Abstract

In scattered-light photoelasticity with unpolarized light, the secondary principal stress direction ψ and the relative phase retardation ρ in a three-dimensional stressed model with rotation of the principal stress axes can be obtained by use of Stokes parameters. For completely automated stress analysis, measurements of the total relative phase retardation and the secondary principal stress direction over the entire field are required, and it is necessary to unwrap ψ and ρ. A phase unwrapping method is thus proposed for the determination of these values based on scattered-light photoelasticity. The values are easily obtained via an arctangent function, overcoming the error associated with the quarter-wave plate by employing an incident light of different wavelengths. The proposed technique provides automated and nondestructive determination of the total relative phase retardation and the secondary principal stress direction in a model exhibiting rotation of the principal stress axes.

© 2006 Optical Society of America

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  1. H. Hurwitz and R. C. Jones, "A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems," J. Opt. Soc. Am. 31, 493-499 (1941).
  2. H. K. Aben, "Optical phenomena in photoelastic models by the rotation of principal axes," Exp. Mech. 6, 13-22 (1966).
    [CrossRef]
  3. L. S. Srinath and A. V. S. S. S. R. Sarma, "Determination of the optically equivalent model in three-dimensional photoelasticity," Exp. Mech. 14, 118-122 (1974).
    [CrossRef]
  4. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).
  5. R. A. Tomlinson and E. A. Patterson, "The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity," Exp. Mech. 42, 43-49 (2002).
    [CrossRef]
  6. H. Aben and A. Errapart, "Photoelastic tomography: possibilities and limitations," in Advances in Experimental Mechanics, C. Pappalettere, ed. (McGraw-Hill, 2004).
  7. R. Weller, "A new method for photoelasticity in three dimensions," J. Appl. Phys. 10, 266 (1939).
    [CrossRef]
  8. H. J. Menges, "Die experimentelle Ermittlung räumlicher Spannungs zustände an durchsichtigen Modellen mit Hilfe des Tyndalleffektes," Z. Angew. Math. Mech. 20, 210-217 (1940).
    [CrossRef]
  9. H. T. Jessop, "The scattered light method of exploration of stresses in two- and three-dimensional models," Br. J. Appl. Phys. 2, 249-260 (1951).
    [CrossRef]
  10. M. M. Frocht and L. S. Srinath, "A non-destructive method for three-dimensional photoelasticity," in Proceedings of the U.S. National Congress of Applied Mechanics, R. M. Haythornthwaite, ed. (American Society of Mechanical Engineering, 1958), pp. 329-337.
  11. Y. F. Cheng, "An automatic system for scattered-light photoelasticity," Exp. Mech. 9, 407-412 (1969).
    [CrossRef]
  12. A. Robert and E. Guillemet, "New scattered light method in three-dimensional photoelasticity," Br. J. Appl. Phys. 15, 567-578 (1964).
    [CrossRef]
  13. A. Robert, "New methods in photoelasticity," Exp. Mech. 7, 224-232 (1967).
    [CrossRef]
  14. J. F. Gross-Petersen, "A scattered-light method in photoelasticity," Exp. Mech. 14, 317-322 (1974).
    [CrossRef]
  15. T. Kihara, H. Kubo, and R. Nagata, "Measurement of 3-D stress distribution by a scattered-light method using depolarized incident light," Appl. Opt. 18, 321-327 (1979).
    [CrossRef] [PubMed]
  16. T. Kihara, M. Unno, C. Kitada, H. Kubo, and R. Nagata, "Three-dimensional stress distribution measurement in a method of the human ankle joint by scattered-light polarizer photoelasticity: Part 2," Appl. Opt. 26, 643-649 (1987).
    [CrossRef] [PubMed]
  17. T. Kihara, "A measurement method of scattered light photoelasticity using unpolarized light," Exp. Mech. 37, 39-44 (1997).
    [CrossRef]
  18. T. Kihara, "A study of measurement method of scattered light photoelasticity using unpolarized light by Poincaré sphere," Proc. Jpn. Soc. Photoelasticity 18, 15-19 (1998).
  19. T. Kihara, "A digital scattered light photoelasticity measurement technique using unpolarized light," Jpn. Soc. Exp. Mech. 4, 22-28 (2004).
  20. T. Kihara, "Photoelastic model measurement with rotated principal axes by scattered-light photoelasticity," Exp. Mech. 44, 455-460 (2004).
    [CrossRef]
  21. R. Desailly, "Visualization of isoclinics and isochromatics in a birefringent slice optically singled out in a three-dimensional model," Opt. Commun. 19, 61-64 (1976).
    [CrossRef]
  22. J. C. Dupré and A. Lagarde, "Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing," Exp. Mech. 37, 393-397 (1997).
    [CrossRef]
  23. M. M. Frocht and R. J. Guernesy, "A special investigation to develop a general method for three-dimensional photoelastic stress analysis," NACA Tech. Note 2822 (National Advisory Committee for Aeronautics, 1952).
  24. K. Ramesh, Digital Photoelasticity (Springer-Verlag, 2000).
  25. T. Kihara, "An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths," Strain 39, 65-71 (2003).
    [CrossRef]
  26. W. H. McMaster, "Matrix representation of polarization," Rev. Mod. Phys. 33, 8-28 (1961).
    [CrossRef]
  27. T. Kihara, "Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer," Opt. Commun. 110, 529-532 (1994).
    [CrossRef]
  28. T. Kihara, "Measurement of Stokes parameters by quarter-wave plate and polarizer," in Advances in Experimental Mechanics, J.M.Dulieu-Barton and S.Quinn, eds. (Trans Tech, 2005), Vol. 4, pp. 235-240.
  29. M. M. Frocht, Photoelasticity (Wiley, 1948), Vol. 2, Chap. 4.

