Abstract

Theory and experiments in the area of computer-generated holograms for geometric transformations are presented. Geometric transform holograms are divided into two categories: (1) those which have a continuous fringe structure, and (2) those which consist of a set of discrete subholograms. Criteria for the realizability of a continuous geometric transform hologram are described. Examples of both types of hologram are to map rings of different radii to a linear sequence of points. They allow the replacement of a ring developed detector by a linear detector array without loss of signal energy and can be applied to optical spectrum analysis and angle-wavelength multiplexing for data transmission through optical fibers.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
    [CrossRef]
  2. O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092 (1974).
    [CrossRef]
  3. S. K. Case, P. R. Haugen, O. J. Løkberg, “Multifacet Holographic Optical Elements for Wave Front Transformations,” Appl. Opt. 20, 2670 (1981).
    [CrossRef] [PubMed]
  4. G. Häusler, N. Streibl, “Optical Compensation of Geometrical Distortion by a Deformable Mirror,” Opt. Commun. 42, 381 (1982).
    [CrossRef]
  5. A. W. Lohmann, N. Streibl, “Map Transformations by Optical Anamorphic Processing,” Appl. Opt. 22, 780 (1983).
    [CrossRef] [PubMed]
  6. G. Herskowitz, H. Kobrinski, U. Levy, “Angular Division Miltiplexing in Optical Fibers,” Laser Focus 19, 83 (Feb.1983).
  7. A. M. Tai, A. A. Friesem, “Transmission of Two-Dimensional Images Through a Single Optical Fiber by Wavelength-Time Encoding,” Opt. Lett. 8, 57 (1983).
    [CrossRef] [PubMed]
  8. W. S. Colburn, R. C. Fairchild, “Design Study for a Low-Distortion Holographic HUD,” AFWAL-TR-81-1263, (1982), pp. 15–17.
  9. W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass., 1959), pp. 244–249.
  10. H. Hamilton, Complex Variables (Wadsworth, Belmont, Calif., 1966), pp. 28–39.
  11. S. K. Case, P. R. Haugen, “Partitioned Holographic Optical Elements,” Opt. Eng. 21, 352 (1982).
  12. H. Bartelt, S. K. Case, “Coordinate Transformations via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
    [CrossRef]
  13. P. Chavel et al.in Technical Digest, Tenth International Optical Computing Conference (IEEE Computer Society and ICO, New York, 1983), pp. 6–12.
    [CrossRef]
  14. R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
    [CrossRef]
  15. Y. Saito et al., “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1983).
    [CrossRef]
  16. C. L. Giles, H. H. Szu, “Optical Implementation of Coordinate Transformations for Pattern Recognition,” J. Opt. Soc. Am. 73, 1860 (1983).

1983 (6)

H. Bartelt, S. K. Case, “Coordinate Transformations via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

Y. Saito et al., “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1983).
[CrossRef]

C. L. Giles, H. H. Szu, “Optical Implementation of Coordinate Transformations for Pattern Recognition,” J. Opt. Soc. Am. 73, 1860 (1983).

G. Herskowitz, H. Kobrinski, U. Levy, “Angular Division Miltiplexing in Optical Fibers,” Laser Focus 19, 83 (Feb.1983).

A. M. Tai, A. A. Friesem, “Transmission of Two-Dimensional Images Through a Single Optical Fiber by Wavelength-Time Encoding,” Opt. Lett. 8, 57 (1983).
[CrossRef] [PubMed]

A. W. Lohmann, N. Streibl, “Map Transformations by Optical Anamorphic Processing,” Appl. Opt. 22, 780 (1983).
[CrossRef] [PubMed]

1982 (3)

R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
[CrossRef]

G. Häusler, N. Streibl, “Optical Compensation of Geometrical Distortion by a Deformable Mirror,” Opt. Commun. 42, 381 (1982).
[CrossRef]

S. K. Case, P. R. Haugen, “Partitioned Holographic Optical Elements,” Opt. Eng. 21, 352 (1982).

1981 (1)

1974 (2)

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
[CrossRef]

O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092 (1974).
[CrossRef]

Bartelt, H.

