Abstract

We present a method of designing a prism to produce an image with a specific orientation. Traditional prism design of this kind is done by trial and error with the aid of geometrical drawing and cannot provide analytical results. Using skew ray tracing sensitivity analysis, we present a merit function that can specify changes in image orientation after the image is reflected by an arbitrary number of flat boundary surfaces. Two design approaches are proposed. One can produce a prism with a minimum number of flat boundary surfaces with the aid of an auxiliary unit vector. The other can produce many configurations of prisms but without the above feature. An illustrative example is used to demonstrate the validity of the proposed approaches. Eight new configurations, which can produce the same change in image orientation, are obtained from the proposed design approaches.

© 2006 Optical Society of America

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References

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  1. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.
  2. E. J. Galvez and C. D. Holmes, "Geometric phase of optical rotators," J. Opt. Soc. Am. A 16, 1981-1985 (1999).
    [CrossRef]
  3. T. T. Liao and P. D. Lin, "Analysis of optical elements with flat boundary surfaces," Appl. Opt. 42, 1191-1202 (2003).
    [CrossRef] [PubMed]
  4. D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976) (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976).
    [CrossRef]
  5. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, "Caustic merit function for optical design," Appl. Opt. 28, 601-606 (1989).
    [CrossRef] [PubMed]
  6. G. W. Stewart, Introduction to Matrix Computations (Academic, 1973).
  7. R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

2003

1999

1989

1976

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976) (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976).
[CrossRef]

Burkhard, D. G.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, "Caustic merit function for optical design," Appl. Opt. 28, 601-606 (1989).
[CrossRef] [PubMed]

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976) (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976).
[CrossRef]

Galvez, E. J.

Holmes, C. D.

Kassim, A. M.

Liao, T. T.

Lin, P. D.

Paul, R. P.

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

Shealy, D. L.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, "Caustic merit function for optical design," Appl. Opt. 28, 601-606 (1989).
[CrossRef] [PubMed]

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976) (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976).
[CrossRef]

Smith, W. J.

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

Stewart, G. W.

G. W. Stewart, Introduction to Matrix Computations (Academic, 1973).

Appl. Opt.

J. Opt. Soc. Am.

D. L. Shealy and D. G. Burkhard, "Caustic surface merit functions in optical design," J. Opt. Soc. Am. 66, 1122 (1976) (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18-22 October 1976).
[CrossRef]

J. Opt. Soc. Am. A

Other

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

G. W. Stewart, Introduction to Matrix Computations (Academic, 1973).

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

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Figures (8)

Fig. 1
Fig. 1

Skew ray tracing at a flat boundary surface.

Fig. 2
Fig. 2

Physical meaning of pose matrix A 0 i .

Fig. 3
Fig. 3

A right-angle prism can invert the image.

Fig. 4
Fig. 4

Minimum number of reflectors.

Fig. 5
Fig. 5

Solid glass corner-cube retroreflector.

Fig. 6
Fig. 6

Flow chart of prism design based on auxiliary unit vector.

Fig. 7
Fig. 7

Eight prism configurations.

Fig. 8
Fig. 8

Different arrangement of Fig. 7(a).

Equations (314)

Equations on this page are rendered with MathJax. Learn more.

