Abstract

Recently several polarimetric techniques have been suggested, designed deliberately for automatic whole-field birefringence imaging in photoelastic models with essentially three-dimensional stresses. In general, these techniques are feasible for mapping three optical parameters that determine birefringence in a given case. However, the difficulty in attaining a high level of data accuracy over the whole image persists. There remains a problem of precise imaging in regions where the mutual interference of three given parameters inevitably causes accuracy deterioration. We show how to correct such imperfections in an imaging polarizer–sample–analyzer (PSA) Fourier polarimetry technique, as suggested earlier [Appl. Opt. 41, 644 (2001)]. The given technique (a method developed so that it maps the phase, the azimuth, and the ellipticity angles of an elliptic retarder) particularly fails to provide precise imaging in regions where the phase is either close to null or approaches π-multiple values and in intervals where the ellipticity angle falls into the proximity of ±π/4 values. These drawbacks can be successfully overcome by incorporation of a compensator into a PSA polarimeter arrangement. Although use of a compensator in the polarimeter makes the original technique more complicated, we demonstrate that the compensator allows two important issues to be resolved. First, it provides precise imaging for each of three optical parameters through the whole accessible intervals of the parameters regardless of the absolute value of the parameter. In addition, it gives a sign of phase that remains undefined in the PSA techniques. Theoretical considerations are presented and are followed by experimental data that illustrate the improved accuracy capabilities of the compensator-enhanced technique.

© 2001 Optical Society of America

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References

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  1. S. K. Mangal, K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by a phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
    [CrossRef]
  2. 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, Houston, 1998 (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.
  3. R. A. Tomlinson, E. A. Patterson, “Evaluating characteristic parameters in integrated photoelasticity,” in Proceedings of the 11th International Conference on Experimental Mechanics, Oxford, UK, 1998 (Balkema, Rotterdam, The Netherlands, 1998), pp. 495–500.
  4. S. Y. Berezhna, I. V. Berezhnyy, M. Takashi, “Integrated photoelasticity through the imaging Fourier polarimetry of an elliptic retarder,” Appl. Opt. 40, 644–651 (2001).
    [CrossRef]
  5. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1987).
  6. S. Y. Berezhna, I. V. Berezhnyy, M. Takashi, A. S. Voloshin, “Full-field automated photoelasticity by Fourier polarimetry with three wavelengths,” Appl. Opt. 40, 52–61 (2001).
    [CrossRef]
  7. P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
    [CrossRef]
  8. D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
    [CrossRef]
  9. 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]
  10. S. Yu. Berezhna, I. V. Berezhnyy, M. Takashi, “Dynamic photometric imaging PSA-polarimeter: instrument for mapping birefringence and optical rotation,” J. Opt. Soc. Am. A 18, 666–672 (2001).
    [CrossRef]
  11. J. L. Pezanniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
    [CrossRef]

2001

1999

S. K. Mangal, K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by a phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

1995

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

1978

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]

1976

P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[CrossRef]

D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
[CrossRef]

Azzam, R. M. A.

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, 1987).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1987).

Berezhna, S. Y.

Berezhna, S. Yu.

Berezhnyy, I. V.

Chipman, R. A.

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

Hauge, P. S.

P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[CrossRef]

Mangal, S. K.

S. K. Mangal, K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by a phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

Patterson, E. 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, Houston, 1998 (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.

R. A. Tomlinson, E. A. Patterson, “Evaluating characteristic parameters in integrated photoelasticity,” in Proceedings of the 11th International Conference on Experimental Mechanics, Oxford, UK, 1998 (Balkema, Rotterdam, The Netherlands, 1998), pp. 495–500.

Pezanniti, J. L.

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

Ramesh, K.

S. K. Mangal, K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by a phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

Takashi, M.

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, Houston, 1998 (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.

R. A. Tomlinson, E. A. Patterson, “Evaluating characteristic parameters in integrated photoelasticity,” in Proceedings of the 11th International Conference on Experimental Mechanics, Oxford, UK, 1998 (Balkema, Rotterdam, The Netherlands, 1998), pp. 495–500.