2004

T. Kihara, "A digital scattered light photoelasticity measurement technique using unpolarized light," Jpn. Soc. Exp. Mech. 4, 22-28 (2004).

T. Kihara, "Photoelastic model measurement with rotated principal axes by scattered-light photoelasticity," Exp. Mech. 44, 455-460 (2004).
[CrossRef]

2003

T. Kihara, "An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths," Strain 39, 65-71 (2003).
[CrossRef]

2002

R. A. Tomlinson and E. A. Patterson, "The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity," Exp. Mech. 42, 43-49 (2002).
[CrossRef]

1998

T. Kihara, "A study of measurement method of scattered light photoelasticity using unpolarized light by Poincaré sphere," Proc. Jpn. Soc. Photoelasticity 18, 15-19 (1998).

1997

J. C. Dupré and A. Lagarde, "Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing," Exp. Mech. 37, 393-397 (1997).
[CrossRef]

T. Kihara, "A measurement method of scattered light photoelasticity using unpolarized light," Exp. Mech. 37, 39-44 (1997).
[CrossRef]

1994

T. Kihara, "Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer," Opt. Commun. 110, 529-532 (1994).
[CrossRef]

1987

1979

1976

R. Desailly, "Visualization of isoclinics and isochromatics in a birefringent slice optically singled out in a three-dimensional model," Opt. Commun. 19, 61-64 (1976).
[CrossRef]

1974

J. F. Gross-Petersen, "A scattered-light method in photoelasticity," Exp. Mech. 14, 317-322 (1974).
[CrossRef]

L. S. Srinath and A. V. S. S. S. R. Sarma, "Determination of the optically equivalent model in three-dimensional photoelasticity," Exp. Mech. 14, 118-122 (1974).
[CrossRef]

1969

Y. F. Cheng, "An automatic system for scattered-light photoelasticity," Exp. Mech. 9, 407-412 (1969).
[CrossRef]

1967

A. Robert, "New methods in photoelasticity," Exp. Mech. 7, 224-232 (1967).
[CrossRef]

1966

H. K. Aben, "Optical phenomena in photoelastic models by the rotation of principal axes," Exp. Mech. 6, 13-22 (1966).
[CrossRef]

