Abstract

A two-dimensional measurement method for a birefringence vector distribution differs from a point-measurement method not only in the two-dimensional system nonuniformity but also in the system reliability. The previously proposed intrinsic vector allowed for the elimination of the influence of the system nonuniformity, whereas the two-dimensional system reliability is ensured by both an online diagnosis technique and an image lock-in processing. It is revealed that the measured intrinsic vector is relevant not only with the birefringence vector distribution in a sample but also with the natural birefringence vector distributions that exist in the optic components. The complete solution from the measured intrinsic vector results in a bidirectional vector for the desired birefringence vector distribution. The correctness of the two-dimensional measurement principle is examined by means of a comparison of the measured data with that calculated from a finite-element analysis based on the photoelastic effect.

© 1999 Optical Society of America

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References

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  1. Y. C. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: measurement principle,” Appl. Opt. 38, 2225–2231 (1999).
    [CrossRef]
  2. 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]
  3. A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by using Pockels effect,” in Proceedings of the 12th International Conference on Gas Discharges and Their Applications, G. Babucke, ed., (KIEBU Druck Gmbh, Greifswald, Germany, 1997), pp. 264–267.
  4. Y. C. Zhu, T. Takada, Y. Inoue, D. M. Tu, “Dynamic observation of needle-plane surface discharge using the electro-optical Pockels effect,” IEEE Trans. Dielectr. Electr. Insul. 3, 460–468 (1996).
    [CrossRef]
  5. 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]
  6. T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).
  7. 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]
  8. U. Gafvert, A. Jaksts, C. Tornkvist, L. Walfridsson, “Electrical field distribution in transformer oil,” IEEE Trans. Electr. Insul. 27, 647–660 (1992).
    [CrossRef]
  9. R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
    [CrossRef]
  10. K. Tanaka, T. Takada, “Measurement of the 2-dimensional electric field vector in dielectric liquid,” IEEE Trans. Dielectr. Electr. Insul. 1, 747–753 (1994).
    [CrossRef]
  11. R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.
  12. 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]
  13. Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  14. J. W. Dally, W. F. Riley, “Theory of photo-elasticity,” in Experimental Stress Analysis (McGraw-Hill, New York, 1978), pp. 406–446.

1999 (1)

1997 (1)

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]

1996 (1)

Y. C. Zhu, T. Takada, Y. Inoue, D. M. Tu, “Dynamic observation of needle-plane surface discharge using the electro-optical Pockels effect,” IEEE Trans. Dielectr. Electr. Insul. 3, 460–468 (1996).
[CrossRef]

1995 (1)

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 (2)

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

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

1992 (3)

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

R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
[CrossRef]

T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).

1991 (1)

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 (1)

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]

Chiba, M.

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by using Pockels effect,” in Proceedings of the 12th International Conference on Gas Discharges and Their Applications, G. Babucke, ed., (KIEBU Druck Gmbh, Greifswald, Germany, 1997), pp. 264–267.

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]

Hayakawa, N.

R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.

Hidaka, K.

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by using Pockels effect,” in Proceedings of the 12th International Conference on Gas Discharges and Their Applications, G. Babucke, ed., (KIEBU Druck Gmbh, Greifswald, Germany, 1997), pp. 264–267.

Hikita, M.

R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.

Hiromoto, N.

T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).

Inoue, Y.

Y. C. Zhu, T. Takada, Y. Inoue, D. M. Tu, “Dynamic observation of needle-plane surface discharge using the electro-optical Pockels effect,” IEEE Trans. Dielectr. Electr. Insul. 3, 460–468 (1996).
[CrossRef]

Itabe, T.

T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).

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]

Kato, K.

R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.

Kawasaki, T.

R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
[CrossRef]

Koyama, T.

Kumada, A.

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by using Pockels effect,” in Proceedings of the 12th International Conference on Gas Discharges and Their Applications, G. Babucke, ed., (KIEBU Druck Gmbh, Greifswald, Germany, 1997), pp. 264–267.

Liu, R.

R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
[CrossRef]

Maeno, T.

T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).

Matsuka, M.

R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.

Murooka, Y.

Nishikata, A.

T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).

Okubo, H.

R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.

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]

Satoh, A.

R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
[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]

Shimizu, R.

R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.

Takada, T.

Y. C. Zhu, T. Koyama, T. Takada, Y. Murooka, “Two-dimensional measurement technique for birefringence vector distributions: measurement principle,” Appl. Opt. 38, 2225–2231 (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, Y. Inoue, D. M. Tu, “Dynamic observation of needle-plane surface discharge using the electro-optical Pockels effect,” IEEE Trans. Dielectr. Electr. Insul. 3, 460–468 (1996).
[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]

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

R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
[CrossRef]

T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).

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]

Tanaka, K.

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

R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
[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]

Tu, D. M.

