Abstract

Optical prisms are most commonly employed because of their ability to output an image of different orientation relative to the input object. Previous papers have presented a systematic but numeric approximation for designing a single prism, which outputs an image with a specific image orientation. Instead of a numerical solution, an exact analytical solution for the same problem is offered. Further, how to design prism systems by using off-the-shelf right-angle prisms and roof prisms as building blocks to obtain an output image with a specific image orientation is addressed. Illustrative examples are given to verify the proposed approach and demonstrate its use.

© 2007 Optical Society of America

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References

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  1. F. J. Duarte and J. A. Piper, "Dispersion theory of multiple-prism beam expanders for pulsed dye lasers," Opt. Commun. 43, 303-307 (1982).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. I. Moreno, "Jones matrix for image-rotation prisms," Appl. Opt. 43, 3373-3381 (2004).
    [CrossRef] [PubMed]
  5. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.
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    [CrossRef]
  7. R. H. Ginsberg, "Image rotation," Appl. Opt. 33, 8105-8108 (1994).
  8. W. Mao, "Adjustment of reflecting prisms," Opt. Eng. 34, 79-82 (1995).
    [CrossRef]
  9. N. Lin, "Orientation conjugation of reflecting prism rotation and second-order approximation of image rotation," Opt. Eng. 33, 2400-2407 (1994).
    [CrossRef]
  10. C. Y. Tsai and P. D. Lin, "Prism design based on image orientation change," Appl. Opt. 45, 3951-3959 (2006).
    [CrossRef] [PubMed]
  11. D. L. Shealy, "Analytical illuminance and caustic surface calculations in geometrical optics," Appl. Opt. 15, 2588-2596 (1976).
    [CrossRef] [PubMed]
  12. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, "Caustic merit function for optical design," Appl. Opt. 28, 601-606 (1989).
    [CrossRef] [PubMed]
  13. R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

2006 (1)

2004 (1)

2002 (1)

1999 (1)

1995 (1)

W. Mao, "Adjustment of reflecting prisms," Opt. Eng. 34, 79-82 (1995).
[CrossRef]

1994 (2)

N. Lin, "Orientation conjugation of reflecting prism rotation and second-order approximation of image rotation," Opt. Eng. 33, 2400-2407 (1994).
[CrossRef]

R. H. Ginsberg, "Image rotation," Appl. Opt. 33, 8105-8108 (1994).

1992 (1)

1989 (1)

1982 (1)

F. J. Duarte and J. A. Piper, "Dispersion theory of multiple-prism beam expanders for pulsed dye lasers," Opt. Commun. 43, 303-307 (1982).
[CrossRef]

1976 (1)

Appel, R. K.

Burkhard, D. G.

Duarte, F. J.

F. J. Duarte, "Beam transmission characteristics of a collinear polarization rotator," Appl. Opt. 31, 3377-3378 (1992).
[CrossRef] [PubMed]

F. J. Duarte and J. A. Piper, "Dispersion theory of multiple-prism beam expanders for pulsed dye lasers," Opt. Commun. 43, 303-307 (1982).
[CrossRef]

Dyer, C. D.

Galvez, E. J.

Ginsberg, R. H.

Holmes, C. D.

Kassim, A. M.

Lin, N.

N. Lin, "Orientation conjugation of reflecting prism rotation and second-order approximation of image rotation," Opt. Eng. 33, 2400-2407 (1994).
[CrossRef]

Lin, P. D.

Mao, W.

W. Mao, "Adjustment of reflecting prisms," Opt. Eng. 34, 79-82 (1995).
[CrossRef]

Moreno, I.

Paul, R. P.

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

Piper, J. A.

F. J. Duarte and J. A. Piper, "Dispersion theory of multiple-prism beam expanders for pulsed dye lasers," Opt. Commun. 43, 303-307 (1982).
[CrossRef]

Shealy, D. L.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

Tsai, C. Y.