1964

A. Robert and E. Guillemet, "New scattered light method in three-dimensional photoelasticity," Br. J. Appl. Phys. 15, 567-578 (1964).
[CrossRef]

1961

W. H. McMaster, "Matrix representation of polarization," Rev. Mod. Phys. 33, 8-28 (1961).
[CrossRef]

1951

H. T. Jessop, "The scattered light method of exploration of stresses in two- and three-dimensional models," Br. J. Appl. Phys. 2, 249-260 (1951).
[CrossRef]

1941

1940

H. J. Menges, "Die experimentelle Ermittlung räumlicher Spannungs zustände an durchsichtigen Modellen mit Hilfe des Tyndalleffektes," Z. Angew. Math. Mech. 20, 210-217 (1940).
[CrossRef]

1939

R. Weller, "A new method for photoelasticity in three dimensions," J. Appl. Phys. 10, 266 (1939).
[CrossRef]

Aben, H.

H. Aben and A. Errapart, "Photoelastic tomography: possibilities and limitations," in Advances in Experimental Mechanics, C. Pappalettere, ed. (McGraw-Hill, 2004).

Aben, H. K.

H. K. Aben, "Optical phenomena in photoelastic models by the rotation of principal axes," Exp. Mech. 6, 13-22 (1966).
[CrossRef]

Cheng, Y. F.

Y. F. Cheng, "An automatic system for scattered-light photoelasticity," Exp. Mech. 9, 407-412 (1969).
[CrossRef]

Desailly, R.

R. Desailly, "Visualization of isoclinics and isochromatics in a birefringent slice optically singled out in a three-dimensional model," Opt. Commun. 19, 61-64 (1976).
[CrossRef]

Dupré, J. C.

J. C. Dupré and A. Lagarde, "Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing," Exp. Mech. 37, 393-397 (1997).
[CrossRef]

Errapart, A.

H. Aben and A. Errapart, "Photoelastic tomography: possibilities and limitations," in Advances in Experimental Mechanics, C. Pappalettere, ed. (McGraw-Hill, 2004).

Frocht, M. M.

M. M. Frocht and L. S. Srinath, "A non-destructive method for three-dimensional photoelasticity," in Proceedings of the U.S. National Congress of Applied Mechanics, R. M. Haythornthwaite, ed. (American Society of Mechanical Engineering, 1958), pp. 329-337.

M. M. Frocht and R. J. Guernesy, "A special investigation to develop a general method for three-dimensional photoelastic stress analysis," NACA Tech. Note 2822 (National Advisory Committee for Aeronautics, 1952).

M. M. Frocht, Photoelasticity (Wiley, 1948), Vol. 2, Chap. 4.

Gdoutos, E. E.

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

Gross-Petersen, J. F.

J. F. Gross-Petersen, "A scattered-light method in photoelasticity," Exp. Mech. 14, 317-322 (1974).
[CrossRef]

Guernesy, R. J.

M. M. Frocht and R. J. Guernesy, "A special investigation to develop a general method for three-dimensional photoelastic stress analysis," NACA Tech. Note 2822 (National Advisory Committee for Aeronautics, 1952).

Guillemet, E.

A. Robert and E. Guillemet, "New scattered light method in three-dimensional photoelasticity," Br. J. Appl. Phys. 15, 567-578 (1964).
[CrossRef]

Hurwitz, H.

Jessop, H. T.

H. T. Jessop, "The scattered light method of exploration of stresses in two- and three-dimensional models," Br. J. Appl. Phys. 2, 249-260 (1951).
[CrossRef]

Jones, R. C.

Kihara, T.

T. Kihara, "A digital scattered light photoelasticity measurement technique using unpolarized light," Jpn. Soc. Exp. Mech. 4, 22-28 (2004).

