Abstract

In this study, a procedure based upon vector analysis methods is outlined for the tracing of optical rays through triple-mirror and tetrahedral prism retrodirective reflectors. This procedure is utilized to determine the light deviation errors of such reflectors in terms of the errors in the three dihedral angles between the reflecting surfaces. It is shown that if the dihedral angles of a given reflector are in error by a tolerance amount ±θ, the resulting deviation errors of that reflector will not exceed 3.26θ for a triple mirror or 3.26 for a tetrahedral prism of refractive index N.

Experimental verifications of the theoretical relationships developed in this study are described. The procedures which are used in this verification are applicable to “in-process” and final inspection operations during the manufacture of prismatic retrodirective reflectors of moderate precision.

© 1958 Optical Society of America

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References

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  1. I. E. S. Lighting Handbook (Illuminating Engineering Society, New York, 1954), p. 7 of Chap. 15.
  2. E. R. Peck, J. Opt. Soc. Am. 38, 1015 (1948).
    [CrossRef] [PubMed]
  3. D. H. Rank, Rev. Sci. Instr. 17, 243 (1946).
    [CrossRef]
  4. F. Twyman, Prism and Lens Making (Hilger and Watts, Ltd., London, 1952), p. 446.

1948 (1)

1946 (1)

D. H. Rank, Rev. Sci. Instr. 17, 243 (1946).
[CrossRef]

Peck, E. R.

Rank, D. H.

D. H. Rank, Rev. Sci. Instr. 17, 243 (1946).
[CrossRef]

Twyman, F.

F. Twyman, Prism and Lens Making (Hilger and Watts, Ltd., London, 1952), p. 446.

J. Opt. Soc. Am. (1)

Rev. Sci. Instr. (1)

D. H. Rank, Rev. Sci. Instr. 17, 243 (1946).
[CrossRef]

Other (2)

F. Twyman, Prism and Lens Making (Hilger and Watts, Ltd., London, 1952), p. 446.

I. E. S. Lighting Handbook (Illuminating Engineering Society, New York, 1954), p. 7 of Chap. 15.

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Figures (4)

Fig. 1
Fig. 1

Front view of a triple mirror or tetrahedral prism showing the division of aperture into six equal triangular areas.

Fig. 2
Fig. 2

Schematic layout of the optical autocollimator used to measure deviation errors and dihedral angle errors of tetrahedral prisms of moderate precision.

Fig. 3
Fig. 3

Photographs of the “spot patterns” produced by three typical tetrahedral prisms. The circle indicates the tolerance on deviation errors for these particular prisms.

Fig. 4
Fig. 4

Top row—photographs of “spot patterns” produced by a triple mirror when adjusted to simulate a “perfect” reflector (“O”) and each of Cases I through III of Table I. Bottom row—“spot patterns”’ similar to those above but produced with modified dihedral angles to display multiple images.

Tables (2)

Tables Icon

Table I Summary of deviation errors for imperfect triple-mirror reflectors.

Tables Icon

Table II Comparison of measured and computed deviation errors for three tetrahedral prisms having various dihedral angle errors.a

Equations (9)

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N 1 = A 1 i + B 1 j + C 1 k N 2 = A 2 i + B 2 j + C 2 k N 3 = A 3 i + B 3 j + C 3 k . }
cos α 1 , 2 = A 1 A 2 + B 1 B 2 + C 1 C 2 = sin θ 1 , 2 cos α 1 , 3 = A 1 A 3 + B 1 B 3 + C 1 C 3 = sin θ 1 , 2 cos α 2 , 3 = A 2 A 3 + B 2 B 3 + C 2 C 3 = sin θ 2 , 3 . }
A 1 = 1 , A 2 = sin θ 1 , 2 , A 3 = sin θ 1 , 3 B 1 = 0 , B 2 = cos θ 1 , 2 , B 3 = ( sin θ 2 , 3 - A 2 A 3 ) / B 2 C 1 = 0 , C 2 = 0 , C 3 = ( 1 - A 3 2 - B 3 2 ) 1 2 . }
I = l i + m j + n k
I = l i + m j + n k .
I = I - 2 ( N · I ) N ,
l = ( 1 - 2 A 2 ) l + ( - 2 A B ) m + ( - 2 A C ) n m = ( - 2 A B ) l + ( 1 - 2 B 2 ) m + ( - 2 B C ) n n = ( - 2 A C ) l + ( - 2 B C ) m + ( 1 - 2 C 2 ) n . }
cos δ = l l + m m + n n ,
cos δ 1 , 2 , 3 = 4 3 ( A 2 2 + A 3 2 + B 3 2 + A 2 A 3 - A 2 B 3 + A 3 B 3 ) - 1 cos δ 2 , 3 , 1 = 4 3 ( A 2 2 + A 3 2 + B 3 2 + A 2 A 3 + A 2 B 3 - A 3 B 3 ) - 1 cos δ 3 , 1 , 2 = 4 3 ( A 2 2 + A 3 2 + B 3 2 - A 2 A 3 + A 2 B 3 + A 3 B 3 ) - 1. }