Abstract

Utilizing an image orientation change merit function, we prove the finding of Schweitzer et al. [Appl. Opt. 37, 5190 (1998)] that only two types of optically stable reflector system exist, namely, preserving or retroreflecting. We also present an analytical method for designing optically stable reflector systems comprising multiple reflectors. It is shown that an infinite number of solutions can be obtained for systems comprising more than three reflectors. Furthermore, it is shown that by adding two parallel refracting flat boundary surfaces at the entrance and exit positions of the light ray in an optical system with multiple reflectors, an optically stable prism can be obtained.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

2007 (1)

2006 (1)

2004 (1)

2002 (1)

2001 (1)

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

1998 (2)

1996 (1)

M. Skop, D. Ben-Ezra, and N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE 19th Convention of Electrical and Electronics Engineers in Israel (IEEE, 1996), pp. 383-386.
[CrossRef]

1992 (1)

1982 (1)

R. P. Paul, Robot Manipulators--Mathematics, Programming and Control (MIT Press, 1982).

1971 (1)

1960 (1)

1958 (1)

Appel, R. K.

Ben-Ezra, D.

M. Skop, D. Ben-Ezra, and N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE 19th Convention of Electrical and Electronics Engineers in Israel (IEEE, 1996), pp. 383-386.
[CrossRef]

Chandler, K. N.

Duarte, F. J.

Dyer, C. D.

Eckhardt, H. D.

Friedman, Y.

Li, W.

Li, X.

Liang, J.

Liang, Z.

Lin, P. D.

Moreno, I.

Paul, R. P.

R. P. Paul, Robot Manipulators--Mathematics, Programming and Control (MIT Press, 1982).

Schweitzer, N.

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

Y. Friedman and N. Schweitzer, “Classification of stable configurations of plane mirrors,” Appl. Opt. 37, 7229-7234 (1998).
[CrossRef]

M. Skop, D. Ben-Ezra, and N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE 19th Convention of Electrical and Electronics Engineers in Israel (IEEE, 1996), pp. 383-386.
[CrossRef]

Skop, M.

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

M. Skop, D. Ben-Ezra, and N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE 19th Convention of Electrical and Electronics Engineers in Israel (IEEE, 1996), pp. 383-386.
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

Sun, D.

Tsai, C. Y.

Wang, W.

Yoder, P. R.

Zhong, Y.

Appl. Opt. (8)

J. Opt. Soc. Am. (2)

Opt. Express (1)

Other (3)

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 100-121.

M. Skop, D. Ben-Ezra, and N. Schweitzer, “Adjustable stabilized reflector for optical communication,” in Proceedings of the IEEE 19th Convention of Electrical and Electronics Engineers in Israel (IEEE, 1996), pp. 383-386.
[CrossRef]

R. P. Paul, Robot Manipulators--Mathematics, Programming and Control (MIT Press, 1982).

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Figures (6)

Fig. 1
Fig. 1

Skew ray tracing at flat boundary surface.

Fig. 2
Fig. 2

Image orientation change induced by a single prism with n = 4 boundary surfaces.

Fig. 3
Fig. 3

Optically stable system comprising four ( n = 6 ) reflectors.

Fig. 4
Fig. 4

Bypassing obstacles in 3-D optical system utilizing optically stable system.

Fig. 5
Fig. 5

Optically stable prism with n = 6 .

Fig. 6
Fig. 6

Optically stable prism with n = 7 .

Equations (14)

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rot ( u , θ ) = [ u x 2 ( 1 C θ ) + C θ u x u y ( 1 C θ ) u z S θ u x u z ( 1 C θ ) + u y S θ u x u y ( 1 C θ ) + u z S θ u y 2 ( 1 C θ ) + C θ u y u z ( 1 C θ ) u x S θ u x u z ( 1 C θ ) u y S θ u y u z ( 1 C θ ) + u x S θ u z 2 ( 1 C θ ) + C θ ] ,
i ( n i ) i 1 = I 2 n i n i T .
Γ = n 0 = n ( n n , N n ) n 1 n 1 ( n n 1 ) n 2 i ( n i ) i 1 2 ( n 2 ) 1 1 ( n 1 , N 1 ) 0 ,
Γ = n 1 1 = n 1 ( n n 1 ) n 2 i ( n i ) i 1 2 ( n 2 ) 1 .
A 0 s = Rot ( q 0 , ψ ) .
Γ s = ( A s 0 ) 1 Γ ( A s 0 ) .
( m 0 x m 0 z ( 1 C Φ 0 ) + m 0 y S Φ 0 ) ( q 0 y q 0 z ( 1 C ψ ) + q 0 x S ψ ) δ ( m y m z ( 1 C Φ 0 ) + m 0 x S Φ 0 ) ( q 0 x q 0 z ( 1 C ψ ) + q 0 y S ψ ) δ ( m 0 x m 0 y ( 1 C Φ 0 ) m 0 z S Φ 0 ) ( q 0 y 2 q 0 x 2 ) ( 1 C ψ ) + ( ( m 0 x 2 m 0 y 2 ) ( 1 C Φ 0 ) + ( 1 δ ) C Φ 0 ) ( q 0 x q 0 y ( 1 C ψ ) q 0 z S ψ ) = 0 ,
( m 0 x m 0 y ( 1 C Φ 0 ) + m 0 z S Φ 0 ) ( q 0 x q 0 y ( 1 C ψ ) q 0 z S ψ ) + ( m 0 y m 0 z ( 1 C Φ 0 ) m 0 x S Φ 0 ) ( q 0 y q 0 z ( 1 C ψ ) + q 0 x S ψ ) δ ( m 0 x m 0 y ( 1 C Φ 0 ) m 0 z S Φ 0 ) ( q 0 x q 0 y ( 1 C ψ ) + q 0 z S ψ ) δ ( m 0 y m 0 z ( 1 C Φ 0 ) + m 0 x S Φ 0 ) ( q 0 y q 0 z ( 1 C ψ ) q 0 x S ψ ) = 0 ,
( m 0 x m 0 z ( 1 C Φ 0 ) m 0 y S Φ 0 ) ( q 0 x q 0 y ( 1 C ψ ) q 0 z S ψ ) δ ( m 0 x m 0 y ( 1 C Φ 0 ) m 0 z S Φ 0 ) ( q 0 x q 0 z ( 1 C ψ ) q 0 y S ψ ) + δ ( m 0 y m 0 z ( 1 C Φ 0 ) + m 0 x S Φ 0 ) ( q 0 y 2 q 0 z 2 ) ( 1 C ψ ) + ( ( m 0 z 2 m 0 y 2 ) ( 1 C Φ 0 ) + ( 1 δ ) C Φ 0 ) ( q 0 y q 0 z ( 1 C ψ ) + q 0 x S ψ ) = 0 ,
I = 3 ( n 3 ) 2 2 ( n 2 ) 1 .
2 ( n 2 ) 1 = 4 ( n 4 ) 3 3 ( n 3 ) 2 .
± ( 2 ( n 2 ) 1 i ( n i ) i 1 n 3 ( n n 3 ) n 4 ) = n 1 ( n n 1 ) n 2 n 2 ( n n 2 ) n 3 .
Γ * = n 1 ( n n 1 ) n 2 n 2 ( n n 2 ) n 3 .
Γ * = 2 ( n 2 ) 1 3 ( n 3 ) 2 = 5 ( n 5 ) 4 4 ( n 4 ) 3 = [ 0.9630 0.2687 0.0217 0.1947 0.6373 0.7456 0.1865 0.7222 0.6660 ] .

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