Abstract

An expression for the required special bandwidth of a computer generated hologram is desired using results from the theory of frequency modulation. The expression is the same as a rule of thumb first presented by Lee. A simple quantization error model is presented for one type of nondetour phase class. It is shown that this hologram may be used to achieve a near optimum simulation of the quantization problem for random phase images.

© 1980 Optical Society of America

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References

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  1. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [Crossref] [PubMed]
  2. B. R. Brown, S. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
    [Crossref]
  3. J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
    [Crossref]
  4. W. J. Dallas, Appl. Opt. 10, 673 (1971).
    [Crossref] [PubMed]
  5. W. J. Dallas, Appl. Opt. 10, 674 (1971).
    [Crossref] [PubMed]
  6. R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
    [Crossref] [PubMed]
  7. W. J. Dallas, A. W. Lohmann, Opt. Commun. 5, 18 (1972).
    [Crossref]
  8. J. A. Bucklew, N. C. Gallagher, Appl. Opt. 18, 2861 (1979).
    [Crossref] [PubMed]
  9. J. P. Hugonin, P. Chavel, Opt. Commun. 16, 342 (1976).
    [Crossref]
  10. W.-H. Lee, Appl. Opt. 13, 1677 (1974).
    [Crossref] [PubMed]
  11. W.-H. Lee, Appl. Opt. 18, 3661 (1979).
    [Crossref] [PubMed]
  12. W.-H. Lee, Appl. Opt. 9, 639 (1970).
    [Crossref] [PubMed]
  13. A. B. Carlson, Communication System (McGraw-Hill, New York, 1968).
  14. C. B. Burckhardt, Appl. Opt. 9, 1949 (1970).
    [Crossref] [PubMed]
  15. L. Franks, Signal Theory (Prentice Hall, Englewood Cliffs, N.J., 1969), p. 73.
  16. N. C. Gallagher, IEEE Trans. Inf. Theory IT-24, 156 (1978).
    [Crossref]
  17. J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 557 (1979).
  18. J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 667 (1979).
    [Crossref]
  19. N. C. Gallagher, B. Liu, Optik (Stuttgart) 42, 65 (1975).
  20. A. Gersho, IEEE Trans. Inf. Theory IT-25, 373 (1979).
    [Crossref]
  21. W. A. Pearlman, Stanford U. Information System Laboratory Technical Report 6503-1, (1974).

1979 (5)

J. A. Bucklew, N. C. Gallagher, Appl. Opt. 18, 2861 (1979).
[Crossref] [PubMed]

W.-H. Lee, Appl. Opt. 18, 3661 (1979).
[Crossref] [PubMed]

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 557 (1979).

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 667 (1979).
[Crossref]

A. Gersho, IEEE Trans. Inf. Theory IT-25, 373 (1979).
[Crossref]

1978 (1)

N. C. Gallagher, IEEE Trans. Inf. Theory IT-24, 156 (1978).
[Crossref]

1976 (1)

J. P. Hugonin, P. Chavel, Opt. Commun. 16, 342 (1976).
[Crossref]

1975 (1)

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

1974 (1)

1972 (1)

W. J. Dallas, A. W. Lohmann, Opt. Commun. 5, 18 (1972).
[Crossref]

1971 (2)

1970 (4)

1969 (1)

B. R. Brown, S. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[Crossref]

1967 (1)

Brown, B. R.

B. R. Brown, S. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[Crossref]

Bucklew, J. A.

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 667 (1979).
[Crossref]

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 557 (1979).

J. A. Bucklew, N. C. Gallagher, Appl. Opt. 18, 2861 (1979).
[Crossref] [PubMed]

Burckhardt, C. B.

Carlson, A. B.

A. B. Carlson, Communication System (McGraw-Hill, New York, 1968).

Chavel, P.

J. P. Hugonin, P. Chavel, Opt. Commun. 16, 342 (1976).
[Crossref]

Dallas, W. J.

Franks, L.

L. Franks, Signal Theory (Prentice Hall, Englewood Cliffs, N.J., 1969), p. 73.

Gabel, R. A.

Gallagher, N. C.

J. A. Bucklew, N. C. Gallagher, Appl. Opt. 18, 2861 (1979).
[Crossref] [PubMed]

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 557 (1979).

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 667 (1979).
[Crossref]

N. C. Gallagher, IEEE Trans. Inf. Theory IT-24, 156 (1978).
[Crossref]

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

Gersho, A.

A. Gersho, IEEE Trans. Inf. Theory IT-25, 373 (1979).
[Crossref]

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
[Crossref]

Hugonin, J. P.

J. P. Hugonin, P. Chavel, Opt. Commun. 16, 342 (1976).
[Crossref]

Lee, W.-H.

Liu, B.

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

R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
[Crossref] [PubMed]

Lohmann, A. W.