Appl. Opt. (7)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

F. J. Duarte and J. A. Piper, "Dispersion theory of multiple-prism beam expanders for pulsed dye lasers," Opt. Commun. 43, 303-307 (1982).
[CrossRef]

Opt. Eng. (2)

W. Mao, "Adjustment of reflecting prisms," Opt. Eng. 34, 79-82 (1995).
[CrossRef]

N. Lin, "Orientation conjugation of reflecting prism rotation and second-order approximation of image rotation," Opt. Eng. 33, 2400-2407 (1994).
[CrossRef]

Other (2)

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

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Figures (8)

Fig. 1
Fig. 1

Minimum number of reflectors.

Fig. 2
Fig. 2

Designed prism of Example A.

Fig. 3
Fig. 3

Pose of roof-pair reflectors.

Fig. 4
Fig. 4

Designed prism of Example B.

Fig. 5
Fig. 5

Single reflector can be replaced by a right-angle prism.

Fig. 6
Fig. 6

Roof-pair reflector can be replaced by a roof prism.

Fig. 7
Fig. 7

Designed prism system of a right-handed merit function.

Fig. 8
Fig. 8

Designed prism system of a left-handed merit function.

Equations (31)

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i ( n i ) i 1 = I 2 n i n i T .
n 0 = n ( n n , N n ) n 1 n 1 ( n n 1 ) n 2 i ( n i ) i 1 2 ( n 2 ) 1 × 1 ( n 1 , N 1 ) 0 .
Γ = [ a 0 b 0 c 0 ] = [ a 0 x b 0 x c 0 x a 0 y b 0 y c 0 y a 0 z b 0 z c 0 z ] = n ( n n , N n ) n 1 n 1 ( n n 1 ) n 2 × i ( n i ) i 1 2 ( n 2 ) 1 1 ( n 1 , N 1 ) 0 ,
Φ 0 = tan 1 { [ ( b 0 z c 0 y ) 2 + ( c 0 x a 0 z ) 2 + ( a 0 y b 0 x ) 2 ] 1 / 2 a 0 x + b 0 y + c 0 z 1 }
for  right-handed   Γ ,
Φ 0 = tan 1 { [ ( b 0 z + c 0 y ) 2 + ( c 0 x a 0 z ) 2 + ( a 0 y + b 0 x ) 2 ] 1 / 2 a 0 x b 0 y + c 0 z 1 }
for  left-handed   Γ ,
m 0 x = ( b 0 z c 0 y ) / ( 2 S Φ 0 ) , m 0 y = ( c 0 x a 0 z ) / ( 2 S Φ 0 ) , m 0 z = ( a 0 y b 0 x ) / ( 2 S Φ 0 ) ,
for   right-handed   Γ ,
m 0 x = ( b 0 z c 0 y ) / ( 2 S Φ 0 ) , m 0 y = ( c 0 x a 0 z ) / ( 2 S Φ 0 ) , m 0 z = ( a 0 y + b 0 x ) / ( 2 S Φ 0 ) ,
