Abstract

A magnetooptic spatial light modulator is used to reconstruct computer-generated Fourier holograms. Different methods for designing the holgrams are considered including binary and complex quantization in conjunction with an iterative algorithm, carrier techniques, phase manipulations, and cell oriented binary coding. The limitations of binary quantization are discussed, and the trade-offs between space–bandwidth and quantization error are considered. Using a device having an array of 48 × 48 elements the best compromise is achieved using carrier techniques in conjunction with phase manipulations and binary quantization.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
    [Crossref]
  2. F. Mok, J. Diep, H.-K. Liu, D. Psaltis, “Real-Time Computer-Generated Hologram by Means of Liquid-Crystal Television Spatial Light Modulator,” Opt. Lett. 11, 748–750 (1986).
    [Crossref] [PubMed]
  3. N. C. Gallagher, B. Liu, “Statistical Properties of the Fourier Transform of Random Phase Diffusers,” Optik 42, 65–86 (1975).
  4. W. J. Dallas, “Deterministic Diffusers for Holography,” Appl. Opt. 12, 1179–1187 (1973).
    [Crossref] [PubMed]
  5. N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that Reduces Image Reconstruction Error,” Appl. Opt. 12, 2328–2335 (1973).
    [Crossref] [PubMed]
  6. J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
    [Crossref]
  7. F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer-Generated Holography: Hologram Repetition and Phase Manipulations,” J. Opt. Soc. Am. A 4, 694–698 (1987).
    [Crossref]
  8. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer Holograms, Generated by Computer,” Appl. Opt. 6, 1739–1748 (1967).
    [Crossref] [PubMed]
  9. D. C. Chu, J. W. Goodman, “Spectrum Shaping with Parity Sequences,” Appl. Opt. 11, 1716–1724 (1972).
    [Crossref] [PubMed]
  10. N. C. Gallagher, “Discrete Spectral Phase Coding,” IEEE Trans. Inf. Theory IT-22, 622–624 (1976).
    [Crossref]
  11. C. K. Hsueh, A. A. Sawchuk, “Computer-Generated Double-Phase Holograms,” Appl. Opt. 17, 3874–3884 (1978).
    [Crossref] [PubMed]
  12. K. D. Rines, N. C. Gallagher, “Reducing Quantization Error in Hsueh-Sawchuk Holograms,” Appl. Opt. 20, 2008–2010 (1981).
    [Crossref] [PubMed]
  13. J. N. Mait, K.-H. Brenner, “Dual-Phase Holograms: Improved Design,” Appl. Opt. 26, 4883–4892 (1987).
    [Crossref] [PubMed]
  14. G. S. Himes, J. N. Mait, “Computer-Generated Hologram Design for a Magneto-Optic Spatial Light Modulator,” Optical Information Processing Systems and Architectures, Bahram Javidi, Editor, Proc. SPIE 1151, to appear (1989).

1987 (2)

1986 (1)

1984 (1)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[Crossref]

1981 (1)

1980 (1)

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
[Crossref]

1978 (1)

1976 (1)

N. C. Gallagher, “Discrete Spectral Phase Coding,” IEEE Trans. Inf. Theory IT-22, 622–624 (1976).
[Crossref]

1975 (1)

N. C. Gallagher, B. Liu, “Statistical Properties of the Fourier Transform of Random Phase Diffusers,” Optik 42, 65–86 (1975).

1973 (2)

1972 (1)

1967 (1)

Brenner, K.-H.

Bryngdahl, O.

Chu, D. C.

Dallas, W. J.

Diep, J.

Fienup, J. R.

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
[Crossref]

Gallagher, N. C.

K. D. Rines, N. C. Gallagher, “Reducing Quantization Error in Hsueh-Sawchuk Holograms,” Appl. Opt. 20, 2008–2010 (1981).
[Crossref] [PubMed]

N. C. Gallagher, “Discrete Spectral Phase Coding,” IEEE Trans. Inf. Theory IT-22, 622–624 (1976).
[Crossref]

N. C. Gallagher, B. Liu, “Statistical Properties of the Fourier Transform of Random Phase Diffusers,” Optik 42, 65–86 (1975).

N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that Reduces Image Reconstruction Error,” Appl. Opt. 12, 2328–2335 (1973).
[Crossref] [PubMed]

Goodman, J. W.