T. Kihara, "Photoelastic model measurement with rotated principal axes by scattered-light photoelasticity," Exp. Mech. 44, 455-460 (2004).
[CrossRef]

T. Kihara, "An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths," Strain 39, 65-71 (2003).
[CrossRef]

T. Kihara, "A study of measurement method of scattered light photoelasticity using unpolarized light by Poincaré sphere," Proc. Jpn. Soc. Photoelasticity 18, 15-19 (1998).

T. Kihara, "A measurement method of scattered light photoelasticity using unpolarized light," Exp. Mech. 37, 39-44 (1997).
[CrossRef]

T. Kihara, "Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer," Opt. Commun. 110, 529-532 (1994).
[CrossRef]

T. Kihara, M. Unno, C. Kitada, H. Kubo, and R. Nagata, "Three-dimensional stress distribution measurement in a method of the human ankle joint by scattered-light polarizer photoelasticity: Part 2," Appl. Opt. 26, 643-649 (1987).
[CrossRef] [PubMed]

T. Kihara, H. Kubo, and R. Nagata, "Measurement of 3-D stress distribution by a scattered-light method using depolarized incident light," Appl. Opt. 18, 321-327 (1979).
[CrossRef] [PubMed]

T. Kihara, "Measurement of Stokes parameters by quarter-wave plate and polarizer," in Advances in Experimental Mechanics, J.M.Dulieu-Barton and S.Quinn, eds. (Trans Tech, 2005), Vol. 4, pp. 235-240.

Kitada, C.

Kubo, H.

Lagarde, A.

J. C. Dupré and A. Lagarde, "Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing," Exp. Mech. 37, 393-397 (1997).
[CrossRef]

McMaster, W. H.

W. H. McMaster, "Matrix representation of polarization," Rev. Mod. Phys. 33, 8-28 (1961).
[CrossRef]

Menges, H. J.

H. J. Menges, "Die experimentelle Ermittlung räumlicher Spannungs zustände an durchsichtigen Modellen mit Hilfe des Tyndalleffektes," Z. Angew. Math. Mech. 20, 210-217 (1940).
[CrossRef]

Nagata, R.

Patterson, E. A.

R. A. Tomlinson and E. A. Patterson, "The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity," Exp. Mech. 42, 43-49 (2002).
[CrossRef]

Ramesh, K.

K. Ramesh, Digital Photoelasticity (Springer-Verlag, 2000).

Robert, A.

A. Robert, "New methods in photoelasticity," Exp. Mech. 7, 224-232 (1967).
[CrossRef]

A. Robert and E. Guillemet, "New scattered light method in three-dimensional photoelasticity," Br. J. Appl. Phys. 15, 567-578 (1964).
[CrossRef]

Sarma, A. V. S. S. S. R.

L. S. Srinath and A. V. S. S. S. R. Sarma, "Determination of the optically equivalent model in three-dimensional photoelasticity," Exp. Mech. 14, 118-122 (1974).
[CrossRef]

Srinath, L. S.

L. S. Srinath and A. V. S. S. S. R. Sarma, "Determination of the optically equivalent model in three-dimensional photoelasticity," Exp. Mech. 14, 118-122 (1974).
[CrossRef]

M. M. Frocht and L. S. Srinath, "A non-destructive method for three-dimensional photoelasticity," in Proceedings of the U.S. National Congress of Applied Mechanics, R. M. Haythornthwaite, ed. (American Society of Mechanical Engineering, 1958), pp. 329-337.

Theocaris, P. S.

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

Tomlinson, R. A.

R. A. Tomlinson and E. A. Patterson, "The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity," Exp. Mech. 42, 43-49 (2002).
[CrossRef]

Unno, M.

Weller, R.

R. Weller, "A new method for photoelasticity in three dimensions," J. Appl. Phys. 10, 266 (1939).
[CrossRef]

Appl. Opt.

Br. J. Appl. Phys.

A. Robert and E. Guillemet, "New scattered light method in three-dimensional photoelasticity," Br. J. Appl. Phys. 15, 567-578 (1964).
[CrossRef]

H. T. Jessop, "The scattered light method of exploration of stresses in two- and three-dimensional models," Br. J. Appl. Phys. 2, 249-260 (1951).
[CrossRef]

Exp. Mech.