H. Bartelt, S. K. Case, “Coordinate Transformations via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092 (1974).
[CrossRef]

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
[CrossRef]

Case, S. K.

H. Bartelt, S. K. Case, “Coordinate Transformations via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

S. K. Case, P. R. Haugen, “Partitioned Holographic Optical Elements,” Opt. Eng. 21, 352 (1982).

S. K. Case, P. R. Haugen, O. J. Løkberg, “Multifacet Holographic Optical Elements for Wave Front Transformations,” Appl. Opt. 20, 2670 (1981).
[CrossRef] [PubMed]

Chavel, P.

P. Chavel et al.in Technical Digest, Tenth International Optical Computing Conference (IEEE Computer Society and ICO, New York, 1983), pp. 6–12.
[CrossRef]

Colburn, W. S.

W. S. Colburn, R. C. Fairchild, “Design Study for a Low-Distortion Holographic HUD,” AFWAL-TR-81-1263, (1982), pp. 15–17.

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
[CrossRef]

W. S. Colburn, R. C. Fairchild, “Design Study for a Low-Distortion Holographic HUD,” AFWAL-TR-81-1263, (1982), pp. 15–17.

Fienup, J. R.

R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
[CrossRef]

Friesem, A. A.

Giles, C. L.

C. L. Giles, H. H. Szu, “Optical Implementation of Coordinate Transformations for Pattern Recognition,” J. Opt. Soc. Am. 73, 1860 (1983).

Hamilton, H.

H. Hamilton, Complex Variables (Wadsworth, Belmont, Calif., 1966), pp. 28–39.

Haugen, P. R.

Häusler, G.

G. Häusler, N. Streibl, “Optical Compensation of Geometrical Distortion by a Deformable Mirror,” Opt. Commun. 42, 381 (1982).
[CrossRef]

Herskowitz, G.

G. Herskowitz, H. Kobrinski, U. Levy, “Angular Division Miltiplexing in Optical Fibers,” Laser Focus 19, 83 (Feb.1983).

Kaplan, W.

W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass., 1959), pp. 244–249.

Kobrinski, H.

G. Herskowitz, H. Kobrinski, U. Levy, “Angular Division Miltiplexing in Optical Fibers,” Laser Focus 19, 83 (Feb.1983).

Levy, U.

G. Herskowitz, H. Kobrinski, U. Levy, “Angular Division Miltiplexing in Optical Fibers,” Laser Focus 19, 83 (Feb.1983).

Lohmann, A. W.

Løkberg, O. J.

Saito, Y.

Y. Saito et al., “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1983).
[CrossRef]

Streibl, N.

A. W. Lohmann, N. Streibl, “Map Transformations by Optical Anamorphic Processing,” Appl. Opt. 22, 780 (1983).
[CrossRef] [PubMed]

G. Häusler, N. Streibl, “Optical Compensation of Geometrical Distortion by a Deformable Mirror,” Opt. Commun. 42, 381 (1982).
[CrossRef]

Szu, H. H.

C. L. Giles, H. H. Szu, “Optical Implementation of Coordinate Transformations for Pattern Recognition,” J. Opt. Soc. Am. 73, 1860 (1983).

Tai, A. M.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092 (1974).
[CrossRef]

C. L. Giles, H. H. Szu, “Optical Implementation of Coordinate Transformations for Pattern Recognition,” J. Opt. Soc. Am. 73, 1860 (1983).

Laser Focus (1)

G. Herskowitz, H. Kobrinski, U. Levy, “Angular Division Miltiplexing in Optical Fibers,” Laser Focus 19, 83 (Feb.1983).

Opt. Commun. (3)

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164 (1974).
[CrossRef]

G. Häusler, N. Streibl, “Optical Compensation of Geometrical Distortion by a Deformable Mirror,” Opt. Commun. 42, 381 (1982).
[CrossRef]

Y. Saito et al., “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8 (1983).
[CrossRef]

Opt. Eng. (3)

R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
[CrossRef]