Voloshin, A. S.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

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.

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

Opt. Lasers Eng.

S. K. Mangal, K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by a phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

Surf. Sci.

P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[CrossRef]

D. E. Aspnes, “A photometric ellipsometer for measuring flux in a general state of polarization,” Surf. Sci. 56, 161–169 (1976).
[CrossRef]

Other

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, Houston, 1998 (Society for Experimental Mechanics, Bethel, Conn., 1998), pp. 118–121.

R. A. Tomlinson, E. A. Patterson, “Evaluating characteristic parameters in integrated photoelasticity,” in Proceedings of the 11th International Conference on Experimental Mechanics, Oxford, UK, 1998 (Balkema, Rotterdam, The Netherlands, 1998), pp. 495–500.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1987).

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Figures (8)

Fig. 1
Fig. 1

Schematic configuration of the Fourier polarimeter: P, rotating linear polarizer; S, sample; C, compensator; A, rotating linear analyzer. A sample and a compensator, considered together, create an optical system, T; r, ratio of the angular rotation speeds of a polarizer and an analyzer.

Fig. 2
Fig. 2

Schematic illustration of a light beam path in the disk, immersed in liquid: a, angle of light incidence; d, sample thickness. In the theoretical calculations of stress distributions along the direction of the beam propagation, the light path was sampled into N = 100 intervals. A detailed explanation of the computation procedure is in Ref. 4.

Fig. 3
Fig. 3

Image of wrapped phase δ obtained when the disk is tilted at α = 30 deg with respect to incidence light. The loading force is 398 N.

Fig. 4
Fig. 4

Image of azimuth angle θ obtained at the experimental geometry described in the text.

Fig. 5
Fig. 5

Image of ellipticity angle ε (absolute value) obtained at the same experimental geometry.

Fig. 6
Fig. 6

Distribution of phase δ along the horizontal scan at 0.6R: 1, theory; 2, by a standard procedure; 3, by an error-corrected procedure. The loading force is 200 N.

Fig. 7
Fig. 7

Distribution of azimuth angle θ along the same scan: 1, theory; 2, by a standard procedure; 3, by an error-corrected procedure.

Fig. 8
Fig. 8

Distribution of ellipticity angle ε along the given scan: 1, theory; 2, by a standard procedure; 3, by an error-corrected procedure.

Equations (13)

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S=cosδ2+i cos 2θ cos 2ε sinδ2sin 2ε sinδ2+i sin 2θ cos 2ε sinδ2-sin 2ε sinδ2+i sin 2θ cos 2ε sinδ2cosδ2-i cos 2θ cos 2ε sinδ2,
ID=Ī1+F2r-1c cos 2a-p+F2r-1s sin 2a-p+F2r+1c cos 2a+p+F2r+1s sin 2a+p,
F2r-1c=14cos2 2ε+1+sin2 2εcos δ, F2r-1s=-12sin 2ε sin δ, F2r+1c=14cos2 2ε cos 4θ1-cos δ, F2r+1s=14cos2 2ε sin 4θ1-cos δ.
cos δ=2F2r-1c-F2r+1c2+F2r+1s21/2,
sin 2ε=-2F2r-1ssin δ,
sin 4θ=4F2r+1scos2 2ε1-cos δ,
cos 4θ=4F2r+1ccos2 2ε1-cos δ,
dδ=-1sin δ dcos δ,
T=SC,
cosδ2=cosδT2cosδC2+cos 2θT-θCcos 2εT sinδT2sinδC2,
sin 2ε=sin 2θC-θTcos 2εT sinδT2sinδC2+sin 2εT sinδT2cosδC2sinδ2,
sin 2θ=-sin 2θC sinδC2cosδT2+cos 2θC sin 2εT sinδT2sinδC2+sin 2θT cos 2εT sinδT2cosδC2cos 2ε sinδ2,
TδT+,-, θT, εT=S±|δ|, θ, εCδC, θC,

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