H. K. Aben, "Optical phenomena in photoelastic models by the rotation of principal axes," Exp. Mech. 6, 13-22 (1966).
[CrossRef]

L. S. Srinath and A. V. S. S. S. R. Sarma, "Determination of the optically equivalent model in three-dimensional photoelasticity," Exp. Mech. 14, 118-122 (1974).
[CrossRef]

R. A. Tomlinson and E. A. Patterson, "The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity," Exp. Mech. 42, 43-49 (2002).
[CrossRef]

A. Robert, "New methods in photoelasticity," Exp. Mech. 7, 224-232 (1967).
[CrossRef]

J. F. Gross-Petersen, "A scattered-light method in photoelasticity," Exp. Mech. 14, 317-322 (1974).
[CrossRef]

Y. F. Cheng, "An automatic system for scattered-light photoelasticity," Exp. Mech. 9, 407-412 (1969).
[CrossRef]

T. Kihara, "A measurement method of scattered light photoelasticity using unpolarized light," Exp. Mech. 37, 39-44 (1997).
[CrossRef]

T. Kihara, "Photoelastic model measurement with rotated principal axes by scattered-light photoelasticity," Exp. Mech. 44, 455-460 (2004).
[CrossRef]

J. C. Dupré and A. Lagarde, "Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing," Exp. Mech. 37, 393-397 (1997).
[CrossRef]

J. Appl. Phys.

R. Weller, "A new method for photoelasticity in three dimensions," J. Appl. Phys. 10, 266 (1939).
[CrossRef]

J. Opt. Soc. Am.

Jpn. Soc. Exp. Mech.

T. Kihara, "A digital scattered light photoelasticity measurement technique using unpolarized light," Jpn. Soc. Exp. Mech. 4, 22-28 (2004).

Opt. Commun.

T. Kihara, "Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer," Opt. Commun. 110, 529-532 (1994).
[CrossRef]

R. Desailly, "Visualization of isoclinics and isochromatics in a birefringent slice optically singled out in a three-dimensional model," Opt. Commun. 19, 61-64 (1976).
[CrossRef]

Proc. Jpn. Soc. Photoelasticity

T. Kihara, "A study of measurement method of scattered light photoelasticity using unpolarized light by Poincaré sphere," Proc. Jpn. Soc. Photoelasticity 18, 15-19 (1998).

Rev. Mod. Phys.

W. H. McMaster, "Matrix representation of polarization," Rev. Mod. Phys. 33, 8-28 (1961).
[CrossRef]

Strain

T. Kihara, "An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths," Strain 39, 65-71 (2003).
[CrossRef]

Z. Angew. Math. Mech.

H. J. Menges, "Die experimentelle Ermittlung räumlicher Spannungs zustände an durchsichtigen Modellen mit Hilfe des Tyndalleffektes," Z. Angew. Math. Mech. 20, 210-217 (1940).
[CrossRef]

Other

M. M. Frocht and L. S. Srinath, "A non-destructive method for three-dimensional photoelasticity," in Proceedings of the U.S. National Congress of Applied Mechanics, R. M. Haythornthwaite, ed. (American Society of Mechanical Engineering, 1958), pp. 329-337.

H. Aben and A. Errapart, "Photoelastic tomography: possibilities and limitations," in Advances in Experimental Mechanics, C. Pappalettere, ed. (McGraw-Hill, 2004).

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

T. Kihara, "Measurement of Stokes parameters by quarter-wave plate and polarizer," in Advances in Experimental Mechanics, J.M.Dulieu-Barton and S.Quinn, eds. (Trans Tech, 2005), Vol. 4, pp. 235-240.

M. M. Frocht, Photoelasticity (Wiley, 1948), Vol. 2, Chap. 4.

M. M. Frocht and R. J. Guernesy, "A special investigation to develop a general method for three-dimensional photoelastic stress analysis," NACA Tech. Note 2822 (National Advisory Committee for Aeronautics, 1952).

