Abstract

Computer-generated holograms with spatial carriers of polar geometry are investigated. Binary Fourier holograms are produced using a point-oriented concept. Two binarization processes are illustrated: simple hardclipping and pulse density coding.

© 1985 Optical Society of America

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References

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  1. W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer, Berlin, 1980), p. 291.
  2. J. Brunol, Y. Fonroget, S. Lowenthal, “Déconvolution analogique en Imagerie par Ouverture Codée Appliquée à la Médecine Nucléaire,” Opt. Acta 25, 113 (1978).
    [CrossRef]
  3. Y. H. Wu, P. Chavel, D. Joyeux, J. Brunol, “Fresnel Detour-Phase Circular Computer-Generated Holograms,” Opt. Commun. 45,155 (1983).
    [CrossRef]
  4. Y. H. Wu, P. Chavel, “Cell-Oriented On-Axis Computer-Generated Holograms for Use in the Fresnel Diffraction Mode,” Appl. Opt. 23, 228 (1984).
    [CrossRef] [PubMed]
  5. W. H. Lee, “Circular-Carrier Holograms,” J. Opt. Soc. Am. 65, 518 (1975).
    [CrossRef]
  6. Y. Ichioka, A. W. Lohmann, “Interferometric Testing of Large Optical Components with Circular Computer Holograms,” Appl. Opt. 11, 2597 (1972).
    [CrossRef] [PubMed]
  7. A. Engel, G. Herziger, “Computer Drawn Modulated Zone Plates,” Appl. Opt. 12, 471 (1973).
    [CrossRef] [PubMed]
  8. O. Bryngdahl, W. H. Lee, “Shearing Interferometry in Polar Coordinates,” J. Opt. Soc. Am. 64, 1606 (1974).
    [CrossRef]
  9. O. Bryngdahl, “Optical Scanner—Light Deflection Using Computer-Generated Diffractive Elements,” Opt. Commun. 15, 237 (1975).
    [CrossRef]
  10. O. Bryngdahl, W. H. Lee, “Laser Beam Scanning Using Computer-Generated Holograms,” Appl. Opt. 15, 183 (1976).
    [CrossRef] [PubMed]
  11. O. Bryngdahl, “Heterodyne Shearing Interferometers Using Diffractive Filters with Rotational Symmetry,” Opt. Commun. 17, 43 (1976).
    [CrossRef]
  12. R. Mercier, S. Lowenthal, “Comparison of In-Line and Carrier Frequency Holograms in Aspheric Testing,” Opt. Commun. 33, 251 (1980).
    [CrossRef]
  13. G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-Generated Circular Carrier Fourier Holograms,” Opt. Commun. 48, 89 (1983).
    [CrossRef]
  14. M. J. Bastiaans, M. J. van Weert, Technische Hogeschool Eindhoven; private communication.
  15. R. Hauck, O. Bryngdahl, “Computer-Generated Holograms with Pulse-Density Modulation,” J. Opt. Soc. Am. A 1, 5 (1984).
    [CrossRef]
  16. R. W. Floyd, L. Steinberg, “An Adaptive Algorithm for Spatial Greyscale,” Proc. of the Soc. for Inf. Disp. 17, 75 (1976).
  17. J. H. McLeod, “The Axicon: a New Type of Optical Element,” J. Opt. Soc. Am. 44, 592 (1954).
    [CrossRef]
  18. W. R. Edmonds, “The Reflaxicon, a New Reflective Optical Element, and Some Applications,” Appl. Opt. 12, 1940 (1973).
    [CrossRef] [PubMed]
  19. W. H. Lee, “Binary Synthetic Holograms,” Appl. Opt. 13, 1677 (1974).
    [CrossRef] [PubMed]

1984 (2)

1983 (2)

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-Generated Circular Carrier Fourier Holograms,” Opt. Commun. 48, 89 (1983).
[CrossRef]

Y. H. Wu, P. Chavel, D. Joyeux, J. Brunol, “Fresnel Detour-Phase Circular Computer-Generated Holograms,” Opt. Commun. 45,155 (1983).
[CrossRef]

1980 (1)

