Abstract

A new approach to the design of computer-generated holograms makes optimal use of the available device resolution. An iterative search algorithm minimizes an error criterion by directly manipulating the binary hologram and observing the effect on the desired reconstruction. Several measures of error and efficiency useful in assessing the optimality of digital holograms are defined. Methods for designing digital holograms that are based on projections and error diffusion are presented as established techniques for comparison to direct binary search.

© 1987 Optical Society of America

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References

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  1. J. P. Allebach, N. C. Gallagher, B. Liu, “Aliasing Error in Digital Holography,” Appl. Opt. 15, 2183 (1976).
    [CrossRef] [PubMed]
  2. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671 (1983).
    [CrossRef] [PubMed]
  3. P. Carnevali, L. Coletti, S. Patarnello, “Image Processing By Simulated Annealing,” IBM J. Res. Dev. 29, 569 (1985).
    [CrossRef]
  4. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
    [CrossRef]
  5. J. P. Allebach, “Design of Antialising Patterns for Time-Sequential Sampling of Spatiotemporal Signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 137 (1984).
    [CrossRef]
  6. J. A. Bucklew, B. E. A. Saleh, “Theorem for High-Resolution High Contrast Image Synthesis,” J. Opt. Soc. Am. A 2, 1233 (1985).
    [CrossRef]
  7. J. P. Allebach, B. Liu, “Minimax Spectrum Shaping with a Bandwidth Constraint,” Appl. Opt. 14, 3062 (1975).
    [CrossRef] [PubMed]
  8. W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), pp. 291–366.
  9. J. P. Allebach, “Representation-Related Errors in Binary Digital Holograms: A Unified Analysis,” Appl. Opt. 20, 290 (1981).
    [CrossRef] [PubMed]
  10. R. H. Squires, J. P. Allebach, “Digital Holograms: A Guide to Reducing Quantization and Phase Encoding Errors,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 12 (1983).
  11. N. C. Gallagher, J. A. Bucklew, “Nondetour Phase Digital Holograms: An Analysis,” Appl. Opt. 19, 4266 (1980).
    [CrossRef] [PubMed]
  12. R. Hauck, O. Bryngdahl, “Computer-Generated Holograms with Pulse-Density Modulation,” J. Opt. Soc. Am. A 1, 5 (1984).
    [CrossRef]
  13. R. W. Floyd, L. Steinberg, “An Adaptive Algorithm for Spatial Greyscale,” Proc. Soc. Inf. Disp. 17, (1976).
  14. N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that Reduces Image Reconstruction Error,” Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  15. W. E. Ross, K. M. Snapp, R. H. Anderson, “Fundamental Characteristics of the Litton Iron Garnet Magneto-Optic Spatial Light Modulator,” Proc. Soc. Photo-opt. Instrum. Eng. 388, 55 (1983).

1985 (2)

P. Carnevali, L. Coletti, S. Patarnello, “Image Processing By Simulated Annealing,” IBM J. Res. Dev. 29, 569 (1985).
[CrossRef]

J. A. Bucklew, B. E. A. Saleh, “Theorem for High-Resolution High Contrast Image Synthesis,” J. Opt. Soc. Am. A 2, 1233 (1985).
[CrossRef]

1984 (2)

J. P. Allebach, “Design of Antialising Patterns for Time-Sequential Sampling of Spatiotemporal Signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 137 (1984).
[CrossRef]

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

1983 (3)

R. H. Squires, J. P. Allebach, “Digital Holograms: A Guide to Reducing Quantization and Phase Encoding Errors,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 12 (1983).

W. E. Ross, K. M. Snapp, R. H. Anderson, “Fundamental Characteristics of the Litton Iron Garnet Magneto-Optic Spatial Light Modulator,” Proc. Soc. Photo-opt. Instrum. Eng. 388, 55 (1983).

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671 (1983).
[CrossRef] [PubMed]

1981 (1)

1980 (1)

1976 (2)

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

J. P. Allebach, N. C. Gallagher, B. Liu, “Aliasing Error in Digital Holography,” Appl. Opt. 15, 2183 (1976).
[CrossRef] [PubMed]

1975 (1)

1973 (1)

1953 (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
[CrossRef]

Allebach, J. P.

J. P. Allebach, “Design of Antialising Patterns for Time-Sequential Sampling of Spatiotemporal Signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 137 (1984).
[CrossRef]

R. H. Squires, J. P. Allebach, “Digital Holograms: A Guide to Reducing Quantization and Phase Encoding Errors,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 12 (1983).

J. P. Allebach, “Representation-Related Errors in Binary Digital Holograms: A Unified Analysis,” Appl. Opt. 20, 290 (1981).
[CrossRef] [PubMed]

J. P. Allebach, N. C. Gallagher, B. Liu, “Aliasing Error in Digital Holography,” Appl. Opt. 15, 2183 (1976).
[CrossRef] [PubMed]

J. P. Allebach, B. Liu, “Minimax Spectrum Shaping with a Bandwidth Constraint,” Appl. Opt. 14, 3062 (1975).
[CrossRef] [PubMed]

Anderson, R. H.

