Abstract

The generalized ray tracing for the extraordinary ray through uniaxial crystals developed by Avendaño-Alejo and Stavroudis [J. Opt. Soc. Am. A 19, 1674 (2002) ] has been applied to derive paraxial refracting equations. Paraxial equations are derived for three cases where the incident, ordinary, and extraordinary rays lie in the incident plane: (a) the crystal axis is parallel to the optical axis, (b) the crystal axis is orthogonal to the optical axis and lies in the plane of incidence, and (c) the crystal axis is orthogonal to both the optical axis and the incident plane. The paraxial ray-tracing equations for the extraordinary ray are represented by matrix operators. The elements of the matrix system give all the information of the focal points and of the principal points. Gaussian formulas are derived, and some examples are presented.

© 2005 Optical Society of America

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References

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  1. O. N. Stavroudis, “Ray tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–191 (1962).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. Z. Shao, C. Yi, “Behavior of extraordinary rays in uniaxial crystals,” Appl. Opt. 33, 1209–1212 (1994).
    [CrossRef] [PubMed]
  4. E. Cojocaru, “Direction cosines and vectorial relations for extraordinary wave propagation in uniaxial media,” Appl. Opt. 36, 302–306 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. H. Kikuta, K. Iwata, “First-order aberration of a double-focus lens made of a uniaxial crystal,” J. Opt. Soc. Am. A 9, 814–819 (1992).
    [CrossRef]
  7. M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [CrossRef]
  8. J. P. Lesso, A. J. Duncan, W. Sibbett, M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” Appl. Opt. 39, 592–598 (2000).
    [CrossRef]
  9. M. Avendaño-Alejo, O. Stavroudis, A. R. Boyain, “Huygens’ principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1668–1673 (2002).
    [CrossRef]
  10. M. Avendaño-Alejo, O. Stavroudis, “Huygens’ principle and rays in uniaxial anisotropic media. II. Crystal axis with arbitrary orientation,” J. Opt. Soc. Am. A 19, 1674–1679 (2002).
    [CrossRef]
  11. E. Hecht, A. Zajac, Optics (Addison-Wesley, London, 1974), Chap. 6.

2002 (2)

2000 (1)

1998 (1)

1997 (1)

1994 (1)

1992 (1)

1990 (1)

1986 (1)

1962 (1)

Avendaño-Alejo, M.

Beyerle, G.

Boyain, A. R.

Cojocaru, E.

Duncan, A. J.

Echarri, R. M.

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, London, 1974), Chap. 6.

Iwata, K.

Kikuta, H.

Lesso, J. P.

Liang, Q. T.

McDermid, I. S.

Padgett, M. J.

Shao, Z.

Sibbett, W.

Simon, M. C.

Stavroudis, O.

Stavroudis, O. N.

Yi, C.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, London, 1974), Chap. 6.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Other (1)

E. Hecht, A. Zajac, Optics (Addison-Wesley, London, 1974), Chap. 6.

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Figures (11)

Fig. 1
Fig. 1

Refraction of the extraordinary ray at a single spherical interface separating two media: an isotropic and a birefringent medium. For simplicity only the Y Z plane is drawn.

Fig. 2
Fig. 2

Refraction of the extraordinary ray at a single spherical interface separating two media: a birefringent and an isotropic medium.

Fig. 3
Fig. 3

Refraction of a paraxial extraordinary ray through a birefringent lens when the optical axis is parallel ( ) to the optical axis.

Fig. 4
Fig. 4

Refraction of a paraxial extraordinary ray through a birefringent lens when the optical axis is orthogonal ( ) to the optical axis and lies in the plane of incidence.

Fig. 5
Fig. 5

Refraction of a paraxial extraordinary ray through a birefringent lens when the optical axis is orthogonal to both the optical axis and the plane of incidence ( ) .

Fig. 6
Fig. 6

Paraxial parameters for ray tracing of the extraordinary ray through a birefringent lens.

Fig. 7
Fig. 7

Cardinal points in a birefringent lens.

Fig. 8
Fig. 8

Gauss image formation for extraordinary rays.

Fig. 9
Fig. 9

(a) Birefringent thin lens with the crystal axis orthogonal to the optical axis; (b) meridional plane of the lens shown in (a); (c) sagittal plane of the lens shown in (a).

Fig. 10
Fig. 10

(a) Birefringent thin lens with the crystal axis parallel to the optical axis, (b) meridional plane of lens (a), but because of the symmetry any plane that contains the optical axis will suffice.

Fig. 11
Fig. 11

Thick birefringent lens showing the second principal focus f e p and the first and second principal planes H 1 and H 2 .

