Abstract

The two-dimensional measurement principle for a birefringence vector distribution in transparent materials is analyzed. The system nonuniformity that results from the system components makes the two-dimensional measurement principle quite different from that of a point-measurement method, and the measurement principle requires a two-dimensional analysis. A pulsed optical phase modulation is employed to simplify the two-dimensional mathematical analysis. As a result, concepts are proposed of the system function that characterizes the system nonuniformity that results from the system components and of the intrinsic function that is related to the birefringence vector distribution in a birefringent sample. The influence of the system nonuniformity on the two-dimensional measurement is eliminated by measurement of the intrinsic function, whereas its two values allow for the mathematical separation of the birefringence vector components. The effectiveness of the two-dimensional analysis is illustrated by measurement of a birefringence vector distribution, which is induced by an internal stress distribution in a poly(methyl methacrylate) plate, owing to the photoelastic effect.

© 1999 Optical Society of America

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References

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  1. J. Tsuji, M. Nishida, K. Kawada, “The basis for photoelastic stress analysis,” in Experimental Methods Using Photoelastic Effect (Japanese Nikkan Kougyo, Tokyo, 1965), pp. 51–170 (in Japanese).
  2. J. W. Dally, W. F. Riley, “Theory of photo-elasticity,” in Experimental Stress Analysis (McGraw-Hill, New York, 1978), pp. 406–446.
  3. M. Zahn, T. Takada, S. Voldman, “Kerr electro-optic field mapping measurement in water using parallel cylindrical electrodes,” J. Appl. Phys. 54, 4749–4761 (1983).
    [CrossRef]
  4. N. O’Flaherty, N. Kiyomoto, I. Srahama, Y. Machida, “A system for high sensitive measurement of birefringence using a photo-elastic modulator and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
    [CrossRef]
  5. H. Sato, M. Yamaguchi, A. Yoshimoto, T. Takada, “High sensitive measurement system of birefringence using optical phase modulator and two Faraday rotators,” Jpn. J. Soc. Mater. Eng. Res. 4, 22–34 (1991).
    [CrossRef]
  6. M. Takayama, T. Takada, T. Yoshino, “Stress distribution using high sensitive birefringence measurement system,” Jpn. J. Nondestructive Tests 44, 249–252 (1995).
  7. M. Ohmi, M. Akatsuka, K. Ishikawa, K. Naito, Y. Yonezawa, Y. Nishida, M. Yamanaka, Y. Izawa, S. Nakai, “High-sensitivity two-dimensional thermal- and mechanical-stress-induced birefringence measurements in a Nd:YAG rod,” Appl. Opt. 33, 6368–6372 (1994).
    [CrossRef] [PubMed]
  8. Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  9. U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
    [CrossRef]
  10. K. Tanaka, T. Takada, “Measurement of the 2-dimensional electric field vector in dielectric liquid,” IEEE Trans. Dieletr. Electr. Insul. 1, 747–753 (1994).
    [CrossRef]
  11. R. M. A. Azzam, N. M. Bashara, “Propagation of polarized light through polarizing optical systems,” in Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 66–152.
  12. Y. C. Zhu, T. Takada, D. M. Tu, “An optical measurement technique for studying residual surface charge distribution,” J. Phys. D 28, 1468–1477 (1995).
    [CrossRef]
  13. Y. C. Zhu, T. Takada, “A 2-dimensional Kerr-effect technique for electric field distribution in liquid dielectrics,” IEEE Trans. Dielectr. Electr. Insul. 4, 748–757 (1997).
    [CrossRef]
  14. T. Kawasaki, Y. Hirata, T. Takada, T. Maeno, “Visualization and quantification technique of latent electric charge image using Pockels effect, high-sensitive measurement using image lock-in amplifier,” Trans. IEE Jpn. 113-C, 1114–1120 (1993).
  15. Y. C. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: data processing and experiment verification,” Appl. Opt. 38, 2216–2224 (1999).
    [CrossRef]

1999

1997

Y. C. Zhu, T. Takada, “A 2-dimensional Kerr-effect technique for electric field distribution in liquid dielectrics,” IEEE Trans. Dielectr. Electr. Insul. 4, 748–757 (1997).
[CrossRef]

1995

M. Takayama, T. Takada, T. Yoshino, “Stress distribution using high sensitive birefringence measurement system,” Jpn. J. Nondestructive Tests 44, 249–252 (1995).

