Abstract

In magnetophotoelasticity, photoelastic models are investigated in a magnetic field in order to initiate rotation of the plane of polarization that is due to the Faraday effect. The method has been used for the measurement of stress distributions that are in equilibrium on the wave normal and therefore cannot be measured with the traditional photoelastic technique. In this category belong bending stresses in plates and shells and residual stresses in glass plates. Two new systems of equations of magnetophotoelasticity are derived. One of them describes evolution of the polarization of light in a magnetophotoelastic medium in terms of eigenvectors, the other in terms of distinctive parameters. For the latter system, an approximate closed-form solution has been found. The integral Wertheim law has been generalized for the case of stress states in equilibrium when rotation of the plane of polarization is present.

© 2002 Optical Society of America

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References

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  1. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  2. H. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
    [CrossRef]
  3. H. K. Aben, “Principles of magnetophotoelasticity,” in Experimental Stress Analysis and Its Influence on Design, M. L. Meyer, ed. (Institution of Mechanical Engineers, London, 1971), pp. 175–182.
  4. H. Aben, S. Idnurm, “Stress concentration in bent plates by magnetophotoelasticity,” in Proceedings of the Fifth International Conference on Experimental Stress Analysis (Udine, Italy, 1974), pp. 4.5–4.10.
  5. G. P. Clarke, H. W. McKenzie, P. Stanley, “The magnetophotoelastic analysis of residual stresses in thermally toughened glass,” Proc. R. Soc. London Ser. A 455, 1149–1173 (1999).
    [CrossRef]
  6. H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).
  7. H. K. Aben, C. I. Idnurm, A. S. Tatarinov, “Magnetophotoelasticity in a strong magnetic field,” in Proceedings of the Fourth National Congress on Theoretical and Applied Mechanics (Bulgarian Academy of Sciences Publishers, Varna, Bulgaria, 1981), Vol. 2, pp. 58–63.
  8. A. E. Puro, “On the tomographic method in magnetophotoelasticity,” Opt. Spectrosc. 81, 119–125 (1996).
  9. A. Puro, “Magnetophotoelasticity as parametric tensor field tomography,” Inverse Probl. 14, 1315–1330 (1998).
    [CrossRef]
  10. P. McIntyre, A. W. Snyder, “Light propagation in twisted anisotropic media: application to photoereceptors,” J. Opt. Soc. Am. 68, 149–157 (1978).
    [CrossRef] [PubMed]
  11. J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.
  12. H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
    [CrossRef]
  13. L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
    [CrossRef]
  14. I. Moreno, C. R. Fernández-Pouza, J. A. Davis, D. J. Franich, “Polarization eigenvectors for reflective twisted nematic liquid crystal displays,” Opt. Eng. 40, 2220–2226 (2001).
    [CrossRef]
  15. H. K. Aben, S. J. Idnurm, E. I. Klabunovskii, M. Uffert, “Photoelasticity with additional physical fields and optically active models,” Exp. Mech. 14, 361–366 (1974).
    [CrossRef]
  16. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  17. P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
  18. L. Ainola, H. Aben, “Transformation equations in polarization optics of inhomogeneous birefringent media,” J. Opt. Soc. Am. A 18, 2164–2170 (2001).
    [CrossRef]
  19. A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 4, 115–126 (1962).
  20. A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).
  21. Encyclopaedia of Mathematics (Kluwer, Dordrecht, The Netherlands, 1995), pp. 945–950.
  22. F. R. Gantmakher, Applications of the Theory of Matrices (Interscience, New York, 1959).
  23. H. Sahsah, S. Djendli, J. Monin, “A new method of birefringence measurements using a Faraday modulator. Application to measurements of stress-optical coefficients,” Meas. Sci. Technol. 11, N46–N50 (2000).
    [CrossRef]
  24. F. Brandi, E. Polacco, G. Ruoso, “Stress optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
    [CrossRef]

2001 (3)

