Abstract

Optical phenomena that occur when polarized light passes through an inhomogeneous birefringent medium are complicated, especially when the principal directions of the dielectric tensor rotate on the light ray. This case is typical in three-dimensional photoelasticity, in particular in integrated photoelasticity by stress analysis on the basis of measured polarization transformations. Analysis of polarization transformations in integrated photoelasticity has been based primarily on a system of two first-order differential equations. Using a transformed coordinate in the direction of light propagation, we have derived a single fourth-order differential equation of three-dimensional photoelasticity. For the case of uniform rotation of the principal directions we have obtained an analytical solution.

© 2006 Optical Society of America

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References

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  1. V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, Oxford, 1971).
  2. R. D. Mindlin and L. E. Goodman, 'The optical equations of three-dimensional photoelasticity,' J. Appl. Phys. 20, 89-95 (1949).
    [CrossRef]
  3. R. C. O'Rourke, 'Three-dimensional photoelasticity,' J. Appl. Phys. 22, 872-878 (1951).
    [CrossRef]
  4. S. E. Segre, 'New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma,' J. Opt. Soc. Am. A 18, 2601-2606 (2001).
    [CrossRef]
  5. L. C. Meira-Belo and U. A. Leitão, 'Singular polarization eigenstates in anisotropic stratified structures,' Appl. Opt. 39, 2695-2704 (2000).
    [CrossRef]
  6. I. Moreno, C. R. Fernándes-Pousa, and J. A. Davis, 'Polarization eigenvectors for reflective twisted nematic liquid crystal displays,' Opt. Eng. (Bellingham) 40, 2220-2226 (2001).
    [CrossRef]
  7. H. Aben, 'Optical phenomena in photoelastic models by the rotation of principal axes,' Exp. Mech. 6, 13-22 (1966).
    [CrossRef]
  8. H. Aben, Integrated Photoelasticity (McGraw-Hill, 1979).
  9. H. K. Aben, J. I. Josepson, and K.-J. Kell, 'The case of weak birefringence in integrated photoelasticity,' Opt. Lasers Eng. 11, 145-157 (1989).
    [CrossRef]
  10. H. Aben and C. Guillemet, Photoelasticity of Glass (Springer-Verlag, 1993).
  11. H. Aben, L. Ainola, and J. Anton, 'Integrated photoelasticity for nondestructive residual stress measurement in glass,' Opt. Lasers Eng. 33, 49-64 (2000).
    [CrossRef]
  12. H. Aben, J. Anton, and A. Errapart, 'Modern photoelastic technology for residual stress measurement in glass,' Verre (Versailles) 9, 44-49 (2003).
  13. A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1998).
  14. L. Ainola and H. Aben, 'Duality in optical theory of twisted birefringent media,' J. Opt. Soc. Am. A 16, 2545-2549 (1999).
    [CrossRef]
  15. L. Ainola and H. Aben, 'Transformation equations in polarization optics of inhomogeneous birefringent media,' J. Opt. Soc. Am. A 18, 2164-2170 (2001).
    [CrossRef]
  16. L. Ainola and H. Aben, 'Principal formulas of integrated photoelasticity in terms of characteristic parameters,' J. Opt. Soc. Am. A 22, 1181-1186 (2005).
    [CrossRef]
  17. H. Aben and J. Josepson, 'Strange interference blots in the interferometry of inhomogeneous birefringent objects,' Appl. Opt. 36, 7172-7179 (1997).
    [CrossRef]
  18. H. Aben and L. Ainola, 'Interference blots and fringe dislocations in optics of twisted birefringent media,' J. Opt. Soc. Am. A 15, 2404-2411 (1998).
    [CrossRef]
  19. S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, 1969).
  20. H. Poincaré, Théorie mathématique de la lumière, II (Carré et Naud, Paris, 1892).

2005 (1)

2003 (1)

H. Aben, J. Anton, and A. Errapart, 'Modern photoelastic technology for residual stress measurement in glass,' Verre (Versailles) 9, 44-49 (2003).