Y. C. Zhu, T. Takada, D. M. Tu, “An optical measurement technique for studying residual surface charge distribution,” J. Phys. D 28, 1468–1477 (1995).
[CrossRef]

1994

K. Tanaka, T. Takada, “Measurement of the 2-dimensional electric field vector in dielectric liquid,” IEEE Trans. Dieletr. Electr. Insul. 1, 747–753 (1994).
[CrossRef]

M. Ohmi, M. Akatsuka, K. Ishikawa, K. Naito, Y. Yonezawa, Y. Nishida, M. Yamanaka, Y. Izawa, S. Nakai, “High-sensitivity two-dimensional thermal- and mechanical-stress-induced birefringence measurements in a Nd:YAG rod,” Appl. Opt. 33, 6368–6372 (1994).
[CrossRef] [PubMed]

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

1993

T. Kawasaki, Y. Hirata, T. Takada, T. Maeno, “Visualization and quantification technique of latent electric charge image using Pockels effect, high-sensitive measurement using image lock-in amplifier,” Trans. IEE Jpn. 113-C, 1114–1120 (1993).

1992

U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
[CrossRef]

1991

H. Sato, M. Yamaguchi, A. Yoshimoto, T. Takada, “High sensitive measurement system of birefringence using optical phase modulator and two Faraday rotators,” Jpn. J. Soc. Mater. Eng. Res. 4, 22–34 (1991).
[CrossRef]

1983

M. Zahn, T. Takada, S. Voldman, “Kerr electro-optic field mapping measurement in water using parallel cylindrical electrodes,” J. Appl. Phys. 54, 4749–4761 (1983).
[CrossRef]

Akatsuka, M.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, “Propagation of polarized light through polarizing optical systems,” in Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 66–152.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Propagation of polarized light through polarizing optical systems,” in Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 66–152.

Dally, J. W.

J. W. Dally, W. F. Riley, “Theory of photo-elasticity,” in Experimental Stress Analysis (McGraw-Hill, New York, 1978), pp. 406–446.

Gafvert, U.

U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
[CrossRef]

Hirata, Y.

T. Kawasaki, Y. Hirata, T. Takada, T. Maeno, “Visualization and quantification technique of latent electric charge image using Pockels effect, high-sensitive measurement using image lock-in amplifier,” Trans. IEE Jpn. 113-C, 1114–1120 (1993).

Ishikawa, K.

Izawa, Y.

Jaksts, A.

U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
[CrossRef]

Kawada, K.

J. Tsuji, M. Nishida, K. Kawada, “The basis for photoelastic stress analysis,” in Experimental Methods Using Photoelastic Effect (Japanese Nikkan Kougyo, Tokyo, 1965), pp. 51–170 (in Japanese).

Kawasaki, T.

T. Kawasaki, Y. Hirata, T. Takada, T. Maeno, “Visualization and quantification technique of latent electric charge image using Pockels effect, high-sensitive measurement using image lock-in amplifier,” Trans. IEE Jpn. 113-C, 1114–1120 (1993).

Kiyomoto, N.

N. O’Flaherty, N. Kiyomoto, I. Srahama, Y. Machida, “A system for high sensitive measurement of birefringence using a photo-elastic modulator and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Koyama, T.

Machida, Y.

N. O’Flaherty, N. Kiyomoto, I. Srahama, Y. Machida, “A system for high sensitive measurement of birefringence using a photo-elastic modulator and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Maeno, T.

