Abstract

In recent years many authors have considered the possibility of using tomography for nondestructive determination of three-dimensional stress fields. A natural starting point for this is integrated photoelasticity. The problem is complicated since the stress field is a tensor field, and in the general case in integrated photoelasticity the relationships between the measurement data and the parameters of the stress field are nonlinear. To elucidate these relationships, we have systematically studied the propagation of polarized light in an inhomogeneous birefringent medium. The inverse problem of integrated photoelasticity is formulated in the general form, and particular cases in which the polarization transformation matrix is exactly determined by integrals of the stress tensor components are considered. The possibility of using the Radon inversion for approximate determination of the normal stress field in an arbitrary section of the test object is outlined.

© 2004 Optical Society of America

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References

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  1. A. Kobayashi, ed., Handbook on Experimental Mechanics, 2nd ed. (VCH, New York, 1993).
  2. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  3. H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).
  4. C. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).
  5. C. A. Kak, M. Slaney, Principles of Computerized Tomography (IEEE Press, Piscataway, N.J., 1988).
  6. H. Kubo, R. Nagata, “Determination of dielectric tensor fields in weakly inhomogeneous anisotropic media,” J. Opt. Soc. Am. 69, 604–610 (1979).
    [CrossRef]
  7. H. Kubo, R. Nagata, “Determination of dielectric tensor fields in weakly inhomogeneous anisotropic media. II,” J. Opt. Soc. Am. 71, 327–333 (1981).
    [CrossRef]
  8. Yu. A. Andrienko, M. S. Dubovikov, A. D. Gladun, “Optical tomography of a birefringent medium,” J. Opt. Soc. Am. A 9, 1761–1764 (1992).
    [CrossRef]
  9. Y. A. Andrienko, M. S. Dubovikov, “Optical tomography of tensor fields: the general case,” J. Opt. Soc. Am. A 11, 1628–1631 (1994).
    [CrossRef]
  10. S. Yu. Berezhna, “Optical tomography of anisotropic inhomogeneous medium,” in Proceedings of the 10th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 431–435.
  11. S. Yu. Berezhna, I. V. Berezhnyi, M. Takashi, “Optical tomographic technique as a part of a hybrid method for elastic analysis,” in Proceedings of the 11th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1998), Vol. 1, pp. 489–494.
  12. M. L. L. Wijerathne, K. Oguni, M. Hori, “Tensor field tomography based on 3D photoelasticity,” Mech. Mater. 34, 533–545 (2002).
    [CrossRef]
  13. H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
    [CrossRef]
  14. J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
    [CrossRef]
  15. H. Aben, L. Ainola, A. Puro, “Photoelastic residual stress measurement in glass articles as a problem of hybrid mechanics,” in Proceedings of the 17th Symposium on Experimental Mechanics of Solids (Instytut Techniki Lotniczej i Mechaniki Stosowanej, Warsaw, 1996), pp. 1–10.
  16. V. Sharafutdinov, “On integrated photoelasticity in case of weak birefringence,” Proc. Estonian Acad. Sci. Phys. Math. 38, 379–389 (1989).
  17. V. A. Sharafutdinov, Integral Geometry of Tensor Fields (VSP, Utrecht, The Netherlands, 1994).
  18. H. Aben, S. Idnurm, J. Josepson, K.-J. Kell, A. Puro, “Optical tomography of the stress tensor field,” in Analytical Methods for Optical Tomography, G. Levin, ed., Proc. SPIE1843, 220–229 (1991).
    [CrossRef]
  19. H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.
  20. T. Abe, Y. Mitsunaga, H. Koga, “Photoelastic computer tomography: a novel measurement method for axial residual stress profile in optical fibers,” J. Opt. Soc. Am. A 3, 133–138 (1986).
    [CrossRef]
  21. A. E. Puro, K.-J. Kell, “Complete determination of stress in fiber preforms of arbitrary cross-section,” J. Lightwave Technol. 10, 1010–1014 (1992).
    [CrossRef]
  22. Y. Park, T.-J. Ahn, Y. H. Kim, W.-T. Han, U.-C. Paek, D. Y. Kim, “Measurement method for profiling the residual stress and the strain-optic coefficient of an optical fiber,” Appl. Opt. 41, 21–26 (2002).
    [CrossRef] [PubMed]
  23. H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414–1421 (1986).
    [CrossRef]
  24. L. Ainola, H. Aben, “Transformation equations in polarization optics of inhomogeneous birefringent media,” J. Opt. Soc. Am. A 18, 2164–2170 (2001).
    [CrossRef]
  25. L. Ainola, H. Aben, “Alternative equations of magnetophotoelasticity and approximate solution of the inverse problem,” J. Opt. Soc. Am. A 19, 1886–1893 (2002).
    [CrossRef]
  26. A. S. Marathay, “Operator formalism in the theory of partial polarization,” J. Opt. Soc. Am. 55, 969–980 (1965).
  27. C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. 61, 1207–1213 (1971).
    [CrossRef]
  28. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  29. K. Mehrany, S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 4, 624–635 (2002).
    [CrossRef]
  30. A. Lakhtakia, “Comment on “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 5, 432–433 (2003).
    [CrossRef]
  31. D. Schupp, “Optische Tensortomographie zur Untersuchung räumlicher Spannungsverteilungen,” Fortsch. Ber. VDI, Reihe 8, No. 858 (VDI Verlag, Düsseldorf, Germany, 2000).

