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

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]

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

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

1978

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

1976

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

1975

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

1974

1972

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

1971

1970

1969

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

1967

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, 557 (1979).

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, 667 (1979).
[CrossRef]

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, IEEE Trans. Inf. Theory IT-25, 557 (1979).

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, 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.

IBM J. Res. Dev.

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

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.

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

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

Optik (Stuttgart)

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

Other

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

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

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

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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; an and bn 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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