Hauck, R.

Himes, G. S.

G. S. Himes, J. N. Mait, “Computer-Generated Hologram Design for a Magneto-Optic Spatial Light Modulator,” Optical Information Processing Systems and Architectures, Bahram Javidi, Editor, Proc. SPIE 1151, to appear (1989).

Hsueh, C. K.

Liu, B.

N. C. Gallagher, B. Liu, “Statistical Properties of the Fourier Transform of Random Phase Diffusers,” Optik 42, 65–86 (1975).

N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that Reduces Image Reconstruction Error,” Appl. Opt. 12, 2328–2335 (1973).
[Crossref] [PubMed]

Liu, H.-K.

Lohmann, A. W.

Mait, J. N.

J. N. Mait, K.-H. Brenner, “Dual-Phase Holograms: Improved Design,” Appl. Opt. 26, 4883–4892 (1987).
[Crossref] [PubMed]

G. S. Himes, J. N. Mait, “Computer-Generated Hologram Design for a Magneto-Optic Spatial Light Modulator,” Optical Information Processing Systems and Architectures, Bahram Javidi, Editor, Proc. SPIE 1151, to appear (1989).

Mok, F.

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[Crossref]

Paris, D. P.

Psaltis, D.

F. Mok, J. Diep, H.-K. Liu, D. Psaltis, “Real-Time Computer-Generated Hologram by Means of Liquid-Crystal Television Spatial Light Modulator,” Opt. Lett. 11, 748–750 (1986).
[Crossref] [PubMed]

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[Crossref]

Rines, K. D.

Sawchuk, A. A.

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[Crossref]

Wyrowski, F.

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

N. C. Gallagher, “Discrete Spectral Phase Coding,” IEEE Trans. Inf. Theory IT-22, 622–624 (1976).
[Crossref]

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

Opt. Eng. (2)

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
[Crossref]

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[Crossref]

Opt. Lett. (1)

Optik (1)

N. C. Gallagher, B. Liu, “Statistical Properties of the Fourier Transform of Random Phase Diffusers,” Optik 42, 65–86 (1975).

Other (1)

G. S. Himes, J. N. Mait, “Computer-Generated Hologram Design for a Magneto-Optic Spatial Light Modulator,” Optical Information Processing Systems and Architectures, Bahram Javidi, Editor, Proc. SPIE 1151, to appear (1989).

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 (9)

Fig. 1
Fig. 1

Rpresentation in the complex plane of binary quantization of the pupil function P(u,υ).

Fig. 2
Fig. 2

(a) Desired ring response; (b) computer simulated reconstruction from a binary hologram; (c) reconstruction from a binary hologram displayed on a 48 × 48 MOSLM.

Fig. 3
Fig. 3

(a) Desired arc response; (b) simulated reconstruction; (c) reconstruction from a MOSLM; (d) reconstruction of an arc using spatial carrier and binary coding.

Fig. 4
Fig. 4

(a) Desired box response; (b) simulated reconstruction; (c) reconstruction from a MOSLM; (d) simulated reconstruction of a box with random phase applied prior to binary coding; (e) reconstruction from a MOSLM corresponding to (d).

Fig. 5
Fig. 5

(a) Desired line segment response; (b) simulated reconstruction of line segments with random phase applied prior to binary coding; (c) reconstruction from a MOSLM corresponding to (b); (d) reconstruction from a MOSLM after application of a random phase, modulation of a spatial carrier, and binary quantization; (e) simulated reconstruction after twenty iterations with initial random phase and binary quantization; (f) reconstruction from a MOSLM corresponding to (e).

Fig. 6
Fig. 6

Binary coding of the dual-phase components (after Ref. 11).

Fig. 7
Fig. 7

Representation of the sets (a) χ2,0, (b) χ2, π /2, (c) χ4,0, and (d) χ4, π /4 showing decision boundaries for minimum distance quantization.

Fig. 8
Fig. 8

(a) Simulated reconstruction of line segments indicating quantization effects using N = 2 and ϕ = 0 for twenty iterations in the error reduction algorithm; (b) simulated reconstruction corresponding to (a) indicating quantization and scaling effects; (c) reconstruction from a MOSLM corresponding to (a) using a 2 × 2 cell for coding; (d) effects of spatial scaling on the reconstruction of the line segment response. Dotted segments indicate regions having a relative intensity of one-half. The box indicates the zeros of the sinc function.

