Abstract

The formulas for ray tracing in monoaxial crystals for discontinuity surfaces of arbitrary shape are generalized. The formulas obtained are applied to the case of an achromatic calcite–calcite doublet.

© 1986 Optical Society of America

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References

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  1. M. C. Simon, “Ray Tracing Formulas for Monoaxial Optical Components,” Appl. Opt. 22, 354 (1983).
    [CrossRef] [PubMed]
  2. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).
  3. J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

1983

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).

Simon, M. C.

Strong, J.

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

Appl. Opt.

Other

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

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Figures (6)

Fig. 1
Fig. 1

Surface coordinate system; its construction from n and z 3.

Fig. 2
Fig. 2

Vector form of the law of reflection.

Fig. 3
Fig. 3

Vector form of Snell’s law.

Fig. 4
Fig. 4

Achromatic calcite–calcite doublet.

Fig. 5
Fig. 5

Spot diagrams of the astigmatic images for λ = 2310 Å. Widths of the lenses are 3-mm (convergent lens) and 2-mm (divergent lens). Six points were excluded from each corner of the square grid of 17 × 17 points.

Fig. 6
Fig. 6

Spot diagrams of the astigmatic images for λ = 6560 Å. Widths of the lenses are 3-mm (convergent lens) and 2-mm (divergent lens). Six points were excluded from each corner of the square grid of 17 × 17 points.

Equations (35)

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S · y = S · y ,             S · z = S · z ,
S · y u = N · y u ,             S · z u = N · z u ,
S · y u = N · y u ,             S · z u = N · z u ,
S = S - 2 ( S · n ) n
N = S + a n 1 + a 2 + 2 a ( S · n ) ,
a = ( u u 0 ) 2 - 1 + ( S · n ) 2 - ( S · n ) .
u 2 = ( 1 - N 3 2 ) u e 2 + N 3 2 u 0 2
u 2 = [ 1 - ( N · z 3 ) 2 ] u e 2 + ( N · z 3 ) 2 u 0 2 .
N x = 1 - ( u u ) 2 [ ( S · y ) 2 + ( S · z ) 2 ] , N y = ( u u ) ( S · y ) , N z = ( u u ) ( S · z ) .
N · z 3 = 1 - ( u u ) 2 [ ( S · y ) 2 + ( S · z ) 2 ] ( x · z 3 ) + ( u u ) ( S · z ) ( z 3 · z ) .
w = u u ,             b = u o 2 - u e 2 u 2 ,
w 4 A - w 2 B + C = 0 ,
A = [ 1 + b ( S · z ) 2 + b ( S · y ) 2 ( x · z 3 ) 2 ] 2 - 4 b ( S · z ) 2 ( z 3 · z ) 2 ,
B = 2 [ 1 + b ( S · z ) 2 + b ( S · y ) 2 ( x · z 3 ) 2 ] [ b ( x · z 3 ) 2 + ( u e u ) 2 ] - 4 b ( u e u ) 2 ( S · z ) 2 ( z 3 · z ) 2 ,
C = [ ( u e u ) 2 + b ( x · z 3 ) 2 ] 2 ,
x = n ,
y = n × z 3 n × z 3 ,
z = n × n × z 3 n × z 3 .
A = ( 1 + b { 1 - S · n ) 2 - [ S · ( n × z 3 ) ] 2 } ) 2 - 4 b { [ 1 - ( S · n ) 2 ] [ 1 - ( z 3 · n ) 2 ] - [ S · ( n × z 3 ) ] 2 } ,
B = 2 ( 1 + b { 1 - ( S · n ) 2 - [ S · ( n × z 3 ) ] 2 } ) × [ b ( z 3 · n ) 2 + ( u e u ) 2 ] - 4 b ( u e u ) 2 × { [ 1 - ( S · n ) 2 ] [ 1 - ( z 3 · n ) 2 ] - [ S · ( n × z 3 ) ] 2 } ,
C = [ b ( z 3 · n ) 2 + ( u e u ) 2 ] 2 .
w + = B + B 2 - 4 A C 2 A ,
w - = B - B 2 - 4 A C 2 A .
( u o - u e ) ( z 3 · z ) ( z 3 · n ) ( S · z ) > 0 ,
( u o - u e ) ( z 3 · z ) ( z 3 · n ) ( S · z ) < 0 ,
N = S + a n 1 + a 2 + 2 a ( S · n ) ,
a = ( 1 w ) 2 - 1 + ( S · n ) 2 - ( S · n ) .
R 1 = N 1 u e 2 ( 1 - N 3 2 ) u e 4 + N 3 u o 4 ,
R 2 = N 2 u e 2 ( 1 - N 3 2 ) u e 4 + N 3 2 u o 4 ,
R 3 = N 3 u o 2 ( 1 - N 3 2 ) u e 4 + N 3 2 u o 4 ,
R 3 = N 3 u e 2 ( 1 - N 3 2 ) u e 4 + N 3 2 u o 4 + ( u o 2 - u e 2 ) N 3 ( 1 - N 3 2 ) u e 4 + N 3 2 u o 4 .
R = N u e 2 ( 1 - N 3 2 ) u e 4 + N 3 2 u 0 4 + ( u o 2 - u e 2 ) N 3 z 3 ( 1 - N 3 2 ) u e 4 + N 3 2 u o 4 ,
N 3 = N · z 3 ,
R = N u e 2 [ 1 - ( N · z 3 ) 2 ] u e 4 + ( N · z 3 ) 2 u o 4 + ( u o 2 - u e 2 ) ( N · z 3 ) z 3 [ 1 - ( N · z 3 ) 2 ] u e 4 + ( N · z 3 ) 2 u o 4 .
λ = 6560 A ˚ , x 1 = 241.6 mm , x 2 = 242.2 mm , λ = 2310 a ˚ , x 2 = 242.1 mm , x 2 = 242.7 mm .

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