Abstract

We have studied the stability of systems of plane mirrors by using a new way to describe ray transformations caused by such systems. All stable systems comprising as many as three mirrors are described and classified. Besides the well-known corner cube, infinitely many stable retroreflecting and direction-preserving three-mirror systems have been found.

© 1998 Optical Society of America

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References

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  1. L. Levi, Applied Optics (Wiley, New York, 1968), pp. 351, 363.
  2. R. C. Spencer, “Optical theory of corner reflector,” MIT Radiation Laboratory Report 433 (Massachusetts Institute of Technology, Cambridge, Mass., 1944).
  3. P. R. Yoder, “Study of light deviation errors in triple mirror and tetrahedral prisms,” J. Opt. Soc. Am. 48, 496–499 (1958).
    [CrossRef]
  4. K. N. Chandler, “On the effects of small errors in angles of corner-cube reflectors,” J. Opt. Soc. Am. 50, 203–206 (1960).
    [CrossRef]
  5. H. D. Eckhardt, “Simple model of corner reflector phenomena,” Appl. Opt. 10, 1559–1566 (1971).
    [CrossRef] [PubMed]
  6. M. Skop, D. Ben-Ezra, N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE Nineteenth Convention of Electrical and Electronics Engineers in Israel, Jerusalem (IEEE, New York, 1996), pp. 383–386.
    [CrossRef]
  7. J. S. Beggs, “Mirror-image kinematics,” J. Opt. Soc. Am. 50, 388–393 (1960).
    [CrossRef]
  8. R. Kingslake, Optical System Design (Academic, New York, 1983), pp. 153–154.
  9. N. Schweitzer, Y. Friedman, M. Skop, “Stability of systems of plane reflecting surfaces,” Appl. Opt. 37, 5190–5192 (1998).
    [CrossRef]
  10. R. E. Hopkins, “Mirror and prism systems,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, San Diego, Calif., 1965), Vol 3, Chap. 7.
  11. R. E. Hopkins, “Mirror and prism systems,” in Military Standardization Handbook: Optical Design, MIL-HDBK 141 (U.S. Defense Supply Agency, Washington, D.C., 1962).
  12. E. A. Skorikov, “Conditions required for the stabilization of the direction of radiation of periscopic antennas,” Radiotekhnika 30, 9–14 (1974).

1998

1974

E. A. Skorikov, “Conditions required for the stabilization of the direction of radiation of periscopic antennas,” Radiotekhnika 30, 9–14 (1974).

1971

1960

1958

Beggs, J. S.

Ben-Ezra, D.

M. Skop, D. Ben-Ezra, N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE Nineteenth Convention of Electrical and Electronics Engineers in Israel, Jerusalem (IEEE, New York, 1996), pp. 383–386.
[CrossRef]

Chandler, K. N.

Eckhardt, H. D.

Friedman, Y.

Hopkins, R. E.

R. E. Hopkins, “Mirror and prism systems,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, San Diego, Calif., 1965), Vol 3, Chap. 7.

R. E. Hopkins, “Mirror and prism systems,” in Military Standardization Handbook: Optical Design, MIL-HDBK 141 (U.S. Defense Supply Agency, Washington, D.C., 1962).

Kingslake, R.

R. Kingslake, Optical System Design (Academic, New York, 1983), pp. 153–154.

Levi, L.

L. Levi, Applied Optics (Wiley, New York, 1968), pp. 351, 363.

Schweitzer, N.

N. Schweitzer, Y. Friedman, M. Skop, “Stability of systems of plane reflecting surfaces,” Appl. Opt. 37, 5190–5192 (1998).
[CrossRef]

M. Skop, D. Ben-Ezra, N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE Nineteenth Convention of Electrical and Electronics Engineers in Israel, Jerusalem (IEEE, New York, 1996), pp. 383–386.
[CrossRef]

Skop, M.