Y. C. Zhu, T. Takada, Y. Inoue, D. M. Tu, “Dynamic observation of needle-plane surface discharge using the electro-optical Pockels effect,” IEEE Trans. Dielectr. Electr. Insul. 3, 460–468 (1996).
[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]

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]

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]

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: measurement principle,” Appl. Opt. 38, 2225–2231 (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, Y. Inoue, D. M. Tu, “Dynamic observation of needle-plane surface discharge using the electro-optical Pockels effect,” IEEE Trans. Dielectr. Electr. Insul. 3, 460–468 (1996).
[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. (1)

IEEE Trans. Dielectr. Electr. Insul. (3)

Y. C. Zhu, T. Takada, Y. Inoue, D. M. Tu, “Dynamic observation of needle-plane surface discharge using the electro-optical Pockels effect,” IEEE Trans. Dielectr. Electr. Insul. 3, 460–468 (1996).
[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]

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

IEEE Trans. Electr. Insul. (2)

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

R. Liu, A. Satoh, T. Kawasaki, K. Tanaka, T. Takada, “High-sensitive Kerr-effect technique for determination of 2-dimensional electric fields,” IEEE Trans. Electr. Insul. 27, 245–254 (1992).
[CrossRef]

J. Appl. Phys. (1)

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 (1)

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. Soc. Mater. Eng. Res. (1)

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. (1)

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. (1)

T. Maeno, A. Nishikata, T. Itabe, N. Hiromoto, T. Takada, “Two-dimensional lock-in amplifier system: a new technique for improving the depth resolution in microscope imaging,” Trans. IEE Jpn. 112-C, 474–479 (1992).

Other (3)

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by using Pockels effect,” in Proceedings of the 12th International Conference on Gas Discharges and Their Applications, G. Babucke, ed., (KIEBU Druck Gmbh, Greifswald, Germany, 1997), pp. 264–267.

R. Shimizu, M. Matsuka, K. Kato, N. Hayakawa, M. Hikita, H. Okubo, “Three-dimensional electric field vector measurements in transformer oil using Kerr effect,” , (Institute of Electrical Engineering of Japan, Tokyo, 1994), pp. 77–85.

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

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the optical measurement system for birefringence vector distributions.

Fig. 2
Fig. 2

(a) U-shaped poly(methyl methacrylate) plate with an external force applied to its upper two sides. (b) Defect distribution D(x, y) with 1 representing the normal positions, 0 the zero-value positions, and 2 the defective positions. (c) Measured retardation distribution θ s (x, y) with marks of A and B indicating the locations where the axis reorientation occurs. (d) Fast-axis direction distribution ϕ s (x, y).

Fig. 3
Fig. 3

Relative relationship between the two principal refractive indices, n 1(x, y) and n 2(x, y), of the birefringence vector for discussing the property of the nonnegative retardation distribution θ s (x, y) and the axis reorientation of the fast-axis direction distribution ϕ s (x, y). (a) n 1(x, y) < n 2(x, y), (b) n 1(x, y) = n 2(x, y), (c) n 1(x, y) > n 2(x, y).

Fig. 4
Fig. 4

(a) Vector subtraction for eliminating the influence of the natural birefringence vector distributions that exist in optic components. (b) Birefringence vector distribution [ϕ s (x, y), θ s (x, y)] calculated with Eq. (7). (c) Birefringence vector distribution [ϕ s (x, y), θ s (x, y)] calculated with the improved equation of Eq. (21). (d) Defect distribution D(x, y) with 1 representing the normal positions, 0 the zero-value positions, and 2 the defective positions.

Fig. 5
Fig. 5

Magnitude distribution data on a poly(methyl methacrylate) plate with experiment condition indicated. (a) Principal stress-difference distribution calculated with a finite-element analysis. (b) Measured retardation distribution θ s (x, y). (c) Defect distribution D(x, y) with 1 representing the normal positions, 0 the zero-value positions, and 2 the defective positions.

Fig. 6
Fig. 6

Direction distribution data on the same sample as in Fig. 5. (a) First-principal-stress direction distribution calculated with a finite-element analysis. (b) Measured fast-axis direction distribution ϕ s (x, y).

Equations (26)

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

Ix, y=1/2γx, yIox, y1+ηx, ycos θmt+ζx, ysin θmt+Ioffx, y,
ηx, y=cos2ΦA-2 sin2ϕsx, ysin 2ϕsx, y-ΦAsin2θsx, y/2,
ζx, y=sin 2ϕsx, y-ΦAsin θsx, y,
2ΦAR-ΦAT=90°,
ζTx, y=sin θsx, ysin 2ϕsx, y-ΦAT,
ζRx, y=sin θsx, ycos 2ϕsx, y-ΦAT.
sin2 θsx, y=ζTx, y2+ζRx, y2,
tan2ϕsx, y-ϕAT=ζTx, yζRx, y,
ΔIx, y=I+x, y-I-x, y=IMx, yζx, y,
IMx, y=γx, yIox, ysin ΘM,
IMx, y=I+x, y+I-x, y-2I0x, ycos ΘM/tanΘM/2.
ζx, y=ΔIx, y/IMx, y.
ΔImx, y=1Ni=1NI+x, y, i-I-x, y, i=1Ni=1N ΔIx, y, i,
IMx, y>Ithreshold,
|ΔIx, y|<IMx, y.
ζTx, y=ζTS+Nx, y-ζTNx, y,
ζRx, y=ζRS+Nx, y-ζRNx, y.
ζTx, y=|sin θsx, y|sin 2ϕsax, y-ΦAT,
ζRx, y=|sin θsx, y|cos 2ϕsax, y-ΦAT,
ζTx, y=|sin θsx, y|sin 2ϕsbx, y+90°-ΦAT,
ζRx, y=|sin θsx, y|cos 2ϕsbx, y+90°-ΦAT.
ϕsax, y=ϕsbx, y+90°
θsx, y=2πL/λΔnx, y,
θsx, y=sin-1ζT2x, y+ζR2x, y1/2.
ϕsx, y=½ tan-1ζTx, y/ζRx, y+ΦAT for ζTx, y>0,
ϕsx, y=½ tan-1ζTx, y/ζRx, y+ΦAT+90° for ζTx, y<0.

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