2001 (3)

2000 (2)

H. Aben, L. Ainola, and J. Anton, 'Integrated photoelasticity for nondestructive residual stress measurement in glass,' Opt. Lasers Eng. 33, 49-64 (2000).
[CrossRef]

L. C. Meira-Belo and U. A. Leitão, 'Singular polarization eigenstates in anisotropic stratified structures,' Appl. Opt. 39, 2695-2704 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

1989 (1)

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

1966 (1)

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

1951 (1)

R. C. O'Rourke, 'Three-dimensional photoelasticity,' J. Appl. Phys. 22, 872-878 (1951).
[CrossRef]

1949 (1)

R. D. Mindlin and L. E. Goodman, 'The optical equations of three-dimensional photoelasticity,' J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

Aben, H.

L. Ainola and H. Aben, 'Principal formulas of integrated photoelasticity in terms of characteristic parameters,' J. Opt. Soc. Am. A 22, 1181-1186 (2005).
[CrossRef]

H. Aben, J. Anton, and A. Errapart, 'Modern photoelastic technology for residual stress measurement in glass,' Verre (Versailles) 9, 44-49 (2003).

L. Ainola and 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, and J. Anton, 'Integrated photoelasticity for nondestructive residual stress measurement in glass,' Opt. Lasers Eng. 33, 49-64 (2000).
[CrossRef]

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

H. Aben and 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 and J. Josepson, 'Strange interference blots in the interferometry of inhomogeneous birefringent objects,' Appl. Opt. 36, 7172-7179 (1997).
[CrossRef]

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

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

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

Aben, H. K.

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

Ainola, L.

Anton, J.

H. Aben, J. Anton, and A. Errapart, 'Modern photoelastic technology for residual stress measurement in glass,' Verre (Versailles) 9, 44-49 (2003).

H. Aben, L. Ainola, and J. Anton, 'Integrated photoelasticity for nondestructive residual stress measurement in glass,' Opt. Lasers Eng. 33, 49-64 (2000).
[CrossRef]

Bhagavantam, S.

S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, 1969).

Davis, J. A.

I. Moreno, C. R. Fernándes-Pousa, and J. A. Davis, 'Polarization eigenvectors for reflective twisted nematic liquid crystal displays,' Opt. Eng. (Bellingham) 40, 2220-2226 (2001).
[CrossRef]

Errapart, A.

H. Aben, J. Anton, and A. Errapart, 'Modern photoelastic technology for residual stress measurement in glass,' Verre (Versailles) 9, 44-49 (2003).

Fernándes-Pousa, C. R.

I. Moreno, C. R. Fernándes-Pousa, and J. A. Davis, 'Polarization eigenvectors for reflective twisted nematic liquid crystal displays,' Opt. Eng. (Bellingham) 40, 2220-2226 (2001).
[CrossRef]

Fuki, A. A.

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1998).

Ginzburg, V. L.

V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, Oxford, 1971).

Goodman, L. E.

R. D. Mindlin and L. E. Goodman, 'The optical equations of three-dimensional photoelasticity,' J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

Guillemet, C.

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

Josepson, J.

Josepson, J. I.

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

Kell, K.-J.

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

Kravtsov, Yu. A.

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1998).

Leitão, U. A.

Meira-Belo, L. C.

Mindlin, R. D.

R. D. Mindlin and L. E. Goodman, 'The optical equations of three-dimensional photoelasticity,' J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

Moreno, I.

I. Moreno, C. R. Fernándes-Pousa, and J. A. Davis, 'Polarization eigenvectors for reflective twisted nematic liquid crystal displays,' Opt. Eng. (Bellingham) 40, 2220-2226 (2001).
[CrossRef]

Naida, O. N.

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1998).

O'Rourke, R. C.

R. C. O'Rourke, 'Three-dimensional photoelasticity,' J. Appl. Phys. 22, 872-878 (1951).
[CrossRef]

Poincaré, H.

H. Poincaré, Théorie mathématique de la lumière, II (Carré et Naud, Paris, 1892).

Segre, S. E.

Venkatarayudu, T.

S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, 1969).