N. Schweitzer, Y. Friedman, M. Skop, “Stability of systems of plane reflecting surfaces,” Appl. Opt. 37, 5190–5192 (1998).
[CrossRef]

M. Skop, D. Ben-Ezra, N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE Nineteenth Convention of Electrical and Electronics Engineers in Israel, Jerusalem (IEEE, New York, 1996), pp. 383–386.
[CrossRef]

Skorikov, E. A.

E. A. Skorikov, “Conditions required for the stabilization of the direction of radiation of periscopic antennas,” Radiotekhnika 30, 9–14 (1974).

Spencer, R. C.

R. C. Spencer, “Optical theory of corner reflector,” MIT Radiation Laboratory Report 433 (Massachusetts Institute of Technology, Cambridge, Mass., 1944).

Yoder, P. R.

Appl. Opt.

J. Opt. Soc. Am.

Radiotekhnika

E. A. Skorikov, “Conditions required for the stabilization of the direction of radiation of periscopic antennas,” Radiotekhnika 30, 9–14 (1974).

Other

R. E. Hopkins, “Mirror and prism systems,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, San Diego, Calif., 1965), Vol 3, Chap. 7.

R. E. Hopkins, “Mirror and prism systems,” in Military Standardization Handbook: Optical Design, MIL-HDBK 141 (U.S. Defense Supply Agency, Washington, D.C., 1962).

L. Levi, Applied Optics (Wiley, New York, 1968), pp. 351, 363.

R. C. Spencer, “Optical theory of corner reflector,” MIT Radiation Laboratory Report 433 (Massachusetts Institute of Technology, Cambridge, Mass., 1944).

R. Kingslake, Optical System Design (Academic, New York, 1983), pp. 153–154.

M. Skop, D. Ben-Ezra, N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE Nineteenth Convention of Electrical and Electronics Engineers in Israel, Jerusalem (IEEE, New York, 1996), pp. 383–386.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Ray transformation by a plane mirror.

Fig. 2
Fig. 2

Representation of a two-mirror system in the complex plane.

Fig. 3
Fig. 3

Two-mirror system with ω = 45°: O, range of the incoming directions; I, range of the first reflected directions; II, range of the second reflected directions; III, range of the third reflected directions; IV, range of the fourth reflected outgoing directions.

Fig. 4
Fig. 4

Two-mirror system with ω = 60°: O, range of the incoming directions; I, range of the first reflected directions; II, range of the second reflected directions; III, range of the third reflected outgoing directions.

Fig. 5
Fig. 5

Stable three-mirror system with ω = 60°.

Equations (19)

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T n r = r - n · r n - n · r n = r - 2 n · r n
T n z = z - 2   Re exp - i φ z exp i φ = - exp 2 i φ z ¯ .
T R k θ n z = - exp 2 i φ + θ z ¯ = exp 2 i θ T n z ,
T m T n z = exp 2 i φ + ω exp - 2 i φ z = exp 2 i ω z = R l 2 ω z ,
s = Int π - ψ ω + 1 ,
T n z = - exp i π z ¯ = z ¯ ,     T m z = exp 2 i ω z ¯ .
T m T n j   exp i ψ + π = exp i ψ + π + 2 j ω .
2 π ψ + π + 2 j ω 2 π + ω , π - ψ ω 2 j π - ψ ω + 1 ,
T n T m T n j   exp i ψ + π = exp - i ψ + π + 2 j ω .
- 2 π - π - ψ - 2 j ω - 2 π + ω , π - ψ ω - 1 2 j π - ψ ω ,
T m T n j z = - z ,
T n T m T n j z = exp i - π + ω z ¯ = - exp i ω z ¯ ,
T m T n j z = exp i 2 ω j z ,
T n T m T n j z = exp - i 2 ω j z ¯
T n T m T n j - 1 z = exp - i 2 ω j - 1 z ¯ .
exp - i 100 z ¯ = - exp i 80 z ¯ ,
T k T m T n j = T k
T k T n T m T n j = T k T n
T k T m T n j = - I ,

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