P i x i + P i y j + P i z k
P j i = [ P i x P i y P i z 1 ] T
P j i
( x y z ) j
P j i
P k i
P k i = A k j P j i
A k j
( x y z ) j
( x y z ) k
l i j = [ l i x l i y l i z 0 ] T
P i
l i
( x y z ) 0
r i i
[ β i 0 0 1 ] T
( β i 0 )
y i
r i i = [ C α i 0 S α i 0 0 1 0 0 S α i 0 C α i 0 0 0 0 1 ] [ β i 0 0 1 ] = [ β i C α i 0 β i S α i 1 ] ,
C α i
S α i
cos ( α i )
sin ( α i )
n i i
n i i = s i ( i r i β i × i r i α i ) / | i r i β i × i r i α i | = s i [ 0 1 0 0 ] T ,
s i
+ 1
1
C θ i > 0
( x y z ) 0
( x y z ) 0
( x y z ) i
A i 0 = [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ] ,
[ I i x I i y I i z 0 ] T
[ J i x J i y J i z 0 ] T
[ K i x K i y K i z 0 ] T
( x y z ) 0
( x y z ) i
[ t i x t i y t i z 0 ] T
( x y z ) 0
( x y z ) i
( x y z ) 0
n i
n i = [ n i x n i y n i z 0 ] T = A 0 i n i i = s i [ I i y J i y K i y 0 ] T .
P i 1 = [ P i 1 x P i 1 y P i 1 z 1 ] T
l i 1 = [ l i 1 x l i 1 y l i 1 z 0 ] T
r i
P i
P i = [ P i x P i y P i z 1 ] T = [ P i 1 x + l i 1 x λ i P i 1 y + l i 1 y λ i P i 1 z + l i 1 z λ i 1 ] T ,
λ i = ( I i y P i 1 x + J i y P i 1 y + K i y P i 1 z + t i y ) / ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z )
α i
β i
θ i
C θ i = l i 1 T n i = s i ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z )
l i
l i = [ l i x l i y l i z 0 ] = [ s i I i y [ 1 N i 2 + N i 2 ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) 2 ] 1 / 2 + N i [ l i 1 x I i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) ] s i J i y [ 1 N i 2 + N i 2 ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) 2 ] 1 / 2 + N i [ l i 1 y J i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) ] s i K i y [ 1 N i 2 + N i 2 ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) 2 ] 1 / 2 + N i [ l i 1 z K i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) ]                               0 ] ,
l i = [ l i x l i y l i z 0 ] = [ l i 1 x 2 I i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) l i 1 y 2 J i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) l i 1 z 2 K i y ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) 0 ] ,
N i
i 1
[ Δ P i Δ l i ] T
[ Δ P i 1 Δ l i 1 ] T
[ Δ P i Δ l i ] T
[ Δ P i 1 Δ l i 1 ] T
[ Δ P i Δ l i ] = M i [ Δ P i 1 Δ l i 1 ] = [ P i P i 1 P i l i 1 0 l i l i 1 ] [ Δ P i 1 Δ l i 1 ] ,
M i
r i . P i / P i 1
P i / l i 1
l i / l i 1
l i ( n i , N i ) / l i 1
l i ( n i ) / l i 1
l i ( n i , N i ) l i 1 = N i [ 1 0 0 0 1 0 0 0 1 ] + ( H i N i ) [ I i y J i y K i y ] [ I i y J i y K i y ] T = N i I + ( H i N i ) n i n i T ,
