Abstract

A general fidelity criterion and a procedure are developed to evaluate the performance of different detour-phase-type computer-generated holograms. Three types of detour-phase hologram are evaluated and compared using both the aforementioned procedure and experimental data.

© 1979 Optical Society of America

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References

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  1. J. W. Goodman, A. M. Silvestri, IBM J. Res. Dev. 14, 478 (1970).
    [CrossRef]
  2. W. J. Dallas, Appl. Opt. 10, 673 (1971).
    [CrossRef] [PubMed]
  3. W. J. Dallas, Appl. Opt. 10, 674 (1971).
    [CrossRef] [PubMed]
  4. P. S. Naidu, Opt. Commun. 15, 361 (1975).
    [CrossRef]
  5. R. S. Powers, J. W. Goodman, Appl. Opt. 14, 1690 (1975).
    [CrossRef] [PubMed]
  6. R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
    [CrossRef] [PubMed]
  7. W. J. Dallas, Appl. Opt. 13, 2274 (1974).
    [CrossRef] [PubMed]
  8. W. J. Dallas, A. W. Lohmann, Opt. Commun. 5, 78 (1972).
    [CrossRef]
  9. B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
    [CrossRef]
  10. J. P. Hugonin, P. Chavel, Opt. Commun. 16, 342 (1976).
    [CrossRef]
  11. P. Chavel, J. P. Hugonin, J. Opt. Soc. Am. 66, 989 (1976).
    [CrossRef]
  12. W. H. Lee, Appl. Opt. 13, 1677 (1974).
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  13. J. A. Bucklew, N. C. Gallagher, Appl. Opt. 18, 575 (1979).
    [CrossRef] [PubMed]
  14. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  15. C. K. Hsueh, A. A. Sawchuk, Appl. Opt. 17, 3874 (1978).
    [CrossRef] [PubMed]
  16. W. H. Lee, Appl. Opt. 9, 639 (1970).
    [CrossRef] [PubMed]
  17. N. C. Gallagher, B. Liu, Appl, Opt. 12, 2328 (1973).
    [CrossRef]
  18. N. C. Gallagher, B. Liu, Optik (Stuttgart) 42, 65 (1975).
  19. J. C. Allebach, N. C. Gallagher, B. Liu, Appl. Opt. 15, 2183 (1976).
    [CrossRef] [PubMed]
  20. J. C. Angus et al., Appl. Opt. 16, 2798 (1977).
    [CrossRef] [PubMed]

1979 (1)

1978 (1)

1977 (1)

1976 (3)

1975 (3)

P. S. Naidu, Opt. Commun. 15, 361 (1975).
[CrossRef]

R. S. Powers, J. W. Goodman, Appl. Opt. 14, 1690 (1975).
[CrossRef] [PubMed]

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

1974 (2)

1973 (1)

N. C. Gallagher, B. Liu, Appl, Opt. 12, 2328 (1973).
[CrossRef]

1972 (1)

W. J. Dallas, A. W. Lohmann, Opt. Commun. 5, 78 (1972).
[CrossRef]

1971 (2)

1970 (3)

1969 (1)

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

1967 (1)

Allebach, J. C.

Angus, J. C.

Brown, B. R.

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

Bucklew, J. A.

Chavel, P.

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

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

Dallas, W. J.

Gabel, R. A.

Gallagher, N. C.

Goodman, J. W.

R. S. Powers, J. W. Goodman, Appl. Opt. 14, 1690 (1975).
[CrossRef] [PubMed]

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

Hsueh, C. K.

Hugonin, J. P.

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

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

Lee, W. H.

Liu, B.

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

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

N. C. Gallagher, B. Liu, Appl, Opt. 12, 2328 (1973).
[CrossRef]

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

Lohmann, A. W.

W. J. Dallas, A. W. Lohmann, Opt. Commun. 5, 78 (1972).
[CrossRef]

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

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

Naidu, P. S.

P. S. Naidu, Opt. Commun. 15, 361 (1975).
[CrossRef]

Paris, D. P.

Powers, R. S.

Sawchuk, A. A.

Silvestri, A. M.

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

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Figures (7)

Fig. 1
Fig. 1

A typical Hsueh-Sawchuk resolution cell (n = 2).

Fig. 2
Fig. 2

A typical Lee resolution cell.

Fig. 3
Fig. 3

Lohmann and Hsueh-Sawchuk (SNR, R).

Fig. 4
Fig. 4

Lee and Lohmann (SNR, R).

Fig. 5
Fig. 5

Lohmann (upper) and Hsueh-Sawchuk (lower) reconstructed images (σ2 = 0.2).

Fig. 6
Fig. 6

Hsueh-Sawchuk reconstructed images [σ2 = 1 (upper), σ2 = ∞(lower).]

