Abstract

The search for fast, precise, and robust testing techniques remains an important problem in automated full-field photoelasticity. The polarizer–sample–analyzer (PSA)–based three-wavelength polarimetric method presented here employs discrete Fourier analysis and a spectral content unwrapping algorithm to provide completely automatic, simple, fast, and accurate determination of both photoelastic parameters. Fourier analysis of experimental data and a three-wavelength approach reduce the effect of noise and efficiently cope with poor accuracy in regions of both isochromatic and isoclinic maps. Because any polarimetric technique yields the phase value in the principal range of the corresponding trigonometric function, the final step in data processing is phase unwrapping. Because of the good quality of the wrapped phase map and because each point is processed independently, our suggested three-wavelength unwrapping algorithm exhibits a high level of robustness. Unlike some other PSA three-wavelength techniques, the given algorithm here solves the problem of phase unwrapping completely. Specifically, it converts experimentally obtained arccosine-type phase maps directly into full phase value distributions, skipping the step of generating an arctangent-type ramped phase map and resorting to other unwrapping routines for final data processing. The accuracy of the new technique has been estimated with a Babinet–Soleil compensator. Test experiments with the disk in diametric compression and a quartz plate have proved that the technique can be used for precise determination of the isoclinic angle and relative retardation, even for large values of the latter.

© 2001 Optical Society of America

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References

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  1. R. K. Muller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 18, 245–251 (1979).
    [CrossRef]
  2. A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the technique,” Exp. Mech. 28, 252–257 (1989).
    [CrossRef]
  3. E. A. Patterson, Z. F. Wang, “Towards full-field automatic photoelastic analysis of complex components,” Strain 27, 49–56 (1991).
    [CrossRef]
  4. 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]
  5. Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
    [CrossRef]
  6. A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
    [CrossRef]
  7. C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
    [CrossRef]
  8. J. A. Quiroga, A. Gonzalez-Cano, “Phase measuring algorithm for extraction of isochromatic of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (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. 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]
  11. M. N. Pacey, X. Z. Wang, S. J. Haake, E. A. Patterson, “The application of evolutionary and maximum entropy algorithms to photoelastic spectral analysis,” Exp. Mech. 39, 265–274 (1999).
    [CrossRef]
  12. N. Plouzennec, A. Lagarde, “Two-wavelength method for full-field automated photoelasticity,” Exp. Mech. 39, 274–278 (1999).
    [CrossRef]
  13. A. Asundi, L. Tong, Ch. G. Boay, “Phase shifting method with a normal polariscope,” Appl. Opt. 38, 5931–5935 (1999).
    [CrossRef]
  14. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  15. 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]
  16. 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]
  17. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  18. J. L. Pezanniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
    [CrossRef]
  19. R. W. Collins, J. Koh, “Dual rotating-compensator multichannel ellipsometer: instrument design for real-time Mueller matrix spectroscopy of surfaces and films,” J. Opt. Soc. Am. A 16, 1997–2006 (1999).
    [CrossRef]
  20. S. Yu Berezhna, I. V. Berezhnyy, M. Takashi, “Photoelastic analysis through Jones matrix imaging Fourier polarimetry,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ’99 (Japan Society of Mechanical Engineering, Tokyo, 1999), Vol. 2, pp. 635–640.
  21. A. S. Kobayashi, ed., Handbook on Experimental Mechanics (Society for Experimental Mechanics, Bethel, Conn.1993).
  22. I. K. Kikoin, Tables of Physical Parameters (Atomizdat, Moscow, 1967).

1999 (4)

M. N. Pacey, X. Z. Wang, S. J. Haake, E. A. Patterson, “The application of evolutionary and maximum entropy algorithms to photoelastic spectral analysis,” Exp. Mech. 39, 265–274 (1999).
[CrossRef]

N. Plouzennec, A. Lagarde, “Two-wavelength method for full-field automated photoelasticity,” Exp. Mech. 39, 274–278 (1999).
[CrossRef]

A. Asundi, L. Tong, Ch. G. Boay, “Phase shifting method with a normal polariscope,” Appl. Opt. 38, 5931–5935 (1999).
[CrossRef]

R. W. Collins, J. Koh, “Dual rotating-compensator multichannel ellipsometer: instrument design for real-time Mueller matrix spectroscopy of surfaces and films,” J. Opt. Soc. Am. A 16, 1997–2006 (1999).
[CrossRef]

1997 (3)

1996 (1)

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

1995 (2)

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

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

1994 (1)

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (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, 49–56 (1991).
[CrossRef]

1989 (2)

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the technique,” Exp. Mech. 28, 252–257 (1989).
[CrossRef]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

1979 (1)

R. K. Muller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 18, 245–251 (1979).
[CrossRef]

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]

Ajovalasit, A.

