Abstract

A vector formulation is used to derive an analytical expression for the complex reflectivity of a corner-cube retroreflector, with use of the complex reflection coefficients for the s and p polarization. This expression shows that the corner-cube retroreflector modifies the incoming electric field according to the angle of incidence and the ray path through the retroreflector. The change in the electric field depends on the errors in the surface finish, the coating nonuniformity, and the nonhomogeneity in the index of refraction in the case of the solid corner-cube retroreflector. The main point is that the corner-cube retroreflector conjugates the incident beam only in the case of a plane wave with a constant amplitude incident upon a perfect corner-cube retroreflector.

© 1995 Optical Society of America

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References

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  1. S. F. Jacobs, “Experiments with retrodirective arrays,” Opt. Eng. 21, 281–293 (1982).
    [CrossRef]
  2. R. A. Chipman, J. Shamir, H. J. Caulfield, Q. B. Zhou, “Wavefront correcting properties of corner-cube arrays,” Appl. Opt. 27, 3203–3209 (1988).
    [CrossRef] [PubMed]
  3. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 81–84.
  4. H. H. Barrett, S. F. Jacobs, “Retroreflective arrays as approximate phase conjugators,” Opt. Lett. 4, 190–192 (1979).
    [CrossRef] [PubMed]
  5. M. Gerrard, J. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), p. 103.

1988 (1)

1982 (1)

S. F. Jacobs, “Experiments with retrodirective arrays,” Opt. Eng. 21, 281–293 (1982).
[CrossRef]

1979 (1)

Appl. Opt. (1)

Opt. Eng. (1)

S. F. Jacobs, “Experiments with retrodirective arrays,” Opt. Eng. 21, 281–293 (1982).
[CrossRef]

Opt. Lett. (1)

Other (2)

M. Gerrard, J. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), p. 103.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 81–84.

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Figures (4)

Fig. 1
Fig. 1

Corner-cube retroreflector, consisting of three plane mirrors placed at right angles to one another.

Fig. 2
Fig. 2

The beam returned from a corner-cube retroreflector appears to be broken into six segments.

Fig. 3
Fig. 3

Arbitrary ray, characterized by the direction of the normalized wave vector K ^ I, incident upon the corner-cube retroreflector.

Fig. 4
Fig. 4

Reflection geometry for an arbitrary ray incident with the unit direction vector K ^ I upon the reflecting surface. The reflecting surface is defined by the unit surface normal Ŝ. The reflected ray K ^ R lies in the plane of incidence, defined by the incident ray and the surface normal.

Equations (27)

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K ^ I = ( l , m , n ) .
K ^ I · i = l , K ^ I · j = m , K ^ I · k = n .
K ^ R = R i j R j k R k i K ^ I .
K ^ R = ( - l , - m , - n ) .
R ( θ I ) = R ( θ I ) exp [ i ϕ ( θ I ) ] ,
R ( θ I ) = R ( θ I ) exp [ i ϕ ( θ I ) ] .
N ^ = K ^ I × S ^ K I × S ^ .
M ^ = K ^ I × N ^ .
E = ( E I · M ^ ) M ^ ,
E = ( E I · N ^ ) N ^ .
E R = R ( θ I ) ( E I · M ^ ) M ^ ,
E R = R ( θ I ) ( E I · N ^ ) N ^ .
R j k i = R k i R j k R i j ,
R j i k = R j k R i k R i j .
E I 12 = E I exp ( i ϕ 12 ) .
E I 23 = ( E I 23 · M ^ 23 ) M ^ 23 ,
E I 23 = ( E I 23 · N ^ 23 ) N ^ 23 .
E I 23 = E R 12 = E 23 + E 23 = R 12 ( θ I ) ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) M ^ 23 + R 12 ( θ I ) ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) N ^ 23 + R 12 ( θ I ) ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) M ^ 23 + R 12 ( θ I ) ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) N ^ 23 .
E R 12 = E R 12 + E R 12 = R 12 ( θ I ) ( E I · M ^ 12 ) M ^ 12 + R 12 ( θ I ) ( E I · M ^ 12 ) N ^ 12 .
E R 23 = R 12 ( θ I ) R 23 ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) M ^ 23 + R 12 ( θ I ) R 23 ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) M ^ 23 + R 12 ( θ I ) R 23 ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) N ^ 23 + R 12 ( θ I ) R 23 ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) N ^ 23 .
E I 31 = E R 23 exp ( i ϕ 23 ) .
E I 31 = ( E I 31 · M ^ 31 ) M ^ 31 ,
E I 31 = ( E I 31 · N ^ 31 ) N ^ 31 .
E I 31 = E R 23 = E I 31 + E I 31 = R 12 ( θ I ) R 23 ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) ( M ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) ( M ^ 23 · N ^ 31 ) N ^ 31 + R 12 ( θ I ) R 23 ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) ( M ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) ( M ^ 23 · N ^ 31 ) N ^ 31 + R 12 ( θ I ) R 23 ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) ( N ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) ( N ^ 23 · N ^ 31 ) N ^ 31 + R 12 ( θ I ) R 23 ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) ( N ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) ( N ^ 23 · N ^ 31 ) N ^ 31 .
E R 31 = R 31 ( E I 23 · M ^ 31 ) M ^ 31 + R 31 ( E I 23 · M ^ 31 ) N ^ 31 .
E R 31 = R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) ( M ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) ( M ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) ( N ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) ( N ^ 23 · M ^ 31 ) M ^ 31 + R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) ( M ^ 23 · N ^ 31 ) N ^ 31 + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) ( M ^ 23 · N ^ 31 ) N ^ 31 + R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) ( N ^ 23 · N ^ 31 ) N ^ 31 + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) ( N ^ 23 · N ^ 31 ) N ^ 31 .
E R 31 = [ R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) ( M ^ 23 · M ^ 31 ) + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) ( M ^ 23 · M ^ 31 ) + R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) ( N ^ 23 · M ^ 31 ) + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) ( N ^ 23 · M ^ 31 ) ] M ^ 31 + [ R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · M ^ 23 ) ( M ^ 23 · N ^ 31 ) + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · M ^ 23 ) ( M ^ 23 · N ^ 31 ) + R 12 ( θ I ) R 23 R 31 ( E I · M ^ 12 ) ( M ^ 12 · N ^ 23 ) ( N ^ 23 · N ^ 31 ) + R 12 ( θ I ) R 23 R 31 ( E I · N ^ 12 ) ( N ^ 12 · N ^ 23 ) ( N ^ 23 · N ^ 31 ) ] N ^ 31 .

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