Abstract

A method of computer generating binary holograms based on the decomposition of a complex value into two phase quantities is described. Each Fourier transform cell is divided into subcells, and phase quantities are encoded by the detour phase technique. Noise due to the displacement of the subcells and the phase coding is discussed. Methods of suppressing this noise are also included.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. C. Gallagher et al., Appl. Opt. 16, 413 (1977).
    [CrossRef] [PubMed]
  2. W. H. Lee, Appl. Opt. 9, 639 (1970).
    [CrossRef] [PubMed]
  3. C. B. Burckhardt, Appl. Opt. 9, 1949 (1970).
    [CrossRef] [PubMed]
  4. J. P. Allebach, N. C. Gallagher, B. Liu, Appl. Opt. 15, 2183 (1976).
    [CrossRef] [PubMed]
  5. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  6. B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [CrossRef] [PubMed]
  7. R. E. Haskell, B. C. Culver, Appl. Opt. 11, 2712 (1972).
    [CrossRef] [PubMed]
  8. M. Severcan, “Computer Generation of Coherent Optical Filters with High Light Efficiency and Large Dynamic Range,” Ph.D. Thesis, Department of Electrical Engineering, Stanford U. (1973).
  9. D. C. Chu, J. W. Goodman, Appl. Opt. 11, 1716 (1972).
    [CrossRef] [PubMed]
  10. D. C. Chu, J. R. Fienup, Optical Engineering 13, 189 (1974).
    [CrossRef]
  11. E. K. Shmarev, Opt. Spectrosc. 41, 535 (1976).
  12. W. J. Dallas, Appl. Opt. 12, 1179 (1973).
    [CrossRef] [PubMed]
  13. N. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  14. J. R. Fienup, “Improved Synthesis and Computational Methods for Computer-Generated Holograms,” Ph.D. Thesis, Department of Electrical Engineering, Stanford U. (1975).
  15. D. Kermisch, J. Opt. Soc. Am. 60, 15 (1970).
    [CrossRef]
  16. N. C. Gallagher, Appl. Opt. 17, 109 (1978).
    [CrossRef] [PubMed]
  17. N. C. Gallagher, B. Liu, Optik 42, 65 (1975).
  18. P. Chavel, J. P. Hugonin, J. Opt. Soc. Am 66, 989 (1976).
    [CrossRef]

1978 (1)

1977 (1)

1976 (3)

J. P. Allebach, N. C. Gallagher, B. Liu, Appl. Opt. 15, 2183 (1976).
[CrossRef] [PubMed]

E. K. Shmarev, Opt. Spectrosc. 41, 535 (1976).

P. Chavel, J. P. Hugonin, J. Opt. Soc. Am 66, 989 (1976).
[CrossRef]

1975 (1)

N. C. Gallagher, B. Liu, Optik 42, 65 (1975).

1974 (1)

D. C. Chu, J. R. Fienup, Optical Engineering 13, 189 (1974).
[CrossRef]

1973 (2)

1972 (2)

1970 (3)

1967 (1)

1966 (1)

Allebach, J. P.

Brown, B. R.

Burckhardt, C. B.

Chavel, P.

P. Chavel, J. P. Hugonin, J. Opt. Soc. Am 66, 989 (1976).
[CrossRef]

Chu, D. C.

D. C. Chu, J. R. Fienup, Optical Engineering 13, 189 (1974).
[CrossRef]

D. C. Chu, J. W. Goodman, Appl. Opt. 11, 1716 (1972).
[CrossRef] [PubMed]

Culver, B. C.

Dallas, W. J.

Fienup, J. R.

D. C. Chu, J. R. Fienup, Optical Engineering 13, 189 (1974).
[CrossRef]

J. R. Fienup, “Improved Synthesis and Computational Methods for Computer-Generated Holograms,” Ph.D. Thesis, Department of Electrical Engineering, Stanford U. (1975).