T. Kawasaki, Y. Hirata, T. Takada, T. Maeno, “Visualization and quantification technique of latent electric charge image using Pockels effect, high-sensitive measurement using image lock-in amplifier,” Trans. IEE Jpn. 113-C, 1114–1120 (1993).

Murooka, Y.

Naito, K.

Nakai, S.

Nishida, M.

J. Tsuji, M. Nishida, K. Kawada, “The basis for photoelastic stress analysis,” in Experimental Methods Using Photoelastic Effect (Japanese Nikkan Kougyo, Tokyo, 1965), pp. 51–170 (in Japanese).

Nishida, Y.

O’Flaherty, N.

N. O’Flaherty, N. Kiyomoto, I. Srahama, Y. Machida, “A system for high sensitive measurement of birefringence using a photo-elastic modulator and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Ohmi, M.

Otani, Y.

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

Riley, W. F.

J. W. Dally, W. F. Riley, “Theory of photo-elasticity,” in Experimental Stress Analysis (McGraw-Hill, New York, 1978), pp. 406–446.

Sato, H.

H. Sato, M. Yamaguchi, A. Yoshimoto, T. Takada, “High sensitive measurement system of birefringence using optical phase modulator and two Faraday rotators,” Jpn. J. Soc. Mater. Eng. Res. 4, 22–34 (1991).
[CrossRef]

Shimada, T.

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

Srahama, I.

N. O’Flaherty, N. Kiyomoto, I. Srahama, Y. Machida, “A system for high sensitive measurement of birefringence using a photo-elastic modulator and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Takada, T.

Y. C. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: data processing and experiment verification,” Appl. Opt. 38, 2216–2224 (1999).
[CrossRef]

Y. C. Zhu, T. Takada, “A 2-dimensional Kerr-effect technique for electric field distribution in liquid dielectrics,” IEEE Trans. Dielectr. Electr. Insul. 4, 748–757 (1997).
[CrossRef]

M. Takayama, T. Takada, T. Yoshino, “Stress distribution using high sensitive birefringence measurement system,” Jpn. J. Nondestructive Tests 44, 249–252 (1995).

Y. C. Zhu, T. Takada, D. M. Tu, “An optical measurement technique for studying residual surface charge distribution,” J. Phys. D 28, 1468–1477 (1995).
[CrossRef]

K. Tanaka, T. Takada, “Measurement of the 2-dimensional electric field vector in dielectric liquid,” IEEE Trans. Dieletr. Electr. Insul. 1, 747–753 (1994).
[CrossRef]

T. Kawasaki, Y. Hirata, T. Takada, T. Maeno, “Visualization and quantification technique of latent electric charge image using Pockels effect, high-sensitive measurement using image lock-in amplifier,” Trans. IEE Jpn. 113-C, 1114–1120 (1993).

H. Sato, M. Yamaguchi, A. Yoshimoto, T. Takada, “High sensitive measurement system of birefringence using optical phase modulator and two Faraday rotators,” Jpn. J. Soc. Mater. Eng. Res. 4, 22–34 (1991).
[CrossRef]

M. Zahn, T. Takada, S. Voldman, “Kerr electro-optic field mapping measurement in water using parallel cylindrical electrodes,” J. Appl. Phys. 54, 4749–4761 (1983).
[CrossRef]

Takayama, M.

M. Takayama, T. Takada, T. Yoshino, “Stress distribution using high sensitive birefringence measurement system,” Jpn. J. Nondestructive Tests 44, 249–252 (1995).

Tanaka, K.

K. Tanaka, T. Takada, “Measurement of the 2-dimensional electric field vector in dielectric liquid,” IEEE Trans. Dieletr. Electr. Insul. 1, 747–753 (1994).
[CrossRef]

Tornkvist, C.

