Abstract

We present a method of error analysis that can be applied for phase-measuring algorithms applied to photoelasticity. We calculate the contributions to the measurement error of the different elements of a circular polariscope as perturbations of the Jones matrices associated with each element. The Jones matrix of the real polariscope can then be calculated as a sum of the nominal matrix and a series of contributions that depend on the errors associated with each element separately. We apply this method to the analysis of phase-measuring algorithms for the determination of isoclinics and isochromatics, including comparisons with real measurements.

© 1998 Optical Society of America

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References

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  1. C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
    [CrossRef]
  2. T. Franz, A. Maidhof, J. Sun, “Verfahren und Vorrichtung zur Bestimmung der Isochromatenwerte in der Spannungsoptik,” German patentDE-195 03 851 A1 (10August1995).
  3. 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]
  4. J. A. Quiroga, A. González-Cano, “Phase-measuring algorithm for the extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
    [CrossRef]
  5. K. Freischlad, C. L. Kouliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am A 7, 542–551 (1990).
    [CrossRef]
  6. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  7. P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
    [CrossRef]

1997 (2)

1995 (1)

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

1991 (1)

1990 (1)

K. Freischlad, C. L. Kouliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am A 7, 542–551 (1990).
[CrossRef]

Buckberry, C.

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

Frankena, H. J.

Franz, T.

T. Franz, A. Maidhof, J. Sun, “Verfahren und Vorrichtung zur Bestimmung der Isochromatenwerte in der Spannungsoptik,” German patentDE-195 03 851 A1 (10August1995).

Freischlad, K.

K. Freischlad, C. L. Kouliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am A 7, 542–551 (1990).
[CrossRef]

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

González-Cano, A.

Kouliopoulos, C. L.

K. Freischlad, C. L. Kouliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am A 7, 542–551 (1990).
[CrossRef]

Maidhof, A.

T. Franz, A. Maidhof, J. Sun, “Verfahren und Vorrichtung zur Bestimmung der Isochromatenwerte in der Spannungsoptik,” German patentDE-195 03 851 A1 (10August1995).

Nurse, A. D.

Quiroga, J. A.

Smorenburg, C.

Sun, J.

T. Franz, A. Maidhof, J. Sun, “Verfahren und Vorrichtung zur Bestimmung der Isochromatenwerte in der Spannungsoptik,” German patentDE-195 03 851 A1 (10August1995).

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Towers, D.

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

van Wingerden, J.

Appl. Opt. (3)

J. Opt. Soc. Am A (1)

K. Freischlad, C. L. Kouliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am A 7, 542–551 (1990).
[CrossRef]

Meas. Sci. Technol. (1)

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

Other (2)

T. Franz, A. Maidhof, J. Sun, “Verfahren und Vorrichtung zur Bestimmung der Isochromatenwerte in der Spannungsoptik,” German patentDE-195 03 851 A1 (10August1995).

P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Evolution of the first moments of the histograms of Δα (error in the value of the isoclinic parameter) with Δθ2 (error in the orientation of the first quarter-wave plate). The standard deviations of the histograms are plotted as error bars.

Fig. 2
Fig. 2

Evolution of the first moments of the histograms of Δδ (error in the value of the isochromatic parameter) with Δθ2 (error in the orientation of the first quarter-wave plate). The standard deviations of the histograms are plotted as error bars.

Fig. 3
Fig. 3

Distributions of (a) isoclinics and (b) isochromatics in a diametrically loaded disk.

Fig. 4
Fig. 4

(a) Experimental values and (b) results of the application of our method obtained for the distribution of isochromatics in a diametrically loaded disk when the following errors are introduced in the polariscope: +10° in the orientation of the polarization of the initial beam, -10° in the orientation of the first quarter-wave plate, +10° in the orientation of the second quarter-wave plate, -10° in the orientation of the analyzer.

Fig. 5
Fig. 5

Histograms of the errors of δ obtained by our method (solid curve) and by experiment (dashed curve) for the distribution of initial errors of Fig. 4.

Fig. 6
Fig. 6

(a) Experimental values and (b) results of our method obtained for δ in the disk with initial errors of +5° (initial beam), -5° (first quarter-wave plate), +5° (second quarter-wave plate), and -5° (analyzer).

