Abstract

The interpretation of fringes observed in photoelastic stress measurements made with coherent well-collimated optical radiation such as a laser beam and slab specimens with parallel surfaces is affected by multiple internal reflections of light within the sample, which are usually negligible when incoherent light is used. An analysis of the multiple-reflection effects in photoelastic measurements involving the plane polariscope configuration is presented. The results show that the isochromatic fringes are modified by the interference of multiply reflected waves. The multipass differential phase accumulations that display oscillatory magnitudes as functions of the model thickness and the optical wavelength result in a shifted and altered intensity profile across the isochromatic fringes. It is shown that for large values of reflectivity, as in the case of samples with reflective coating or partial mirrors, the bright fringes split into multiple peaks.

© 2001 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2000 (1)

A. K. Bhowmik, “Algorithm for whole-field determination of rotation, azimuth, and phase shift in dielectric helicoidal bianisotropic media,” Optik 111, 103–106 (2000).

1999 (1)

1998 (1)

K. Shimizu, M. Suetsugu, T. Nakamura, S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. Ser. A 41, 134–141 (1998).
[CrossRef]

1997 (2)

1996 (1)

P. Gemeiner, D. Yang, J. C. Canit, “Multiple reflections in a photoelastic modulators: errors in polarization measurement,” J. Opt. (Paris) 27, 202–210 (1996).
[CrossRef]

1995 (3)

T. C. Oakberg, “Modulated interference effects: use of photoelastic modulators with lasers,” Opt. Eng. 34, 1545–1550 (1995).
[CrossRef]

Y. Niitsu, K. Ichinose, K. Ikegami, “Stress measurement of transparent materials by polarized laser,” JSME Int. J. Ser. A 38, 68–72 (1995).

M. A. Bukhshtab, “The influence of surface reflections on computation and measurement of retardance,” Meas. Sci. Technol. 6, 910–920 (1995).
[CrossRef]

1994 (1)

D. Hu, C. Wang, S. Gu, Z. Cao, “Application of laser electro-optic modulation method in measuring principal stress direction,” Appl. Laser Technol. 14, 72–74 (1994).

1991 (2)

N. G. Theofanous, A. T. Arapoyianni, “Effect of multiple reflections on retardation-based electro-optic measurements,” J. Opt. Soc. Am. A 8, 1746–1754 (1991).
[CrossRef]

S. Heinemann, “Method for birefringence measurements with correction of errors due to multiple reflection in anisotropic plates,” J. Mod. Opt. 38, 1721–1729 (1991).
[CrossRef]

1988 (1)

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

1967 (1)

1964 (2)

1954 (1)

D. Post, “Isochromatic fringe sharpening and fringe multiplication,” Proc. Soc. Stress Anal. 36, 8259–8263 (1954).

Arapoyianni, A. T.

Asundi, A.

Bhowmik, A. K.

A. K. Bhowmik, “Algorithm for whole-field determination of rotation, azimuth, and phase shift in dielectric helicoidal bianisotropic media,” Optik 111, 103–106 (2000).

Boay, C. G.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999).

Bukhshtab, M. A.

M. A. Bukhshtab, “The influence of surface reflections on computation and measurement of retardance,” Meas. Sci. Technol. 6, 910–920 (1995).
[CrossRef]

Canit, J. C.

P. Gemeiner, D. Yang, J. C. Canit, “Multiple reflections in a photoelastic modulators: errors in polarization measurement,” J. Opt. (Paris) 27, 202–210 (1996).
[CrossRef]

Cao, Z.

D. Hu, C. Wang, S. Gu, Z. Cao, “Application of laser electro-optic modulation method in measuring principal stress direction,” Appl. Laser Technol. 14, 72–74 (1994).

Dally, J. W.

J. W. Dally, W. F. Riley, Experimental Stress Analysis (McGraw-Hill, New York, 1978).

Gao, W.

G. Kou, C. Tao, W. Gao, L. Hao, X. Yang, T. Li, “New method for measuring photoelastic properties of optical materials by using laser heterodyne interferometry,” in Automated Optical Inspection for Industry: Theory, Technology, and Applications II, S. Ye, ed., Proc. SPIE3558, 462–469 (1998).

Gemeiner, P.

P. Gemeiner, D. Yang, J. C. Canit, “Multiple reflections in a photoelastic modulators: errors in polarization measurement,” J. Opt. (Paris) 27, 202–210 (1996).
[CrossRef]

Gonzalez-Cano, A.

Gu, S.

D. Hu, C. Wang, S. Gu, Z. Cao, “Application of laser electro-optic modulation method in measuring principal stress direction,” Appl. Laser Technol. 14, 72–74 (1994).