Appl. Opt. (2)

Exp. Mech. (1)

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

J. Appl. Phys. (2)

R. D. Mindlin and L. E. Goodman, 'The optical equations of three-dimensional photoelasticity,' J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

R. C. O'Rourke, 'Three-dimensional photoelasticity,' J. Appl. Phys. 22, 872-878 (1951).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Eng. (Bellingham) (1)

I. Moreno, C. R. Fernándes-Pousa, and J. A. Davis, 'Polarization eigenvectors for reflective twisted nematic liquid crystal displays,' Opt. Eng. (Bellingham) 40, 2220-2226 (2001).
[CrossRef]

Opt. Lasers Eng. (2)

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

H. Aben, L. Ainola, and J. Anton, 'Integrated photoelasticity for nondestructive residual stress measurement in glass,' Opt. Lasers Eng. 33, 49-64 (2000).
[CrossRef]

Verre (Versailles) (1)

H. Aben, J. Anton, and A. Errapart, 'Modern photoelastic technology for residual stress measurement in glass,' Verre (Versailles) 9, 44-49 (2003).

Other (6)

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1998).

S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, 1969).

H. Poincaré, Théorie mathématique de la lumière, II (Carré et Naud, Paris, 1892).

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

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

V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, Oxford, 1971).

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Equations (96)