R. Mercier, S. Lowenthal, “Comparison of In-Line and Carrier Frequency Holograms in Aspheric Testing,” Opt. Commun. 33, 251 (1980).
[CrossRef]

1978 (1)

J. Brunol, Y. Fonroget, S. Lowenthal, “Déconvolution analogique en Imagerie par Ouverture Codée Appliquée à la Médecine Nucléaire,” Opt. Acta 25, 113 (1978).
[CrossRef]

1976 (3)

O. Bryngdahl, W. H. Lee, “Laser Beam Scanning Using Computer-Generated Holograms,” Appl. Opt. 15, 183 (1976).
[CrossRef] [PubMed]

O. Bryngdahl, “Heterodyne Shearing Interferometers Using Diffractive Filters with Rotational Symmetry,” Opt. Commun. 17, 43 (1976).
[CrossRef]

R. W. Floyd, L. Steinberg, “An Adaptive Algorithm for Spatial Greyscale,” Proc. of the Soc. for Inf. Disp. 17, 75 (1976).

1975 (2)

O. Bryngdahl, “Optical Scanner—Light Deflection Using Computer-Generated Diffractive Elements,” Opt. Commun. 15, 237 (1975).
[CrossRef]

W. H. Lee, “Circular-Carrier Holograms,” J. Opt. Soc. Am. 65, 518 (1975).
[CrossRef]

1974 (2)

1973 (2)

1972 (1)

1954 (1)

Bastiaans, M. J.

M. J. Bastiaans, M. J. van Weert, Technische Hogeschool Eindhoven; private communication.

Brunol, J.

Y. H. Wu, P. Chavel, D. Joyeux, J. Brunol, “Fresnel Detour-Phase Circular Computer-Generated Holograms,” Opt. Commun. 45,155 (1983).
[CrossRef]

J. Brunol, Y. Fonroget, S. Lowenthal, “Déconvolution analogique en Imagerie par Ouverture Codée Appliquée à la Médecine Nucléaire,” Opt. Acta 25, 113 (1978).
[CrossRef]

Bryngdahl, O.

R. Hauck, O. Bryngdahl, “Computer-Generated Holograms with Pulse-Density Modulation,” J. Opt. Soc. Am. A 1, 5 (1984).
[CrossRef]

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-Generated Circular Carrier Fourier Holograms,” Opt. Commun. 48, 89 (1983).
[CrossRef]

O. Bryngdahl, “Heterodyne Shearing Interferometers Using Diffractive Filters with Rotational Symmetry,” Opt. Commun. 17, 43 (1976).
[CrossRef]

O. Bryngdahl, W. H. Lee, “Laser Beam Scanning Using Computer-Generated Holograms,” Appl. Opt. 15, 183 (1976).
[CrossRef] [PubMed]

O. Bryngdahl, “Optical Scanner—Light Deflection Using Computer-Generated Diffractive Elements,” Opt. Commun. 15, 237 (1975).
[CrossRef]

O. Bryngdahl, W. H. Lee, “Shearing Interferometry in Polar Coordinates,” J. Opt. Soc. Am. 64, 1606 (1974).
[CrossRef]

Chavel, P.

Y. H. Wu, P. Chavel, “Cell-Oriented On-Axis Computer-Generated Holograms for Use in the Fresnel Diffraction Mode,” Appl. Opt. 23, 228 (1984).
[CrossRef] [PubMed]

Y. H. Wu, P. Chavel, D. Joyeux, J. Brunol, “Fresnel Detour-Phase Circular Computer-Generated Holograms,” Opt. Commun. 45,155 (1983).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer, Berlin, 1980), p. 291.

Edmonds, W. R.

Engel, A.

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An Adaptive Algorithm for Spatial Greyscale,” Proc. of the Soc. for Inf. Disp. 17, 75 (1976).

Fonroget, Y.

J. Brunol, Y. Fonroget, S. Lowenthal, “Déconvolution analogique en Imagerie par Ouverture Codée Appliquée à la Médecine Nucléaire,” Opt. Acta 25, 113 (1978).
[CrossRef]

Hauck, R.