W. J. Dallas, A. W. Lohmann, Opt. Commun. 5, 18 (1972).
[Crossref]

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
[Crossref] [PubMed]

Lohmann, S. W.

B. R. Brown, S. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[Crossref]

Paris, D. P.

Pearlman, W. A.

W. A. Pearlman, Stanford U. Information System Laboratory Technical Report 6503-1, (1974).

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
[Crossref]

Appl. Opt. (9)

IBM J. Res. Dev. (2)

B. R. Brown, S. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[Crossref]

J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
[Crossref]

IEEE Trans. Inf. Theory (4)

N. C. Gallagher, IEEE Trans. Inf. Theory IT-24, 156 (1978).
[Crossref]

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 557 (1979).

J. A. Bucklew, N. C. Gallagher, IEEE Trans. Inf. Theory IT-25, 667 (1979).
[Crossref]

A. Gersho, IEEE Trans. Inf. Theory IT-25, 373 (1979).
[Crossref]

Opt. Commun. (2)

J. P. Hugonin, P. Chavel, Opt. Commun. 16, 342 (1976).
[Crossref]

W. J. Dallas, A. W. Lohmann, Opt. Commun. 5, 18 (1972).
[Crossref]

Optik (Stuttgart) (1)

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

Other (3)

A. B. Carlson, Communication System (McGraw-Hill, New York, 1968).

W. A. Pearlman, Stanford U. Information System Laboratory Technical Report 6503-1, (1974).

L. Franks, Signal Theory (Prentice Hall, Englewood Cliffs, N.J., 1969), p. 73.

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Figures (7)

Fig. 1
Fig. 1

Representation of a fringe cross section. Δ is the minimum pen plotter increment; x indicates possible pen plotter positions; a n and b n are the desired pen plotter positions.

Fig. 2
Fig. 2

Typical sampling cell from a projection type NDPH.

Fig. 3
Fig. 3

Representation of a 1-D quantizer. If a data sample falls in the ith input interval then the output of the quantizer is the ith output level (represented by heavy dots).

Fig. 4
Fig. 4

Representation of a 2-D quantizer. If a data vector falls within some tile then the output is chosen to be the output level associated with that tile (heavy dots).

Fig. 5
Fig. 5

Graph of the probability density r(x); the arrows indicate Dirac delta functions of area l/t.

Fig. 6
Fig. 6

Reconstructed image from a projection hologram with thirteen subcells.

Fig. 7
Fig. 7

Reconstructed image from a detour phase version of a projection hologram where a single transform sample value is used for each subcell.

Tables (1)

Tables Icon

Table I Comparison of Predicted Error with Simulation Error for Lee Type Holograms

Equations (35)