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d E d z = V E ,
E = ( E 1 E 2 ) ,
V = i C [ 1 2 ( σ 11 σ 22 ) σ 12 σ 12 1 2 ( σ 11 σ 22 ) ] .
E = U E 0 ,
s 0 = [ 1 0 0 1 ]
s 1 = [ i 0 0 i ] , s 2 = [ 0 1 1 0 ] , s 3 = [ 0 i i 0 ] ,
U = β 0 s 0 + β 1 s 1 + β 2 s 2 + β 3 s 3 .
β 0 2 + β 1 2 + β 2 2 + β 3 2 = 1 .
σ = 1 2 C ( σ 11 σ 22 ) , τ = C σ 12 ,
V = σ s 1 τ s 3 .
d U d z = V U .
U ( z 0 ) = s 0 .
β 0 s 0 + β 1 s 1 + β 2 s 2 + β 3 s 3 = ( σ s 1 τ s 3 ) ( β 0 s 0 + β 1 s 1 + β 2 s 2 + β 3 s 3 ) .
s 0 s j = s j s 0 = s j , s j 2 = s 0 , j = 1 , 2 , 3 ,
s 1 s 2 = s 2 s 1 = s 3 , s 2 s 3 = s 3 s 2 = s 1 ,
s 3 s 1 = s 1 s 3 = s 2 .
β 0 = σ β 1 + τ β 3 ,
β 1 = σ β 0 + τ β 2 ,
β 2 = σ β 3 τ β 1 ,
β 3 = σ β 2 τ β 0 .
β 0 ( z 0 ) = 1 , β 1 ( z 0 ) = β 2 ( z 0 ) = β 3 ( z 0 ) = 0 .
σ τ , τ σ , β 0 β 1 , β 1 β 2 , β 2 β 3 , β 3 β 0 ;
σ σ , τ τ , β 0 β 2 , β 1 β 3 , β 2 β 0 , β 3 β 1 ;
σ τ , τ σ , β 0 β 3 , β 1 β 0 , β 2 β 1 , β 3 β 2 .
U = S ( α * ) G ( γ ) S ( α 0 ) ,
S ( ϑ ) = [ cos ϑ sin ϑ sin ϑ cos ϑ ] .
G ( γ ) = [ exp ( i γ ) 0 0 exp ( i γ ) ] .
β 1 = cos γ cos ( α * α 0 ) ,
β 2 = sin γ cos ( α * + α 0 ) ,
β 3 = cos γ sin ( α * α 0 ) ,
β 4 = sin γ sin ( α * + α 0 ) .
β 1 = 1 σ  2 + τ  2 ( σ β 0 τ β 2 ) ,
β 3 = 1 σ  2 + τ  2 ( τ β 0 + σ β 2 ) .
( σ σ  2 + τ  2 β 0 ) + σ β 0 ( τ σ  2 + τ  2 β 2 ) τ β 2 = 0 ,
( τ σ  2 + τ  2 β 0 ) + τ β 0 + ( σ σ  2 + τ  2 β 2 ) + σ β 2 = 0 .
β 0 ( z 0 ) = 1 , β 0 ( z 0 ) = 0 ,
β 2 ( z 0 ) = 0 , β 2 ( z 0 ) = 0 .
β 0 σ σ + τ τ σ 2 + τ 2 β 0 + ( σ 2 + τ 2 ) β 0 + σ τ σ τ σ 2 + τ 2 β 2 = 0 ,
β 2 σ σ + τ τ σ 2 + τ 2 β 2 + ( σ 2 + τ 2 ) β 2 σ τ σ τ σ 2 + τ 2 β 0 = 0 .
( 1 σ 2 + τ 2 β 0 ) + σ 2 + τ 2 β 0 + σ τ σ τ ( σ 2 + τ 2 ) 3 2 β 2 = 0 ,
( 1 σ 2 + τ 2 β 2 ) + σ 2 + τ 2 β 2 σ τ σ τ ( σ 2 + τ 2 ) 3 2 β 0 = 0 .
u = z 0 z σ 2 + τ 2 d z .
σ 2 + τ 2 = C 2 ( σ 1 σ 2 ) ,
u = C 2 z 0 z ( σ 1 σ 2 ) d z ;
d 2 β 0 d u 2 + β 0 R ( u ) d β 2 d u = 0 ,
d 2 β 2 d u 2 + β 2 + R ( u ) d β 0 d u = 0 .
R ( u ) = σ ( u ) ( d τ d u ) τ ( u ) ( d τ d u ) [ σ ( u ) ] 2 + [ τ ( u ) ] 2 .
β 0 u = 0 = 1 , d β 0 d u u = 0 = 0 ,
β 2 u = 0 = 0 , d β 2 d u u = 0 = 0 .
d 2 β 1 d u 2 + β 1 + R ( u ) d β 3 d u = 0 ,
d 2 β 3 d u 2 + β 3 R ( u ) d β 1 d u = 0 .
β 1 u = 0 = 0 , d β 1 d u u = 0 = σ 0 σ 0 2 + τ 0 2 ,
β 3 u = 0 = 0 , d β 3 d u u = 0 = τ 0 σ 0 2 + τ 0 2 .
σ 0 = σ ( z 0 ) , τ 0 = τ ( z 0 ) .
σ d τ d u τ d σ d u = σ 2 d d u ( τ σ ) ,
R ( u ) = 1 1 + τ 2 / σ 2 d d u ( τ σ )
R ( u ) = d d u arctan ( τ σ ) .
