Abstract

A phase unwrapping method that employs scattered-light photoelasticity with unpolarized light was proposed for automated three-dimensional stress analysis [Appl. Opt. 45, 8848 (2006)]. I now demonstrate the validity of this method by performing nondestructive measurements at three different wavelengths of the secondary principal stress direction ψj and the total relative phase retardation ρjtot in the plane that contains the rotated principal stress directions in a spherical frozen stress model and compare the results obtained with mechanically sliced models. The parameters ψj and ρjtot were measured nondestructively over the entire field of view for the first time, to the best of my knowledge.

© 2007 Optical Society of America

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  1. T. Kihara, "Phase unwrapping method for three dimensional stress analysis by scattered-light photoelasticity with unpolarized light," Appl. Opt. 45, 8848-8854 (2006).
    [CrossRef] [PubMed]
  2. H. Hurwitz, Jr. 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).
  3. H. K. Aben, "Optical phenomena in photoelastic models by the rotation of principal axes," Exp. Mech. 6, 13-22 (1966).
    [CrossRef]
  4. 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]
  5. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).
  6. T. Kihara, "Automatic whole-field measurement of principal stress directions using three wavelengths," in Recent Advances in Experimental Mechanics, J. F. Silva-Gomes, F. B. Branco, F. Martinsde Brito, J. Gil-Saraiva, M. Lurdes-Eusébio, J. Sousa-Cirne, and A. Correia da Cruz, eds. (A. A. Balkema, 1994), pp. 95-99.
  7. C. Buckberry and D. Towers, "Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase measuring techniques," Meas. Sci. Technol. 6, 1227-1235 (1995).
    [CrossRef]
  8. A. D. Nurse, "Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping," Appl. Opt. 36, 5781-5786 (1997).
    [CrossRef] [PubMed]
  9. T. Kihara, "An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths," Strain 39, 65-71 (2003).
    [CrossRef]
  10. A. Ajovalasit, S. Barone, and G. Petrucci, "Toward RGB photoelasticity: full-field automated photoelasticity in white light," Exp. Mech. 35, 193-200 (1995).
    [CrossRef]
  11. K. Ramesh and S. S. Deshmukh, "Automation of white light photoelasticity by phase-shifting technique using colour image processing hardware," Opt. Lasers Eng. 28, 47-60 (1997).
    [CrossRef]
  12. A. Ajovalasit, G. Petrucci, and M. Scafidi, "Phase shifting photoelasticity in white light," Opt. Lasers Eng. 45, 596-611 (2007).
    [CrossRef]
  13. A. Zenina, J.-C. Dupré, and A. Lagarde, "Separation of isochromatic and isoclinic patterns of a slice optically isolated in a 3-D photoelastic medium," Eur. J. Mech. A Solids 18, 633-640 (1999).
    [CrossRef]
  14. T. Oi and M. Takashi, "An approach to general 3-D stress analysis by multidirectional scattered light technique," in IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics, A. Lagarde, ed. Vol. 82 of Solid Mechanics and Its Applications (Springer, 2002), pp. 57-84.
    [CrossRef]
  15. T. Kihara, "A measurement method of scattered light photoelasticity using unpolarized light," Exp. Mech. 37, 39-44 (1997).
    [CrossRef]
  16. T. Kihara, "A digital scattered light photoelasticity measurement technique using unpolarized light," Jpn. Soc. Exp. Mech. 4, 22-28 (2004).
  17. T. Kihara, "Photoelastic model measurement with rotated principal axes by scattered-light photoelasticity," Exp. Mech. 44, 455-460 (2004).
    [CrossRef]
  18. W. H. McMaster, "Matrix representation of polarization," Rev. Mod. Phys. 33, 8-28 (1961).
    [CrossRef]
  19. K. T. Ramesh, Digital Photoelasticity: Advanced Technologies and Applications (Springer-Verlag, 2000).
    [CrossRef]
  20. 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]
  21. 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.
  22. 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).

2007

A. Ajovalasit, G. Petrucci, and M. Scafidi, "Phase shifting photoelasticity in white light," Opt. Lasers Eng. 45, 596-611 (2007).
[CrossRef]

2006

2005

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.

