1995).
    [CrossRef]
  18. D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
    [CrossRef]
  19. J. L. Pezanniti, R. A. Chipman, “Imaging polarimeters for optical metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 280–294 (1990).
  20. I. K. Kikoin, Tables of Physical Parameters (Atomizdat, Moscow, 1967).

1997 (5)

1996 (1)

M. H. Chiu, C. D. Chen, D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1930 (1996).
[CrossRef]

1995 (2)

J. L. Pezanniti, R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng. 34, 1569–1573 (1995).
[CrossRef]

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

1994 (1)

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

1992 (2)

A. V. S. S. R. Sarma, S. A. Pillai, G. Subramanian, T. K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Exp. Mech. 31, 24–29 (1992).
[CrossRef]

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

1991 (1)

E. A. Patterson, Z. F. Wang, “Towards full-field automatic photoelastic analysis of complex components,” Strain 27(2), 49–56 (1991).
[CrossRef]

1990 (1)

1978 (2)

R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
[CrossRef] [PubMed]

R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
[CrossRef] [PubMed]

R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1997).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1997).

Chen, C. D.

M. H. Chiu, C. D. Chen, D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1930 (1996).
[CrossRef]

Chipman, R. A.

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

J. L. Pezanniti, R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng. 34, 1569–1573 (1995).
[CrossRef]

D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
[CrossRef]

J. L. Pezanniti, R. A. Chipman, “Imaging polarimeters for optical metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 280–294 (1990).

Chiu, M. H.

M. H. Chiu, C. D. Chen, D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1930 (1996).
[CrossRef]

deBoer, J. F.

Goldstein, D. H.

Gonzalez-Cano, A.

Han, A.

T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
[CrossRef]

Kikoin, I. K.

I. K. Kikoin, Tables of Physical Parameters (Atomizdat, Moscow, 1967).

McMurray, T.

T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
[CrossRef]

Milner, T. E.

Nelson, J. S.

Ng, T. W.

Nurse, A. D.

Otani, Yu.

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

Patterson, E. A.

E. A. Patterson, Z. F. Wang, “Towards full-field automatic photoelastic analysis of complex components,” Strain 27(2), 49–56 (1991).
[CrossRef]

R. A. Tomlinson, E. A. Patterson, “Determination of characteristic parameters for integrated photoelasticity using phase stepping,” in Proceedings of the SEM Annual Meeting on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.

Pearce, J. A.

T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
[CrossRef]

Pezanniti, J. L.

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

J. L. Pezanniti, R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng. 34, 1569–1573 (1995).
[CrossRef]

J. L. Pezanniti, R. A. Chipman, “Imaging polarimeters for optical metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 280–294 (1990).

Pillai, S. A.

A. V. S. S. R. Sarma, S. A. Pillai, G. Subramanian, T. K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Exp. Mech. 31, 24–29 (1992).
[CrossRef]

Quiroga, J. A.

Rose, A. H.

Sarma, A. V. S. S. R.

A. V. S. S. R. Sarma, S. A. Pillai, G. Subramanian, T. K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Exp. Mech. 31, 24–29 (1992).
[CrossRef]

Shimada, T.

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

Su, D. C.

M. H. Chiu, C. D. Chen, D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1930 (1996).
[CrossRef]

Subramanian, G.

A. V. S. S. R. Sarma, S. A. Pillai, G. Subramanian, T. K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Exp. Mech. 31, 24–29 (1992).
[CrossRef]

Tomlinson, R. A.

R. A. Tomlinson, E. A. Patterson, “Determination of characteristic parameters for integrated photoelasticity using phase stepping,” in Proceedings of the SEM Annual Meeting on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.

Umeda, N.

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

van Gemert, M. J. C.

Varadan, T. K.

A. V. S. S. R. Sarma, S. A. Pillai, G. Subramanian, T. K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Exp. Mech. 31, 24–29 (1992).
[CrossRef]

Wang, C. M.

Wang, Z. F.

E. A. Patterson, Z. F. Wang, “Towards full-field automatic photoelastic analysis of complex components,” Strain 27(2), 49–56 (1991).
[CrossRef]

Williams, R. A.

Yoshizawa, T.

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

Appl. Opt. (5)

Exp. Mech. (1)

A. V. S. S. R. Sarma, S. A. Pillai, G. Subramanian, T. K. Varadan, “Computerized image processing for whole-field determination of isoclinics and isochromatics,” Exp. Mech. 31, 24–29 (1992).
[CrossRef]

J. Opt. Soc. Am. A (1)

M. H. Chiu, C. D. Chen, D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924–1930 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

J. L. Pezanniti, R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng. 34, 1569–1573 (1995).
[CrossRef]

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

Opt. Commun. (1)

R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (2)

Strain (1)

E. A. Patterson, Z. F. Wang, “Towards full-field automatic photoelastic analysis of complex components,” Strain 27(2), 49–56 (1991).
[CrossRef]

Other (5)

T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
[CrossRef]

R. A. Tomlinson, E. A. Patterson, “Determination of characteristic parameters for integrated photoelasticity using phase stepping,” in Proceedings of the SEM Annual Meeting on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1997).