S. K. Case, P. R. Haugen, “Partitioned Holographic Optical Elements,” Opt. Eng. 21, 352 (1982).

H. Bartelt, S. K. Case, “Coordinate Transformations via Multifacet Holographic Optical Elements,” Opt. Eng. 22, 497 (1983).
[CrossRef]

Opt. Lett. (1)

Other (4)

P. Chavel et al.in Technical Digest, Tenth International Optical Computing Conference (IEEE Computer Society and ICO, New York, 1983), pp. 6–12.
[CrossRef]

W. S. Colburn, R. C. Fairchild, “Design Study for a Low-Distortion Holographic HUD,” AFWAL-TR-81-1263, (1982), pp. 15–17.

W. Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass., 1959), pp. 244–249.

H. Hamilton, Complex Variables (Wadsworth, Belmont, Calif., 1966), pp. 28–39.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Ring-to-point geometric transformation.

Fig. 2
Fig. 2

Angle multiplexing in optical fibers.

Fig. 3
Fig. 3

Angle-wavelength multiplexing for data transmission through optical fibers.

Fig. 4
Fig. 4

One hologram system for geometric transformation.

Fig. 5
Fig. 5

Ring-to-point transform using discrete phase CGH: (a) portion of CGH; (b) all output points; (c) first eleven points.

Fig. 6
Fig. 6

Two hologram system for geometric transformation.

Fig. 7
Fig. 7

Input light distributions: (a) uniform circular input; (b) input with opaque ring.

Fig. 8
Fig. 8

Ring-to-line transform using continuous phase CGH: (a) Portion of CGH; (b)–(e) output intensity profiles for different input light levels for input of Fig. 7(a); (f) intensity profile for input of Fig. 7(b).

Fig. 9
Fig. 9

Ring-to-point transform using two hologram system with uniform input [Fig. 7(a)]: (a) output photograph; (b) isometric plot.

Fig. 10
Fig. 10

Ring-to-point transform for input with opaque ring [Fig. 7(b)]: (a) output photograph; (b) isometric plot.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

a 2 ( u , v ) = 1 i λ f - a 1 ( x , y ) exp { i [ ϕ ( x , y ) - 2 π λ f ( x u + y v ) ] } × d x d y ,
[ x y ] [ u v ] = [ λ f 2 π ϕ ( x , y ) x λ f 2 π ϕ ( x , y ) y ] .
[ x y ] [ u v ] = [ g ( x , y ) h ( x , y ) ]
λ f 2 π ϕ x = g ( x , y ) ,
λ f 2 π ϕ y = h ( x , y )
g y = h x .
y ( ϕ x ) = x ( ϕ y ) ,
f ( z * ) = g ( x , y ) + i h ( x , y )
f ( z ) = g ( x , - y ) + i h ( x , - y )
g ( x , - y ) y = - h ( x , - y ) x ,
λ f 2 π ϕ ( x , y ) = g ( x m , y n ) x + h ( x m , y n ) y
λ f 2 π ϕ x = g ( x m , y n ) ,
λ f 2 π ϕ y = h ( x m , y n ) ,
( u m n , v m n ) = [ g ( x m , y n ) , h ( x m , y n ) ] .
W [ ( u - u m n ) Δ x λ f , ( v - v m n ) Δ y λ f ] ,
sinc [ ( u - u m n ) Δ x λ f , ( v - v m n ) Δ y λ f ] .
p λ f Δ x by q λ f Δ y ,
t ( x , y ) = k 1 + k 2 cos [ ϕ ( x , y ) + α x + β y ] ,
[ x y ] [ u v ] = [ x 2 + y 2 0 ] .
ϕ ( x , y ) = 2 π r n x / λ f
[ x 1 y 1 ] [ x 2 y 2 ] = [ x 0 ln ( x 1 2 + y 1 2 ) 1 / 2 - x 0 tan - 1 ( y 1 / x 1 ) ] ,
ϕ ( x 1 , y 1 ) = 2 π x 0 λ f [ x 1 ln ( x 1 2 + y 1 2 ) 1 / 2 - y 1 tan - 1 ( y 1 / x 1 ) - x 1 ] .

Metrics