2003

A. Lakhtakia, “Comment on “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 5, 432–433 (2003).
[CrossRef]

2002

M. L. L. Wijerathne, K. Oguni, M. Hori, “Tensor field tomography based on 3D photoelasticity,” Mech. Mater. 34, 533–545 (2002).
[CrossRef]

K. Mehrany, S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

Y. Park, T.-J. Ahn, Y. H. Kim, W.-T. Han, U.-C. Paek, D. Y. Kim, “Measurement method for profiling the residual stress and the strain-optic coefficient of an optical fiber,” Appl. Opt. 41, 21–26 (2002).
[CrossRef] [PubMed]

L. Ainola, H. Aben, “Alternative equations of magnetophotoelasticity and approximate solution of the inverse problem,” J. Opt. Soc. Am. A 19, 1886–1893 (2002).
[CrossRef]

2001

1994

1992

A. E. Puro, K.-J. Kell, “Complete determination of stress in fiber preforms of arbitrary cross-section,” J. Lightwave Technol. 10, 1010–1014 (1992).
[CrossRef]

Yu. A. Andrienko, M. S. Dubovikov, A. D. Gladun, “Optical tomography of a birefringent medium,” J. Opt. Soc. Am. A 9, 1761–1764 (1992).
[CrossRef]

1989

V. Sharafutdinov, “On integrated photoelasticity in case of weak birefringence,” Proc. Estonian Acad. Sci. Phys. Math. 38, 379–389 (1989).

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

1986

1981

1979

1978

J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
[CrossRef]

1971

1965

Abe, T.

Aben, H.

L. Ainola, H. Aben, “Alternative equations of magnetophotoelasticity and approximate solution of the inverse problem,” J. Opt. Soc. Am. A 19, 1886–1893 (2002).
[CrossRef]

L. Ainola, H. Aben, “Transformation equations in polarization optics of inhomogeneous birefringent media,” J. Opt. Soc. Am. A 18, 2164–2170 (2001).
[CrossRef]

H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414–1421 (1986).
[CrossRef]

H. Aben, L. Ainola, A. Puro, “Photoelastic residual stress measurement in glass articles as a problem of hybrid mechanics,” in Proceedings of the 17th Symposium on Experimental Mechanics of Solids (Instytut Techniki Lotniczej i Mechaniki Stosowanej, Warsaw, 1996), pp. 1–10.

H. Aben, S. Idnurm, J. Josepson, K.-J. Kell, A. Puro, “Optical tomography of the stress tensor field,” in Analytical Methods for Optical Tomography, G. Levin, ed., Proc. SPIE1843, 220–229 (1991).
[CrossRef]

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).

Aben, H. K.

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

Ahn, T.-J.

Ainola, L.