U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
[CrossRef]

Tsuji, J.

J. Tsuji, M. Nishida, K. Kawada, “The basis for photoelastic stress analysis,” in Experimental Methods Using Photoelastic Effect (Japanese Nikkan Kougyo, Tokyo, 1965), pp. 51–170 (in Japanese).

Tu, D. M.

Y. C. Zhu, T. Takada, D. M. Tu, “An optical measurement technique for studying residual surface charge distribution,” J. Phys. D 28, 1468–1477 (1995).
[CrossRef]

Umeda, N.

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

Voldman, S.

M. Zahn, T. Takada, S. Voldman, “Kerr electro-optic field mapping measurement in water using parallel cylindrical electrodes,” J. Appl. Phys. 54, 4749–4761 (1983).
[CrossRef]

Walfridsson, L.

U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
[CrossRef]

Yamaguchi, M.

H. Sato, M. Yamaguchi, A. Yoshimoto, T. Takada, “High sensitive measurement system of birefringence using optical phase modulator and two Faraday rotators,” Jpn. J. Soc. Mater. Eng. Res. 4, 22–34 (1991).
[CrossRef]

Yamanaka, M.

Yonezawa, Y.

Yoshimoto, A.

H. Sato, M. Yamaguchi, A. Yoshimoto, T. Takada, “High sensitive measurement system of birefringence using optical phase modulator and two Faraday rotators,” Jpn. J. Soc. Mater. Eng. Res. 4, 22–34 (1991).
[CrossRef]

Yoshino, T.

M. Takayama, T. Takada, T. Yoshino, “Stress distribution using high sensitive birefringence measurement system,” Jpn. J. Nondestructive Tests 44, 249–252 (1995).

Yoshizawa, T.

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

Zahn, M.

M. Zahn, T. Takada, S. Voldman, “Kerr electro-optic field mapping measurement in water using parallel cylindrical electrodes,” J. Appl. Phys. 54, 4749–4761 (1983).
[CrossRef]

Zhu, Y. C.

Y. C. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: data processing and experiment verification,” Appl. Opt. 38, 2216–2224 (1999).
[CrossRef]

Y. C. Zhu, T. Takada, “A 2-dimensional Kerr-effect technique for electric field distribution in liquid dielectrics,” IEEE Trans. Dielectr. Electr. Insul. 4, 748–757 (1997).
[CrossRef]

Y. C. Zhu, T. Takada, D. M. Tu, “An optical measurement technique for studying residual surface charge distribution,” J. Phys. D 28, 1468–1477 (1995).
[CrossRef]

Appl. Opt.

IEEE Trans. Dielectr. Electr. Insul.

Y. C. Zhu, T. Takada, “A 2-dimensional Kerr-effect technique for electric field distribution in liquid dielectrics,” IEEE Trans. Dielectr. Electr. Insul. 4, 748–757 (1997).
[CrossRef]

IEEE Trans. Dieletr. Electr. Insul.

K. Tanaka, T. Takada, “Measurement of the 2-dimensional electric field vector in dielectric liquid,” IEEE Trans. Dieletr. Electr. Insul. 1, 747–753 (1994).
[CrossRef]

IEEE Trans. Electr. Insul.

U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
[CrossRef]

J. Appl. Phys.

M. Zahn, T. Takada, S. Voldman, “Kerr electro-optic field mapping measurement in water using parallel cylindrical electrodes,” J. Appl. Phys. 54, 4749–4761 (1983).
[CrossRef]

J. Phys. D

Y. C. Zhu, T. Takada, D. M. Tu, “An optical measurement technique for studying residual surface charge distribution,” J. Phys. D 28, 1468–1477 (1995).
[CrossRef]

Jpn. J. Nondestructive Tests

M. Takayama, T. Takada, T. Yoshino, “Stress distribution using high sensitive birefringence measurement system,” Jpn. J. Nondestructive Tests 44, 249–252 (1995).