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m = sin [ m sin 1 A ( ω , λ ) ] m π exp { i m [ ϕ ( ω , λ ) + 2 π α ω ] } ,
1 / T 1.5 π max x , y | ϕ ( x , y ) x | Δ 1.5 B x
B x = 1 π max x , y | ϕ ( x , y ) x | .
f Δ = W = 1 2 π max x , y | ϕ ( x , y ) x | .
β F M 4 W = 2 π max x , y | ϕ ( x , y ) x | = 2 B x ,
[ u ( ω a n ) u ( ω + b n ) ] exp ( i 2 π ω x ) d ω .
[ u ( ω a n ϕ n ) u ( ω + b n + ϕ n ) ] exp ( i 2 π ω x ) d ω
e n ( x ) = sin π x θ n π x exp ( i 2 π x a n i π x θ n ) sin π x ϕ n π x exp ( i 2 π x b n i π x ϕ n ) .
E { | n e n ( x ) | 2 } = MSE .
E { | e n ( x ) | 2 } = 1 π 2 x 2 ( 1 sin π x Δ π x Δ ) cos 2 π x ( b n a n ) 2 π 2 x 2 ( 1 sin π x Δ π x Δ ) 2 .
E { | n e n ( x ) | 2 } = n | e n ( x ) | 2 + 1 π 2 x 2 ( 1 sin π x Δ π x Δ ) 2 × { m n sin π x ( b n a n ) sin π x ( b m a m ) × exp [ i π x ( a m + b m a n b n ) ] } .
K ( π x ) 2 ( 1 sin π Δ π Δ x ) ,
( 1 sin π x Δ π x Δ ) 2 ( 1 sin π x Δ π x Δ ) .
A ( ω , λ ) π max [ A ( ω , λ ) ] exp [ i ϕ ( ω , λ ) ] exp ( i 2 π α ω ) .
u ˆ ( x , y ) = m n k = 0 N 1 A n m k ( T 2 N ) sin c ( x T N ) sin c ( y A n m k ) × exp [ i 2 π ( x m T + y n T ) ] exp [ i 2 π ( k + 1 2 ) T N x ] ,
u ˆ ( x , y ) = m n k = 0 N 1 A n m k ( T 2 N ) sin c [ ( x + 1 T ) T N ] sin c ( y A n m k ) × exp [ i 2 π ( x m T + y n T ) ] exp [ i 2 π ( K + 1 2 ) T N ( x + 1 T ) ] .
u ˆ ( x , y ) T 2 N exp ( i π / N ) exp ( π T x / N ) k = 0 N 1 exp ( i 2 π k l / N ) × exp ( i 2 π k x T / N ) m n A n m k exp [ i 2 π ( x m T + y n T ) ] .
A ( u + k T / N , υ ) u { cos [ θ ( u + k T / N , υ ) 2 π k / N ] } × cos [ θ ( u + k T / N , υ ) 2 π k / N ] ,
m , n = A n m k exp [ i 2 π ( x m T + y n T ) ] = 1 T 2 × p , q = a k ( x p T , y q T ) × exp [ i 2 π k N ( x p T ) ] ,
a k ( x , y ) = F 1 { A ( u , υ ) u [ cos [ θ ( u , υ ) 2 π k N ] ] } ,
u ˆ ( x , y ) = 1 N exp [ i π N ( T x + 1 ) ] k = 0 N 1 exp [ i 2 π k N ( 1 + p ) ] × p , q = a k ( x p T , y q T ) = exp [ i π N ( T x + 1 ) ] p , q = F 1 1 N k = 0 N 1 × exp [ i 2 π k N ( 1 + p ) ] A ( u , υ ) × u { cos [ θ ( u , υ ) 2 π k N ] cos [ θ ( u , υ ) 2 π k N ] } × exp [ i 2 π ( u p + υ q ) / T ] .
u [ cos ( x ) ] = 1 2 + 2 π n = 0 ( 1 ) n 2 n + 1 cos [ ( 2 n + 1 ) x ] .
A 2 N k = 0 N 1 exp [ i 2 π k N ( 1 + p ) ] cos ( θ 2 π k N ) = A 2 N exp ( i θ ) × [ exp ( i 2 π p ) 1 exp ( i 2 π p N ) 1 ] + A 2 N exp ( i θ ) [ exp [ i 2 π ( 2 + p ) ] 1 exp [ ι ˙ 2 π N ( 2 + p ) ] 1 ] .
u ˆ ( x , y ) = 1 2 exp [ π N ( T x + 1 ) ] F 1 A ( u , υ ) exp [ i θ ( u , υ ) ] .
m s e = C R R 2 p ( x ) λ ( x ) d x ,
1 N k = 0 N 1 A k exp ( i 2 π k / N ) = X 1 + i X 2 ,
{ X k } N = 1 k = 0 ;
R N + i I N = 1 N k = 0 N 1 X k exp ( i 2 π k / N ) .
R N T / 24 N N ( 0,1 ) , I N T / 24 N N ( 0,1 ) ,
k = 0 N 1 A m n k exp ( i 2 π k / N ) ,
K = 0 N 1 Z K exp ( i 2 π K / N ) ,
E { | e n ( x ) | 2 } = 1 4 π 2 x 2 E { | exp ( i π x θ n ) exp ( i π x θ n ) | 2 } + 1 4 π 2 x 2 E { | exp ( i π x ϕ n ) exp ( i π x ϕ n ) | 2 } + Cross Terms = 1 π 2 x 2 [ 1 E { cos 2 π x θ n } ] + Cross Terms = 1 π 2 x 2 [ 1 sin π x Δ π x Δ ] + Cross Terms . Cross Terms = E { 1 π 2 x 2 sin ( π x θ n ) sin ( π x ϕ n ) ( exp [ i 2 π x ( b n a n ) ] × exp [ i π x ( ϕ n θ n ] + exp [ i 2 π x ( a n b n ) ] × exp [ i π x ( θ n ϕ n ) ] ) } .
Cross Terms = E { 1 π 2 x 2 [ cos π x ( θ n ϕ n ) cos π x ( ϕ n + θ n ) ] × [ C cos π x ( ϕ n θ n ) S sin π x ( ϕ n θ n ) ] } .
Cross Terms = cos 2 π x ( b n a n ) 2 π 2 x 2 ( 1 sin π x Δ π x Δ ) 2 .
E { | n e n ( x ) | 2 } = E { n | e n ( x ) | 2 + m n sin π x θ n π 2 x 2 sin π x θ m exp [ i 2 π x ( a m a n + θ m θ n 2 ) ] m n sin π x θ m sin π x ϕ n π 2 x 2 exp [ i 2 π x ( b m a n + ϕ m θ n 2 ) ] m n sin π x ϕ m sin π x θ n π 2 x 2 exp [ i 2 π x ( b m a n + ϕ m θ n 2 ) ] + m n sin π x ϕ m sin π x ϕ n π 2 x 2 exp [ i 2 π x ( b m ϕ n + ϕ m ϕ n 2 ) ] } .

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