L. Ainola, H. Aben, “Alternative equations of magnetophotoelasticity and approximate solution of the inverse problem,” J. Opt. Soc. Am. A 19, 1886–1893 (2002).
[CrossRef]

L. Ainola, H. Aben, “Transformation equations in polarization optics of inhomogeneous birefringent media,” J. Opt. Soc. Am. A 18, 2164–2170 (2001).
[CrossRef]

H. Aben, L. Ainola, A. Puro, “Photoelastic residual stress measurement in glass articles as a problem of hybrid mechanics,” in Proceedings of the 17th Symposium on Experimental Mechanics of Solids (Instytut Techniki Lotniczej i Mechaniki Stosowanej, Warsaw, 1996), pp. 1–10.

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

Andrienko, Y. A.

Andrienko, Yu. A.

Anton, J.

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Berezhna, S. Yu.

S. Yu. Berezhna, “Optical tomography of anisotropic inhomogeneous medium,” in Proceedings of the 10th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 431–435.

S. Yu. Berezhna, I. V. Berezhnyi, M. Takashi, “Optical tomographic technique as a part of a hybrid method for elastic analysis,” in Proceedings of the 11th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1998), Vol. 1, pp. 489–494.

Berezhnyi, I. V.

S. Yu. Berezhna, I. V. Berezhnyi, M. Takashi, “Optical tomographic technique as a part of a hybrid method for elastic analysis,” in Proceedings of the 11th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1998), Vol. 1, pp. 489–494.

Danyluk, H. T.

J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
[CrossRef]

Doyle, J. F.

J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
[CrossRef]

Dubovikov, M. S.

Errapart, A.

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

Gladun, A. D.

Guillemet, C.

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).

Han, W.-T.

Herman, C. T.

C. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

Hori, M.

M. L. L. Wijerathne, K. Oguni, M. Hori, “Tensor field tomography based on 3D photoelasticity,” Mech. Mater. 34, 533–545 (2002).
[CrossRef]

Idnurm, S.

H. Aben, S. Idnurm, J. Josepson, K.-J. Kell, A. Puro, “Optical tomography of the stress tensor field,” in Analytical Methods for Optical Tomography, G. Levin, ed., Proc. SPIE1843, 220–229 (1991).
[CrossRef]

Josepson, J.

H. Aben, S. Idnurm, J. Josepson, K.-J. Kell, A. Puro, “Optical tomography of the stress tensor field,” in Analytical Methods for Optical Tomography, G. Levin, ed., Proc. SPIE1843, 220–229 (1991).
[CrossRef]

Josepson, J. I.

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

Kak, C. A.

C. A. Kak, M. Slaney, Principles of Computerized Tomography (IEEE Press, Piscataway, N.J., 1988).

Kell, K.-J.

A. E. Puro, K.-J. Kell, “Complete determination of stress in fiber preforms of arbitrary cross-section,” J. Lightwave Technol. 10, 1010–1014 (1992).
[CrossRef]

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

H. Aben, S. Idnurm, J. Josepson, K.-J. Kell, A. Puro, “Optical tomography of the stress tensor field,” in Analytical Methods for Optical Tomography, G. Levin, ed., Proc. SPIE1843, 220–229 (1991).
[CrossRef]

Khorasani, S.

K. Mehrany, S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

Kim, D. Y.

Kim, Y. H.

Koga, H.

Kubo, H.

Lakhtakia, A.

A. Lakhtakia, “Comment on “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 5, 432–433 (2003).
[CrossRef]

Marathay, A. S.

Mehrany, K.

K. Mehrany, S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

Mitsunaga, Y.

Nagata, R.

Oguni, K.

M. L. L. Wijerathne, K. Oguni, M. Hori, “Tensor field tomography based on 3D photoelasticity,” Mech. Mater. 34, 533–545 (2002).
[CrossRef]

Paek, U.-C.

Park, Y.

Puro, A.

H. Aben, S. Idnurm, J. Josepson, K.-J. Kell, A. Puro, “Optical tomography of the stress tensor field,” in Analytical Methods for Optical Tomography, G. Levin, ed., Proc. SPIE1843, 220–229 (1991).
[CrossRef]

H. Aben, L. Ainola, A. Puro, “Photoelastic residual stress measurement in glass articles as a problem of hybrid mechanics,” in Proceedings of the 17th Symposium on Experimental Mechanics of Solids (Instytut Techniki Lotniczej i Mechaniki Stosowanej, Warsaw, 1996), pp. 1–10.