Jpn. J. Soc. Mater. Eng. Res.

H. Sato, M. Yamaguchi, A. Yoshimoto, T. Takada, “High sensitive measurement system of birefringence using optical phase modulator and two Faraday rotators,” Jpn. J. Soc. Mater. Eng. Res. 4, 22–34 (1991).
[CrossRef]

Opt. Eng.

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

Trans. IEE Jpn.

T. Kawasaki, Y. Hirata, T. Takada, T. Maeno, “Visualization and quantification technique of latent electric charge image using Pockels effect, high-sensitive measurement using image lock-in amplifier,” Trans. IEE Jpn. 113-C, 1114–1120 (1993).

Other

J. Tsuji, M. Nishida, K. Kawada, “The basis for photoelastic stress analysis,” in Experimental Methods Using Photoelastic Effect (Japanese Nikkan Kougyo, Tokyo, 1965), pp. 51–170 (in Japanese).

J. W. Dally, W. F. Riley, “Theory of photo-elasticity,” in Experimental Stress Analysis (McGraw-Hill, New York, 1978), pp. 406–446.

R. M. A. Azzam, N. M. Bashara, “Propagation of polarized light through polarizing optical systems,” in Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 66–152.

N. O’Flaherty, N. Kiyomoto, I. Srahama, Y. Machida, “A system for high sensitive measurement of birefringence using a photo-elastic modulator and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagrams of optical systems. (a) Rotation method. The angle Φ A of the transmission axis of the analyzer A is set twice at angles of Φ AR and Φ AT by a mechanical rotation. (b) Two-beam separation method. Two sets of an analyzer, an optical lens, and a camera are used for the two light beams separated at the NPBS (half mirror).

Fig. 2
Fig. 2

Square-pulsed modulation of the optical phase retardation distribution.

Fig. 3
Fig. 3

(a) Equivalent vector OA. (b) Ideal vector OA with ζ T (x, y) and ζ R (x, y) the lengths of its vertical and horizontal components, respectively. (c) Rotation of the vector from OA to OC with a rotation angle of 90° such that |CD| = |OB| holds. (d) Angular relationship for the transmission axes of the polarizer P and the analyzer(s) A, where OA is perpendicular (crossed) to OP, whereas OA R and OA T are symmetric with respect to OA.

Fig. 4
Fig. 4

(a) PMMA plate with a central load of 2.2 kg (b) Image calculation showing the effectiveness of the measurement principle to eliminate the influence of the two-dimensional system nonuniformity.

Equations (16)

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

Ix, y=1/2γx, yIox, y1+ηx, ycos θmt+ζx, ysin θmt,
ηx, y=cos2ΦA-2 sin2ϕsx, ysin 2ϕsx, y-ΦAsin2θsx, y/2,
ζx, y=sin 2ϕsx, y-ΦAsin θsx, y,
I+x, y=1/2γx, yIox, y1+ηx, ycos ΘM+ζx, ysin ΘM,
I-x, y=1/2γx, yIox, y1+ηx, ycos ΘM-ζx, ysin ΘM,
I0x, y=1/2γx, yIox, y1+ηx, y,
ΔIx, y=I+x, y-I-x, y=IMx, yζx, y,
IMx, y=γx, yIox, ysin ΘM,
ζx, y=ΔIx, y/IMx, y.
IMx, y=I+x, y+I-x, y-2I0x, ycos ΘM/tanΘM/2.
2ΦAR-ΦAT=90°,
ζTx, y=sin 2ϕsx, y-ΦATsin θsx, y,
ζRx, y=sin 2ϕsx, y-ΦARsin θsx, y=cos 2ϕsx, y-ΦATsin θsx, y.
ΦAR+ΦAT/2=90°,
sin2 θsx, y=ζTx, y2+ζRx, y2,
tan2ϕsx, y-ΦAT=ζTx, yζRx, y

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