Abstract

Flexibility is desirable in optical systems for transfer, processing, and display of pictorial information. Two important operations in this respect are image rotation and magnification. Simple means for their realization by one single system are discussed. Experimental verifications using diffractive and refractive elements are described.

© 1980 Optical Society of America

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References

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  1. O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
    [CrossRef]
  2. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]

1974 (2)

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

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Figures (9)

FIG. 1
FIG. 1

System for performing geometrical transformations between planes O and D. Optical elements: L lens of focal length fL and P combination of rotatable components.

FIG. 2
FIG. 2

Notations used to indicate orientations of optical elements in plane O of Fig. 1.

FIG. 3
FIG. 3

Relative locations of frequency-spectra terms arising from a combination of three one-dimensional zone plates. Arrows indicate orientations of the zone plates.

FIG. 4
FIG. 4

Recordings of Fraunhofer diffraction configurations obtained from superposed two-dimensional Fresnel zone plates.

FIG. 5
FIG. 5

Results using two crossed one-dimensional Fresnel zone plates as element P of Fig. 1. Rotations introduced: a, 0°; b, 30°; c, 45°; and d, 90°.

FIG. 6
FIG. 6

Results using two one-dimensional Fresnel zone plates turned in opposite directions as element P of Fig. 1. Angle between the plates: a, 90°; b, 45°; c, 135°; and d, 30°.

FIG. 7
FIG. 7

Results using three superposed one-dimensional Fresnel zone plates as element P of Fig. 1 with orientations as indicated in Fig. 3.

FIG. 8
FIG. 8

System version using refractive elements. Indication of operation with different settings of the cylinder lens combination.

FIG. 9
FIG. 9

Results using two cylinder lenses as element P of Fig. 1; a illustrates image configuration, b multiple exposures for different Ω, c for different ω, and d for different Ω and ω.

Equations (22)

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A ( r ) = a ( r ) P ( r ) = a ( r ) exp { i ϕ ( r ) } ,
A ˜ ( r ) = + A ( r ) exp { i ( k / f L ) r r } d r = + a ( r ) exp { i ψ ( r ) } d r ,
r = ( f L / k ) r ϕ ( r ) ,
P ( r ) = m = 1 M p m ( r ) .
p m ( r ) = p [ ϕ m ( r ) ] = p [ ϕ m ( r ) + n 2 π ] n = 0 , ± 1 , ± 2 , .
p m ( r ) = n m = + C n m exp { i n m ϕ m ( r ) } ,
C n m = 1 2 π 0 2 π p m ( r ) exp { i n m ϕ m ( r ) } d ϕ m .
P ( r ) = κ = 0 B κ exp { i ϕ κ } = κ = 0 p κ .
B κ = m = 1 M C m m , κ
ϕ κ = m = 1 M n m , κ ϕ m ,
[ n 1 , , n M ] .
r κ = ( f L / k ) r ϕ κ ( r )
p ( x ) = ( 1 / 2 ) + ( 1 / 2 ) cos ϕ ( x )
ξ = b + x cos φ + y sin φ ,
ϕ = ( k / 2 f z ) ( x 2 cos 2 φ + y 2 sin 2 φ + b 2 + 2 x b cos φ + 2 y b sin φ + 2 x y cos φ sin φ ) .
ϕ κ = ( k / 2 f z ) [ x 2 ( cos 2 φ ) κ + y 2 ( sin 2 φ ) κ + ( b 2 ) κ + 2 × ( b cos φ ) κ + 2 y ( b sin φ ) κ + 2 x y ( cos φ sin φ ) κ ] ,
r κ = ( f L / f z ) ( b κ + D κ r ) ,
b κ = b ( ( cos φ ) κ ( sin φ ) κ )
D κ = { ( cos 2 φ ) κ ( cos φ sin φ ) κ ( cos φ sin φ ) κ ( sin 2 φ ) κ } .
D [ 0 , 0 ] , D [ 1 , 0 ] , D [ 1 , 1 ] , and D [ 1 , 1 ] .
D [ 1 , 1 ] = ( f L / f z ) sin 2 ω { sin 2 Ω cos 2 Ω cos 2 Ω sin 2 Ω } .
D [ 1 , 1 ] = { 2 ( cos 2 Ω cos 2 ω + sin 2 Ω sin 2 ω ) sin 2 Ω cos 2 ω sin 2 Ω cos 2 ω 2 ( cos 2 Ω sin 2 ω + sin 2 Ω cos 2 ω ) } .