I. Moreno, C. R. Fernández-Pouza, J. A. Davis, D. J. Franich, “Polarization eigenvectors for reflective twisted nematic liquid crystal displays,” Opt. Eng. 40, 2220–2226 (2001).
[CrossRef]

F. Brandi, E. Polacco, G. Ruoso, “Stress optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

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

2000 (1)

H. Sahsah, S. Djendli, J. Monin, “A new method of birefringence measurements using a Faraday modulator. Application to measurements of stress-optical coefficients,” Meas. Sci. Technol. 11, N46–N50 (2000).
[CrossRef]

1999 (2)

L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
[CrossRef]

G. P. Clarke, H. W. McKenzie, P. Stanley, “The magnetophotoelastic analysis of residual stresses in thermally toughened glass,” Proc. R. Soc. London Ser. A 455, 1149–1173 (1999).
[CrossRef]

1998 (2)

A. Puro, “Magnetophotoelasticity as parametric tensor field tomography,” Inverse Probl. 14, 1315–1330 (1998).
[CrossRef]

H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
[CrossRef]

1996 (1)

A. E. Puro, “On the tomographic method in magnetophotoelasticity,” Opt. Spectrosc. 81, 119–125 (1996).

1978 (1)

1974 (1)

H. K. Aben, S. J. Idnurm, E. I. Klabunovskii, M. Uffert, “Photoelasticity with additional physical fields and optically active models,” Exp. Mech. 14, 361–366 (1974).
[CrossRef]

1970 (1)

H. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
[CrossRef]

1962 (2)

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 4, 115–126 (1962).

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).

Aben, H.

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

L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
[CrossRef]

H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
[CrossRef]

H. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
[CrossRef]

H. Aben, S. Idnurm, “Stress concentration in bent plates by magnetophotoelasticity,” in Proceedings of the Fifth International Conference on Experimental Stress Analysis (Udine, Italy, 1974), pp. 4.5–4.10.

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

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

Aben, H. K.

H. K. Aben, S. J. Idnurm, E. I. Klabunovskii, M. Uffert, “Photoelasticity with additional physical fields and optically active models,” Exp. Mech. 14, 361–366 (1974).
[CrossRef]

H. K. Aben, C. I. Idnurm, A. S. Tatarinov, “Magnetophotoelasticity in a strong magnetic field,” in Proceedings of the Fourth National Congress on Theoretical and Applied Mechanics (Bulgarian Academy of Sciences Publishers, Varna, Bulgaria, 1981), Vol. 2, pp. 58–63.

H. K. Aben, “Principles of magnetophotoelasticity,” in Experimental Stress Analysis and Its Influence on Design, M. L. Meyer, ed. (Institution of Mechanical Engineers, London, 1971), pp. 175–182.

Ainola, L.

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).

Brandi, F.

F. Brandi, E. Polacco, G. Ruoso, “Stress optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Clarke, G. P.

G. P. Clarke, H. W. McKenzie, P. Stanley, “The magnetophotoelastic analysis of residual stresses in thermally toughened glass,” Proc. R. Soc. London Ser. A 455, 1149–1173 (1999).
[CrossRef]

Davis, J. A.

I. Moreno, C. R. Fernández-Pouza, J. A. Davis, D. J. Franich, “Polarization eigenvectors for reflective twisted nematic liquid crystal displays,” Opt. Eng. 40, 2220–2226 (2001).
[CrossRef]

Djendli, S.

H. Sahsah, S. Djendli, J. Monin, “A new method of birefringence measurements using a Faraday modulator. Application to measurements of stress-optical coefficients,” Meas. Sci. Technol. 11, N46–N50 (2000).
[CrossRef]

Fernández-Pouza, C. R.

I. Moreno, C. R. Fernández-Pouza, J. A. Davis, D. J. Franich, “Polarization eigenvectors for reflective twisted nematic liquid crystal displays,” Opt. Eng. 40, 2220–2226 (2001).
[CrossRef]

Franich, D. J.

