Abstract

A screen consisting of a number of tiny triple mirrors (cubic corners) is an autocollimating screen which reflects back the incident ray of light always into the original direction. Such a screen can be used in a projection-type three-dimensional display. Proposed in this paper is a curved triple-mirror screen which is composed of many tiny triple mirrors having one cylindrical and two flat reflecting surfaces. Such a screen has a diffusing property in the vertical direction in addition to the autocollimating property in the horizontal direction. These properties enable us to furnish a wider observable field of view without reducing the image quality when it is used in a 3–D display system. Another advantage of this screen is that it has high reflection efficiency. It is found that the design theory and experiment show a good agreement, and that a horizontal directivity (autocollimation) of 0.5° is attainable with a screen having 1-mm pitch.

© 1971 Optical Society of America

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References

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  1. C. B. Burckhardt, R. J. Collier, E. T. Doherty, Appl. Opt. 7, 627 (1968).
    [CrossRef] [PubMed]
  2. A. W. Lohmann, IBM Tech. Disclosure Bulletin 10, 1452 (1968).
  3. H. E. Ives, J. Opt. Soc. Amer. 21, 109 (1931).
    [CrossRef]
  4. T. Okoshi, T. Hasegawa, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.
  5. A. Yano, T. Okoshi, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.
  6. P. R. Yoder, J. Opt. Soc. Amer. 48, 496 (1958).
    [CrossRef]
  7. Y. Fujii, K. Nakajima, Seisan Kenkyu (Monthly J. Inst. Industrial Science, Univ. Tokyo) 21, 611 (1969).

1968 (2)

C. B. Burckhardt, R. J. Collier, E. T. Doherty, Appl. Opt. 7, 627 (1968).
[CrossRef] [PubMed]

A. W. Lohmann, IBM Tech. Disclosure Bulletin 10, 1452 (1968).

1958 (1)

P. R. Yoder, J. Opt. Soc. Amer. 48, 496 (1958).
[CrossRef]

1931 (1)

H. E. Ives, J. Opt. Soc. Amer. 21, 109 (1931).
[CrossRef]

Burckhardt, C. B.

Collier, R. J.

Doherty, E. T.

Fujii, Y.

Y. Fujii, K. Nakajima, Seisan Kenkyu (Monthly J. Inst. Industrial Science, Univ. Tokyo) 21, 611 (1969).

Hasegawa, T.

T. Okoshi, T. Hasegawa, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.

Ives, H. E.

H. E. Ives, J. Opt. Soc. Amer. 21, 109 (1931).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, IBM Tech. Disclosure Bulletin 10, 1452 (1968).

Nakajima, K.

Y. Fujii, K. Nakajima, Seisan Kenkyu (Monthly J. Inst. Industrial Science, Univ. Tokyo) 21, 611 (1969).

Okoshi, T.

A. Yano, T. Okoshi, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.

T. Okoshi, T. Hasegawa, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.

Yano, A.

A. Yano, T. Okoshi, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.

Yoder, P. R.

P. R. Yoder, J. Opt. Soc. Amer. 48, 496 (1958).
[CrossRef]

Appl. Opt. (1)

IBM Tech. Disclosure Bulletin (1)

A. W. Lohmann, IBM Tech. Disclosure Bulletin 10, 1452 (1968).

J. Opt. Soc. Amer. (2)

H. E. Ives, J. Opt. Soc. Amer. 21, 109 (1931).
[CrossRef]

P. R. Yoder, J. Opt. Soc. Amer. 48, 496 (1958).
[CrossRef]

Other (3)

Y. Fujii, K. Nakajima, Seisan Kenkyu (Monthly J. Inst. Industrial Science, Univ. Tokyo) 21, 611 (1969).

T. Okoshi, T. Hasegawa, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.

A. Yano, T. Okoshi, Spring General Meeting of Japan Society of Applied Physics, 2 April 1970.

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Figures (13)

Fig. 1
Fig. 1

Projection-type 3–D display system.