Puro, A. E.

A. E. Puro, K.-J. Kell, “Complete determination of stress in fiber preforms of arbitrary cross-section,” J. Lightwave Technol. 10, 1010–1014 (1992).
[CrossRef]

Schupp, D.

D. Schupp, “Optische Tensortomographie zur Untersuchung räumlicher Spannungsverteilungen,” Fortsch. Ber. VDI, Reihe 8, No. 858 (VDI Verlag, Düsseldorf, Germany, 2000).

Sharafutdinov, V.

V. Sharafutdinov, “On integrated photoelasticity in case of weak birefringence,” Proc. Estonian Acad. Sci. Phys. Math. 38, 379–389 (1989).

Sharafutdinov, V. A.

V. A. Sharafutdinov, Integral Geometry of Tensor Fields (VSP, Utrecht, The Netherlands, 1994).

Slaney, M.

C. A. Kak, M. Slaney, Principles of Computerized Tomography (IEEE Press, Piscataway, N.J., 1988).

Takashi, M.

S. Yu. Berezhna, I. V. Berezhnyi, M. Takashi, “Optical tomographic technique as a part of a hybrid method for elastic analysis,” in Proceedings of the 11th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1998), Vol. 1, pp. 489–494.

Whitney, C.

Wijerathne, M. L. L.

M. L. L. Wijerathne, K. Oguni, M. Hori, “Tensor field tomography based on 3D photoelasticity,” Mech. Mater. 34, 533–545 (2002).
[CrossRef]

Appl. Opt.

Exp. Mech.

J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
[CrossRef]

J. Lightwave Technol.

A. E. Puro, K.-J. Kell, “Complete determination of stress in fiber preforms of arbitrary cross-section,” J. Lightwave Technol. 10, 1010–1014 (1992).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

K. Mehrany, S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

A. Lakhtakia, “Comment on “Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices,” J. Opt. A, Pure Appl. Opt. 5, 432–433 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mech. Mater.

M. L. L. Wijerathne, K. Oguni, M. Hori, “Tensor field tomography based on 3D photoelasticity,” Mech. Mater. 34, 533–545 (2002).
[CrossRef]

Opt. Lasers Eng.

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

Proc. Estonian Acad. Sci. Phys. Math.

V. Sharafutdinov, “On integrated photoelasticity in case of weak birefringence,” Proc. Estonian Acad. Sci. Phys. Math. 38, 379–389 (1989).

Other

V. A. Sharafutdinov, Integral Geometry of Tensor Fields (VSP, Utrecht, The Netherlands, 1994).

H. Aben, S. Idnurm, J. Josepson, K.-J. Kell, A. Puro, “Optical tomography of the stress tensor field,” in Analytical Methods for Optical Tomography, G. Levin, ed., Proc. SPIE1843, 220–229 (1991).
[CrossRef]

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

A. Kobayashi, ed., Handbook on Experimental Mechanics, 2nd ed. (VCH, New York, 1993).

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).

C. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

C. A. Kak, M. Slaney, Principles of Computerized Tomography (IEEE Press, Piscataway, N.J., 1988).

S. Yu. Berezhna, “Optical tomography of anisotropic inhomogeneous medium,” in Proceedings of the 10th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 431–435.

S. Yu. Berezhna, I. V. Berezhnyi, M. Takashi, “Optical tomographic technique as a part of a hybrid method for elastic analysis,” in Proceedings of the 11th International Conference on Experimental Mechanics (A. A. Balkema, Rotterdam, The Netherlands, 1998), Vol. 1, pp. 489–494.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

H. Aben, L. Ainola, A. Puro, “Photoelastic residual stress measurement in glass articles as a problem of hybrid mechanics,” in Proceedings of the 17th Symposium on Experimental Mechanics of Solids (Instytut Techniki Lotniczej i Mechaniki Stosowanej, Warsaw, 1996), pp. 1–10.

