Abstract

The alignment of a corner cube affects the measurement of its dihedral angle. For 5 deg of tilt, the error is up to 7%, depending on the orientation of the tilt. A vector model is devised to derive formulas that take misalignment into account for both solid and hollow corner cubes. When the wave-front tilt caused by the dihedral angle error is not much greater than that caused by the surface figure, because of vignetting for a tilting illumination, the surface figure of the cube facet makes varying contributions to the wave-front tilt for different incident angles. Simulations and experimental results are presented.

© 1992 Optical Society of America

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References

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  1. P. R. Yoder, “Study of the light deviation errors in triple mirrors and tetrahedral prisms,” J. Opt. Soc. Am. 48, 496–499 (1958).
    [CrossRef]
  2. B. W. Joseph, R. J. Donohue, “Dot patterns from imperfect cube-corner reflections,” J. Opt. Soc. Am. 62, 727 (A) (1972).
  3. D. A. Thomas, J. C. Wyant, “Determination of the dihedral angle errors of a corner cube from its Twyman–Green interferogram,” J. Opt. Soc. Am. 67, 467–472 (1977).
    [CrossRef]

1977 (1)

1972 (1)

B. W. Joseph, R. J. Donohue, “Dot patterns from imperfect cube-corner reflections,” J. Opt. Soc. Am. 62, 727 (A) (1972).

1958 (1)

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Figures (9)

Fig. 1
Fig. 1

(a) Corner cube coordinates and reflections.3 (b) Prism apertures with the associated reflection sequence.

Fig. 2
Fig. 2

Coordinate system xsyszs with the zs axis coincident with the axis of symmetry of the cube. The symmetry xs and x axes are in the same plane, and xyz is the cube coordinate system that is defined in Fig. 1.

Fig. 3
Fig. 3

Front surface coordinates XYZ and the cube coordinates xyz. The Z axis is normal to the front surface of the cube. The symmetry X and the x axes are in the same plane. The heavy line is the symmetry axis of the cube. The direction cosine of the symmetry axis of the cube is (Lc, Mc, Nc) in terms of the front surface coordinates.

Fig. 4
Fig. 4

Four paths of the ray. A′, B′, and C′ are shown with respect to the front surface coordinates, and a′, b′, and c′ are shown with respect to the cube coordinates. The subscripts represent the sequence of reflections by the surfaces i, j, and k, where i, j, k = 1, 2, 3 and ijk.

Fig. 5
Fig. 5

Interferograms of a hollow cube tested at three angles: (a) β = 0°, (b) β = 15°, γ = 0°, (c) β = 10°; γ = 180°. The distortion of the pupil is caused by the tilt of the cube. The shape of each segment varies depending on the angle of tilt.

Fig. 6
Fig. 6

Interferograms of a regular solid cube tested at three angles: (a) β = 0°, (b) β = 15°, γ = 0°, (c) β = 10°; γ = 180°.

Fig. 7
Fig. 7

Top view of the ray paths in a regular cube at the normal incident angle. The ray entering the cube at N, for example, is reflected by the three facets at N, N′, and N″ sequentially. The three facets are divided by lines OP, OW, and OT.

Fig. 8
Fig. 8

Vignetting of a tilted cube: (a) the side view of a tilted cube and ray paths; (b) the apparent and the actual apertures.

Fig. 9
Fig. 9

Relationship of the incident ray direction to the alignment angle γ and the entrance angle β.

Equations (26)