W. E. Ross, K. M. Snapp, R. H. Anderson, “Fundamental Characteristics of the Litton Iron Garnet Magneto-Optic Spatial Light Modulator,” Proc. Soc. Photo-opt. Instrum. Eng. 388, 55 (1983).

Bryngdahl, O.

Bucklew, J. A.

Carnevali, P.

P. Carnevali, L. Coletti, S. Patarnello, “Image Processing By Simulated Annealing,” IBM J. Res. Dev. 29, 569 (1985).
[CrossRef]

Coletti, L.

P. Carnevali, L. Coletti, S. Patarnello, “Image Processing By Simulated Annealing,” IBM J. Res. Dev. 29, 569 (1985).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), pp. 291–366.

Floyd, R. W.

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

Gallagher, N. C.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671 (1983).
[CrossRef] [PubMed]

Hauck, R.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671 (1983).
[CrossRef] [PubMed]

Liu, B.

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
[CrossRef]

Patarnello, S.

P. Carnevali, L. Coletti, S. Patarnello, “Image Processing By Simulated Annealing,” IBM J. Res. Dev. 29, 569 (1985).
[CrossRef]

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
[CrossRef]

Ross, W. E.

W. E. Ross, K. M. Snapp, R. H. Anderson, “Fundamental Characteristics of the Litton Iron Garnet Magneto-Optic Spatial Light Modulator,” Proc. Soc. Photo-opt. Instrum. Eng. 388, 55 (1983).

Saleh, B. E. A.

Snapp, K. M.

W. E. Ross, K. M. Snapp, R. H. Anderson, “Fundamental Characteristics of the Litton Iron Garnet Magneto-Optic Spatial Light Modulator,” Proc. Soc. Photo-opt. Instrum. Eng. 388, 55 (1983).

Squires, R. H.

R. H. Squires, J. P. Allebach, “Digital Holograms: A Guide to Reducing Quantization and Phase Encoding Errors,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 12 (1983).

Steinberg, L.

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

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671 (1983).
[CrossRef] [PubMed]

Appl. Opt. (5)

IBM J. Res. Dev. (1)

P. Carnevali, L. Coletti, S. Patarnello, “Image Processing By Simulated Annealing,” IBM J. Res. Dev. 29, 569 (1985).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

J. P. Allebach, “Design of Antialising Patterns for Time-Sequential Sampling of Spatiotemporal Signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 137 (1984).
[CrossRef]

J. Chem. Phys. (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21, 1087 (1953).
[CrossRef]

J. Opt. Soc. Am. A (2)

Proc. Soc. Inf. Disp. (1)

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

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. H. Squires, J. P. Allebach, “Digital Holograms: A Guide to Reducing Quantization and Phase Encoding Errors,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 12 (1983).

W. E. Ross, K. M. Snapp, R. H. Anderson, “Fundamental Characteristics of the Litton Iron Garnet Magneto-Optic Spatial Light Modulator,” Proc. Soc. Photo-opt. Instrum. Eng. 388, 55 (1983).

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671 (1983).
[CrossRef] [PubMed]

Other (1)

W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), pp. 291–366.

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Figures (20)

Fig. 1
Fig. 1

Relationship between efficiency measures.

Fig. 2
Fig. 2

Flow chart of the direct binary search algorithm.

Fig. 3
Fig. 3

Set of 16 × 16 objects used to generate experimental results.

Fig. 4
Fig. 4

Fraction of addressable hologram points changed during each iteration of the DBS algorithm.

Fig. 5
Fig. 5

Normalized rms error after each iteration of the DBS algorithm.

Fig. 6
Fig. 6

Binarization efficiency after each iteration of the DBS algorithm.

Fig. 7
Fig. 7

Normalized rms error for each of the four biases implemented in the error diffusion algorithm.

Fig. 8
Fig. 8

Binarization efficiency for each of the four biases implemented in the error diffusion algorithm.

Fig. 9
Fig. 9

Comparison of the normalized rms errors of the three methods investigated.

Fig. 10
Fig. 10

Comparison of the binarization efficiencies of the three methods investigated.

Fig. 11
Fig. 11

Binary transmittance functions of 64 × 64 holograms for the complex amplitude-based error criterion. The left column was synthesized by the projection method, the middle column by error diffusion, and the right column by DBS. The letter R is the object for the top row; the diagonal test pattern is the object for the bottom row.

Fig. 12
Fig. 12

Diffraction patterns of 64 × 64 holograms for the complex amplitude-based error criterion. The left column was synthesized by the projection method, the middle column by error diffusion, and the right column by DBS. The letter R is the object for the top row; the diagonal test pattern is the object for the bottom row.

Fig. 13
Fig. 13

Digitally reconstructed images of 64 × 64 holograms for the complex amplitude-based error criterion. The left column was synthesized by the projection method, the middle column by error diffusion, and the right column by DBS. The letter R is the object for the top row; the diagonal test pattern is the object for the bottom row.