D. Schupp, “Optische Tensortomographie zur Untersuchung räumlicher Spannungsverteilungen,” Fortsch. Ber. VDI, Reihe 8, No. 858 (VDI Verlag, Düsseldorf, Germany, 2000).

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Figures (1)

Fig. 1
Fig. 1

Schema of the coordinate systems.

Equations (141)

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d2Edz2+ω2c2E=0,
E=E1E2,
ε=ε11ε12ε21ε22.
E=Eexp(-ikz),
k=ωcε.
iκ d2Edz2-dEdz-iC0(ε-εs0)E=0,
κ=1/2k,C0=k/2ε,
s0=1001.
iκ d2Edξ2-L dEdξ-iC0L2(ε-εs0)E=0.
E=E(0)+κE(1)+ .
dE(0)dz+iC0(ε-εs0)E(0)=0,
dE(1)dz+iC0(ε-εs0)E(1)=i d2E(0)dz2, .
dE(1)dz+iC0(ε-εs0)E(1)
=C0dεdz-iC02(ε-εs0)2dE(0).
dE(1)dz+iC0(ε11-ε)E(11)
=C0dε11dz-iC02(ε11-ε)2E(0).
E(1)=E(0)(0)exp-i0zC0(ε11-ε)dz×0zC0dε11dz-iC02(ε11-ε)2dz.
κE(1)E(0)=κ0zC0dε11dz-iC02(ε11-ε)2dz.
κE(1)E(0)-κLC02(ε11-ε)2.
s1=i00-i,s2=01-10,s3=0ii0.
s0sj=sjs0=sj,sj2=-s0,j=1, 2, 3,
s1s2=-s2s1=s3,s2s3=-s3s2=s1,
s3s1=-s1s3=s2.
E(0)=Eexp[-if(z)].
dEdz=VE,
V=-iCo(ε-εs0)-df(z)dzs0.
f(z)=C012 (ε11+ε22)-εdz.
V=-12 C0(ε11-ε22)s1-C0ε12s3.
ε=εs0+2C1εσ+2C2εtrσ,
V=-12 C(σ11-σ22)s1-Cσ12s3.
C=2C0C1ε.
E=UE0,
E(z0)=E0.
V=-V*,
U-1=U*,
|det U|=1.
U=β0s0+βpsp,p=1,2,3,
β02+βpβp=1.
U-1=β0s0-βpsp.
dU/dz=VU.
U(z0)=U0,
U0=s0.
E(z*)=E*,
U(z*)=U*.
E*=U*E0.
U*=s0+z0z*Vdz+z0z*VVdzdz+z0z*VVVdzdzdz+ .
z0z*VVdzdz
=12z0z*Vdz2+12z0z*V,Vdzdz,
z0z*VVVdzdzdz
=16z0z*Vdz3+12z0z*VV,Vdzdzdz+13z0z*V,Vdz,Vdzdz .
V,Vdz=VVdz-Vdz V.
W=V,Vdz,
U*=expz0z*Vdz+G(W),
expz0z*Vdz=s0+n=11n!z0z*Vdzn,
G(W)=12z0z*Wdz+12z0z*VWdzdz+13z0z*W,Vdzdz .
G(0)=0.
W=0,
U*=expz0z*Vdz.
U=expz0z*Vdz
expz0zVdz V=V expz0zVdz.
W=12 C(σ11-σ22)σ12dz-σ12(σ11-σ22)dzs2.
σ12=12 (σ11-σ22)tan 2φ.
W=14 Ctan 2φ(σ11-σ22)dz-tan 2φ×(σ11-σ22)dzs2,
W=12 C(σ11-σ22)1cos2 2φdφdz×(σ11-σ22)dzdz s2.
σ12=A2 (σ11-σ22),
A=tan 2φ0.
V=-12C(σ11-σ22)(s1+As3).
U*=cos F s0+11+A2sin F (s1+As3),