Fig. 7
Fig. 7

Histograms of the errors of δ obtained by our method (solid curve) and by experiment (dashed curve) for the disk with the distribution of initial errors of Fig. 6.

Fig. 8
Fig. 8

Distributions of (a) the isoclinics and (b) the isochromatics in a plate with a hole and a cut.

Fig. 9
Fig. 9

(a) Experimental values and (b) the results of our method for α in a plate with a distribution of initial errors of +5°, -5°, +5°, and -5°.

Fig. 10
Fig. 10

(a) Experimental values and (b) the results of our method for δ in a plate with a distribution of initial errors of +5°, -5°, +5°, and -5°.

Fig. 11
Fig. 11

Histograms of the errors in α obtained with our method (solid curve) and by experiment (dashed curve) for the plate with the distribution of initial errors of Fig. 9.

Fig. 12
Fig. 12

Histograms of the errors in δ obtained with our method (solid curve) and by experiment (dashed curve) for the plate with the distribution of initial errors of Fig. 10.

Equations (25)

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J = j 11 j 12 j 21 j 22 .
J * θ = J θ + Δ θ = S - Δ θ J θ S Δ θ ,
S β = cos   β sin   β - sin   β cos   β .
S Δ θ = 1 Δ θ - Δ θ 1 ,
J * θ = 1 - Δ θ Δ θ 1 J θ 1 Δ θ - Δ θ 1 .
j 11 * j 12 * j 21 * j 22 * = j 11 j 12 j 21 j 22 + Δ θ - j 12 - j 21 j 11 - j 22 j 11 - j 22 j 12 + j 21 ,
J * θ = J θ + Δ θ J O θ ,
J O θ = - j 12 - j 21 j 11 - j 22 j 11 - j 22 j 12 + j 21
D η ,   θ = exp i η cos 2   θ + sin 2   θ exp i η - 1 sin   θ   cos   θ exp i η - 1 sin   θ   cos   θ exp i η sin 2   θ + cos 2   θ .
exp i η + Δ η = exp i η 1 + i Δ η ,
D * η ,   θ = D η ,   θ + i   exp i η Δ η D R θ ,
D R θ = cos 2   θ sin   θ   cos   θ sin   θ   cos   θ sin 2   θ .
D * η ,   θ = D η ,   θ + Δ θ D O η ,   θ + i   exp i η Δ η D R θ ,
D O η ,   θ = 1 - exp i η sin   2 θ 1 + exp i η cos   2 θ 1 + exp i η cos   2 θ exp i η - 1 sin   2 θ .
P * θ = P θ + Δ θ P O θ ,
P θ = cos 2   θ sin   θ   cos   θ sin   θ   cos   θ sin 2   θ ,
P O θ = - sin   2 θ cos   2 θ cos   2 θ sin   2 θ .
M = P θ 4 D π / 2 ,   θ 3 D δ ,   α D π / 2 ,   θ 2 P θ 1 ,
a = cos   θ 1 sin   θ 1 .
a * = cos   θ 1 sin   θ 1 + Δ θ 1 - sin   θ 1 cos   θ 1 = a + Δ θ 1 a O .
M * = P * θ 4 D * π / 2 ,   θ 3 D δ ,   α D * π / 2 ,   θ 2 ,
M * = M + Δ θ 4 P O θ 4 D π / 2 ,   θ 3 D δ ,   α D π / 2 ,   θ 2 + Δ θ 3 P θ 4 D O π / 2 ,   θ 3 D δ ,   α D π / 2 ,   θ 2 + Δ η 3 P θ 4 D R π / 2 ,   θ 3 D δ ,   α D π / 2 ,   θ 2 + Δ θ 2 P θ 4 D π / 2 ,   θ 3 D δ ,   α D O π / 2 ,   θ 2 + Δ η 2 P θ 4 D π / 2 ,   θ 3 D δ ,   α D R π / 2 ,   θ 2 .
M * = M + j = 1 5   j E j ,
b = M * a * = Ma + Δ θ 1 Ma O + j = 1 5   j E j a ,
I * = b + b ,

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