Gallagher, N. C.

Goodman, J. W.

Haskell, R. E.

Hugonin, J. P.

P. Chavel, J. P. Hugonin, J. Opt. Soc. Am 66, 989 (1976).
[CrossRef]

Kermisch, D.

Lee, W. H.

Liu, B.

Lohmann, A. W.

Paris, D. P.

Severcan, M.

M. Severcan, “Computer Generation of Coherent Optical Filters with High Light Efficiency and Large Dynamic Range,” Ph.D. Thesis, Department of Electrical Engineering, Stanford U. (1973).

Shmarev, E. K.

E. K. Shmarev, Opt. Spectrosc. 41, 535 (1976).

Appl. Opt. (11)

J. Opt. Soc. Am (1)

P. Chavel, J. P. Hugonin, J. Opt. Soc. Am 66, 989 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Spectrosc. (1)

E. K. Shmarev, Opt. Spectrosc. 41, 535 (1976).

Optical Engineering (1)

D. C. Chu, J. R. Fienup, Optical Engineering 13, 189 (1974).
[CrossRef]

Optik (1)

N. C. Gallagher, B. Liu, Optik 42, 65 (1975).

Other (2)

J. R. Fienup, “Improved Synthesis and Computational Methods for Computer-Generated Holograms,” Ph.D. Thesis, Department of Electrical Engineering, Stanford U. (1975).

M. Severcan, “Computer Generation of Coherent Optical Filters with High Light Efficiency and Large Dynamic Range,” Ph.D. Thesis, Department of Electrical Engineering, Stanford U. (1973).

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 (22)

Fig. 1
Fig. 1

Decomposition of a vector H(u,υ) in the unit circle into two constant magnitude vectors H1(u,υ), H2(u,υ), and the parity term P(u,υ).

Fig. 2
Fig. 2

Implementation of one cell in a DPH.

Fig. 3
Fig. 3

Expression of one cell in terms of H1(u,υ) and H2(u,υ).

Fig. 4
Fig. 4

Noise weighting function WP(x) for two subcells.

Fig. 5
Fig. 5

Configuration of one cell with 2n subcells.

Fig. 6
Fig. 6

Noise weighting function WP(x) for four subcells.

Fig. 7
Fig. 7

Noise weighting function WP(x) for eight subcells.

Fig. 8
Fig. 8

Noise weighting function WP(x) for sixteen subcells.

Fig. 9
Fig. 9

Inverse Fourier transform of parity term P(u,υ) associated with the object h(x,y), letter P.

Fig. 10
Fig. 10

Decomposed vectors for H(u,υ) and P(u,υ).

Fig. 11
Fig. 11

Illustration of the speckle: (a) ideal object; (b) reconstruction in amplitude; (c) reconstruction in intensity.

Fig. 12
Fig. 12

Constrained bandwidth used in space–bandwidth iterative method: (a) a rect function having width of one half of the period of the discrete Fourier transform; (b) the result after several

Fig. 13
Fig. 13

Reconstruction of a DPH with zero phase.

Fig. 14
Fig. 14

Reconstruction of a DPH with random phase.

Fig. 15
Fig. 15

Reconstruction of a Lohmann hologram with random phase.

Fig. 16
Fig. 16

Reconstruction of a DPH with the space-transform iterative method (constant spectrum).

Fig. 17
Fig. 17

Reconstruction of a DPH with the space-transform iterative method (constrained bandwidth, 32 × 32 out of 64 × 64).

Fig. 18
Fig. 18

Reconstruction of a DPH with the space-transform iterative method (constrained bandwidth, 64 × 64 out of 128 × 128, and plotting 64 × 64).

Fig. 19
Fig. 19

Reconstruction of a DPH with two subcells.

Fig. 20
Fig. 20

Reconstruction of a DPH with eight subcells.

Fig. 21
Fig. 21

Reconstruction of a DPH with sixteen subcells.