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R 1 = [ 1 0 0 0 1 0 0 0 1 ] , R 2 = [ 1 2 12 0 2 12 1 0 0 0 1 ] , R 3 = [ 1 0 2 13 0 1 2 23 2 13 2 23 1 ] .
S ˆ i j k = R k R j R i S ˆ 0 ,
S ˆ 123 S ˆ 213 = R 3 R 2 R 1 S ˆ 0 R 3 R 1 R 2 S ˆ 0 = R 3 ( R 2 R 1 R 1 R 2 ) S ˆ 0 = R 3 [ 0 4 12 0 4 12 0 0 0 0 0 ] S ˆ 0 = 4 12 ( b , a , 0 ) , S ˆ 132 S ˆ 123 = R 2 R 3 R 1 S ˆ 0 R 3 R 2 R 1 S ˆ 0 = ( R 2 R 3 R 3 R 2 ) R 1 S ˆ 0 = [ 0 0 0 0 0 4 23 0 4 23 0 ] R 1 S ˆ 0 = 4 23 ( 0 , c , b ) , S ˆ 312 S ˆ 132 = R 2 R 1 R 3 S ˆ 0 R 2 R 3 R 1 S ˆ 0 = R 2 ( R 1 R 3 R 3 R 1 ) S ˆ 0 = R 2 [ 0 0 4 13 0 0 0 4 13 0 0 ] S ˆ 0 = 4 13 ( c , 0 , a ) .
S ˆ 321 S ˆ 312 = 4 12 ( b , a , 0 ) , S ˆ 231 S ˆ 321 = 4 23 ( 0 , c , b ) , S ˆ 213 S ˆ 231 = 4 13 ( c , 0 , a ) .
12 = ± | S ˆ 123 S ˆ 213 | / 4 ( a 2 + b 2 ) 1 / 2 , 23 = ± | S ˆ 132 S ˆ 123 | / 4 ( b 2 + c 2 ) 1 / 2 , 12 = ± | S ˆ 312 S ˆ 132 | / 4 ( a 2 + c 2 ) 1 / 2 .
[ x y z ] = [ 2 / 6 0 1 / 3 1 / 6 1 / 2 1 / 3 1 / 6 1 / 2 1 / 3 ] [ x s y s z s ] .
12 = ± | S ˆ 321 S ˆ 312 | / 4 2 / 3 , 23 = ± | S ˆ 231 S ˆ 321 | / 4 2 / 3 , 13 = ± | S ˆ 213 S ˆ 231 | / 4 2 / 3 .
[ x y z ] = [ 2 / 6 0 1 / 3 1 / 6 1 / 2 1 / 3 1 / 6 1 / 2 1 / 3 ] [ q L c M c / q L c N c / q 0 N c / q M c / q L c M c N c ] [ X Y Z ] ,
[ a b c ] = [ 2 / 6 0 1 / 3 1 / 6 1 / 2 1 / 3 1 / 6 1 / 2 1 / 3 ] × [ q L c M c / q L c N c / q 0 N c / q M c / q L c M c N c ] [ A / n B / n ( C 2 + n 2 1 ) 1 / 2 / n ] .
A ˆ = A / n , B ˆ r = B / n , C ˆ r = ( C 2 + n 2 1 ) 1 / 2 / n ,
A ˆ r = A ˆ i j k / n , B ˆ r = B ˆ i j k / n , C ˆ r = ( C ˆ i j k 2 + n 2 1 ) 1 / 2 / n .
d A r = d A / n , d B r = d B / n , d C r = | C | ( C 2 + n 2 1 ) 1 / 2 d C / n .
A ˆ r A ˆ r = ( A ˆ 123 A ˆ 213 ) / n , B ˆ r B ˆ r = ( B ˆ 123 B ˆ 213 ) / n , C ˆ r C ˆ r = ( C ˆ 123 C ˆ 213 ) | C ˆ | ( C ˆ 2 + n 2 1 ) 1 / 2 / n ,
12 = ± [ ( A ˆ 123 A ˆ 213 ) 2 + ( B ˆ 123 B ˆ 213 ) 2 + k ( C ˆ 123 C ˆ 213 ) 2 ] 1 / 2 / 4 n ( a 2 + b 2 ) 1 / 2 , 23 = ± [ ( A ˆ 132 A ˆ 123 ) 2 + ( B ˆ 132 B ˆ 123 ) 2 + k ( C ˆ 132 C ˆ 123 ) 2 ] 1 / 2 / 4 n ( b 2 + c 2 ) 1 / 2 , 13 = ± [ ( A ˆ 312 A ˆ 132 ) 2 + ( B ˆ 312 B ˆ 132 ) 2 + k ( C ˆ 312 C ˆ 132 ) 2 ] 1 / 2 / 4 n ( a 2 + c 2 ) 1 / 2 ,
( 1 ) a regular solid cube ( 0 . 0000 , 0 . 0000 , 1 . 0000 ) ; ( 2 ) a skewed solid cube ( 0 . 2588 , 0 . 0000 , 0 . 9659 ) ; ( 3 ) a skewed solid cube ( 0 . 0000 , 0 . 2588 , 0 . 9659 ) ;