R. Hauck, O. Bryngdahl, “Computer-Generated Holograms with Pulse-Density Modulation,” J. Opt. Soc. Am. A 1, 5 (1984).
[CrossRef]

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-Generated Circular Carrier Fourier Holograms,” Opt. Commun. 48, 89 (1983).
[CrossRef]

Herziger, G.

Ichioka, Y.

Joyeux, D.

Y. H. Wu, P. Chavel, D. Joyeux, J. Brunol, “Fresnel Detour-Phase Circular Computer-Generated Holograms,” Opt. Commun. 45,155 (1983).
[CrossRef]

Lee, W. H.

Lohmann, A. W.

Lowenthal, S.

R. Mercier, S. Lowenthal, “Comparison of In-Line and Carrier Frequency Holograms in Aspheric Testing,” Opt. Commun. 33, 251 (1980).
[CrossRef]

J. Brunol, Y. Fonroget, S. Lowenthal, “Déconvolution analogique en Imagerie par Ouverture Codée Appliquée à la Médecine Nucléaire,” Opt. Acta 25, 113 (1978).
[CrossRef]

McLeod, J. H.

Mercier, R.

R. Mercier, S. Lowenthal, “Comparison of In-Line and Carrier Frequency Holograms in Aspheric Testing,” Opt. Commun. 33, 251 (1980).
[CrossRef]

Neugebauer, G.

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-Generated Circular Carrier Fourier Holograms,” Opt. Commun. 48, 89 (1983).
[CrossRef]

Steinberg, L.

R. W. Floyd, L. Steinberg, “An Adaptive Algorithm for Spatial Greyscale,” Proc. of the Soc. for Inf. Disp. 17, 75 (1976).

van Weert, M. J.

M. J. Bastiaans, M. J. van Weert, Technische Hogeschool Eindhoven; private communication.

Wu, Y. H.

Y. H. Wu, P. Chavel, “Cell-Oriented On-Axis Computer-Generated Holograms for Use in the Fresnel Diffraction Mode,” Appl. Opt. 23, 228 (1984).
[CrossRef] [PubMed]

Y. H. Wu, P. Chavel, D. Joyeux, J. Brunol, “Fresnel Detour-Phase Circular Computer-Generated Holograms,” Opt. Commun. 45,155 (1983).
[CrossRef]

Appl. Opt. (6)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

J. Brunol, Y. Fonroget, S. Lowenthal, “Déconvolution analogique en Imagerie par Ouverture Codée Appliquée à la Médecine Nucléaire,” Opt. Acta 25, 113 (1978).
[CrossRef]

Opt. Commun. (5)

Y. H. Wu, P. Chavel, D. Joyeux, J. Brunol, “Fresnel Detour-Phase Circular Computer-Generated Holograms,” Opt. Commun. 45,155 (1983).
[CrossRef]

O. Bryngdahl, “Optical Scanner—Light Deflection Using Computer-Generated Diffractive Elements,” Opt. Commun. 15, 237 (1975).
[CrossRef]

O. Bryngdahl, “Heterodyne Shearing Interferometers Using Diffractive Filters with Rotational Symmetry,” Opt. Commun. 17, 43 (1976).
[CrossRef]

R. Mercier, S. Lowenthal, “Comparison of In-Line and Carrier Frequency Holograms in Aspheric Testing,” Opt. Commun. 33, 251 (1980).
[CrossRef]

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-Generated Circular Carrier Fourier Holograms,” Opt. Commun. 48, 89 (1983).
[CrossRef]

Proc. of the Soc. for Inf. Disp. (1)

R. W. Floyd, L. Steinberg, “An Adaptive Algorithm for Spatial Greyscale,” Proc. of the Soc. for Inf. Disp. 17, 75 (1976).

Other (2)

W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer, Berlin, 1980), p. 291.

M. J. Bastiaans, M. J. van Weert, Technische Hogeschool Eindhoven; private communication.

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Figures (7)

Fig. 1
Fig. 1

Spectra of linear, circular, and radial gratings due to different types of illuminating wave.