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

Asundi, A.

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]

Barone, S.

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

Berezhnyy, I. V.

S. Yu Berezhna, I. V. Berezhnyy, M. Takashi, “Photoelastic analysis through Jones matrix imaging Fourier polarimetry,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ’99 (Japan Society of Mechanical Engineering, Tokyo, 1999), Vol. 2, pp. 635–640.

Boay, Ch. G.

Buckberry, C.

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

Chipman, R. A.

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

Collins, R. W.

Fujisawa, M.

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Goldstein, D. H.

Gonzalez-Cano, A.

Haake, S. J.

M. N. Pacey, X. Z. Wang, S. J. Haake, E. A. Patterson, “The application of evolutionary and maximum entropy algorithms to photoelastic spectral analysis,” Exp. Mech. 39, 265–274 (1999).
[CrossRef]

Huntley, J. M.

Kikoin, I. K.

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

Koh, J.

Lagarde, A.

N. Plouzennec, A. Lagarde, “Two-wavelength method for full-field automated photoelasticity,” Exp. Mech. 39, 274–278 (1999).
[CrossRef]

Morimoto, Y.

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Muller, R. K.

R. K. Muller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 18, 245–251 (1979).
[CrossRef]

Ng, T. W.

Nurse, A. D.

Pacey, M. N.

M. N. Pacey, X. Z. Wang, S. J. Haake, E. A. Patterson, “The application of evolutionary and maximum entropy algorithms to photoelastic spectral analysis,” Exp. Mech. 39, 265–274 (1999).
[CrossRef]

Patterson, E. A.

M. N. Pacey, X. Z. Wang, S. J. Haake, E. A. Patterson, “The application of evolutionary and maximum entropy algorithms to photoelastic spectral analysis,” Exp. Mech. 39, 265–274 (1999).
[CrossRef]

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

Petrucci, G.

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

Pezanniti, J. L.

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

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]

Plouzennec, N.

N. Plouzennec, A. Lagarde, “Two-wavelength method for full-field automated photoelasticity,” Exp. Mech. 39, 274–278 (1999).
[CrossRef]

Quiroga, J. A.

Redner, A. S.

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the technique,” Exp. Mech. 28, 252–257 (1989).
[CrossRef]

Saackel, L. R.

R. K. Muller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 18, 245–251 (1979).
[CrossRef]

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]

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]

Takashi, M.

S. Yu Berezhna, I. V. Berezhnyy, M. Takashi, “Photoelastic analysis through Jones matrix imaging Fourier polarimetry,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ’99 (Japan Society of Mechanical Engineering, Tokyo, 1999), Vol. 2, pp. 635–640.

Tong, L.

Towers, D.

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

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]

Voloshin, A. S.

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the technique,” Exp. Mech. 28, 252–257 (1989).
[CrossRef]

Wang, X. Z.

M. N. Pacey, X. Z. Wang, S. J. Haake, E. A. Patterson, “The application of evolutionary and maximum entropy algorithms to photoelastic spectral analysis,” Exp. Mech. 39, 265–274 (1999).
[CrossRef]

Wang, Z. F.

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

Yu Berezhna, S.

S. Yu Berezhna, I. V. Berezhnyy, M. Takashi, “Photoelastic analysis through Jones matrix imaging Fourier polarimetry,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ’99 (Japan Society of Mechanical Engineering, Tokyo, 1999), Vol. 2, pp. 635–640.

Appl. Opt. (6)

Exp. Mech. (6)

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

M. N. Pacey, X. Z. Wang, S. J. Haake, E. A. Patterson, “The application of evolutionary and maximum entropy algorithms to photoelastic spectral analysis,” Exp. Mech. 39, 265–274 (1999).
[CrossRef]

N. Plouzennec, A. Lagarde, “Two-wavelength method for full-field automated photoelasticity,” Exp. Mech. 39, 274–278 (1999).
[CrossRef]

R. K. Muller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 18, 245–251 (1979).
[CrossRef]

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the technique,” Exp. Mech. 28, 252–257 (1989).
[CrossRef]

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)

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. (2)

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

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Opt. Lasers Eng. (1)

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

Opt. Lett. (1)

Strain (1)

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

Other (3)

S. Yu Berezhna, I. V. Berezhnyy, M. Takashi, “Photoelastic analysis through Jones matrix imaging Fourier polarimetry,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ’99 (Japan Society of Mechanical Engineering, Tokyo, 1999), Vol. 2, pp. 635–640.