F=12 C1+A2z0z*(σ11-σ22)dz.
dUdz=VU
U=expz0zVdz.
dUdz=expz0zVdz V,
dUdz=UV.
dUdz U=UVU.
dUdz U=U dUdz.
Y=dUdz,U,
Y=0.
dUdz U-1=V.
(β0s0+βpsp)(β0s0+βrsr)=12 C(σ11-σ22)s1+Cσ12s3.
β0β0+βpβp=0,
-β1β0+β0β1-β3β2+β2β3=-12 C(σ11-σ22),
-β2β0+β2β1+β0β2-β1β3=0,
-β3β0-β2β1+β1β2+β0β3=-Cσ12.
β0(z0)=1,βp(z0)=0,p=1,2,3.
Y=(β0s0+βpsp)(β0s0+βrsr)-(β0s0+βrsr)(β0s0+βpsp),
Y=2[(β2β3-β2β3)s1+(β3β1-β3β1)s2+(β1β2-β1β2)s3].
β2β3-β2β3=0,
β3β1-β3β1=0,
β1β2-β1β2=0.
β0β1-β1β0=-12C(σ11-σ22),
β0β2-β2β0=0,
β0β3-β3β0=-Cσ12.
β2=0,β3*=Aβ1*,
11+A2arcsin1+A2β1*=-12 Cz0z*(σ11-σ22)dz.
cos 2φ0arcsinβ1*cos 2φ0=-12 Cz0z*(σ11-σ22)dz,
sin 2φ0arcsinβ1*cos 2φ=-Cz0z*σ12dz.
φ0=12arctanβ3*β1*.
arcsin β1*=-12 Cz0z*(σ1-σ2)dz.
U=G(λ)S(ϑ)G(-μ),
G(λ)=cos λs0+sin λs1,
S(ϑ)=cos ϑs0-sin ϑs2.
β0=cos ϑ cos(λ-μ),β1=cos ϑ sin(λ-μ),
β2=-sin ϑ cos(λ+μ),β3=-sin ϑ sin(λ+μ).
β1*=cos ϑ*cos 2μ*,
φ0=-12arctantan ϑ*cos 2μ*.
U=S(α)G(γ)S(-α0),
β0=cos γ cos(α-α0),β1=sin γ cos(α+α0),
β2=cos γ sin(α-α0),β3=sin γ sin(α+α0).
α=α0,β1*=sin γ*cos 2α*,β3*=sin γ*sin 2α*.
φ0=α*.
γ*cos 2α*=-12 Cz0z*(σ11-σ22)dz,
γ*sin 2α*=-Cz0z*σ12dz.
U=S(θ)T(ε)G(δ/2)T(-ε)S(-θ),
T(ν)=cos ε s0+sin ε s3.
β0=cosδ2,β1=sinδ2cos 2θ cos 2ε,
β2=sinδ2sin 2ε,β3=sinδ2sin 2θ cos 2ε.
ν=0,β1*=sinδ*2cos 2θ*,β3*1=sinδ*2sin 2θ*.
θ*=φ0.
δ*cos 2θ*=-Cz0z*(σ11-σ22)dz,
δ*sin 2θ*=-2Cz0z*σ12dz.
σ11-σ22=Bσ12,B=2/A,
x1=x1,x2=α22x2+α23x3,
x3=α32x2+α33x3,
σ11-σ22=Bσ12.
σpr=αpsαrtσst,
σ11=σ11,
σ22=α222σ22+α232σ33+2α22α23σ23,
σ12=α22σ12+α23σ13.
σ11-α222σ22-α232σ33-2α22α23σ23
=B(α22σ12+α23σ13).
σ22=σ33,σ13=σ23=0.
σ11=σ22,σ12=σ31=0,
σ11=σ33,σ12=σ32=0.
σ11=σ22=σ33=const.
G1=-Cz0z*(σ11-σ22)dz,
G2=-2Cz0z*σ12dz,
σ12x1+σ22x2+σ23z=0
xz0x2x1z0z*σ12dzdx2+z0z*σ22dz=0.
z0z*σ22dz=-12Cxz0x2G2x1dx2.
z0z*σ11dz=12Cz0x2G2x1dx2-G1C.
σ^11(l, θ)=--σ11(x2, x3)×δ(x2cos θ+x3sin θ-l)dx2dx3.

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