Fig. 7
Fig. 7

Lee reconstructed images (σ2 = 0.43).

Equations (63)

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f ( x , y ) = m = 0 N - 1 k = 0 N - 1 A m k exp ( i θ m k ) exp [ j ( x m + y k ) d ] ,
g ( x , y ) = m = 0 N - 1 k = 0 N - 1 L ( A m k , x , y ) exp [ i θ m k ( x + 1 / d ) ] × exp [ i ( x m + y k ) d ] ,
SNR = f ( x , y ) 2 d x d y f ( x , y ) - R g ( x , y ) 2 d x d y ,
f ( x , y ) 2 d x d y = R 2 g ( x , y ) 2 d x d y .
f ( x , y ) = m n A m n exp ( i θ m n ) exp [ i 2 π ( x n d + y m d ) ] ,
f ^ ( x , y ) = sinc ( x d 2 + 1 2 ) d 2 2 m n ( sin π A m n y / π y ) × exp [ i θ m n ( 1 + x d ) ] exp [ i 2 π ( x n d + y m d ) ] ,
P A ^ m n ( z )
P A ^ m n ( z ) = z σ 2 exp ( - z 2 / 2 σ 2 ) rect [ ( z - d / 2 ) / d ] + exp ( - ½ σ 2 ) δ ( z - d ) ,
mse = E ( | m n { A m n exp ( i θ m n ) - R sin [ π ( A ^ m n + δ 1 m n ) y π y ] × exp [ i ( θ m n + δ 2 m n ) ( 1 + x d ) ] } × exp [ i 2 π ( x n d + y m d ) ] | 2 ) ,
e { | m = 0 M - 1 n = 0 M - 1 b m n exp [ i 2 π ( x n d + y m d ) ] } 2 = M 2 [ E { b m n 2 } - E { b m n } 2 ]
E { b m n 2 } = E { | A exp ( i θ ) - R sin π ( A + δ 1 ) y π y × exp [ i ( θ + δ 2 ) ( 1 + x d ) ] | 2 } ,
E { b m n 2 } = E ( A 2 + R 2 π 2 y 2 { sinc ( Δ 1 , y ) sin 2 ( π y A ^ ) + 1 2 [ 1 - sinc ( Δ 1 , y ) ] } - 2 A R π y sinc ( y Δ 1 / 2 ) × sin ( π y A ^ ) sinc ( x T Δ 2 2 π ) cos ( x d θ ) ) .
E { b m n } 2 = | E { R sin [ π ( A ^ + δ 1 ) y π y ] exp [ i ( θ + δ 2 ) ( 1 + x d ) ] } | 2 .
E { b m n } 2 = R 2 | E { exp [ i θ ( 1 + x d ) ] × sin ( π y A ^ ) π y sinc ( π y Δ 1 2 π ) sinc [ ( 1 + x d ) Δ 2 2 π ] } | 2 .
mse = M { A 2 ¯ + sin 2 π A ^ y ¯ π 2 y 2 K 1 R 2 - 2 A sin π A y π y cos ( x d θ ) K 2 R ¯ - | sin π y A ^ π y exp [ i θ ( 1 + x d ) ] | 2 ¯ K 3 R 2 + K 4 R 2 } ,
K 1 - sinc ( y / N ) ,
K 2 = sinc ( y / N ) sinc ( x d / N ) ,
K 3 = { sinc ( y / 2 N ) sinc [ ( 1 + x d ) / N ] } 2 ,
K 4 = [ 1 - sinc ( y / N ) ] / 2 π 2 y 2 ,
X 1 ( y ) = 0 d sin π y A π y f ( A ) d A ,
X 2 ( y ) = 0 d sin 2 π y A π 2 y 2 f ( A ) d A ,
X 3 ( y ) = 0 d A sin ( π y A ) π y f ( A ) d A ,
f ( A ) = ( A / σ 2 ) exp ( - A 2 / 2 σ 2 ) .
mse = M [ 2 σ 2 + X 2 ( y ) + sinc 2 ( y ) exp ( - ½ σ 2 ) K 1 R 2 - 2 sinc ( x d ) ( X 3 ( y ) + sinc ( y ) { exp ( - ½ σ 2 ) + ( π 2 σ 2 / 2 ) 1 / 2 [ 1 - erf ( 1 / 2 σ 2 ) ] } ) K 2 R - K 3 R 2 sinc 2 ( 1 + x d ) [ X 1 ( y ) + sinc ( y ) exp ( - ½ σ 2 ) ] 2 + K 4 R 2 ] ,
d 1 n m = d 2 π ( θ n m + ψ n m ) ,
d 2 n m = d 2 π ( θ n m - ψ n m ) ,
