Abstract

A general solution of the tomographic problem for weakly refracting inhomogeneous birefringent media is presented. The solution requires an unusual tomographic experiment in which the considered medium is probed with three series of standard tomographic sets of light rays, when the light is polarized. Data for the tomographic reconstruction are produced by recording the interferometric data and measuring birefringent parameters in the standard way.

© 1994 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. M. Hertz, “Kerr effect tomography for nonintrusive spatially resolved measurements of asymmetric electrical field distributions,” Appl. Opt. 25, 914–921 (1986).
    [CrossRef] [PubMed]
  3. H. Aben, K.-J. Kell, “Integrated photoelasticity as tensor field tomography,” Z. Angew. Math. Mech. 66, T118–T119 (1986).
  4. H. Aben, “Kerr effect tomography for general axisymmetric field,” Appl. Opt. 26, 2921–2924 (1987).
    [CrossRef] [PubMed]
  5. L. S. Srinath, “Principal-stress differences in transverse planes of symmetry,” Exp. Mech. 11, 130–137 (1971).
    [CrossRef]
  6. J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
    [CrossRef]
  7. H. Aben, S. Indurn, “Integrated photoelasticity of the general three-dimensional stress state,” presented at the International Conference on Measurement of Static and Dynamic Parameters of Structures and Materials, Plzen, Czechoslovakia, 1987.
  8. Y. A. Andrienko, M. S. Dubovikov, A. D. Gladun, “Optical tensor field tomography: the Kerr effect and axisymmetric integrated photoelasticity,” J. Opt. Soc. Am. A 9, 1765–1768 (1992).
    [CrossRef]
  9. Y. A. Andrienko, M. S. Dubovikov, A. D. Gladun, “Optical tomography of a birefringent medium,” J. Opt. Soc. Am. A 9, 1761–1764 (1992).
    [CrossRef]

1992 (2)

1987 (1)

1986 (2)

H. M. Hertz, “Kerr effect tomography for nonintrusive spatially resolved measurements of asymmetric electrical field distributions,” Appl. Opt. 25, 914–921 (1986).
[CrossRef] [PubMed]

H. Aben, K.-J. Kell, “Integrated photoelasticity as tensor field tomography,” Z. Angew. Math. Mech. 66, T118–T119 (1986).

1978 (1)

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

1971 (1)

L. S. Srinath, “Principal-stress differences in transverse planes of symmetry,” Exp. Mech. 11, 130–137 (1971).
[CrossRef]

Aben, H.

H. Aben, “Kerr effect tomography for general axisymmetric field,” Appl. Opt. 26, 2921–2924 (1987).
[CrossRef] [PubMed]

H. Aben, K.-J. Kell, “Integrated photoelasticity as tensor field tomography,” Z. Angew. Math. Mech. 66, T118–T119 (1986).

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

H. Aben, S. Indurn, “Integrated photoelasticity of the general three-dimensional stress state,” presented at the International Conference on Measurement of Static and Dynamic Parameters of Structures and Materials, Plzen, Czechoslovakia, 1987.

Andrienko, Y. A.

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.

Gladun, A. D.

Hertz, H. M.

Indurn, S.

H. Aben, S. Indurn, “Integrated photoelasticity of the general three-dimensional stress state,” presented at the International Conference on Measurement of Static and Dynamic Parameters of Structures and Materials, Plzen, Czechoslovakia, 1987.

Kell, K.-J.

H. Aben, K.-J. Kell, “Integrated photoelasticity as tensor field tomography,” Z. Angew. Math. Mech. 66, T118–T119 (1986).

Srinath, L. S.

L. S. Srinath, “Principal-stress differences in transverse planes of symmetry,” Exp. Mech. 11, 130–137 (1971).
[CrossRef]

Appl. Opt. (2)

Exp. Mech. (2)

L. S. Srinath, “Principal-stress differences in transverse planes of symmetry,” Exp. Mech. 11, 130–137 (1971).
[CrossRef]

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

J. Opt. Soc. Am. A (2)

Z. Angew. Math. Mech. (1)

H. Aben, K.-J. Kell, “Integrated photoelasticity as tensor field tomography,” Z. Angew. Math. Mech. 66, T118–T119 (1986).

Other (2)

H. Aben, S. Indurn, “Integrated photoelasticity of the general three-dimensional stress state,” presented at the International Conference on Measurement of Static and Dynamic Parameters of Structures and Materials, Plzen, Czechoslovakia, 1987.

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

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Figures (5)

Fig. 1
Fig. 1

Geometry of the auxiliary tomographic problem.

Fig. 2
Fig. 2

Diagrams of the tomographic experiment in (a) traditional tomography and (b) tensor field tomography.

Fig. 3
Fig. 3

Geometry of the onion-peeling algorithm in the axisymmetric case.

Fig. 4
Fig. 4

Geometry of the reconstruction of the tensor inside the ring in the axisymmetric case.

Fig. 5
Fig. 5

Geometry of the reconstruction of the tensor inside the slice in the asymmetric case.

Equations (26)

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n ( z ) = n k k ( z ) / 2 ,
f i j ( z ) = n i j ( z ) δ i j n ( z ) , i , j = 1 , 2 .
V i j ( z ) = exp [ i k Φ ( z ) ] U i j ( z ) ,
K = ω / c ,
Φ ( z ) = n ( z ) d z
Û ( z ) = Û M Û M 1 Û 2 Û 1 .
Û m = exp [ i K f ˆ ( z m ) Δ z m ] .
Û C D Û B C Û A B = Û ray 1 ,
Φ A B + Φ B C + Φ C D = Φ ray 1 ,
Û B C = Û C D 1 Û ray 1 Û A B 1 ,
Φ B C = Φ ray 1 Φ A B Φ C D .
f ˆ B C = ( i / K l B C ) ln Û B C ,
n ( B C ) = Δ Φ B C / l B C ,
F 1 ray 1 2 f 11 ( B C ) = 2 f 22 ( B C ) = n x 1 x 1 n y 1 y 1 ,
F 2 ray 1 f 12 ( B C ) = f 21 ( B C ) = n x 1 , y 1 ,
F 3 ray 1 2 n ( B C ) = n x 1 x 1 + n y 1 y 1 .
F 1 ray m = n x x ( n y y sin 2 α m + n z z cos 2 α m n y z sin 2 α m ) , m = 1 3 ,
F 2 ray m = n x y sin α m n x z cos α m , m = 1 3 ,
F 3 ray 1 = n x x + n y y + sin 2 α 1 + n z z cos 2 α 1 n y z sin 2 α 1 .
F 1 ray 1 = n r r n y y ,
F 2 ray 1 = n r y ,
F 3 ray 1 = n r r + n y y .
R 1 ray 2 = n r r ( n y y sin 2 α + n φ φ cos 2 α ) ,
F 1 ray m = n r r ( n y y sin 2 α m + n φ φ cos 2 α m + n y φ sin 2 α m ) , m = 1 3 ,
F 2 ray m = n r φ cos α m + n r y sin α m , m = 1 3 ,
F 3 ray 1 = n r r + n φ φ cos 2 α 1 + n y y sin 2 α 1 + n y φ sin 2 α 1 ,

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