for   left-handed   Γ .
rot ( m i , Φ i ) = [ m i x 2 ( 1 C Φ i ) + C Φ i m i x m i y ( 1 C Φ i ) m i z S Φ i m i x m i z ( 1 C Φ i ) + m i y S Φ i m i x m i y ( 1 C Φ i ) + m i z S Φ i m i y 2 ( 1 C Φ i ) + C Φ i m i y m i z ( 1 C Φ i ) m i x S Φ i m i x m i z ( 1 C Φ i ) m i y S Φ i m i y m i z ( 1 C Φ i ) + m i x S Φ i m i z 2 ( 1 C Φ i ) + C Φ i ] .
n n = Γ n 1 ,
Γ = n 1 1 = n 1 ( n n 1 ) n 2 i ( n i ) i 1 2 ( n 2 ) 1 .
3 ( n 3 ) 2 = Γ [ 2 ( n 2 ) 1 ] 1 .
{ [ m 0 x m 0 z ( 1 C Φ 0 ) m 0 y S Φ 0 ] m 2 x + ( m 0 y 2 m 0 x 2 ) × ( 1 C Φ 0 ) m 2 z } S Φ 2 + { [ m 0 y m 0 x ( 1 C Φ 0 ) m 0 z S Φ 0 ] m 2 x 2 + [ m 0 y m 0 z ( 1 C Φ 0 ) + m 0 x S Φ 0 ] × m 2 x m 2 z 2 m 0 y m 0 x ( 1 C Φ 0 ) } C Φ 2 + [ m 0 x m 0 y ( 1 C Φ 0 ) m 0 z S Φ 0 ] m 2 x 2 + [ m 0 y m 0 z ( 1 C Φ 0 ) + m 0 x S Φ 0 ] m 2 x m 2 z = 0 ,
{ ( m 0 y 2 + m 0 z 2 ) ( 1 C Φ 0 ) m 2 x [ m 0 x m 0 z ( 1 C Φ 0 ) + m 0 y S Φ 0 ] m 2 z } S Φ 2 { [ m 0 y m 0 z ( 1 C Φ 0 ) + m 0 x S Φ 0 ] m 2 z 2 + [ m 0 x m 0 y ( 1 C Φ 0 ) m 0 z S Φ 0 ] m 2 x m 2 z 2 m 0 y m 0 z ( 1 C Φ 0 ) } C Φ 2 + [ m 0 y m 0 z ( 1 C Φ 0 ) + m 0 x S Φ 0 ] m 2 z 2 + [ m 0 y m 0 x ( 1 C Φ 0 ) m 0 z S Φ 0 ] m 2 x m 2 z = 0 ,
{ [ m 0 y m 0 x ( 1 C Φ 0 ) + m 0 z S Φ 0 ] m 2 x + [ m 0 y m 0 z ( 1 C Φ 0 ) m 0 x S Φ 0 ] m 2 z } S Φ 2 + [ ( m 0 x m 2 x + m 0 z m 2 x ) ( m 0 x m 2 z m 0 z m 2 x ) ( 1 C Φ 0 ) + m 0 y S Φ 0 ] C Φ 2 ( m 0 x m 2 x + m 0 z m 2 z ) ( m 0 x m 2 z m 0 z m 2 x ) ( 1 C Φ 0 ) + m 0 y S Φ 0 = 0.
tan   Φ 2 = 2 m 0 y ( m 0 x m 2 z m 0 z m 2 x ) ( m 0 x m 2 z m 0 z m 2 x ) 2 m 0 y 2 .
m 2 z m 2 x = m 0 x m 0 z C 2 ( Φ 2 / 2 ) ± C ( Φ 2 / 2 ) m 0 y S 2 ( Φ 2 / 2 ) m 0 y 2 m 0 x 2 C 2 ( Φ 2 / 2 ) m 0 y 2 S 2 ( Φ 2 / 2 ) .
Γ = [ a 0 b 0 c 0 ] = [ 0.229316 0.843802 0.485192 0.842691 0.421574 0.334883 0.487120 0.332073 0.807739 ] .
2 ( n 2 ) 1 = [ 0.920296 0.340305 0.192997 0.340305 0.939693 0.034202 0.192997 0.034202 0.980603 ] .
3 ( n 3 ) 2 = Γ [ 2 ( n 2 ) 1 ] 1 = [ 0.169752 0.854358 0.491179 0.854358 0.120832 0.505443 0.491179 0.505443 0.709416 ] .
