Abstract

Equations of light propagation through a slightly anisotropic medium are derived. They are applied to the analysis of the tomographic problem of dielectric tensor fields. Two particular tomographic problems are also considered: (1) the axisymmetric problem and (2) the problem of the tensor with slight (but not negligible) rotations of the principal axes.

© 1992 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 electric 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, T.118 (1986).
  4. J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
    [CrossRef]
  5. H. Aben, “Kerr effect tomography for general axisymmetric field,” Appl. Opt. 26, 2921–2924 (1987).
    [CrossRef] [PubMed]
  6. Yu. 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]
  7. E. A. Dubovikova, M. S. Dubovikov, “Regularization, experimental errors and accuracy estimation in tomography and interferometry,” J. Opt. Soc. Am. A 4, 2033–2038 (1987).
    [CrossRef]

1992 (1)

1987 (2)

1986 (2)

1978 (1)

J. F. Doyle, H. T. Danyluk, “Integrated photoelasticity for axisymmetric problems,” Exp. Mech. 18, 215–220 (1978).
[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, T.118 (1986).

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

Andrienko, Yu. 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.

Dubovikova, E. A.

Gladun, A. D.

Hertz, H. M.

Kell, K.-J.

H. Aben, K.-J. Kell, “Integrated photoelasticity as tensor field tomography,”Z. Angew. Math. Mech. 66, T.118 (1986).

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Figures (1)

Fig. 1
Fig. 1

Geometry of the onion-peeling reconstruction.

Equations (42)

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ɛ i j ( r ) = ɛ 0 [ δ i j + Δ ɛ i j ( r ) ] ,
Δ ɛ i j 1.
E i z = i ω c n i j E j ,
n i j ( r ) = n 0 [ δ i j + Δ n i j ( r ) ] ,
n 0 = ɛ 0 1 / 2 ,             Δ n i j = Δ ɛ i j / 2.
E i ( z , t ) = V i j ( z ) E j ( 0 ) ( t ) ,
V i j z = i ω c n i k V k j .
V ^ ( z ) = { L exp [ i ω c n ^ ( z ) d z ] } ,
{ L exp [ i ω c n ^ ( z ) d z ] } = lim M m = M 1 exp [ i ω c n ^ ( z m ) Δ z ] .
V i j ( z ) = exp { i K [ z + Δ Φ ( z ) ] } U i j ( z ) ,
K = n 0 ω / c ,
Δ Φ ( z ) = Δ n ( z ) d z
Δ n ( z ) = tr [ Δ n ^ ( z ) ] / 2 Δ n k k ( z ) / 2 ,
f i j ( z ) = Δ n i j ( z ) - δ i j Δ n ( z ) .
U ^ ( z ) = { L exp [ i K f ^ ( z ) d z ] } .
U i j z = i K f i k U k j .
U ^ = [ U 0 + i U 3 U 2 + i U 1 - U 2 + i U 1 U 0 - i U 3 ] ,
U 0 2 + U α U α = 1
f ^ = [ f 3 f 1 - i f 2 f 1 + i f 2 - f 3 ] .
U 0 z = - K f β U β ,
U α z = K ( f α U 0 - τ α β γ f β U γ ) ,
Δ n ^ = [ δ n r r cos 2 φ + Δ n φ φ sin 2 φ Δ n r y cos φ Δ n r y cos φ Δ n y y ] .
Δ n = ( Δ n r r cos 2 φ + Δ n φ φ sin 2 φ + Δ n y y ) / 2 , f 1 = Δ n r y cos φ , f 2 = 0 , f 3 = ( Δ n r r cos 2 φ + Δ n φ φ sin 2 φ - Δ n y y ) / 2.
U ^ C D U ^ B C U ^ A B = U ^ ( x ) ,
Δ Φ A B + Δ Φ B C + Δ Φ C D = Δ Φ ( x ) ,
U ^ B C = U ^ A B * U ^ ( x ) U ^ A B + ,
Δ Φ B C = Δ Φ ( x ) - 2 Δ Φ A B .
U ^ B C = exp ( i K f ^ B C Δ z ) ,
Δ Φ B C = Δ n ( B C ) Δ z ,
( Δ n r r - Δ n y y ) / 2 = f 3 ( B C ) = { - 2 U 0 ( x ) × [ U 0 ( A B ) U 3 ( A B ) + U 1 ( A B ) U 2 ( A B ) ] + U 3 ( x ) × [ 2 U 0 ( A B ) 2 + 2 U 1 ( A B ) 2 - 1 ] + 2 U 1 ( x ) × [ U 0 ( A B ) U 2 ( A B ) - U 1 ( A B ) U 3 ( A B ) ] } / K Δ z ,
Δ n r y = f 1 ( B C ) = { 2 U 0 ( x ) × [ - U 0 ( A B ) U 1 ( A B ) + U 2 ( A B ) U 3 ( A B ) ] + U 1 ( x ) × [ 2 U 0 ( A B ) 2 + 2 U 3 ( A B ) 2 - 1 ] - 2 U 3 ( x ) × [ U 0 ( A B ) U 2 ( A B ) + U 1 ( A B ) U 3 ( A B ) ] } / K Δ z ,
( Δ n r r + Δ n y y ) / 2 = Δ n ( B C ) = [ Δ Φ ( x ) - 2 Δ Φ A B ] / Δ z .
Δ n φ φ = { Δ n r r + Δ n y y - [ ( Δ n r r - Δ n y y ) 2 + 4 Δ n r y 2 ] 1 / 2 } / 2.
( Δ F 2 ) ( Δ r ) 2 const .
( Δ F ) 2 ( Δ r ) 4 const .
v ^ ( m ) ( r ) = u ( m ) + p , q = 1 3 τ m p q γ p ( r ) u ( q ) .
n i j ( r ) = m = 1 3 n m ( r ) v i ( m ) ( r ) v j ( m ) ( r ) ,
n ^ = [ n 1 0 ( γ ) 0 ( γ ) n 2 sin 2 ϑ + n 3 cos 2 ϑ + γ 1 ( n 2 - n 3 ) sin 2 ϑ ] .
Φ ( 1 ) = n 1 ( r ) d l .
Φ ( 1 ) = { n 2 ( r ) sin 2 ϑ + n 3 ( r ) cos 2 ϑ + γ 1 ( r ) × [ n 2 ( r ) - n 3 ( r ) ] sin 2 ϑ } d l .
Γ 1 = [ Φ ( 1 ) - sin 2 ϑ n 2 ( r ) d l - cos 2 ϑ n 3 ( r ) d l ] / sin 2 ϑ .
Γ 1 = γ 1 ( r ) [ n 2 ( r ) - n 3 ( r ) ] d l .

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