K. Ramesh, Digital Photoelasticity (Springer-Verlag, 2000).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the optical arrangement and image processing system for experiments. EO, electro-optic modulation device; CL 1 and CL 2 , cylindrical lenses; B, immersion cell; Q 2 , quarter-wave plate; P 1 and P 2 , linear polarizers; BS, beam splitter.

Fig. 2
Fig. 2

Schematic diagram illustrating the stratified model consisting of the frozen stress models 1, 2, and 3, with corresponding incident unpolarized light sheets 1, 2, and 3. Each light sheet is recorded for 0° and 45° orientations.

Fig. 3
Fig. 3

Images of ρ j for models (a) 1, (b) 2, and (c) 3.

Fig. 4
Fig. 4

Images of (a) ψ 3 and (b) ρ 3 tot , and (c) distribution of ρ 3 tot along the horizontal diameter for model 3.

Fig. 5
Fig. 5

Images for simulated Stokes parameters (a) s 3 ( y 2 , y 0 ; 45 ) and (b) s 3 ( y 3 , y 0 ; 45 ) .

Fig. 6
Fig. 6

Simulated ψ 3 for the stratified model. (a) ψ 3 for entire model and (b) unwrapped ψ 3 .

Fig. 7
Fig. 7

Images of simulated (a) sin 2 ω 2 , 0 and (b) cos 2 ω 2 , 0 .

Fig. 8
Fig. 8

Simulated ρ 3 for the stratified model. (a) ρ 3 for entire model, (b) unwrapped ρ 3 tot , (c) distribution of ρ 3 tot along the horizontal diameter.

Tables (1)

Tables Icon

Table 1 Parameters Used in the Numerical Simulation of Three Circular Disks under Diametral Compressive Load P of Difference Directions α a

Equations (21)