h ( u , v ) = m k j = 0 n - 1 rect [ u - ( m d + d 2 - d 4 n - j d n ) d / 2 n , v - ( k d - d 2 + d 1 k m ) d / 2 ] + rect [ u - ( m d + d 2 - d 4 n - j d n ) d / 2 n , v - ( k d - d 2 + d 2 k m ) d / 2 ] .
H ( x , y ) = m k r = 0 n - 1 d 2 4 n sinc ( x d 2 n ) sinc ( y d 2 ) × exp ( i π d y ) exp ( - i y d θ m k ) exp [ - i 2 π ( x m d + y k d ) ] × 2 cos [ 2 π x ( d 2 - d 4 n - r d n ) - y d ψ m k 2 π ] .
H ( x , y ) = m k d 2 4 n sinc ( x d 2 n ) sinc ( y d 2 - 1 2 ) × exp [ - i 2 π ( x m d + y k d ) ] exp ( i π d y ) exp [ i θ m k ( 1 - y d ) ] × r = 0 n - 1 cos [ 2 π x ( d 2 - d 4 n - r d n ) + ψ m k ( 1 - y d ) ] .
sinc ( x d 2 n ) 1 ,
sin ( π x d ) sin ( π x d n ) n sin π x d π x d = n sinc ( x d ) ,
cos [ ψ k m ( y d - 1 ) - π x d 2 n ] cos [ ψ k m ( 1 - y d ) ] .
H ( x , y ) = - d 2 4 n sinc ( y d 2 - 1 2 ) exp [ i 2 π ( x m d + y k d ) ] × exp ( i π d y ) exp [ i θ k m ( 1 - y d ) ] cos [ ψ k m ( 1 - y d ) ] n sinc ( x d ) .
H ( x , y ) = m k cos [ ψ k m ( 1 - y d ) ] exp [ i θ k m ( 1 - y d ) ] × exp [ - i 2 π ( x m d + y k d ) ] .
cos ψ ( 1 - y d ) exp [ i θ ( 1 - y d ) ] = ( exp { i 2 π d ( 1 - y d ) [ ( ψ + θ ) d 2 π ] } + exp [ i 2 π d ( 1 - y d ) { ( θ - ψ ) d 2 π } ] ) 2
= { exp [ i 2 π d ( 1 - y d ) d 1 ] + exp [ i 2 π d ( 1 - y d ) d 2 ] } 2 .
exp [ i 2 π d ( 1 - y d ) ( δ 1 + δ 2 2 ) ] exp [ i θ ( 1 - y d ) ] cos [ π ( 1 - y d ) d ( δ 1 - δ 2 ) + ψ ( 1 - y d ) ] .
E { b m n 2 } = E { A exp ( i θ ) - exp [ i θ ( 1 - y d ) ] × exp [ i 2 π d ( 1 - y d ) δ 1 + δ 2 2 ) ] × cos [ π ( 1 - y d ) d ( δ 1 - δ 2 ) + ψ ( 1 - y d ) ] | 2 } .
E { A 2 + 1 2 + cos 2 ψ ( 1 - y d ) 2 S N C 2 - 2 A cos [ ( 1 - y d ) ψ ] × cos ( θ y d ) S N C } ,
E { b m n } 2 = | E { exp [ i θ ( 1 - y d ) ] exp { i 2 π d ( 1 - y d ) δ 1 + δ 2 2 ] × cos [ π d ( 1 - y d ) ( δ 1 - δ 2 ) + ψ ( 1 - y d ) ] } | 2 = E { S N C exp [ i θ ( 1 - y d ) ] cos ψ ( 1 - y d ) } 2 .
f ( z ) = 0 T x σ 2 exp [ - x 2 / 2 σ 2 ] cos ( z cos - 1 x ) d x .
2 σ 2 + 1 2 + S N C 2 2 { f [ 2 ( 1 - y d ) ] + exp ( - ½ σ 2 ) } + 2 S N C sinc ( y d ) ( f ( 2 - y d ) 2 + f ( y d ) 2 + exp ( - ½ σ 2 ) + ( π σ 2 / 2 ) 1 / 2 { 1 - erf [ 1 / ( 2 σ 2 ) 1 / 2 ] } ) .
S N C sinc ( 1 - y d ) [ f ( 1 - y d ) + exp ( - ½ σ 2 ) ] 2 .
mse = M [ 2 σ 2 + { 1 2 + SNC 2 2 [ f ( 2 - 2 y d ) + exp ( - ½ σ 2 ) ] } R 2 - 2 S N C R sinc ( y d ) ( f ( 2 - y d ) 2 + f ( y d ) 2 + exp ( - ½ σ 2 ) + ( π σ 2 / 2 ) 1 / 2 { 1 - erf [ 1 / ( 2 σ 2 ) 1 / 2 ] } ) - R 2 S N C 2 sinc 2 ( 1 - y d ) [ f ( 1 - y d ) + exp ( - ½ σ 2 ) ] 2 ] ,