Fig. 2
Fig. 2

Curved triple mirror.

Fig. 3
Fig. 3

New coordinate system [0′(XYZ)] for computing the reflecting pattern.

Fig. 4
Fig. 4

Diffusing angle θ vs maximum tilt angle δ of Szc surface.

Fig. 5
Fig. 5

Plane XY′ where the reflection pattern is observed.

Fig. 6
Fig. 6

Reflection pattern of a curved triple mirror screen.

Fig. 7
Fig. 7

Unit screen of a curved triple mirror screen.

Fig. 8
Fig. 8

Effective reflectance Ke vs incident angle θ.

Fig. 9
Fig. 9

Optical setup for the photographic recording of the reflection pattern.

Fig. 10
Fig. 10

Setup for the electronic recording of the reflection pattern.

Fig. 11
Fig. 11

Over-all reflection pattern of sample 4 for normal and skew incidences in the case of ϕ = 90°: (1) θ = 0°, (b) θ = 10°, (c) θ = 20°, and (d) θ = 30°.

Fig. 12
Fig. 12

Energy distribution of the reflection pattern in the same cases as in Fig. 11 (desirable component).

Fig. 13
Fig. 13

Projection-type 3–D display system using a CTM screen.

Tables (2)

Tables Icon

Table I The Samples of TM and CTM

Tables Icon

Table II The Angle Characteristic of TM and CTM

Equations (48)