I. Moreno, C. R. Fernández-Pouza, J. A. Davis, D. J. Franich, “Polarization eigenvectors for reflective twisted nematic liquid crystal displays,” Opt. Eng. 40, 2220–2226 (2001).
[CrossRef]

Gantmakher, F. R.

F. R. Gantmakher, Applications of the Theory of Matrices (Interscience, New York, 1959).

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).

Guillemet, C.

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

Idnurm, C. I.

H. K. Aben, C. I. Idnurm, A. S. Tatarinov, “Magnetophotoelasticity in a strong magnetic field,” in Proceedings of the Fourth National Congress on Theoretical and Applied Mechanics (Bulgarian Academy of Sciences Publishers, Varna, Bulgaria, 1981), Vol. 2, pp. 58–63.

Idnurm, S.

H. Aben, S. Idnurm, “Stress concentration in bent plates by magnetophotoelasticity,” in Proceedings of the Fifth International Conference on Experimental Stress Analysis (Udine, Italy, 1974), pp. 4.5–4.10.

Idnurm, S. J.

H. K. Aben, S. J. Idnurm, E. I. Klabunovskii, M. Uffert, “Photoelasticity with additional physical fields and optically active models,” Exp. Mech. 14, 361–366 (1974).
[CrossRef]

Josepson, J.

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

Klabunovskii, E. I.

H. K. Aben, S. J. Idnurm, E. I. Klabunovskii, M. Uffert, “Photoelasticity with additional physical fields and optically active models,” Exp. Mech. 14, 361–366 (1974).
[CrossRef]

Kuske, A.

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 4, 115–126 (1962).

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).

McIntyre, P.

McKenzie, H. W.

G. P. Clarke, H. W. McKenzie, P. Stanley, “The magnetophotoelastic analysis of residual stresses in thermally toughened glass,” Proc. R. Soc. London Ser. A 455, 1149–1173 (1999).
[CrossRef]

Monin, J.

H. Sahsah, S. Djendli, J. Monin, “A new method of birefringence measurements using a Faraday modulator. Application to measurements of stress-optical coefficients,” Meas. Sci. Technol. 11, N46–N50 (2000).
[CrossRef]

Moreno, I.

I. Moreno, C. R. Fernández-Pouza, J. A. Davis, D. J. Franich, “Polarization eigenvectors for reflective twisted nematic liquid crystal displays,” Opt. Eng. 40, 2220–2226 (2001).
[CrossRef]

Polacco, E.

F. Brandi, E. Polacco, G. Ruoso, “Stress optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Puro, A.

A. Puro, “Magnetophotoelasticity as parametric tensor field tomography,” Inverse Probl. 14, 1315–1330 (1998).
[CrossRef]

Puro, A. E.

A. E. Puro, “On the tomographic method in magnetophotoelasticity,” Opt. Spectrosc. 81, 119–125 (1996).

Ruoso, G.

F. Brandi, E. Polacco, G. Ruoso, “Stress optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Sahsah, H.

H. Sahsah, S. Djendli, J. Monin, “A new method of birefringence measurements using a Faraday modulator. Application to measurements of stress-optical coefficients,” Meas. Sci. Technol. 11, N46–N50 (2000).
[CrossRef]

Snyder, A. W.

Stanley, P.

G. P. Clarke, H. W. McKenzie, P. Stanley, “The magnetophotoelastic analysis of residual stresses in thermally toughened glass,” Proc. R. Soc. London Ser. A 455, 1149–1173 (1999).
[CrossRef]

Tatarinov, A. S.

H. K. Aben, C. I. Idnurm, A. S. Tatarinov, “Magnetophotoelasticity in a strong magnetic field,” in Proceedings of the Fourth National Congress on Theoretical and Applied Mechanics (Bulgarian Academy of Sciences Publishers, Varna, Bulgaria, 1981), Vol. 2, pp. 58–63.