Fig. 22
Fig. 22

Reconstruction of a DPH with method II (image appears at the fourth order).

Tables (1)

Tables Icon

Table I Comparisons of Diffraction Efficiency and S/N for Methods I and II

Equations (55)

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

H = A exp ( j θ ) = 1 2 exp [ ( j ( θ + ψ ) ] + 1 2 exp [ j ( θ ψ ) ] ,
ψ = cos 1 A , 0 ψ π / 2 .
d 1 = d ( θ + ψ ) / 2 π ,
d 2 = d ( θ ψ ) / 2 π ,
H = A exp ( j θ ) = ( n / 2 ) exp [ j ( θ + ψ ) ] + ( n / 2 ) exp [ j ( θ + ψ ) ] ,
ψ = cos 1 ( A / n ) , 0 ψ π / 2 ,
1 2 exp [ j ( θ + ψ ) ] + 1 2 exp [ j ( θ ψ ) ] = ( A / n ) exp ( j θ ) = H / n
H / 2 = ( A / 2 ) exp ( j θ ) = 1 2 exp [ j ( θ + ψ ) ] + 1 2 exp [ j ( θ ψ ) ] ,
ψ = cos 1 ( A / 2 ) , π / 3 ψ π / 2.
A = cos ψ 0 ψ π / 2 ,
( d A ) / ( d ψ ) = sin ψ 0 ψ π / 2.
A ˆ = cos ψ ˆ
θ ˆ = θ + ( Δ 1 + Δ 2 ) / 2 ,
ψ ˆ = ψ + ( Δ 1 Δ 2 ) / 2 .
H s ( u , υ ) = [ H ( u , υ ) · m , n = δ ( u m d ) δ ( υ n d ) ] * rect ( u / d , υ / d ) = [ H ( u , υ ) · 1 d 2 comb ( u / d , υ / d ) ] * rect ( u / d , υ / d ) ,
comb ( u , υ ) = m , n = δ ( u m ) δ ( υ n ) ,
rect ( u , υ ) = { 1 | u | , | υ | 1 2 0 othewise ,
h s ( x , y ) = [ h ( x , y ) * comb ( d x , d y ) ] · d 2 sinc ( d x , d y ) = d 2 W h ( x , y ) [ h ( x , y ) * comb ( d x , d y ) ] ,
W h ( x , y ) = sinc ( d x , d y )
sinc ( x , y ) = sin π x π x · sin π y π y .
H + P = 2 H 1 ,
H P = 2 H 2 ,
H = H 1 + H 2 = 1 2 ( H + P ) + 1 2 ( H P ) .
H s ( u , υ ) = { [ H ( u , υ ) + P ( u , υ ) ] 1 d 2 comb ( u / d , υ / d ) } * rect ( u + d / 4 d / 2 , υ d ) + { [ H ( u , υ ) P ( u , υ ) ] 1 d 2 comb ( u / d , υ / d ) } * rect ( u d / 4 d / 2 , υ d ) = [ H ( u , υ ) · 1 d 2 comb ( u / d , υ / d ) ] * rect ( u / d , υ / d ) + [ P ( u , υ ) · 1 d 2 comb ( u / d , υ / d ) ] * [ rect ( u + d / 4 d / 2 , υ d ) rect ( u d / 4 d / 2 , υ d ) ] ,
P s ( x , y ) = [ p ( x , y ) * comb ( d x , d y ) ] d 2 2 sinc ( d x / 2 , d y ) × [ exp ( j 2 π x d / 4 ) exp ( j 2 π x d / 4 ) ] = j d 2 W p ( x , y ) [ p ( x , y ) * comb ( d x , d y ) ] ,