A. S. Kobayashi, ed., Handbook on Experimental Mechanics (Society for Experimental Mechanics, Bethel, Conn.1993).

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

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Figures (11)

Fig. 1
Fig. 1

Schematic configuration of the polarimeter: 1, light source; 2, spatial filter; 3, collimator; 4, rotating polarizer; 5, 8, rotary devices; 6, sample; 7, rotating analyzer; 9, objective; 10, CCD camera; 11, computer; 12, camera controller; 13, rotary stage controller.

Fig. 2
Fig. 2

Retardation φ along several horizontal lines chosen from the full-field image of the compensator’s usable aperture (R = 5 mm). The retardation value was set at 90 deg. Orientation of the crystallographic axes was at 0 deg in a laboratory coordinate system. Curves demonstrate that the compensator is not perfectly uniform within its aperture. Results of averaging over 3 × 9 images.

Fig. 3
Fig. 3

Retardation along the central line of the compensator’s aperture collected at various orientations of the compensator’s crystallographic axes. The value of the retardation was set at 90 deg. Orientation of the birefringence axes: 1, at 45 deg; 2, at 25 deg; 3, at 0 deg; 4, at -25 deg; 5, at -45 deg; and 6, at -90 deg in the laboratory coordinate system.

Fig. 4
Fig. 4

Error in measured retardation versus retardation value. Results obtained for the central line of the compensator’s aperture. Crystallographic axes of the compensator are set at 0-deg orientation in the laboratory coordinate system.

Fig. 5
Fig. 5

Mean absolute error for the azimuth angle of the compensator’s crystallographic axes versus retardation. The orientation of the axes was set at 10 deg in the laboratory coordinate system.

Fig. 6
Fig. 6

Dependence of the measurement accuracy of the azimuth angle on the value of the azimuth angle. Measurements were made at 90-deg retardation.

Fig. 7
Fig. 7

Dependence of the retardation error on the number of images collected in one measurement series. Data obtained for a Babinet–Soleil compensator at 90-deg retardation and 0-deg azimuth angle.

Fig. 8
Fig. 8

Dependence of phase error on the number of measurement series. Each measurement series includes nine images. First, nine images in each measurement sequence are collected by use of a 0-deg orientation of aligned polarizer and analyzer. Before collection, a next set of nine images, the aligned polarizers and analyzers, are simultaneously rotated in the corresponding angle. The value of the rotation angle is 360 deg/p, where p is the number of series. For example, for a sequence, which includes three measurement series, the rotation angle is 120 deg. Then a usual procedure of collecting nine images is repeated. Retardation of the compensator is set at 90 deg; its crystallographic axes are oriented at 0 deg in the laboratory coordinate system.

Fig. 9
Fig. 9

Phase distribution along the horizontal line at 0.83R from the disk center for the model under a 1342-N load: (a)–(c) at wavelengths marked; (d) result of phase unwrapping for at λ1 = 435.8 nm obtained at n = 6.

Fig. 10
Fig. 10

Results for the isoclinic angle obtained for the model loaded by 441 N (a) along the horizontal diameter and in its immediate proximity at λ = 435.8 nm, where plots 1 and 2 were obtained at 0.004R below and above the center, respectively; and (b) along the horizontal line at 0.47R to the top from the center, where the plot, obtained at λ = 579 nm, is given relative to averaged and theoretical distributions.

Fig. 11
Fig. 11

(a) Phase distribution along the central line of the quartz plate, measured at three values of λ. (b) Result of phase unwrapping at λ1 = 435.8 nm.

Tables (2)

Tables Icon

Table 1 Stress-Optic Coefficients of BK7 Glass

Tables Icon

Table 2 Theoretical Data for Birefringence in Quartz along with the Plate’s Retardationa

Equations (14)

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ID=a0+a4 cos2A+2P+b4 sin2A+2P,
a2/a0=1/41+cos Φ,
a4/a0=1/4 cos 4θ1-cos Φ,
b4/a0=1/4 sin 4θ1-cos Φ.
φ=cos-12a2/a0-a4/a02+b4/a021/2,
θ=1/4 cos-1a4/a0/a4/a02+b4/a021/2,
θ=1/4 sin-1b4/a0a4/a02+b4/a021/2,
Iα=1/2 a0+a2 cos 2α+b2 sin 2α,
Δ=n1-n2d=Φλ2π,
Δ=n+φλ2πλ,
Δ=n+1-φλ2πλ,
n1-n2=Cλσ1-σ2,
Δ=Cλσ1-σ2d=n+φλ2πλ.
Φ=2πλne-nod,

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