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).

Uffert, M.

H. K. Aben, S. J. Idnurm, E. I. Klabunovskii, M. Uffert, “Photoelasticity with additional physical fields and optically active models,” Exp. Mech. 14, 361–366 (1974).
[CrossRef]

Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. (1)

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 4, 115–126 (1962).

Exp. Mech. (2)

H. K. Aben, S. J. Idnurm, E. I. Klabunovskii, M. Uffert, “Photoelasticity with additional physical fields and optically active models,” Exp. Mech. 14, 361–366 (1974).
[CrossRef]

H. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
[CrossRef]

Inverse Probl. (1)

A. Puro, “Magnetophotoelasticity as parametric tensor field tomography,” Inverse Probl. 14, 1315–1330 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (2)

H. Sahsah, S. Djendli, J. Monin, “A new method of birefringence measurements using a Faraday modulator. Application to measurements of stress-optical coefficients,” Meas. Sci. Technol. 11, N46–N50 (2000).
[CrossRef]

F. Brandi, E. Polacco, G. Ruoso, “Stress optic modulator: a novel device for high sensitivity linear birefringence measurements,” Meas. Sci. Technol. 12, 1503–1508 (2001).
[CrossRef]

Opt. Eng. (1)

I. Moreno, C. R. Fernández-Pouza, J. A. Davis, D. J. Franich, “Polarization eigenvectors for reflective twisted nematic liquid crystal displays,” Opt. Eng. 40, 2220–2226 (2001).
[CrossRef]

Opt. Spectrosc. (1)

A. E. Puro, “On the tomographic method in magnetophotoelasticity,” Opt. Spectrosc. 81, 119–125 (1996).

Optik (1)

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).

Proc. R. Soc. London Ser. A (1)

G. P. Clarke, H. W. McKenzie, P. Stanley, “The magnetophotoelastic analysis of residual stresses in thermally toughened glass,” Proc. R. Soc. London Ser. A 455, 1149–1173 (1999).
[CrossRef]

Other (10)

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

H. K. Aben, C. I. Idnurm, A. S. Tatarinov, “Magnetophotoelasticity in a strong magnetic field,” in Proceedings of the Fourth National Congress on Theoretical and Applied Mechanics (Bulgarian Academy of Sciences Publishers, Varna, Bulgaria, 1981), Vol. 2, pp. 58–63.

H. K. Aben, “Principles of magnetophotoelasticity,” in Experimental Stress Analysis and Its Influence on Design, M. L. Meyer, ed. (Institution of Mechanical Engineers, London, 1971), pp. 175–182.

H. Aben, S. Idnurm, “Stress concentration in bent plates by magnetophotoelasticity,” in Proceedings of the Fifth International Conference on Experimental Stress Analysis (Udine, Italy, 1974), pp. 4.5–4.10.

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

Encyclopaedia of Mathematics (Kluwer, Dordrecht, The Netherlands, 1995), pp. 945–950.

F. R. Gantmakher, Applications of the Theory of Matrices (Interscience, New York, 1959).

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

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

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).

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Equations (129)