cube Normal Incident 15 ° Incident 10 ° Incident ( 1 ) ( 0 . 5774 0 . 5774 0 . 5774 ) ( 0 . 7083 0 . 4991 0 . 4991 ) ( 0 . 4800 0 . 6203 0 . 6203 ) ( 2 ) ( 0 . 3464 0 . 6633 0 . 6633 ) ( 0 . 5015 0 . 6117 0 . 6117 ) ( 0 . 2365 0 . 6870 0 . 6870 ) ( 3 ) ( 0 . 5577 0 . 3747 0 . 7407 ) ( 0 . 6890 0 . 2994 0 . 6601 ) ( 0 . 4604 0 . 4190 0 . 7826 )
Cube Normal Incident (%) 15 ° Incident (%) 10 ° Incident (%) ( 1 ) ( 0 , 0 , 0 ) ( 6 , 14 , 6 ) ( 4 , 7 , 4 ) ( 2 ) ( 8 , 15 , 8 ) ( 3 , 6 , 3 ) ( 11 , 19 , 11 ) ( 3 ) ( 18 , 2 , 14 ) ( 8 , 11 , 16 ) ( 24 , 9 , 11 )
[ X Y Z ] = [ ( B ˆ 2 + C ˆ 2 ) 1 / 2 0 A ˆ A ˆ B ˆ / ( B ˆ 2 + C ˆ 2 ) 1 / 2 C ˆ / ( B ˆ 2 + C ˆ 2 ) 1 / 2 B ˆ A ˆ C ˆ / ( B ˆ 2 + C ˆ 2 ) 1 / 2 B ˆ / ( B ˆ 2 + C ˆ 2 ) 1 / 2 C ˆ ] [ x o y o z o ] ,
[ A i j k B i j k C i j k ] = [ ( B 2 + C 2 ) 1 / 2 0 A A B / ( B 2 + C 2 ) 1 / 2 C / ( B 2 + C 2 ) 1 / 2 B A C / ( B 2 + C 2 ) 1 / 2 B / ( B 2 + C 2 ) 1 / 2 C ] [ ( t i j k ) x ( t i j k ) y 1 ] .
( A 123 A 213 ) 2 + ( B 123 B 213 ) 2 + k ( C 123 C 213 ) 2 ( A 123 A 213 ) 2 + ( B 123 B 213 ) 2 .
12 = ± [ ( A 123 A 213 ) 2 + ( B 123 B 213 ) ] 1 / 2 / 4 ( a 2 + b 2 ) 1 / 2 , 23 = ± [ ( A 132 A 123 ) 2 + ( B 132 B 123 ) ] 1 / 2 / 4 ( b 2 + c 2 ) 1 / 2 , 13 = ± [ ( A 312 A 132 ) 2 + ( B 312 B 132 ) ] 1 / 2 / 4 ( a 2 + c 2 ) 1 / 2 ,
12 = ± [ ( A 123 A 213 ) 2 + ( B 123 B 213 ) 2 ] 1 / 2 / 4 n ( a 2 + b 2 ) 1 / 2 , 23 = ± [ ( A 132 A 123 ) 2 + ( B 132 B 123 ) 2 ] 1 / 2 / 4 n ( b 2 + c 2 ) 1 / 2 , 13 = ± [ ( A 312 A 132 ) 2 + ( B 312 B 132 ) 2 ] 1 / 2 / 4 n ( a 2 + c 2 ) 1 / 2 .
Eq . Normal Incident 15 ° Incident 10 ° Incident ( 5 ) ( 0 . 17 9 . 14 1 . 58 ) ( 0 . 31 9 . 00 1 . 61 ) ( 0 . 17 9 . 08 1 . 75 ) ( 7 ) ( 0 . 17 9 . 14 1 . 58 ) ( 0 . 38 7 . 02 1 . 77 ) ( 0 . 11 9 . 95 1 . 64 )
Eq . Normal Incident 15 ° Incident 10 ° Incident ( 14 ) ( 0 . 43 38 . 63 3 . 40 ) ( 0 . 29 38 . 58 3 . 22 ) ( 0 . 58 38 . 71 3 . 41 ) ( 7 ) ( 0 . 43 38 . 63 3 . 40 ) ( 0 . 30 33 . 36 3 . 46 ) ( 0 . 55 41 . 59 3 . 30 )
A = sin ( β ) cos ( γ ) , B = sin ( β ) sin ( γ ) , C = cos ( β ) .
[ a b c ] = [ 2 / 6 0 1 / 3 1 / 6 1 / 2 1 / 3 1 / 6 1 / 2 1 / 3 ] [ A B C ] .

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