3 ( n 3 ) 2 2 ( n 2 ) 1 = [ 4 ( n 4 ) 3 ] 1 Γ .
{ [ m 0 x m 0 z ( 1 C Φ 0 ) m 0 y S Φ 0 ] m 2 x + [ ( m 0 y 2 + m 0 x 2 ) × ( 1 C Φ 0 ) + 2 C Φ 0 ] m 2 z } S Φ 2 + { [ m 0 x m 0 y ( 1 C Φ 0 ) + m 0 z S Φ 0 ] m 2 x 2 + [ m 0 z m 0 y ( 1 C Φ 0 ) m 0 x S Φ 0 ] m 2 x m 2 z 2 m 0 z S Φ 0 } C Φ 2 { [ m 0 x m 0 y ( 1 C Φ 0 ) + m 0 z S Φ 0 ] m 2 x 2 + [ m 0 z m 0 y ( 1 C Φ 0 ) m 0 x S Φ 0 ] m 2 x m 2 z } = 0 ,
{ [ ( m 0 y 2 + m 0 z 2 ) ( 1 C Φ 0 ) + 2 C Φ 0 ] m 2 x + [ m 0 x m 0 z × ( 1 C Φ 0 ) + m 0 y S Φ 0 ] m 2 z } S Φ 2 + { [ m 0 y m 0 z ( 1 + C Φ 0 ) + m 0 x S Φ 0 ] m 2 z 2 + [ m 0 x m 0 y ( 1 + C Φ 0 ) m 0 z S Φ 0 ] m 2 x m 2 z 2 m 0 x S Φ 0 } C Φ 2 { [ m 0 y m 0 z ( 1 + C Φ 0 ) + m 0 x S Φ 0 ] m 2 z 2 + [ m 0 x m 0 y ( 1 + C Φ 0 ) m 0 z S Φ 0 ] m 2 x m 2 z } = 0 ,
{ [ ( m 0 x m 0 y ) ( 1 C Φ 0 ) m 0 z S Φ 0 ] m 2 x + [ m 0 y m 0 z × ( 1 C Φ 0 ) + m 0 x S Φ 0 ] m 2 z } S Φ 2 + { m 0 x m 0 z ( 1 C Φ 0 ) × ( m 2 x 2 m 2 z 2 ) + ( m 0 z 2 m 0 x 2 ) ( 1 C Φ 0 ) m 2 x m 2 z + m 0 y S Φ 0 } C Φ 2 { m 0 x m 0 z ( 1 C Φ 0 ) ( m 2 x 2 m 2 z 2 ) + ( m 0 z 2 m 0 x 2 ) ( 1 C Φ 0 ) m 2 x m 2 z m 0 y S Φ 0 } = 0.
tan   Φ 2 = 2 S Φ 0 ( m 0 x m 2 x + m 0 z m 2 z ) ( 1 C Φ 0 ) [ ( m 0 z m 2 x m 0 x m 2 z ) 2 + m 0 y 2 ] + 2 C Φ 0 .
m 2 z m 2 x = [ tan 2 ( Φ 0 / 2 ) tan 2 ( Φ 2 / 2 ) ] × [ m 0 x m 0 z ± tan 2 ( Φ 0 / 2 ) tan 2 ( Φ 2 / 2 ) ( m 0 x 2 + m 0 z 2 ) 1 tan 2 ( Φ 0 / 2 ) tan 2 ( Φ 2 / 2 ) m 0 z 2 1 ] .
4 ( n 4 ) 3 3 ( n 3 ) 2 = Γ [ 2 ( n 2 ) 1 ] 1 = [ m r x 2 ( C Φ r + 1 ) + C Φ r m r x m r y ( C Φ r 3 ) + m r z S Φ r m r x m r z ( C Φ r + 1 ) m r x m r y ( C Φ r 3 ) + m r z S Φ r m r y 2 ( C Φ r 3 ) C Φ r m r y m r z ( C Φ r + 1 ) m r x S Φ r m r x m r z ( C Φ r + 1 ) m r y m r z ( C Φ r + 1 ) m r x S Φ r m r z 2 ( C Φ r + 1 ) + C Φ r ] ,
Γ = [ a 0 b 0 c 0 ] = [ 0.338508 0.910799 0.236342 0.938177 0.346001 0.010341 0.072356 0.225231 0.971615 ] .

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