Abstract

The principle of cube corner reflector operation is simple, and a careful examination of operational details is valuable in promoting an understanding of the potentials and limitations in reflector use. This paper describes an easily visualized model by which these principles may be understood. From this model, simple procedures may be derived for determining the effective aperture area variation of conventional corner reflector configurations. The variation of the effective aperture of some conventional configurations is then computed. The effects of angular errors in the alignment of the reflecting surfaces are also modeled.

© 1971 Optical Society of America

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References

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  1. W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).
  2. R. W. Schmieder, Appl. Opt. 6, 537 (1967.).
    [CrossRef] [PubMed]
  3. J. L. Synge, Quart. Appl. Math. 4, 166 (1946).
  4. N. E. Rityn, Sov. J. Opt. Tech. 34, 198 (1967).

1967 (2)

R. W. Schmieder, Appl. Opt. 6, 537 (1967.).
[CrossRef] [PubMed]

N. E. Rityn, Sov. J. Opt. Tech. 34, 198 (1967).

1946 (1)

J. L. Synge, Quart. Appl. Math. 4, 166 (1946).

Brouwer, W.

W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).

Rityn, N. E.

N. E. Rityn, Sov. J. Opt. Tech. 34, 198 (1967).

Schmieder, R. W.

Synge, J. L.

J. L. Synge, Quart. Appl. Math. 4, 166 (1946).

Appl. Opt. (1)

Quart. Appl. Math. (1)

J. L. Synge, Quart. Appl. Math. 4, 166 (1946).

Sov. J. Opt. Tech. (1)

N. E. Rityn, Sov. J. Opt. Tech. 34, 198 (1967).

Other (1)

W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, New York, 1964).

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Figures (12)

Fig. 1
Fig. 1

Geometry of reflection.

Fig. 2
Fig. 2

Geometry of reflections in a corner reflector.

Fig. 3
Fig. 3

Locations of images of reflecting surfaces.

Fig. 4
Fig. 4

Corner reflector effective aperture visualization models.

Fig. 5
Fig. 5

Geometry of the effective aperture computation.

Fig. 6
Fig. 6

Variations in effective aperture area for index of refraction = 1.

Fig. 7
Fig. 7

Variations in effective aperture area for index of refraction = 1.5.

Fig. 8
Fig. 8

Variations in effective aperture area for index of refraction = 1.7.

Fig. 9
Fig. 9

Geometry affecting the nature of the returned beam.

Fig. 10
Fig. 10

Constructions showing variations of irradiance at the receiving surfaces.

Fig. 11
Fig. 11

Angles used in specifying the alignment of the reflecting surfaces.

Fig. 12
Fig. 12

Spreading of the return beam by diffraction.

Equations (27)