Equations (75)

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( ξ o η o ζ o ) = [ n i n o 0 0 0 n i n o 0 0 0 1 ζ i [ 1 ( n i n o ) 2 ( 1 ζ i 2 ) ] 1 2 ] ( ξ i η i ζ i ) ,
( ξ e 1 η e 1 ζ e 1 )
= 1 δ e g { [ n o 2 [ n e 2 + ( n o 2 n e 2 ) β 1 2 ] n o 2 ( n o 2 n e 2 ) α 1 β 1 0 n o 2 ( n o 2 n e 2 ) α 1 β 1 n o 2 [ n e 2 + ( n o 2 n e 2 ) α 1 2 ] 0 0 0 Γ Δ ζ o ] ( ξ o η o ζ o ) γ 1 ( n o 2 n e 2 ) Δ [ 1 0 0 0 1 0 0 0 0 ] ( α 1 β 1 γ 1 ) } ,
δ e g 2 = n e 2 { n o 2 Γ 2 ( n o 2 n e 2 ) [ γ 1 Δ + n o 2 ( α 1 ξ o + β 1 η o ) ] 2 } ,
Δ 2 = Γ δ o 2 + n o 2 ( n o 2 n e 2 ) ( α 1 ξ o + β 1 η o ) 2 ,
Γ = n e 2 + ( n o 2 n e 2 ) ( 1 γ 1 2 ) = n o 2 ( n o 2 n e 2 ) γ 1 2 ,
δ o 2 = n e 2 n o 2 ( 1 ζ o 2 ) ,
( ξ o f η o f ζ o f ) = 1 n o Γ e { [ n e 2 + ( n o 2 n e 2 ) α 2 2 ( n o 2 n e 2 ) α 2 β 2 0 ( n o 2 n e 2 ) α 2 β 2 n e 2 + ( n o 2 n e 2 ) β 2 2 0 0 0 Δ e ζ e 1 ] ( ξ e 1 η e 1 ζ e 1 ) + ζ e 1 γ 2 ( n o 2 n e 2 ) [ 1 0 0 0 1 0 0 0 0 ] ( α 2 β 2 γ 2 ) } ,
Γ e 2 = n e 2 sin 2 ψ + n o 2 cos 2 ψ ,
Δ e 2 = n e 2 ( n o 2 n e 2 ) sin 2 ψ + [ ( n o 2 n e 2 ) γ 2 cos ψ + n e 2 ζ e 1 ] 2 ,
cos ψ = α 2 ξ e 1 + β 2 η e 1 + γ 2 ζ e 1 .
( ξ e 2 η e 2 ζ e 2 ) = 1 n i 2 [ n o 0 0 0 n o 0 0 0 1 ζ o f [ n i 2 2 n o 2 ( 1 ζ o f 2 ) ] 1 2 ] ( ξ o f η o f ζ o f ) .
( ξ e 1 η e 1 ζ e 1 ) = ( 0 n 0 n i θ 1 n e 2 + ( n e 2 n o 2 ) ϕ 1 n e 2 1 ) ,
( ξ o f η o f ζ o f ) = ( 0 n e 2 θ 2 n o 2 + ( n o 2 n e 2 ) ϕ 2 n o 2 1 ) ,
( ξ e 2 η e 2 ζ e 2 ) = ( 0 n e 2 θ 2 n i 2 n o + ( n o 2 n e 2 ) ϕ 2 n i 2 n o 1 ) ,
( ξ e 1 η e 1 ζ e 1 ) = ( 0 n e n i θ 1 n o 2 + ( n e 2 n o 2 ) ϕ 1 n o 2 1 ) .
( ξ e 2 η e 2 ζ e 2 ) = ( 0 n o 2 θ 2 n e n i + ( n o 2 n e 2 ) ϕ 2 n e n i 1 ) ,
( ξ e 1 η e 1 ζ e 1 ) = ( 0 n i θ 1 n e 1 ) .
( ξ e 2 η e 2 ζ e 2 ) = ( 0 n e θ 2 n i 1 ) .
( n e 2 u t 1 n o y t 1 ) = R i b ( n i u i 1 y i 1 ) , where R i b = [ 1 n o n i R 1 0 1 ] .
D i b = n o n i R 1 .
( n e 2 u i 2 n o y i 2 ) = T b ( n e 2 u t 1 n o y t 1 ) , where T b = [ 1 0 n o t n e 2 1 ] .
( n i 2 u t 2 y t 2 ) = R b i ( n e 2 u i 2 n o y i 2 ) , where R b i = [ 1 n i 2 n o R 2 0 1 ] ,
D b i = n i 2 n o R 2 .
( n i 2 u t 2 y t 2 ) = M ( n i u i 1 y i 1 ) ,
where M = R b i T b R i b = [ m 11 m 12 m 21 m 22 ] .
( n o u t 1 y t 1 ) = R i b ( n e n i u i 1 n o y i 1 ) ,
where R i b = [ 1 2 n o 2 n e ( n e + n i ) n o R 1 0 1 ] .