Hao, L.

G. Kou, C. Tao, W. Gao, L. Hao, X. Yang, T. Li, “New method for measuring photoelastic properties of optical materials by using laser heterodyne interferometry,” in Automated Optical Inspection for Industry: Theory, Technology, and Applications II, S. Ye, ed., Proc. SPIE3558, 462–469 (1998).

Harris, J.

Heinemann, S.

S. Heinemann, “Method for birefringence measurements with correction of errors due to multiple reflection in anisotropic plates,” J. Mod. Opt. 38, 1721–1729 (1991).
[CrossRef]

Holmes, D. A.

Hu, D.

D. Hu, C. Wang, S. Gu, Z. Cao, “Application of laser electro-optic modulation method in measuring principal stress direction,” Appl. Laser Technol. 14, 72–74 (1994).

Ichinose, K.

Y. Niitsu, K. Ichinose, K. Ikegami, “Stress measurement of transparent materials by polarized laser,” JSME Int. J. Ser. A 38, 68–72 (1995).

Ikegami, K.

Y. Niitsu, K. Ichinose, K. Ikegami, “Stress measurement of transparent materials by polarized laser,” JSME Int. J. Ser. A 38, 68–72 (1995).

Ito, R.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Kaneda, Y.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Kondo, T.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Kou, G.

G. Kou, C. Tao, W. Gao, L. Hao, X. Yang, T. Li, “New method for measuring photoelastic properties of optical materials by using laser heterodyne interferometry,” in Automated Optical Inspection for Industry: Theory, Technology, and Applications II, S. Ye, ed., Proc. SPIE3558, 462–469 (1998).

Kruger, J.

Li, T.

G. Kou, C. Tao, W. Gao, L. Hao, X. Yang, T. Li, “New method for measuring photoelastic properties of optical materials by using laser heterodyne interferometry,” in Automated Optical Inspection for Industry: Theory, Technology, and Applications II, S. Ye, ed., Proc. SPIE3558, 462–469 (1998).

Morita, R.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Nakamura, T.

K. Shimizu, M. Suetsugu, T. Nakamura, S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. Ser. A 41, 134–141 (1998).
[CrossRef]

Ng, T. W.

Niitsu, Y.

Y. Niitsu, K. Ichinose, K. Ikegami, “Stress measurement of transparent materials by polarized laser,” JSME Int. J. Ser. A 38, 68–72 (1995).

Oakberg, T. C.

T. C. Oakberg, “Modulated interference effects: use of photoelastic modulators with lasers,” Opt. Eng. 34, 1545–1550 (1995).
[CrossRef]

Ogasawara, N.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Post, D.

D. Post, “Isochromatic fringe sharpening and fringe multiplication,” Proc. Soc. Stress Anal. 36, 8259–8263 (1954).

Quiroga, J. A.

Riley, W. F.

J. W. Dally, W. F. Riley, Experimental Stress Analysis (McGraw-Hill, New York, 1978).

Shimizu, K.

K. Shimizu, M. Suetsugu, T. Nakamura, S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. Ser. A 41, 134–141 (1998).
[CrossRef]

Suetsugu, M.

K. Shimizu, M. Suetsugu, T. Nakamura, S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. Ser. A 41, 134–141 (1998).
[CrossRef]

Sugihashi, A.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Takahashi, S.

K. Shimizu, M. Suetsugu, T. Nakamura, S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. Ser. A 41, 134–141 (1998).
[CrossRef]

Tao, C.

G. Kou, C. Tao, W. Gao, L. Hao, X. Yang, T. Li, “New method for measuring photoelastic properties of optical materials by using laser heterodyne interferometry,” in Automated Optical Inspection for Industry: Theory, Technology, and Applications II, S. Ye, ed., Proc. SPIE3558, 462–469 (1998).

Theofanous, N. G.

Tong, L.

Umegaki, S.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Wang, C.

D. Hu, C. Wang, S. Gu, Z. Cao, “Application of laser electro-optic modulation method in measuring principal stress direction,” Appl. Laser Technol. 14, 72–74 (1994).

Waxler, R. M.

Weinberger, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999).

Yang, D.

P. Gemeiner, D. Yang, J. C. Canit, “Multiple reflections in a photoelastic modulators: errors in polarization measurement,” J. Opt. (Paris) 27, 202–210 (1996).
[CrossRef]

Yang, X.

G. Kou, C. Tao, W. Gao, L. Hao, X. Yang, T. Li, “New method for measuring photoelastic properties of optical materials by using laser heterodyne interferometry,” in Automated Optical Inspection for Industry: Theory, Technology, and Applications II, S. Ye, ed., Proc. SPIE3558, 462–469 (1998).