H i = s i N i 2 ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) / [ 1 N i 2 + N i 2 ( I i y l i 1 x + J i y l i 1 y + K i y l i 1 z ) 2 ] 1 / 2 ,
l i ( n i ) l i 1 = [ a i b i c i ]
= [ 1 2 I i y 2 2 I i y J i y 2 I i y K i y 2 J i y I i y 1 2 I i y 2 2 J i y K i y 2 K i y I i y 2 K i y J i y 1 2 K i y 2 ]
= I 2 n i n i T .
l i ( n i ) / l i 1
n i
r i
r i
i = 1 , i = 2 , ,
i = n
[ Δ P n Δ l n ] T
[ Δ P 0 Δ l 0 ] T
[ Δ P n Δ l n ] = M n M n 1 M 1 [ Δ P 0 Δ l 0 ] = [ P n P 0 P n l 0 0 l n l 0 ] [ Δ P 0 Δ l 0 ] .
l n / P 0 = 0
l n
P 0
l n / l 0
l i / l i 1 ( i = 1 , 2 , , n )
( i = 1 )
( i = n )
l n l 0 = l n ( n n , N n ) l n 1 l n 1 ( n n 1 ) l n 2 l i ( n i ) l i 1 l 2 ( n 2 ) l 1 × l 1 ( n 1 , N 1 ) l 0 .
( x y z ) 0
( x y z ) 0
y 0
( x y z ) 0
n 1 = [ 0 1 0 0 ] T
n 2 = [ 0 1 / 2 1 / 2 0 ] T
n 3 = [ 0 1 / 2 1 / 2 0 ] T
n 4 = [ 0 1 0 0 ] T
l 4 l 0 = l 4 ( n 4 , N 4 ) l 3 l 3 ( n 3 ) l 2 l 2 ( n 2 ) l 1 l 1 ( n 1 , N 1 ) l 0 = [ 1 0 0 0 1 0 0 0 1 ] .
( n 2 )
l i ( n i ) / l i 1
( i = 2 , 3 , , n 1 )
Γ = l n 1 l 1 = l n 1 ( n n 1 ) l n 2 l i ( n i ) l i 1 l 2 ( n 2 ) l 1 .
( n 2 )
l 1 ( n 1 , N 1 ) / l 0
l n ( n n , N n ) / l n 1
r 1
r n
r 1
r n
( n 2 )
[ l n ( n n , N n ) / l n 1 ] Γ [ l 1 ( n 1 , N 1 ) / l 0 ] = Γ
n 1 = [ 0 1 0 0 ] T
I 1 y l 0 x + J 1 y l 0 y + K 1 y l 0 z = 1
I n y l n 1 x + J n y l n 1 y + K n y l n 1 z = 1
N n = 1 / N 1
n n = [ I n y J n y K n y 0 ] T
r n
l i ( n i ) / l i 1
1 .   A   3 × 3
| a i | = | b i | = 1
a i b i = 0
c i
b i × a i
a i × b i
l i ( n i ) / l i 1
l i ( n i ) / l i 1
l i ( n i ) / l i 1
l j ( n j ) / l j 1
n i
n j
( x y z ) 0
y 0
i = 2
l 1
n 2
l 2
m 2 = n 2 × l 1 / S θ 2
θ 2
l 2
m 2
l 1
l 2
l 1
m 2
π + 2 θ 2
l 2 = r o t ( m 2 , π + 2 θ 2 ) l 1
n 2
l 1
m 2
π + θ 2
n 2 = r o t ( m 2 , π + θ 2 ) l 1
Γ 2
Γ 2
Γ 2 = [ m 2 x 2 ( 1 C Φ 2 ) + C Φ 2 m 2 x m 2 y ( 1 C Φ 2 ) - m 2 z S Φ 2 m 2 x m 2 z ( 1 C Φ 2 ) + m 2 y S Φ 2 m 2 x m 2 y ( 1 C Φ 2 ) + m 2 z S Φ 2 m 2 y 2 ( 1 C Φ 2 ) + C Φ 2 m 2 y m 2 z ( 1 C Φ 2 ) - m 2 x S Φ 2 m 2 x m 2 z ( 1 C Φ 2 ) - m 2 y S Φ 2 m 2 y m 2 z ( 1 C Φ 2 ) + m 2 x S Φ 2 m 2 z 2 ( 1 C Φ 2 ) + C Φ 2 ] .
m 2 = [ m 2 x m 2 y m 2 z 0 ] T
Φ 2
m 2 y = 0
l 1
0 Φ 2 < π
n 2 = r o t ( m 2 , π + Φ 2 / 2 ) l 1
Γ 2
m 2
l 1
m 2 = [ m 2 x 0 m 2 z 0 ] / ( m 2 x 2 + m 2 z 2 ) 1 / 2
m 2 z m 2 x m 2 x m 2 z
l 1
n 2
m 2
0 < Φ 2 < π
n 2 = r o t ( m 2 , π + Φ 2 / 2 ) l 1 = [ m 2 z S ( Φ 2 / 2 )
C ( Φ 2 / 2 )
m 2 x S ( Φ 2 / 2 ) ]
n 2
l 2 ( n 2 ) l 1 = I 2 n 2 n 2 T = [ 1 2 m 2 z 2 S 2 ( Φ 2 / 2 ) 2 m 2 z S ( Φ 2 / 2 ) C ( Φ 2 / 2 ) 2 m 2 x m 2 z S 2 ( Φ 2 / 2 ) 2 m 2 z S ( Φ 2 / 2 ) C ( Φ 2 / 2 ) 1 2 C 2 ( Φ 2 / 2 ) 2 m 2 x S ( Φ 2 / 2 ) C ( Φ 2 / 2 ) 2 m 2 x m 2 z S 2 ( Φ 2 / 2 ) 2 m 2 x S ( Φ 2 / 2 ) C ( Φ 2 / 2 ) 1 2 m 2 x 2 S 2 ( Φ 2 / 2 ) ] .