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dEdz=VE,
E=E1E2,
V=-i12C(σ1-σ2)d(ϕ-ψ)dz-d(ϕ-ψ)dzi12C(σ1-σ2).
dψdz=KH,
E=UE0,
U=S(α*)G(γ)S(-α0).
S(α*)=cos α*-sin α*sin α*cos α*,
G(γ)=exp(iγ)00exp-iγ.
E=G(γ)E0,
E=S(-α*)E,E0=S(-α0)E0.
α=α*-α0,
cos 2α* dγdz-sin 2γ sin 2α* dα0dz=-12C(σ1-σ2),
dα*dz-cos 2γ dα0dz=-d(ϕ-ψ)dz,
sin 2α* dγdz+sin 2γ cos 2α* dα0dz=0.
dγdz=-12C(σ1-σ2)cos 2α*,
dα*dz=12C(σ1-σ2)cot 2γ sin 2α*-d(ϕ-ψ)dz,
dαdz=-12C(σ1-σ2)tan γ sin 2α*-d(ϕ-ψ)dz.
γ(z0)=0,α*(z0)=0,α(z0)=0,
γ(z1)=-12Cz0z1(σ1-σ2)dz.
U=S(-θ)T(ϵ)G(δ/2)T(-ϵ)S(θ),
T(ϵ)=cos ϵi sin ϵi sin ϵcos ϵ.
E=G(δ/2)E0,
E=T(-ϵ)S(θ)E,E0=T(-ϵ)S(θ)E0.
12 dδdz+2 sin δ2 sin δ2 sin 2ϵ-cos δ2 tan 2θ dθdz-2 sin δ2 tan 2ϵcos δ2+sin δ2 sin 2ϵ dϵdz=-12C(σ1-σ2),
12 sin 2ϵ dδdz-2 sin2 δ2 cos2 2ϵ dθdz+sin δ cos 2ϵ dϵdz=d(ϕ-ψ)dz,
12 cos 2ϵ dδdz+2 sin δ2×cos 2ϵsin δ2 sin 2ϵ+cos δ2 cot 2θ dθdz+2 sin δ2 sin δ2 cot 2θ-cos δ2 sin 2ϵ dϵdz=0.
U=G(λ)S(ϑ)G(-μ).
dλdz-cot 2ϑ dμdz=-12C(σ1-σ2),
cos 2λ dϑdz-sin 2λ sin 2ϑ dμdz=-d(ϕ-ψ)dz,
sin 2λ dϑdz+cos 2λ sin 2ϑ dμdz=0
dλdz=d(ϕ-ψ)dz cot 2ϑ sin 2λ-12C(σ1-σ2),
dϑdz=-d(ϕ-ψ)dz cos 2λ,
dμdz=d(ϕ-ψ)dz sin 2λsin 2ϑ.
λ(z0)=0,μ(z0)=0,ϑ(z0)=0.
S(α*)G(γ)S(-α0)=G(λ)S(ϑ)G(-μ).
cos α cos γ=cos κ cos ϑ,
cos β sin γ=sin κ cos ϑ,
sin α cos γ=cos ρ sin ϑ,
sin β sin γ=-sin ρ sin ϑ,
β=α0+α*,κ=λ-μ,ρ=λ+μ.
tan α=cos ρcos κ tan ϑ,
cos 2γ=cos 2κ cos2 ϑ+cos 2ρ sin2 ϑ.
dκdz=-d(ϕ-ψ)dz sin 2λ tan ϑ-12C(σ1-σ2),
dρdz=d(ϕ-ψ)dz sin 2λ cot ϑ-12C(σ1-σ2),
κ(z0)=0,ρ(z0)=0.
S(-θ)T(ϵ)G(δ/2)T(-ϵ)S(θ)=G(λ)S(ϑ)G(-μ).
cos δ2=cos κ cos ϑ,
sin δ2 cos 2θ cos 2ϵ=sin κ cos ϑ,
sin δ2 sin 2ϵ=cos ρ sin ϑ,
sin δ2 sin 2θ cos 2ϵ=-sin ρ sin ϑ.
cos δ2=cos κ cos ϑ,
tan 2θ=-sin ρsin κ tan ϑ,