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

T. Oi and M. Takashi, "An approach to general 3-D stress analysis by multidirectional scattered light technique," in IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics, A. Lagarde, ed. Vol. 82 of Solid Mechanics and Its Applications (Springer, 2002), pp. 57-84.
[CrossRef]

2000

K. T. Ramesh, Digital Photoelasticity: Advanced Technologies and Applications (Springer-Verlag, 2000).
[CrossRef]

1999

A. Zenina, J.-C. Dupré, and A. Lagarde, "Separation of isochromatic and isoclinic patterns of a slice optically isolated in a 3-D photoelastic medium," Eur. J. Mech. A Solids 18, 633-640 (1999).
[CrossRef]

1997

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

A. D. Nurse, "Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping," Appl. Opt. 36, 5781-5786 (1997).
[CrossRef] [PubMed]

K. Ramesh and S. S. Deshmukh, "Automation of white light photoelasticity by phase-shifting technique using colour image processing hardware," Opt. Lasers Eng. 28, 47-60 (1997).
[CrossRef]

1995

C. Buckberry and D. Towers, "Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase measuring techniques," Meas. Sci. Technol. 6, 1227-1235 (1995).
[CrossRef]

A. Ajovalasit, S. Barone, and G. Petrucci, "Toward RGB photoelasticity: full-field automated photoelasticity in white light," Exp. Mech. 35, 193-200 (1995).
[CrossRef]

1994

T. Kihara, "Automatic whole-field measurement of principal stress directions using three wavelengths," in Recent Advances in Experimental Mechanics, J. F. Silva-Gomes, F. B. Branco, F. Martinsde Brito, J. Gil-Saraiva, M. Lurdes-Eusébio, J. Sousa-Cirne, and A. Correia da Cruz, eds. (A. A. Balkema, 1994), pp. 95-99.

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]

1979

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

1974

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]

1966

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

1961

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

1952

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).

1941

Aben, H. K.

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

Ajovalasit, A.

A. Ajovalasit, G. Petrucci, and M. Scafidi, "Phase shifting photoelasticity in white light," Opt. Lasers Eng. 45, 596-611 (2007).
[CrossRef]

A. Ajovalasit, S. Barone, and G. Petrucci, "Toward RGB photoelasticity: full-field automated photoelasticity in white light," Exp. Mech. 35, 193-200 (1995).
[CrossRef]

Barone, S.

A. Ajovalasit, S. Barone, and G. Petrucci, "Toward RGB photoelasticity: full-field automated photoelasticity in white light," Exp. Mech. 35, 193-200 (1995).
[CrossRef]

Buckberry, C.

C. Buckberry and D. Towers, "Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase measuring techniques," Meas. Sci. Technol. 6, 1227-1235 (1995).
[CrossRef]

Deshmukh, S. S.

K. Ramesh and S. S. Deshmukh, "Automation of white light photoelasticity by phase-shifting technique using colour image processing hardware," Opt. Lasers Eng. 28, 47-60 (1997).
[CrossRef]

Dupré, J.-C.

A. Zenina, J.-C. Dupré, and A. Lagarde, "Separation of isochromatic and isoclinic patterns of a slice optically isolated in a 3-D photoelastic medium," Eur. J. Mech. A Solids 18, 633-640 (1999).
[CrossRef]

Frocht, M. M.

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).

Gdoutos, E. E.

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

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).

Hurwitz, H.

Jones, R. C.

Kihara, T.

T. Kihara, "Phase unwrapping method for three dimensional stress analysis by scattered-light photoelasticity with unpolarized light," Appl. Opt. 45, 8848-8854 (2006).
[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.

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 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, "Automatic whole-field measurement of principal stress directions using three wavelengths," in Recent Advances in Experimental Mechanics, J. F. Silva-Gomes, F. B. Branco, F. Martinsde Brito, J. Gil-Saraiva, M. Lurdes-Eusébio, J. Sousa-Cirne, and A. Correia da Cruz, eds. (A. A. Balkema, 1994), pp. 95-99.

Lagarde, A.

A. Zenina, J.-C. Dupré, and A. Lagarde, "Separation of isochromatic and isoclinic patterns of a slice optically isolated in a 3-D photoelastic medium," Eur. J. Mech. A Solids 18, 633-640 (1999).
[CrossRef]

McMaster, W. H.

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

Nurse, A. D.

Oi, T.

T. Oi and M. Takashi, "An approach to general 3-D stress analysis by multidirectional scattered light technique," in IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics, A. Lagarde, ed. Vol. 82 of Solid Mechanics and Its Applications (Springer, 2002), pp. 57-84.
[CrossRef]

Petrucci, G.

A. Ajovalasit, G. Petrucci, and M. Scafidi, "Phase shifting photoelasticity in white light," Opt. Lasers Eng. 45, 596-611 (2007).
[CrossRef]

A. Ajovalasit, S. Barone, and G. Petrucci, "Toward RGB photoelasticity: full-field automated photoelasticity in white light," Exp. Mech. 35, 193-200 (1995).
[CrossRef]

Ramesh, K.

K. Ramesh and S. S. Deshmukh, "Automation of white light photoelasticity by phase-shifting technique using colour image processing hardware," Opt. Lasers Eng. 28, 47-60 (1997).
[CrossRef]

Ramesh, K. T.