Fig. 14
Fig. 14

Digitally reconstructed images of 128 × 128 holograms for the complex amplitude-based error criterion. The left column was synthesized by the projection method, the middle column by error diffusion, and the right column by DBS. The letter R is the object for the top row; the diagonal test pattern is the object for the bottom row.

Fig. 15
Fig. 15

Digitally reconstructed images of 64 × 64 holograms for the magnitude only-based error criterion. The left column was synthesized by the projection method, the middle column by error diffusion, and the right column by DBS. The letter R is the object for the top row; the diagonal test pattern is the object for the bottom row.

Fig. 16
Fig. 16

Optical reconstruction of a 64 × 64 hologram synthesized by DBS under the complex amplitude-based error criterion.

Fig. 17
Fig. 17

Optical reconstruction of 128 × 128 hologram synthesized by DBS under the complex amplitude-based error criterion.

Fig. 18
Fig. 18

Optical reconstruction of 64 × 64 hologram synthesized by DBS under the magnitude only-based error criterion.

Fig. 19
Fig. 19

Optical reconstruction of 64 × 64 hologram synthesized by DBS under the complex amplitude-based error criterion replicated 2 × 2 times on the Litton device.

Fig. 20
Fig. 20

Optical reconstruction of 64 × 64 hologram synthesized by DBS under the magnitude only-based error criterion replicated 2 × 2 times on the Litton device.

Tables (2)

Tables Icon

Table I Effect of Random Seeds on DBS

Tables Icon

Table II Parameters Used for Generating Results of Figs. 7 and 8

Equations (31)

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h ( x , y ) = H ( u , v ) exp [ i 2 π ( u x + v y ) ] d x d y .
H ( u , v ) = k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 H k l rect ( u - k R R , v - l s S ) ,
rect ( a , b ) = { 1 , if a , b > 1 / 2 , 0 , else .
h ( x , y ) = R S sinc ( R x , S y ) k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 H k l exp [ i 2 π ( R x k + S y l ) ] ,
h ( m X , n Y ) = ( R S X Y ) 1 / 2 sinc ( m M , n N ) h m n ,
h m n = 1 M N k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 H k l exp { i 2 π ( m k M + n l N ) } .
g m n = { h m - m 0 , n - n 0 , if ( m - m 0 , n - n 0 ) R , 0 , otherwise ,
h ( x , y ) = ( R S X Y ) 1 / 2 sinc ( R x , S y ) m = - M / 2 M / 2 - 1 n = - N / 2 N / 2 - 1 h m n × s M N ( x - m X X , y - n Y Y ) ,
s M N ( x , y ) = sinc ( x , y ) sinc ( x / M , y / N ) exp [ - i π ( x / M + y / N ) ] .
e = 1 A B m = - A / 2 A / 2 - 1 n = - B / 2 B / 2 - 1 f ¯ m n - λ g m n 2 ,
λ = m = - A / 2 A / 2 - 1 n = - B / 2 B / 2 - 1 f ¯ m n g m n * m = - A / 2 A / 2 - 1 n = - B / 2 B / 2 - 1 g m n 2 .
e ¯ rms = e 1 / 2 max m n f ¯ m n ,
η obj = 1 M N m = - A / 2 A / 2 - 1 n = - B / 2 B / 2 - 1 f ˜ m n 2 .
η tot = 1 M N m = - A / 2 A / 2 - 1 n = - B / 2 B / 2 - 1 g m n 2 .
η bin = η tot η obj .
e = 1 A B { m = - A / 2 A / 2 - 1 n = - B / 2 B / 2 - 1 f ¯ m n 2 - λ 2 m = - A / 2 A . / 2 - 1 n = - B / 2 B / 2 - 1 g m n 2 } .
e = M N A B η obj ( 1 - λ 2 η bin ) .
η bin = 1 λ 2 .
g m n = { g ^ m n + 1 M N exp [ i 2 π ( k ^ m M + l ^ n N ) ] , if H k ^ l ^ = 0 , g ^ m n - 1 M N exp [ i 2 π ( k ^ m M + l ^ n N ) ] , if H k ^ l ^ = 1 ,
T p = ( T 1 / T 0 ) p T 0 .
( k q , l q ) = ( Δ k q mod M , Δ l q mod N ) ,
e = 1 A B m = - A / 2 A / 2 - 1 n = - B / 2 B / 2 - 1 ( f ¯ m n - λ g m n ) 2 .
F ^ k l = F k l cos [ 2 π ( k m 0 / M + l n 0 / N ) - θ k l ) ] ,
F ˜ k l = F ¯ k l + B k l
B k l = max k l F k l .
I k l = | F k l c + c 2 exp [ i 2 π ( k m 0 / M + l n 0 / N ) ] | 2 ,
I k l = F ^ k l + F k l 2 c c + c 4 .
B k l = F k l 2 c + c 4
c = 2 max k l F k l .
B k l = F k l .
B k l = { 0 , if F ^ k l 0 , - F ^ k l , else .

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