H ( u , v ) = m k l r m k rect [ u - ( m d - 3 d 8 ) d / 4 , v - k d A 1 k m ] + ( 1 - l r m k ) rect [ u - ( m d + d 8 ) d / 4 , v - k d A 3 k m ] + l i m k rect [ u - ( m d - d 8 ) d / 4 , v - k d A 2 k m ] + ( 1 - l i m k ) rect [ u - ( m d + 3 d 8 ) d / 4 , v - k d A 4 k m ] ,
f ^ ( x , y ) = d 4 sinc ( x d 4 + 1 4 ) exp ( i 3 π / 4 ) m k exp [ i 2 π ( x m d + y k d ) ] × [ A 1 m k sinc ( y A 1 m k ) l r m k exp ( i 3 π d x / 4 ) - A 3 m k sinc ( y A 3 m k ) ( 1 - l r m k ) exp ( - i π d x / 4 ) - j A 2 m k sinc ( y A 2 m k ) l i m k exp ( i π d x / 4 ) + i A 4 m k sinc ( y A 4 m k ) ( 1 - l i m k ) exp ( - i 3 π d x / 4 ) ] .
M ( 2 A 1 2 ¯ + 2 R 2 sin 2 π y A ^ 1 ¯ π 2 y 2 - 4 A 1 sin π y A ^ 1 ¯ π y R cos ( π d x / 2 ) cos ( π d x / 4 ) - sin 3 ( π d x / 2 ) ( sin π y A ^ 1 ¯ π y ) 2 - { 2 R sin π y A ^ 1 ¯ π y sin ( π d x / 2 ) cos [ π ( d x + 1 ) / 4 ] } 2 ) ,
S 0 ( y ) = 0 T sin π y A 1 π y f ( A 1 ) d A 1 ,
S 1 ( y ) = 0 T sin 2 π y A 1 π 2 y 2 f ( A 1 ) d A 1 ,
S 2 ( y ) = 0 T A 1 sin π y A 1 π y f ( A 1 ) d A 1 ,
f ( A 1 ) = ( 2 / π σ 2 ) 1 / 2 exp ( - A 1 2 / 2 σ 2 ) .
sin 2 π y A ^ 1 ¯ π 2 y 2 = ( S 1 ( y ) + erf [ 1 / ( 2 σ 2 ) 1 / 2 ] sinc 2 ( y ) ,
A 1 sin π y A ^ 1 ¯ π y = S 2 ( y ) + sin c ( y ) exp ( - ½ σ 2 ) ( 2 σ 2 / π ) 1 / 2 ,
sin π y A ^ 1 ¯ π y = S 0 ( y ) + sinc ( y ) erfc [ 1 / ( 2 σ 2 ) 1 / 2 ] ,
mse = M [ 2 σ 2 + R 2 [ 1 - sinc ( y / N ) π 2 y 2 ] + sinc ( y / N ) 2 R 2 { S 1 ( y ) + erfc [ 1 / ( 2 σ 2 ) 1 / 2 ] sinc 2 ( y ) } - R 4 sinc ( y / 2 N ) cos ( π d x / 2 ) cos ( π d x / 4 ) [ S 2 ( y ) + sinc ( y ) ( 2 σ 2 ) 1 / 2 / π exp ( - ½ σ 2 ) ] - ( 2 R sinc ( y / 2 N ) sin ( π d x / 2 ) cos [ π ( d x + 1 ) / 4 ] × { S 0 ( y ) + sinc ( y ) erfc [ 1 / ( 2 σ 2 ) 1 / 2 ] } ) 2 ] .
E g ( x , y ) 2 = N 2 [ E { L ( A , x , y ) 2 } - E { L ( A , x , y ) } 2 sinc 2 ( 1 + x d ) ] .
E | m = 0 N - 1 n = 0 N - 1 a m n exp [ i 2 π ( x n T + y m T ) ] | 2 ,
E m n p q a m n a p q * exp { i 2 π [ x ( n - q ) T + y ( m - p ) T } ,
m n E a m n 2 + E a m n 2 m n p q exp { i 2 π [ x ( n - q ) T + y ( m - p ) T ] } ,             ( p , q ) ( m , n )
N 2 E a m n 2 + E a m n 2 m p exp [ i 2 π y ( m - p ) T ] n q exp [ i 2 π x ( n - q ) T ] ,             ( p , q ) ( m , n )
N 2 E a m n 2 - f ( N ) E a m n 2 ,
f ( N ) = N 4 - N 2 i f k = l = 0 , N 2 otherwise .
N 2 ( E a m n 2 - E a m n 2 ) .

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