K. T. Ramesh, Digital Photoelasticity: Advanced Technologies and Applications (Springer-Verlag, 2000).
[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]

Scafidi, M.

A. Ajovalasit, G. Petrucci, and M. Scafidi, "Phase shifting photoelasticity in white light," Opt. Lasers Eng. 45, 596-611 (2007).
[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]

Takashi, M.

T. Oi and M. Takashi, "An approach to general 3-D stress analysis by multidirectional scattered light technique," in IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics, A. Lagarde, ed. Vol. 82 of Solid Mechanics and Its Applications (Springer, 2002), pp. 57-84.
[CrossRef]

Theocaris, P. S.

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

Towers, D.

C. Buckberry and D. Towers, "Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase measuring techniques," Meas. Sci. Technol. 6, 1227-1235 (1995).
[CrossRef]

Zenina, A.

A. Zenina, J.-C. Dupré, and A. Lagarde, "Separation of isochromatic and isoclinic patterns of a slice optically isolated in a 3-D photoelastic medium," Eur. J. Mech. A Solids 18, 633-640 (1999).
[CrossRef]

Appl. Opt.

Eur. J. Mech. A Solids

A. Zenina, J.-C. Dupré, and A. Lagarde, "Separation of isochromatic and isoclinic patterns of a slice optically isolated in a 3-D photoelastic medium," Eur. J. Mech. A Solids 18, 633-640 (1999).
[CrossRef]

Exp. Mech.

A. Ajovalasit, S. Barone, and G. Petrucci, "Toward RGB photoelasticity: full-field automated photoelasticity in white light," Exp. Mech. 35, 193-200 (1995).
[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]

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]

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).

Meas. Sci. Technol.

C. Buckberry and D. Towers, "Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase measuring techniques," Meas. Sci. Technol. 6, 1227-1235 (1995).
[CrossRef]

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]

Opt. Lasers Eng.

K. Ramesh and S. S. Deshmukh, "Automation of white light photoelasticity by phase-shifting technique using colour image processing hardware," Opt. Lasers Eng. 28, 47-60 (1997).
[CrossRef]

A. Ajovalasit, G. Petrucci, and M. Scafidi, "Phase shifting photoelasticity in white light," Opt. Lasers Eng. 45, 596-611 (2007).
[CrossRef]

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]

Other

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

T. Kihara, "Automatic whole-field measurement of principal stress directions using three wavelengths," in Recent Advances in Experimental Mechanics, J. F. Silva-Gomes, F. B. Branco, F. Martinsde Brito, J. Gil-Saraiva, M. Lurdes-Eusébio, J. Sousa-Cirne, and A. Correia da Cruz, eds. (A. A. Balkema, 1994), pp. 95-99.

K. T. Ramesh, Digital Photoelasticity: Advanced Technologies and Applications (Springer-Verlag, 2000).
[CrossRef]

T. Oi and M. Takashi, "An approach to general 3-D stress analysis by multidirectional scattered light technique," in IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics, A. Lagarde, ed. Vol. 82 of Solid Mechanics and Its Applications (Springer, 2002), pp. 57-84.
[CrossRef]

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 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).

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Figures (10)

Fig. 1
Fig. 1

Schematic of the optical arrangement and image processing system used for the experiments: EO, electro-optic modulation device; CL1, CL2, cylindrical lenses; B, immersion cell; Q2, quarter-wave plate; P1, P2, linear polarizers; BS, beam splitter.

Fig. 2
Fig. 2

Isochromatics and isoclinics of a frozen stress sphere under diametral compression as measured with a crossed-plane polariscope.

Fig. 3
Fig. 3

Thick 6.3 mm slices obtained by optically slicing a frozen stress sphere.

Fig. 4
Fig. 4

Original images of I j ( 45 , 135 , 135 ) λ 1 measured from each section: (a) I 4 ( 45 , 135 , 135 ) λ 1 , (b) I 3 ( 45 , 135 , 135 ) λ 1 , (c) I 2 ( 45 , 135 , 135 ) λ 1 , (d) I 1 ( 45 , 135 , 135 ) λ 1 .

Fig. 5
Fig. 5

Images of ψ 4 for wavelengths of (a) λ 1 , (b) λ 2 , (c) λ 3 . (d) Calculation of three wavelengths by use of Eq. (10).

Fig. 6
Fig. 6

(a)–(d) Unwrapped ψ j for optical slices 4, 3, 2, and 1, respectively.

Fig. 7
Fig. 7

Images obtained for (a) sin   2 ω 3 , 0 and (b) cos   2 ω 3 , 0 .

Fig. 8
Fig. 8

Images of ρ 4 λ 1 for optical slice 4 and (b) distribution of ρ 4 λ 1 on the equatorial plane.