Fig. 9
Fig. 9

(a) Simulated reconstruction of line segments indicating quantization effects using N = 4 and ϕ = 0 for twenty iterations in the error reduction algorithm; (b) simulated reconstruction corresponding to (a) indicating quantization and scaling effects; (c) reconstruction from MOSLM corresponding to (a) and coding with a 4 × 4 cell.

Equations (24)

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

P ( u , υ ) = | P ( u , υ ) | exp [ j θ ( u , υ ) ] ,
P b ( u , υ ) = { + 1 , Re { P ( u , υ ) } 0 , 1 , otherwise .
p b ( x , y ) = P b ( u , υ ) exp [ j 2 π ( x u + y υ ) ] dxdy .
P c ( u , υ ) = | P ( u , υ ) | cos [ 2 π u x 0 + θ ( u , υ ) ] ,
p i ( x , y ) = P m P q p i 1 ( x , y ) ,
p 0 ( x , y ) = | p ( x , y ) | exp [ j θ 0 ( x , y ) ]
P m g ( x , y ) = | p ( x , y ) | exp [ j θ g ( x , y ) ] .
P q G ( u , υ ) = { ξ : ξ χ } .
P ( u , υ ) = ( ½ ) { exp [ j θ + ( u , υ ) ] exp ( j ϕ ) + exp [ j θ ( u , υ ) ] } ,
θ + ( u , υ ) = θ ( u , υ ) + cos 1 | P ( u , υ ) | ϕ ,
θ ( u , υ ) = θ ( u , υ ) cos 1 | P ( u , υ ) | .
d + ( u , υ ) = d y 2 π θ + ( u , υ ) ,
d ( u , υ ) = d y 2 π θ ( u , υ ) .
θ N + ( u , υ ) = 2 π m ( u , υ ) / N ,
θ N ( u , υ ) = 2 π n ( u , υ ) / N ,
P m , n ( u , υ ) = cos { [ m ( u , υ ) n ( u , υ ) ] π / N + ϕ / 2 } × exp ( j { [ m ( u , υ ) + n ( u , υ ) ] π / N + ϕ / 2 } ) .
χ N , ϕ = { ξ m , n : ξ m , n = cos [ ( m n ) π / N + ϕ / 2 ] × exp { j [ ( m + n ) π / N + ϕ / 2 ] } ,
m = [ 0 , N 1 ] , n = [ 0 , N 1 ] , and ϕ = [ 0 , 2 π ) } .
P χ , MD P ( u , υ ) = { ξ m , n : | P ( u , υ ) ξ m , n | | P ( u , υ ) ξ | for all ξ χ N , ϕ } .
P ( k , l ) = i = 0 N FFT 1 j = 0 N FFT 1 p ( i , j ) exp [ j 2 π N FFT ( i k + j l ) ] .
P h ( u , υ ) = k = 0 ( N SLM / i u ) 1 l = 0 ( N SLM / i υ ) 1 P ( k , l ) rect ( u i u Δ s k i u Δ w , υ i υ Δ s l i υ Δ w ) ,
p h ( x , y ) = i u i υ Δ w 2 sinc ( i u Δ w x , i υ Δ w y ) × i = 0 N FFT 1 j = 0 N FFT 1 p ( i , j ) sin π ( N SLM ) ( Δ s x i i u N FFT ) sin π i u ( Δ s x i i u N FFT ) × sin π ( N SLM ) ( Δ s y j i υ N FFT ) sin π i υ ( Δ s y j i υ N FFT ) .
e c ( x , y ) = i u i υ Δ w 2 sinc ( i u Δ w x 2 , i υ Δ w y ) sin ( π i u Δ w x 2 ) × i = 0 N FFT 1 j = 0 N FFT 1 p ( i , j ) sin π ( N SLM ) ( Δ s x i i u N FFT ) sin π i u ( Δ s x i i u N FFT ) × sin π ( N SLM ) ( Δ s y j i υ N FFT ) sin π i υ ( Δ s y j i υ N FFT ) .
e q ( s ) = 1 2 S π + 2 S π 0 π / 2 cos θ erf ( S 2 cos θ ) d θ ,

Metrics