J. L. Pezanniti, R. A. Chipman, “Imaging polarimeters for optical metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 280–294 (1990).

I. K. Kikoin, Tables of Physical Parameters (Atomizdat, Moscow, 1967).

Cited By

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

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic configuration of the polarimeter. 1, light source; 2, rotating polarizer; 3, sample; 4, rotating analyzer; 5, CCD camera; 6, frame grabber; 7, computer; 8, stepping motors.

Fig. 2
Fig. 2

Dependence of phase on ellipticity of the polarizers, obtained by computer simulation through formulas (2), (16), and (17). The plots reveal a stronger influence of the polarizers’ ellipticity on the phase data at the rotation ratio of 1:2. An interval for the ellipticity angle of polarizers fully covers values that may occur in practice.

Fig. 3
Fig. 3

Results of simulations of the polarimeter with elliptic polarizers for azimuth angle. As seen from the plot, azimuth angle turns out to be insensitive to the actual ellipticity of polarizers, since the two lines completely coincide. For an explanation of this behavior, an analytical expression is necessary.

Fig. 4
Fig. 4

Simulated dependencies of a sample’s ellipticity angle on the ellipticity of polarizers, obtained by using formulas (4), (16), and (17) at 1:2 and 1:3 rotation ratios for polarizer and analyzer. The plot proves that using the rotation ratio of 1:3 is a better choice.

Fig. 5
Fig. 5

Phase distribution over the crystal quartz plate, measured by use of the suggested technique. An image was obtained after applying a misalignment correction routine.

Fig. 6
Fig. 6

Azimuth angle distribution across the given quartz plate. A misalignment correction procedure was applied during the measurement.

Fig. 7
Fig. 7

Ellipticity angle distribution over the studied quartz plate. The image was obtained by using a measurement sequence of four series (each contains nine intensity images), which allows for simultaneous application of both beam wander and misalignment error correction routines.

Fig. 8
Fig. 8

Application of a beam wander correction algorithm to phase measurement in an experiment with a nonuniform beam. Data are provided for every second pixel. By comparing data scatters in the two curves, one can see the usefulness of the correction algorithm.

Fig. 9
Fig. 9

Results of measuring azimuth angle for the phase plate (randomly oriented) before (curve 1) and after (curve 2) applying a beam wander correction procedure.

Fig. 10
Fig. 10

Uncertainty of the phase data versus number of start positions P used in the beam wander correction routine. The vertical axis gives a maximum scatter of the phase data over the whole image, the result of comparing 262,144 pixels.

Fig. 11
Fig. 11

Scatter of the azimuth angle data over the image against number of start positions P used in the beam wander correction algorithm.

Equations (17)

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

I D = a 0 + a 2 cos ( 2 A - 2 P ) + b 2 sin ( 2 A - 2 P ) + b 4 cos ( 2 A + 2 P ) + b 4 sin ( 2 A + 2 P ) ,
δ = cos - 1 ( 2 { ( a 2 / a 0 ) - [ ( a 4 / a 0 ) 2 + ( b 4 / a 0 ) 2 ] 1 / 2 } ( ,
θ = 1 4 cos - 1 { ( a 4 / a 0 ) / [ ( a 4 / a 0 ) 2 + ( b 4 / a 0 ) 2 ] 1 / 2 } ,
θ = 1 4 sin - 1 b 4 / a 0 [ ( a 4 / a 0 ) 2 + ( b 4 / a 0 ) 2 ] 1 / 2
= 1 2 sin - 1 - 2 b 2 / a 0 sin   δ
= ± 1 2 sin - 1 { 1 - 4 [ ( a 4 / a 0 ) 2 + ( b 4 / a 0 ) 2 ] 1 / 2 / ( 1 - cos   δ ) } 1 / 2 .
I D = a 0 + a 2 m cos ( 2 A - 2 P ) + b 2 m sin ( 2 A - 2 P ) + a 4 m cos ( 2 A + 2 P ) + b 4 m sin ( 2 A + 2 P ) ,
2 μ = ( b 2 , 0 mN + b 2 , 90 mN ) / ( a 2 , 0 mN + a 2 , 90 mN ) ,
δ = cos - 1 a 2 , 0 mN + a 2 , 90 mN - 2 ( a 4 mN ) 2 + ( b 4 mN ) 2 1 + 4 μ 2 1 / 2 ,
θ = 1 4 sin - 1 b n mN 1 + 4 μ 2 [ ( a 4 mN ) 2 + ( b 4 mN ) 2 ] 1 / 2 - 1 2   μ
= - 1 2 sin - 1 b 2 , 0 mN - b 2 , 90 mN sin   δ
a j real = 2 m k = 0 m - 1 ( I k + Δ I k ) cos j   2 π k m = a j ideal + a j error ,
a j ideal = 2 m k = 0 m - 1   i k cos j   2 π k m
a j error = 2 m k = 0 m - 1 Δ i k cos j   2 π k m
a j averaged = 1 P p = 1 P a j p ideal + p = 1 P a j p error .
I D E o + E o ,
E o = T A T S T P E i ,

Metrics