τ σ = 2 σ 12 σ 11 σ 22 .
2 σ 12 σ 11 σ 22 = tan 2 φ ,
τ σ = tan 2 φ .
R ( u ) = 2 ( d φ d u ) .
d 2 β 0 d u 2 + β 0 2 d φ d u d β 2 d u = 0 ,
d 2 β 2 d u 2 + β 2 + 2 d φ d u d β 0 d u = 0 ,
d 2 β 1 d u 2 + β 1 + 2 d φ d u d β 3 d u = 0 ,
d 2 β 3 d u 2 + β 3 2 d φ d u d β 1 d u = 0 .
β 1 u = 0 = 0 , d β 1 d u u = 0 = cos 2 φ 0 ,
β 3 u = 0 = 0 , d β 3 d u u = 0 = sin 2 φ 0 ,
d 4 β 0 d u 4 2 ( d φ d u ) 1 d 2 φ d u 2 d 3 β 0 d u 3 [ ( d φ d u ) 1 d 3 φ d u 3 2 ( d φ d u ) 2 ( d 2 φ d u 2 ) 2 2 4 ( d φ d u ) 2 ] d 2 β 0 d u 2 [ 2 ( d φ d u ) 1 d 2 φ d u 2 4 d φ d u d 2 φ d u 2 ] d β 0 d u [ ( d φ d u ) 1 d 3 φ d u 3 2 ( d φ d u ) 2 ( d 2 φ d u 2 ) 2 1 ] β 0 = 0 .
β 0 u = 0 = 1 , d β 0 d u u = 0 = 0 , d 2 β 0 d u 2 u = 0 = 1 , d 3 β 0 d u 3 u = 0 = 0 .
β 1 u = 0 = 0 , d β 1 d u u = 0 = cos 2 φ 0 ,
d 2 β 1 d u 2 u = 0 = 2 sin 2 φ 0 d φ d u u = 0 ,
d 3 β 1 d u 3 u = 0 = cos 2 φ 0 [ 1 + 4 ( d φ d u ) u = 0 2 ] + 2 sin 2 φ 0 d 2 φ d u 2 u = 0 ;
β 2 u = 0 = 0 , d β 2 d u u = 0 = 0 ,
d 2 β 2 d u 2 u = 0 = 0 , d 3 β 2 d u 3 u = 0 = 2 d φ d u u = 0 ;
β 3 u = 0 = 0 , d β 3 d u u = 0 = sin 2 φ 0 ,
d 2 β 3 d u 2 u = 0 = 2 cos 2 φ 0 d φ d u u = 0 ,
d 3 β 3 d u 3 u = 0 = sin 2 φ 0 [ 1 + 4 ( d φ d u ) u = 0 2 ] 2 cos 2 φ d 2 φ d u 2 u = 0 .
d φ d u = c ,
d 4 β k d u 4 + ( 2 + 4 c 2 ) d 2 β k d u 2 + β k = 0 , k = 0 , 1 , 2 , 3 ,
β 0 u = 0 = 1 , d β 0 d u u = 0 = 0 , d 2 β 0 d u 2 u = 0 = 1 , d 3 β 0 d u 3 = 0 ,
β 1 u = 0 = 0 , d β 1 d u u = 0 = cos 2 φ 0 , d 2 β 1 d u 2 u = 0 = 2 c sin 2 φ 0 ,
d 3 β 1 d u 3 u = 0 = ( 1 + 4 c 2 ) cos 2 φ 0 ,
β 2 u = 0 = 0 , d β 2 d u u = 0 = 0 , d 2 β 2 d u 2 u = 0 = 0 , d 3 β 2 d u 3 u = 0 = 2 c ,
β 3 u = 0 = 0 , d β 3 d u u = 0 = sin 2 φ 0 , d 2 β 3 d u 2 u = 0 = 2 c cos 2 φ 0 ,
d 3 β 3 d u 3 u = 0 = ( 1 + 4 c 2 ) sin 2 φ 0 .
β 0 = 1 K 2 K 1 K 2 cos K 1 u + K 1 1 K 1 K 2 cos K 2 u ,
β 1 = 1 K 1 K 4 cos 2 φ 0 sin K 1 u + K 3 sin 2 φ 0 cos K 1 u + 1 K 2 K 5 cos 2 φ 0 sin K 2 u K 3 sin 2 φ 0 cos K 1 u ,
β 2 = 1 K 1 K 3 sin K 1 u 1 K 2 K 3 sin K 2 u ,
β 3 = 1 K 1 K 4 sin 2 φ 0 sin K 1 u K 3 cos 2 φ 0 cos K 1 u + 1 K 2 K 5 sin 2 φ 0 sin K 2 u + K 3 cos 2 φ 0 cos K 2 u ,
K 1 = 1 + 2 c 2 + 2 c 1 + c 2 , K 2 = 1 + 2 c 2 2 c 1 + c 2 ,
K 3 = 2 c K 1 K 2 , K 4 = K 2 ( 1 + 4 c 2 ) K 1 K 2 , K 5 = 1 + 4 c 2 K 1 K 1 K 2 .
K 1 = 1 , K 2 = 1 , K 3 = 1 2 , K 4 = 1 2 , K 5 = 1 2 .
β 0 = cos u ,
β 1 = cos 2 φ 0 sin u ,
β 2 = 0 ,
β 3 = sin 2 φ 0 sin u .

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