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

S ( y j , y 0 ; θ ) = U ( y j - 1 , y 0 ) M j S ( y j ; θ ) ,
U ( y j - 1 , y 0 ) = R ( 2 ω j - 1 , 0 ) M j 1 , 0 ,
R ( 2 ω j - 1 , 0 ) = [ 1 0 0 0 0 cos 2 ω j - 1 , 0 - sin 2 ω j - 1 , 0 0 0 sin 2 ω j - 1 , 0 cos 2 ω j - 1 , 0 0 0 0 0 1 ] ,
M j - 1 , 0 = [ 1 0 0 0 0 1 - ( 1 - cos ρ j - 1 , 0 ) sin 2 2 ψ j - 1 , 0 ( 1 - cos ρ j - 1 , 0 ) sin 2 ψ j - 1 , 0 cos 2 ψ j - 1 , 0 - sin ρ j - 1 , 0 sin 2 ψ j - 1 , 0 0 ( 1 - cos ρ j - 1 , 0 ) sin 2 ψ j - 1 , 0 cos 2 ψ j - 1 , 0 1 - ( 1 - cos ρ j - 1 , 0 ) cos 2 2 ψ j - 1 , 0 sin ρ j - 1 , 0  cos  2 ψ j - 1 , 0 0 sin ρ j - 1 , 0 sin 2 ψ j - 1 , 0 sin ρ j 1 , 0 cos 2 ψ j - 1 , 0 cos ρ j - 1 , 0 ] ,
M j = [ 1 0 0 0 0 1 - ( 1 - cos ρ j ) sin 2 2 ψ j ( 1 - cos ρ j ) sin 2 ψ j cos 2 ψ j - sin ρ j sin 2 ψ j 0 ( 1 - cos ρ j ) sin 2 ψ j cos 2 ψ j 1 - ( 1 - cos ρ j ) cos 2 2 ψ j sin ρ j cos 2 ψ j 0 sin ρ j sin 2 ψ j - sin ρ j cos 2 ψ j cos ρ j ] ,
S ( y j ; θ ) = S 0 ( y j ; θ ) [ 1 cos 2 θ sin 2 θ 0 ] .
S ( y j - 1 , y 0 ; θ ) = U ( y j - 1 , y 0 ) S ( y j - 1 ; θ ) = [ 1 0 0 cos 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) + cos ρ j - 1 , 0 sin 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) 0 cos 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) - cos ρ j - 1 , 0 sin 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) 0 sin ρ j - 1 , 0 sin 2 ψ j - 1 , 0 0 0 sin 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) - cos ρ j - 1 , 0 cos 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) - sin ρ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) sin 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) + cos ρ j - 1 , 0 cos 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) sin ρ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) - sin ρ j - 1 , 0 cos 2 ψ j - 1 , 0 cos ρ j - 1 , 0 ] S ( y j - 1 ; θ ) .
U ( y j 1 , y 0 ) 1 S ( y j , y 0 ; θ ) / S 0 ( y j , y 0 ; θ ) = M j S ( y j ; θ ) / S 0 ( y j ; θ ) ,
U ( y j - 1 , y 0 ) - 1 = [ 1 0 0 cos 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) + cos ρ j - 1 , 0 sin 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) 0 sin 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) - cos ρ j - 1 , 0 cos 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) 0 - sin ρ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) 0 0 cos 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) - cos ρ j - 1 , 0 sin 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) sin ρ j - 1 , 0 sin 2 ψ j - 1 , 0 sin 2 ψ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) + cos ρ j - 1 , 0 cos 2 ψ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) - sin ρ j - 1 , 0 cos 2 ψ j - 1 , 0 sin ρ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) cos ρ j - 1 , 0 ] ,