Yolken, H. T.

Appl. Laser Technol. (1)

D. Hu, C. Wang, S. Gu, Z. Cao, “Application of laser electro-optic modulation method in measuring principal stress direction,” Appl. Laser Technol. 14, 72–74 (1994).

Appl. Opt. (3)

J. Mod. Opt. (1)

S. Heinemann, “Method for birefringence measurements with correction of errors due to multiple reflection in anisotropic plates,” J. Mod. Opt. 38, 1721–1729 (1991).
[CrossRef]

J. Opt. (Paris) (1)

P. Gemeiner, D. Yang, J. C. Canit, “Multiple reflections in a photoelastic modulators: errors in polarization measurement,” J. Opt. (Paris) 27, 202–210 (1996).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Jpn. J. Appl. Phys. (1)

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

JSME Int. J. Ser. A (2)

K. Shimizu, M. Suetsugu, T. Nakamura, S. Takahashi, “Evaluation of concentrated load by caustics and its application in the measurement of optical constant,” JSME Int. J. Ser. A 41, 134–141 (1998).
[CrossRef]

Y. Niitsu, K. Ichinose, K. Ikegami, “Stress measurement of transparent materials by polarized laser,” JSME Int. J. Ser. A 38, 68–72 (1995).

Meas. Sci. Technol. (1)

M. A. Bukhshtab, “The influence of surface reflections on computation and measurement of retardance,” Meas. Sci. Technol. 6, 910–920 (1995).
[CrossRef]

Opt. Eng. (1)

T. C. Oakberg, “Modulated interference effects: use of photoelastic modulators with lasers,” Opt. Eng. 34, 1545–1550 (1995).
[CrossRef]

Optik (1)

A. K. Bhowmik, “Algorithm for whole-field determination of rotation, azimuth, and phase shift in dielectric helicoidal bianisotropic media,” Optik 111, 103–106 (2000).

Proc. Soc. Stress Anal. (1)

D. Post, “Isochromatic fringe sharpening and fringe multiplication,” Proc. Soc. Stress Anal. 36, 8259–8263 (1954).

Other (3)

J. W. Dally, W. F. Riley, Experimental Stress Analysis (McGraw-Hill, New York, 1978).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999).

G. Kou, C. Tao, W. Gao, L. Hao, X. Yang, T. Li, “New method for measuring photoelastic properties of optical materials by using laser heterodyne interferometry,” in Automated Optical Inspection for Industry: Theory, Technology, and Applications II, S. Ye, ed., Proc. SPIE3558, 462–469 (1998).

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Figures (5)

Fig. 1
Fig. 1

Optical configuration of the plane-polariscope setup: P, S, and A, polarizer, sample, and analyzer, respectively; f, s, stress-induced fast and slow principal axes, respectively, in the medium.

Fig. 2
Fig. 2

Transmittance across the first- and second-order bright isochromatic fringes as a function of stress-induced relative phase retardation. Continuous and dashed curves, the intensities calculated with and without the multiple internal reflections calculated, respectively.

Fig. 3
Fig. 3

Contribution from the multiple internal reflections to the relative phase shift of light across the first- and second-order bright isochromatic fringes.

Fig. 4
Fig. 4

Intensity distribution across the first- and second-order bright isochromatic fringes for various surface reflectivities with the multiple reflections taken into account. The values of the reflectivity are a, 0.1; b, 0.3; c, 0.5; d, 0.7.

Fig. 5
Fig. 5

Variations in the multipass contributions to differential phase shift across the first-order bright isochromatic fringes for surface reflectivities of a, 0.1; b, 0.3; c, 0.5; d, 0.7.

Tables (1)

Tables Icon

Table 1 Properties of the Epoxy Resin Model Used in Calculations for This Paper

Equations (13)

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nf=n-½ n3q11σ,
ns=n-½ n3q12σ,
Δ=π/λ n3q11-q12σt,
I=I0 sin2 2α sin2Δ/2,
E=E0 expiδ,
E=E0T expiδ1-R expi2δ=E0 expiδ,
E0=E01-R1-R2+4R sin2 δ1/2,
δ=δ+ϕ,
ϕ=tan-1R sin 2δ1-R cos 2δ.
I=I0 sin2 2α14Af-As2+AfAssin2Δ+Φ2,
Φ=tan-1Rs sin 2δs-Rf sin 2δf-RfRs sin 2Δ1-Rs cos 2δs-Rf cos 2δf+RfRs cos 2Δ,
Rf,s=nf,s-1nf,s+12.
Af,s=1+4Rf,s1-Rf,s2sin2 δf,s-1.

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