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R 0 X 1 = R ^ 0 · X ^ 1 , R 1 X 1 = R ^ 1 · X ^ 1 , R 0 X 2 = R 0 · X ^ 2 , R 1 X 2 = R ^ 1 · X ^ 2 R 0 X 3 = R ^ 0 · X ^ 3 , R 1 X 3 = R ^ 1 · X ^ 3 ,
R ^ 1 = - R 0 X 1 X ^ + R 0 X 2 X ^ 2 + R 0 X 3 X ^ 3
R ^ 0 = R ^ 0 X 1 X ^ 1 + R 0 X 2 X ^ 2 + R 0 X 3 X ^ 3 , R ^ 1 = - R 0 X 1 X ^ 1 + R 0 X 2 X ^ 2 + R 0 X 3 X ^ 3 , R ^ 2 = - R 0 X 1 X ^ 1 - R 0 X 2 X ^ 2 + R 0 X 3 X ^ 3 , R ^ 3 = - R 0 X 1 X ^ 1 - R 0 X 2 X ^ 2 - R 0 X 3 X ^ 3 = - R ^ 0 .
R ^ 1 = T X 1 R ^ 0 ,             R 2 = T X 2 R ^ 1 ,             R ^ 3 = T X 3 R ^ 2
T X 1 | - 1 0 0 0 1 0 0 0 0 | , T X 2 | 1 0 0 0 - 1 0 0 0 1 | , T X 3 | 1 0 0 0 1 0 0 0 - 1 | .
R = 1 2 [ sin 2 ( θ i - θ e ) sin 2 ( θ i + θ e ) + tan 2 ( θ i - θ e ) tan 2 ( θ i + θ e ) ] .
H P ( x p , y p ) = 1 4 R 2 A S T ( x r , y r ) N ( x s , y s ) d A S ,
H P ( x p , y p ) = 1 4 R 2 A S T ( x s + x p 2 , y s + y p 2 ) N ( x s , y s ) d A S .
H D ( x d , y d ) = 1 4 R 2 A S T ( x s - x d 2 , y s - y d 2 ) N ( x s , y s ) d A .
R ^ 1 = T X 1 R ^ 0 ,
T θ 2 = | cos 2 θ 2 0 sin 2 θ 2 0 1 0 - sin 2 θ 2 0 cos 2 θ 2 | .
R ^ 3 = T θ 1 T X 3 T θ 3 T X 2 T θ 2 T X 1 R ^ 0 ,
T θ 3 = | cos 2 θ 3 - sin 2 θ 3 0 sin 2 θ 3 cos 2 θ 3 0 0 0 1 |
T θ 1 = | 1 0 0 0 cos 2 θ 1 - sin 2 θ 1 0 sin 2 θ 1 cos 2 θ 1 | .
T X 1 ( T θ 2 R ^ 0 ) Rotated incident ray = T - θ 2 ( T X 1 R ^ 0 ) Reflected ray
T X 1 T θ 2 = T - θ 2 T X 1 .
T X 1 T θ 3 = - T - θ 3 T X 1 .
T X 1 ( T θ 1 R ^ 0 ) = T θ 1 ( T X 1 R ^ 0 ) or             T X 1 T θ 1 = T θ 1 T X 1 .
T X i T θ j = { T θ j T X i if i = j , T - θ j T X i if i j ,
[ R ^ 3 ] 1 = T θ 1 T θ 3 T - θ 2 T X 3 T X 2 T X 1 R ^ 0 = T θ 1 T θ 3 T - θ 2 R ^ 3 .
1 , 3 , 2 ; 2 , 3 , 1 ; 2 , 1 , 3 ; 3 , 1 , 2 ; 3 , 2 , 1.
[ R ^ 3 ] 2 = T θ 3 T - θ 1 T - θ 2 R ^ 3 , [ R ^ 3 ] 3 = T θ 2 T θ 1 T - θ 3 R ^ 3 , [ R ^ 3 ] 4 = T θ 1 T - θ 2 T - θ 3 R ^ 3 , [ R ^ 3 ] 5 = T θ 3 T θ 2 T - θ 1 R ^ 3 , [ R ^ 3 ] 6 = T θ 2 T - θ 3 T - θ 1 R ^ 3 .
ρ 1 = T θ 1 T θ 3 T - θ 2 , ρ 2 = T θ 3 T - θ 1 T - θ 2 , ρ 3 = T θ 2 T θ 1 T - θ 3 , ρ 4 = T θ 1 T - θ 2 T - θ 3 , ρ 5 = T θ 3 T θ 2 T - θ 1 , ρ 6 = T θ 2 T - θ 3 T - θ 1 .
ρ 1 ρ 1 = T θ 1 T - θ 2 T θ 3 , ρ 2 ρ 2 = T - θ 1 T - θ 2 T θ 3 , ρ 4 ρ 4 = T θ 1 T - θ 2 T - θ 3 , ρ 6 ρ 6 = T - θ 1 T θ 2 T - θ 3 , ρ 3 ρ 3 = T θ 1 T θ 2 T - θ 3 , ρ 5 ρ 5 = T - θ 1 T θ 2 T θ 3 .
4 R θ 1 sin β 1 perpendicular to the projection X 1 , 4 R θ 2 sin β 2 perpendicular to the projection X 2 , and             4 R θ 3 sin β 3 perpendicular to the projection X 3 .
1.22 ( λ / W r ) = ( 2 / W r ) / R or             R = W r 2 / ( 2.44 λ ) .
H r = H o [ A e ( ϕ ) ] 2 / ( R 2 λ 2 ) ,

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