= [ 1 0 0 0 0 s 1 ( y j - 1 , y 0 ; 0 ) s 2 ( y j - 1 , y 0 ; 0 ) s 3 ( y j - 1 , y 0 ; 0 ) 0 s 1 ( y j - 1 , y 0 ; 45 ) s 2 ( y j - 1 , y 0 ; 45 ) s 3 ( y j - 1 , y 0 ; 45 ) 0 U ( y j - 1 , y 0 ) 42 1 U ( y j - 1 , y 0 ) 43 1 U ( y j - 1 , y 0 ) 44 1 ] .
M j S ( y j ; 0 ) / S 0 ( y j ; 0 ) = [ 1 1 - ( 1 - cos ρ j ) sin 2 2 ψ j ( 1 - cos ρ j ) sin 2 ψ j cos 2 ψ j sin ρ j sin 2 ψ j ] ,
M j S ( y j ; 45 ) / S 0 ( y j ; 45 ) = [ 1 ( 1 - cos ρ j ) sin 2 ψ j cos 2 ψ j 1 - ( 1 - cos ρ j ) cos 2 2 ψ j - sin ρ j cos 2 ψ j ] .
U ( y j - 1 , y 0 ) 42 1 = sin ρ j - 1 , 0 sin ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) = s 3 ( y j - 1 , y 0 ; 0 ) cos 2 ω j - 1 , 0 + s 3 ( y j - 1 , y 0 ; 45 ) sin 2 ω j - 1 , 0 ,
U ( y j - 1 , y 0 ) 43 1 = sin ρ j - 1 , 0 cos ( 2 ψ j - 1 , 0 + 2 ω j - 1 , 0 ) = s 3 ( y j - 1 , y 0 ; 45 ) cos 2 ω j - 1 , 0 s 3 ( y j - 1 , y 0 ; 0 ) sin 2 ω j - 1 , 0 ,
U ( y j - 1 , y 0 ) 44 1 = cos ρ j - 1 , 0 = [ s 1 ( y j - 1 , y 0 ; 0 ) + s 2 ( y j - 1 , y 0 ; 45 ) ] × cos 2 ω j - 1 , 0 + [ s 2 ( y j - 1 , y 0 ; 0 ) s 1 ( y j - 1 , y 0 ; 45 ) ] sin 2 ω j - 1 , 0 - 1.
sin 2 ω j - 1 , 0 = { 2 [ s 2 ( y j - 1 , y 0 ; 0 ) - s 1 ( y j - 1 , y 0 ; 45 ) ] } / { [ s 2 ( y j - 1 , y 0 ; 0 ) s 1 ( y j - 1 , y 0 ; 45 ) ] 2 + [ s 1 ( y j - 1 , y 0 ; 0 ) + s 2 ( y j - 1 , y 0 ; 45 ) ] 2 } 0.5 = [ 2 sin 2 ω j - 1 , 0 cos 2 ( ρ j - 1 , 0 / 2 ) ] / [ 2 cos 2 ( ρ j - 1 , 0 / 2 ) ] ,
cos 2 ω j - 1 , 0 = { 2 [ s 1 ( y j - 1 , y 0 ; 0 ) + s 2 ( y j - 1 , y 0 ; 45 ) ] } / { [ s 2 ( y j - 1 , y 0 ; 0 ) - s 1 ( y j - 1 , y 0 ; 45 ) ] 2 + [ s 1 ( y j - 1 , y 0 ; 0 ) + s 2 ( y j - 1 , y 0 ; 45 ) ] 2 } 0.5 = [ 2 cos 2 ω j - 1 , 0 cos 2 ( ρ j - 1 , 0 / 2 ) ] / [ 2 cos 2 ( ρ j - 1 , 0 / 2 ) ] ,
ω j - 1 , 0 = 0.5 tan - 1 { [ s 2 ( y j - 1 , y 0 ; 0 ) - s 1 ( y j - 1 , y 0 ; 45 ) ] / [ s 1 ( y j - 1 , y 0 ; 0 ) + s 2 ( y j - 1 , y 0 ; 45 ) ] } = 0.5 tan - 1 { [ 2 sin 2 ω j - 1 , 0 cos 2 ( ρ j - 1 , 0 / 2 ) ] / [ 2 cos 2 ω j - 1 , 0 cos 2 ( ρ j - 1 , 0 / 2 ) ] } .
ψ j = 0.25 tan - 1 { [ s ( y j - 1 , y 0 ; 45 ) s ( y j , y 0 ; 0 ) + s ( y j - 1 , y 0 ; 0 ) s ( y j , y 0 ; 45 ) ] / [ s ( y j - 1 , y 0 ; 0 ) s ( y j , y 0 ; 0 ) - s ( y j - 1 , y 0 ; 45 ) s ( y j , y 0 ; 45 ) ] } = 0.25 tan - 1 { [ 2 sin 2 ( ρ j / 2 ) sin 4 ψ j ] / [ 2 sin 2 ( ρ j / 2 ) cos 4 ψ j ] } ,
ρ j = tan - 1 { [ ( U ( y j - 1 , y 0 ) 42 1 , U ( y j - 1 , y 0 ) 43 1 , U ( y j - 1 , y 0 ) 44 1 ) s ( y j , y 0 ; 0 ) sin 2 ψ j - ( U ( y j - 1 , y 0 ) 42 1 , U ( y j - 1 , y 0 ) 43 1 , U ( y j - 1 , y 0 ) 44 1 ) s ( y j , y 0 ; 45 ) cos 2 ψ j ] / [ s ( y j - 1 , y 0 ; 0 ) s ( y j , y 0 ; 0 ) + s ( y j - 1 , y 0 ; 45 ) s ( y j , y 0 ; 45 ) - 1 ] } = tan - 1 [ sin ρ j ( sin 2 2 ψ j + cos 2 2 ψ j ) / cos ρ j ] ,
S ( y j , y 0 ; θ ) = [ [ S 1 ( y j , y 0 ; θ ) 2 + S 2 ( y j , y 0 ; θ ) 2 + S 3 ( y j , y 0 ; θ ) 2 ] 0.5 l j ( θ , 0 , 0 ) l j ( θ , 90 , 90 ) l j ( θ , 45 , 45 ) l j ( θ , 135 , 135 ) [ l j ( θ , −45 , 0 ) l j ( θ , 45 , 0 ) ] / cos Δ ρ ] ,

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