Fig. 9
Fig. 9

(a)–(d) Unwrapped ρ j λ 1 for optical slices 4, 3, 2, and 1, respectively. (e)–(h) Distributions of unwrapped ρ j λ 1 on the equatorial plane.

Fig. 10
Fig. 10

Images of (a) ψ 4 and (b) ρ 4 λ 1 for mechanical slice 4. (c) Distribution of ρ 4 λ 1 on the equatorial plane.

Equations (17)

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

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 ) / S 0 ( y j , y 0 ; θ ) , S 2 ( y j , y 0 ; θ ) / S 0 ( y j , y 0 ; θ ) , S 3 ( y j , y 0 ; θ ) / S 0 ( y j , y 0 ; θ ) }
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 ) ] .
ψ 1 = 0.25 tan 1 { [ s 2 ( y 1 , y 0 ; 0 ) + s 1 ( y 1 , y 0 ; 45 ) ] / [ s 1 ( y 1 , y 0 ; 0 ) s 2 ( y 1 , y 0 ; 45 ) ] } ,
ρ 1 = tan 1 { [ s 3 ( y 1 , y 0 ; 0 ) sin   2 ψ 1 s 3 ( y 1 , y 0 ; 45 ) cos   2 ψ 1 ] / [ s 1 ( y 1 , y 0 ; 0 ) + s 2 ( y 1 , y 0 ; 45 ) 1 ] } ,
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 ) ] λ 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 ) ] λ 2 + [ 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 ) ] λ 3 } / { [ 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 ) ] λ 1 + [ 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 ) ] λ 2 + [ 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 ) ] λ 3 } )
= 0.25 tan 1 ( { [ sin 2 ( ρ j λ 1 / 2 ) + sin 2 ( ρ j λ 2 / 2 )
+ sin 2 ( ρ j λ 3 / 2 ) ] 2   sin   4 ψ j } / { [ sin 2 ( ρ j λ 1 / 2 )
+ sin 2 ( ρ j λ 2 / 2 ) + sin 2 ( ρ j λ 3 / 2 ) ] 2   cos   4 ψ j } ) .
sin   2 ω j 1 , 0 = { 2 [ s 2 ( y j 1 , y 0 ; 0 ) s 1 ( y j 1 , y 0 ; 45 ) ] λ 1 + 2 [ s 2 ( y j 1 , y 0 ; 0 ) s 1 ( y j 1 , y 0 ; 45 ) ] λ 2 + 2 [ s 2 ( y j 1 , y 0 ; 0 ) s 1 ( y j 1 , y 0 ; 45 ) ] λ 3 } / ( { [ 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 } λ 1 0.5 + { } λ 2 0.5 + { } λ 3 0.5 ) = { 2   sin   2 ω j 1 , 0 [ cos 2 ( ρ j 1 , 0 λ 1 / 2 ) + cos 2 ( ρ j 1 , 0 λ 2 / 2 ) + cos 2 ( ρ j 1 , 0 λ 3 / 2 ) ] } / { 2 [ cos 2 ( ρ j 1 , 0 λ 1 / 2 ) + cos 2 ( ρ j 1 , 0 λ 2 / 2 ) + cos 2 ( ρ j 1 , 0 λ 3 / 2 ) ] } ,
cos 2 ω j 1 , 0 = { 2 [ s 1 ( y j 1 , y 0 ; 0 ) + s 2 ( y j 1 , y 0 ; 45 ) ] λ 1 + 2 [ s 1 ( y j 1 , y 0 ; 0 ) + s 2 ( y j 1 , y 0 ; 45 ) ] λ 2 + 2 [ s 1 ( y j 1 , y 0 ; 0 ) + s 2 ( y j 1 , y 0 ; 45 ) ] λ 3 } / ( { [ 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 } λ 1 0.5 + { } λ 2 0.5 + { } λ 3 0.5 ) = { 2 cos 2 ω j 1 , 0 [ cos 2 ( ρ j 1 , 0 λ 1 / 2 ) + cos 2 ( ρ j 1 , 0 λ 2 / 2 ) + cos 2 ( ρ j 1 , 0 λ 3 / 2 ) ] } / { 2 [ cos 2 ( ρ j 1 , 0 λ 1 / 2 ) + cos 2 ( ρ j 1 , 0 λ 2 / 2 ) + cos 2 ( ρ j 1 , 0 λ 3 / 2 ) ] } .
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 ° ) λ i l j ( θ , 90 ° , 9 0 ° ) λ i l j ( θ , 45 ° , 4 5 ° ) λ i l j ( θ , 135 ° , 1 3 5 ° ) λ i [ l j ( θ , 45 ° , 0 ° ) λ i l j ( θ , 45 ° , 0 ° ) λ i ] / cos   Δ ρ λ i ] ,

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