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I = I - ( N · I ) N .
[ m n ] = [ R ] [ m n ] ,
[ R ] = [ 1 - 2 a 2 - 2 a b - 2 a c - 2 a b 1 - 2 b 2 - 2 b c - 2 a c - 2 b c 1 - 2 c 2 ] ,
N x = ( 1 , 0 , 0 ) , N y = ( 0 , 1 , 0 ) , N z c = ( 1 2 sin δ , 1 2 sin δ , cos δ ) .
[ R x ] = [ - 1 0 0 0 1 0 0 0 1 ] ,
[ R y ] = [ 1 0 0 0 - 1 0 0 0 1 ] ,
[ R z c ] = [ cos 2 δ - sin 2 δ - 2 sin δ cos δ - sin 2 δ cos 2 δ - 2 sin δ cos δ - 2 sin δ cos δ - 2 sin δ cos δ 1 - 2 cos 2 δ ] .
[ R x ] [ R y ] = [ R y ] [ R x ] .
[ m n ] = [ R z c ] [ R y ] [ R x ] [ m n ] = [ ( - 1 + sin 2 δ ) + m sin 2 δ - 2 n sin δ cos δ l sin 2 δ + n ( - 1 + sin 2 δ ) - 2 n sin δ cos δ 2 sin δ cos δ + 2 m sin δ cos δ + n ( 1 - 2 cos 2 δ ) ] .
[ X Y Z ] = [ T ] [ x - a / 3 y - a / 3 z - a / 3 ] ,
[ T ] = [ 1 / 6 1 / 6 - 2 / 6 - 1 / 2 1 / 2 0 1 / 3 1 / 3 1 / 3 ] .
[ x - a / 3 y - a / 3 z - a / 3 ] = [ T ] - 1 [ X Y Z ] ,
[ T ] - 1 = [ 1 / 6 - 1 / 2 1 / 3 1 / 6 1 / 2 1 / 3 - 2 / 6 0 1 / 3 ] .
[ L M N ] = [ T ] [ m n ]
[ m n ] = [ T ] - 1 [ L M N ]
[ L M N ] = [ T ] [ R z c ] [ R y ] [ R x ] [ T ] - 1 [ L M N ] = [ R ] [ L M N ] , [ R ] = [ - cos 2 δ 0 - sin 2 δ 0 - 1 0 sin 2 δ 0 - cos 2 δ ] .
[ R ] = [ T ] [ R x ] [ R y ] [ R z c ] [ T ] - 1 = [ - cos 2 δ 0 sin 2 δ 0 - 1 0 - sin 2 δ 0 - cos 2 δ ] .
[ R ] = [ T ] [ R y ] [ R z c ] [ R x ] [ T ] - 1 = [ - 1 3 - 2 3 cos 2 δ             2 6 sin 2 δ             - 2 3 + 2 3 cos 2 δ - 2 6 sin 2 δ             - cos 2 δ             1 3 sin 2 δ - 2 3 + 2 3 cos 2 δ             - 1 3 sin 2 δ             - 2 3 - 1 3 cos 2 δ ] .
[ R ] = [ T ] [ R x ] [ R z c ] [ R y ] [ T ] - 1 = [ - 1 3 - 2 3 cos 2 δ         - 2 6 sin 2 δ             - 2 3 + 2 3 cos 2 δ 2 6 sin 2 δ             - cos 2 δ             - 1 3 sin 2 δ - 2 3 + 2 3 cos 2 δ             1 3             sin 2 δ - 2 3 - 1 3 cos 2 δ ] .
M = - M
cos Θ = - [ L , M , N ] [ L M N ] = ( 1 - M 2 ) cos 2 δ + M 2 .
Θ = 2 δ max ,
X / d = L / N             Y / d = M / N .
X / d = L / N ,             Y / d = M / N .
L 2 + M 2 + N 2 = 1.
Y = 0.
( X / d ) max = 2 δ max .
X 2 - ( 1 M 2 - 1 ) Y 2 + d 2 = 0 ,
X 2 + Y 2 + d 2 = ( X + 2 d ) 2 / ( L + 2 N ) 2 ,
1 2 ( X / d - 2 ) 2 + ( Y / d ) 2 = 1.
Y d = - 1 3 sin 2 δ [ ( 2 / 3 ) + ( 1 / 3 ) cos 2 δ ] .
( Y / d ) max = ( 1 / 3 ) ( 2 δ max ) .
1 2 ( X / d + 2 ) 2 + ( Y / d ) 2 = 1.
= - sin θ cos ϕ , m = - sin θ sin ϕ , n = - cos θ .
X 1 = ( 3 / 3 ) ( / n ) a = ( 3 / 3 ) a · tan θ cos ϕ , Y 1 = ( 3 / 3 ) ( m / n ) a = ( 3 / 3 ) a · tan θ sin ϕ .
tan θ cos ( ϕ + 2 π / 3 ) - 1 / 2 2
S = ( a 2 / 3 3 ) cos θ × [ 4 ( 1 - 2 tan θ cos ϕ ) [ 1 - 2 tan θ cos ( ϕ - 2 π / 3 ) ] - [ 1 + 2 2 tan θ cos ( ϕ + 2 π / 3 ) ] 2 ] .
tan θ cos ( ϕ + 2 π / 3 ) - 1 / 2 2 ,
S = ( 4 a 2 / 3 3 ) cos θ ( 1 - 2 tan θ cos ϕ ) [ 1 - 2 tan θ cos ( ϕ - 2 π / 3 ) ] .
K e = S / ( S o cos θ ) · r 3 ,
S o = ( 3 / 2 ) a 2 .
θ tan - 1 ( 1 / 6 ) = 22 ° .
S = a 2 [ cos θ ( 1 - 2 tan 2 θ ) ] / 3.
22 ° θ tan - 1 ( 2 / 3 ) = 39 ° ,
S = ( 4 a 2 / 3 3 ) cos θ ( 1 - 3 tan θ ) / 2 ) .
[ K e ] max = 2 / 3             ( 67 % ) .
K e ( CTM ) = ( 2 / 3 ) S / ( S o cos θ ) r 3 .
[ K e ( CTM ) ] max = 4 / 9             ( 44 % ) .

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