D i b = 2 n o 2 n e ( n e + n i ) n o R 1
( n o u i 2 y i 2 ) = T b ( n o u t 1 y t 1 ) , where T b = [ 1 0 t n o 1 ] .
( n e n i 2 u t 2 n o y t 2 ) = R b i ( n o u i 2 y i 2 ) ,
where R b i = [ 1 n e ( n e + n i 2 ) 2 n o 2 n 0 R 2 0 1 ] .
D b i = n e ( n e + n i 2 ) 2 n o 2 n o R 2
( n e n i 2 u t 2 n o y t 2 ) = M ( n e n i u i 1 n o y i 1 ) ,
where M = R b i T b R i b = [ m 11 m 12 m 21 m 22 ] .
( n e u t 1 y t 1 ) = R i b ( n i u i 1 y i 1 ) , where R i b = [ 1 n e n i R 1 0 1 ] .
D i b = n e n i R 1
( n e u i 2 y i 2 ) = T b ( n e u t 1 y t 1 ) , where T b = [ 1 0 t n e 1 ] .
( n i 2 u t 2 y t 2 ) = R b i ( n e u i 2 y i 2 ) , where R b i = [ 1 n i 2 n e R 2 0 1 ] .
D b i = n i 2 n e R 2
( n i 2 u t 2 y t 2 ) = M ( n i u i 1 y i 1 ) ,
where M = R b i T b R i b = [ m 11 m 12 m 21 m 22 ] .
M = [ 1 n o t D b i n e 2 D i b + n o t D i b D b i n e 2 D b i n o t n e 2 1 n o t D i b n e 2 ] .
( n i 2 u t 2 y t 2 ) = [ m 11 m 12 m 21 m 22 ] ( 0 1 ) = ( m 12 m 22 ) .
f e p = 1 u t 2 = n i 2 m 12 .
V 2 H 2 ¯ = ( 1 u t 2 + y t 2 u t 2 ) = n i 2 m 12 ( m 22 1 ) .
( 0 1 ) = [ m 11 m 12 m 21 m 22 ] ( n i u i 1 y i 1 ) .
f e a = 1 u i 1 = n i m 12 .
V 1 H 1 ¯ = ( 1 u i 1 + y i 1 u i 1 ) = n i m 12 ( 1 m 11 ) .
f e a = f e p = 1 m 12 ,
V 1 H 1 ¯ = 1 m 12 ( 1 m 11 ) ,
V 2 H 2 ¯ = 1 m 12 ( m 22 1 ) .
M = [ 1 t D b i n o D i b + t D i b D b i n 0 D b i t n o 1 t D i b n o ] .
f e a = n e n i n o m 12 ,
f e p = n e n i 2 n o m 12 ,
V 1 H 1 ¯ = n e n i n o m 12 ( 1 m 11 ) ,
V 2 H 2 ¯ = n e n i 2 n o m 12 ( m 22 1 ) ,
M = [ 1 t D b i n e D i b + t D i b D b i n e D b i t n e 1 t D i b n e ] .
f e a = n i m 12 ,
f e p = n i 2 m 12 ,
V 1 H 1 ¯ = n i m 12 ( 1 m 11 ) ,
V 2 H 2 ¯ = n i 2 m 12 ( m 22 1 ) ,
u i 1 = y i 1 L 1 , u t 1 = y t 1 L 1 ,
u i 2 = y i 2 L 2 , u t 2 = y t 2 L 2 .
n e 2 n o L 1 n i L 1 = D i b ,
n i 2 L 2 n e 2 n o L 2 = D b i ,
n o L 1 n e n i n o L 1 = D i b ,
n e n i 2 L 2 n o L 2 = D b i ,
n e L 1 n i L 1 = D i b ,
n i 2 L 1 n e L 2 = D b i ,
D 1 f e p = 1 f e a = ( n o n i n i ) ( 1 R 1 1 R 2 ) ,
D 1 f e p = 1 f e a = [ 2 n o 2 n e ( n e + n i ) n e n i ] ( 1 R 1 1 R 2 ) ,
D 1 f e p = 1 f e a = ( n e n i n i ) ( 1 R 1 1 R 2 ) .
M = [ 0.9723 0.0184 1.5078 1 ] .
M = [ 0.9862 0.0082 1.6824 1 ] .

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