Fig. 2
Fig. 2

Illustration indicates the direction of error diffusion in the generation process of the CGH: (a) circular carrier CGH, (b) radial carrier CGH.

Fig. 3
Fig. 3

(a) Circular carrier and (c) radial carrier CGHs and (b) and (d) the corresponding computer-simulated reconstructions produced with a binarization process based on pulse-density modulation.

Fig. 4
Fig. 4

(a) Circular carrier and (c) radial carrier CGHs and (b) and (d) the corresponding computer-simulated reconstructions produced with a binarization process based on hardclipping.

Fig. 5
Fig. 5

Setup for optical reconstruction of CGHs with carriers in polar geometry.

Fig. 6
Fig. 6

Optical reconstructions of a circular carrier CGH (a) binarized by hardclipping and (b) binarized by pulse-density modulation; a radial carrier CGH (c) binarized by hardclipping and (d) binarized by pulse-density modulation.

Fig. 7
Fig. 7

Optical reconstructions of a circular carrier CGH illuminated by (a) a conical wave and (b) a conical wave and its complex conjugate; a radial carrier CGH illuminated by (c) a helical wave and (d) a helical wave and its complex conjugate.

Equations (26)

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S H ( r ) = | S H ( r ) | exp [ i φ H ( r ) ] ,
2 cos [ φ H ( r ) ] C H ( r ) + C H * ( r ) ,
H ( r ) = | S H ( r ) | 2 cos ( ϕ H + φ H ) + B ( r ) = S H ( r ) C H ( r ) + S H * ( r ) C H * ( r ) + B ( r ) ,
C W ( r ) = exp [ i ϕ W ( r ) ]
S W ( r ) = | S W ( r ) | exp [ i φ W ( r ) ] .
W ( r ) H ( r ) = S W ( r ) C W ( r ) [ S H ( r ) C H ( r ) + S H * ( r ) C H * ( r ) + B ] .
C W C H = 1.
R ( ρ ) = S H ( ρ ) * C W ( ρ ) * C H ( ρ ) + S H * ( ρ ) * C W ( ρ ) * C H * ( ρ ) + B C W ( ρ ) .
R ( ρ ) = S H ( ρ ) + S H * ( ρ ) * C W ( ρ ) * C W ( ρ ) + B C W ( ρ ) .
1 2 ( C H + C H * ) = 1 2 [ exp ( 2 π i r / r H ) + exp ( 2 π i r / r H ) ] ,
C W ( r ) = exp ( 2 π i r / r W ) ,
HT [ exp ( 2 π i r / r W ) ] = 0 exp ( 2 π i r / r W ) r J 0 ( 2 π r ρ ) d r = { 1 / [ 4 π 2 r W ( 1 / r W 2 ρ 2 ) 3 / 2 ] 0 < ρ < 1 / r W i / [ 4 π 2 r W ( ρ 2 1 / r W 2 ) 3 / 2 ] ρ > 1 / r W ,
HT [ cos ( 2 π r / r H ) ] = { 1 / [ 4 π 2 | r H | ( 1 / r H 2 ρ 2 ) 3 / 2 ] 0 < ρ < 1 / | r H | 0 ρ > 1 / | r H | .
1 2 ( C H + C H * ) = 1 2 [ exp ( 2 π i θ / θ H ) + exp ( 2 π i θ / θ H ) ] ,
C W ( r ) = exp ( 2 π i θ / θ W ) ] ,
FT [ exp ( i m θ ) ] = 2 π m / ρ 2 exp [ i m ( ζ + π / 2 ) ] ,
FT [ cos ( m θ ) ] = 2 π ( i ) m m / ρ 2 cos ( m ζ ) .
ρ min = 1 / ( | θ H | r max ) .
ρ max = 1 / ( | θ H | r min ) .
r ρ 0 = | λ f / r H | .
r H < 1 / Δ ρ .
r H < 1 / ( 3 Δ ρ ) .
θ H r max < 1 / Δ ρ .
θ H r max < 1 / ( 3 Δ ρ ) .
ρ max = 1 / ( θ H r min ) = 1 / ( 2 δ x )
C H ( r ) + C H * ( r ) ,

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