0 < Φ 2 < π
l 3 ( n 3 ) / l 2 = Γ 2 l 2 ( n 2 ) / l 1
n 2
n 3
( x y z ) 0
y 0
Γ 2 = Γ = [ a 2 b 2 c 2 ] = [ a 2 x b 2 x c 2 x a 2 y b 2 y c 2 y a 2 z b 2 z c 2 z ]
| a 2 | = | b 2 | = 1
a 2 b 2 = 0
c 2 = a 2 × b 2
c 2 = a 2 × b 2
i = 2
Γ i
c i = a i × b i
b i
b i
Γ i *
Φ i
0 Φ i π
m i = [ m i x m i y m i z 0 ] T
Γ i
Γ i *
Φ i = tan 1 { [ ( b i z c i y ) 2 + ( c i x a i z ) 2 + ( a i y b i x ) 2 ] 1 / 2 a i x + b i y + c i z 1 } ,
m i x = ( b i z c i y ) / ( 2 S Φ i ) ,
m i y = ( c i x a i z ) / ( 2 S Φ i ) ,
m i z = ( a i y b i x ) / ( 2 S Φ i ) .
Γ i
c i = a i × b i
m i
l i 1
0 Φ i < π
m i
l i 1
l i
n i
l i 1
m i
π + Φ i / 2
n i = r o t ( m i , π + Φ i / 2 ) l i 1
l i
l i 1
π + Φ i
m i
l i = r o t ( m i , π + Φ i ) l i 1
m i
l i 1
0 < Φ i < π
m i
m i + 1
l i
l i 1
l i
n i = r o t ( m i , π + Φ i / 2 ) l i 1 , l i = r o t ( m i , π + Φ i ) l i 1
l i ( n i ) / l i 1
Γ i + 1
Γ i + 1 = Γ i l i ( n i ) / l i 1
i = i + 1
m i
l i 1
m i = [ m i x 0 m i z 0 ] / ( m i x 2 + m i z 2 ) 1 / 2
m i z m i x m i x m i z
l i 1
l i
n i
l i 1
π + Φ i / 2
n i = r o t ( m i , π + Φ i / 2 ) l i 1
l i
l i 1
π + Φ i
m i
l i = r o t ( m i , π + Φ i ) l i 1
l i ( n i ) / l i 1
0 < Φ i < π
l i + 1 ( n i + 1 ) / l i = Γ i l i ( n i ) / l i 1
m i
l i 1
0 < Φ i < π
m i
l i 1
l i
n i = r o t ( m i , π + Φ i / 2 ) l i 1
l i = r o t ( m i , π + Φ i ) l i 1
l i ( n i ) / l i 1
( i + 1 )
Γ i + 1
Γ i + 1 = Γ i l i ( n i ) / l i 1
i = i + 1
Γ 2 = Γ = [ 1 0 0 0 1 0 0 0 1 ]
( x y z ) 0
l 0 = [ 0 1 0     0 ] T
y 0
l 0
l 1 = [ 0 1     0 0 ] T
Γ 2 * = [ 1 0 0 0 1 0 0 0 1 ] .
Φ 2 = π
m 2 = [ 0 1 0 0 ] T
l 1
m 2 = [ 1 / 2 0 1 / 2 0 ]
Φ 2 = 2 π / 3
n 2 = r o t ( m 2 , 2 π / 3 ) l 1 = [ 3 / 8 1 / 2 3 / 8 0 ] T
l 2 ( n 2 ) l 1 [ 1 / 4 3 / 8 3 / 4 3 / 8 1 / 2 3 / 8 3 / 4 3 / 8 1 / 4 ] .
Γ 3 = Γ 2 l 2 ( n 2 ) l 1 = [ 1 / 4 3 / 8 3 / 4 3 / 8 1 / 2 3 / 8 3 / 4 3 / 8 1 / 4 ] .
Φ 3 = 104.476 °
m 3 = [ 0 0.6 0.4 0 ] T
n 3 = r o t ( m 3 , π + Φ 3 / 2 ) l 2 = [ 1 / 4 3 / 8 3 / 4 0 ] T
l 3 ( n 3 ) l 2 [ 7 / 8 3 / 32 3 / 8 3 / 32 1 / 4 27 / 32 3 / 8 27 / 32 1 / 8 ] .
Γ 4 = Γ 3 l 3 ( n 3 ) l 2 = [ 1 / 8 27 / 32 3 / 8 27 / 32 1 / 4 3 / 32 3 / 8 3 / 32 7 / 8 ] .
Γ 4
Γ 4
n 4
n 4 = ( 3 / 4 3 / 8 1 / 4 )
l 4 ( n 4 ) / l 3
l 3 ( n 3 ) / l 2
l 2 ( n 2 ) / l 1
n 2
n 3
n 4
Γ 2 = Γ
i = 2
| a i | = | b i | = 1
a i b i = 0
c i = b i × a i
Γ i
n i
n i
l i ( n i ) / l i 1
i = i + 1
Γ i + 1 = Γ i l i ( n i ) / l i 1
n = 7
n 4
n 5
l 4 ( n 4 ) / l 3
l 5 ( n 5 ) / l 4
n 4
n 5
A 0 i

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