W p ( x , y ) = sinc ( d x / 2 , d y ) sin ( π d x / 2 )
H s ( u , υ ) = H s ( u , υ ) + P s ( u , υ ) ,
P s ( u , υ ) = [ P ( u , υ ) · 1 d 2 comb ( u / d , υ / d ) ] * ( rect ( υ / d ) · { rect [ u + ( 2 n 1 ) d / 4 n d / 2 n ] rect [ u + ( 2 n 3 ) d / 4 n d / 2 n ] + + rect [ u ( 2 n 3 ) d / 4 n d / 2 n ] rect [ u ( 2 n 1 ) d / 4 n d / 2 n ] } ) .
P s ( x , y ) = j d 2 W p ( x , y ) [ p ( x , y ) * comb ( d x , d y ) ] ,
W p ( x , y ) = ( 1 ) n n sinc ( d x / 2 n , d y ) × { sin ϕ sin 3 ϕ + + ( 1 ) n 1 sin ( 2 n 1 ) ϕ } ,
W p ( x , y ; n = 2 ) = sinc ( d x / 4 , d y ) sin ( π d x / 4 ) cos ( π d x / 2 ) ,
W p ( x , y ; n = 3 ) = sin c ( d x / 6 , d y ) × sin ( π d x / 6 ) [ 1 4 3 sin 2 ( π d x / 3 ) ] ,
W p ( x , y ; n = 4 ) = sinc ( d x / 8 , d y ) sin ( π d x / 8 ) cos ( π d x / 4 ) · [ 2 sin 2 ( π d x / 4 ) 1 ] .
O ( x , y ) = | h s ( x , y ) + p s ( x , y ) | 2 = d 4 | W h ( x , y ) [ h ( x , y ) * comb ( d x , d y ) ] + j W p ( x , y ) · [ p ( x , y ) * comb ( d x , d y ) ] | 2 .
j p ( x , y ) h ( x , y ) .
P = ( 1 A 2 ) 1 / 2 exp [ j ( θ + π / 2 ) ] .
Q = { 1 [ ( 1 A 2 ) 1 / 2 ] 2 } 1 / 2 exp { j [ ( θ + π / 2 ) + π / 2 ] } = A exp ( j θ ) = H .
q = F 1 { Q } = F 1 { H } = h ,
| q | 2 = | h | 2 .
P = 1 2 exp [ j ( θ + π / 2 + ψ ) ] + 1 2 exp [ j ( θ + π / 2 ψ ) ] ,
ψ = cos 1 | P | = cos 1 ( 1 A 2 ) 1 / 2 = π / 2 ψ .
θ 1 = θ + π / 2 + ψ = θ ψ + π = θ 2 + π ,
θ 2 = θ + π / 2 ψ = θ + ψ = θ 1 ,
W h ( x ) = sinc ( d x ) ,
W p ( x ) = ( 1 ) n n sinc ( d x / 2 n ) [ sin ϕ sin 3 ϕ + + ( 1 ) n 1 · sin ( 2 n 1 ) ϕ ] ,
η 1 | W h ( 0 ) | = 1.
η 2 | W p ( n / d ) | = sinc ( 1 2 ) = 2 / π 0.64.
S / N = η / max | N | ,
| N 1 | | W p ( 1 2 d ) | = | 1 n sinc ( 1 4 n ) { sin ( π / 4 n ) sin ( 3 π / 4 n ) + + ( 1 ) n 1 sin [ ( 2 n 1 ) π / 4 n ] } | ,
( S / N ) 1 = η 1 / | N 1 | .
| N 2 | = max [ | W h ( n / d 1 2 d ) | , | W h ( n / d + 1 2 d ) | ]
| W h ( n / d 1 2 d ) | = | sinc ( n 1 2 ) | = 1 / [ π ( n 1 2 ) ] .
| W h ( n / d + 1 2 d ) | = 1 / [ π ( n + 1 2 ) ] .
| N 2 | 1 / [ π ( n 1 2 ) ]
( S / N ) 2 = η 2 / | N 2 | 2 n 1.

Metrics