tan2 2ϵ=cos2 ρsin2 κ cot2 ϑ+sin2 ρ.
cΣ(z)=12C(σ1-σ2),aΨ(z)=dψdz.
12C(σ1-σ2)1,dψdz1
c1,a1.
|λ|1,|ϑ|1,|κ|1,|ρ|1,
dλdz=-aΨ(z) λϑ0-cΣ(z),
dϑ0dz=aΨ(z),
dκdz=2aΨ(z)λϑ0-cΣ(z),
dρdz=-2aΨ(z) λϑ0-cΣ(z).
dϑ1dz=aΨ(z)(1-2λ2).
ϑ0=aΩ(z),
Ω(z)=z0zΨ(z)dz
Ω(z)=1a[ψ(z)-ψ(z0)].
dλdz+Ψ(z)Ω(z)λ+cΣ(z)=0.
λ=-c (z)Ω(z),
(z)=z0zΣ(z)Ω(z)dz.
dκdz=-cΣ(z)-2a2cΨ(z)(z).
κ(z1)=-c(I0+2a2I1),
I0=z0z1Σ(z)dz,
I1=z0z1(z)Ψ(z)dz.
ρ(z1)=-c(I0-2I2),
ϑ1(z1)=a(Φ0-2c2I3),
I2=z0z1(z) Ψ(z)[Ω(z)]2 dz,
I3=z0z1[(z)]2 Ψ(z)[Ω(z)]2 dz,
Φ0=1a[ψ(z1)-ψ(z0)].
Jn=z0z1Σ(z)[Ω(z)]ndz
J*=z0z1z0zΣ(z)dz2Ψ(z)dz.
I0=J0,
I1=Φ0J1-J2,
I2=J0-J1Φ0,
I3=-Φ0J02+2J0J1-J12Φ0+J*.
α=1-12ρ21-12κ2ϑ1,
1-2γ2=(1-2κ2)(1-12ϑ12)2+(1-2ρ2)ϑ12
α=(1-12ρ2+12κ2)ϑ1,
γ2=κ2+(ρ2-κ2)ϑ12.
α(z1)=a(Φ0-c2K1),
[γ(z1)]2=c2(I02+a2K2+a4K3+c2a2K4),
K1=2[I3+Φ0I2(I2-I0)],
K2=4[I0I1+Φ02I2(I2-I0)],
K3=4I1(I1-I0),
K4=16Φ0I2I3(I2-I0).
α(z1)=aΦ0,
γ(z1)=cI0.
α(z1)=ψ(z1)-ψ(z0),
K1=2J*,
K2=4J12,
K3=4(Φ0J1-J2)2,
K4=16J1-J12Φ0 J12Φ0.
α(z1)=a(Φ0-2c2J*),
γ(z1)=2acJ1.
γ(z1)=Cz0z1(σ1-σ2)ψ(z)dz.
K1=2J*,K2=0,K3=4J22,K4=0.
γ(z1)=2a2cJ2
γ(z1)=Cz0z1(σ1-σ2)[ψ(z)]2dz.
ψ(z)=az,
γ(z1)=aCz0z1(σ1-σ2)zdz,
γ(z1)=aCz0z1(σ1-σ2)z2dz,
α(z1)=a[Φ0-c2K1(b)],
γ(z1)=acK2(b)
γ(z1)=a2cK3(b),
c=γ(z1)aK2(b).
K1(b)K2(b)=[2a-α(z1)]a[γ(z1)]2.
c=γ(z1)K3(b)a2,
K1(b)K3(b)=[2b-α(z1)]a3[γ(z1)]2.
Ψ(z)=1,Ω(z)=z+1.
cΣ(z)=czn,n=2s+1,s=0, 1, 2 . . . .
K1(n)=8(2n+3)(n+2),K2(n)=16(n+2)2.
n+22(2n+3)=[2a-α(1)]a[γ(1)]2,
c=(n+2)γ(1)4a.
cΣ(z)=c[1-(n+1)zn],n=2s,s=1, 2, . . . .
K1=8n23(n+3)(2n+3),K3=64n29(n+3)2.
3(n+3)8(2n+3)=[2a-α(1)]a3[γ(1)]2,
c=3(n+3)γ(1)8na2.
cΣ(z)=c[z+b(1-3z2)].
K1(b)=215(4+b2),K2=169,
340(4+b2)=[2a-α(1)]a[γ(1)]2,
c=3γ(1)4a.

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