Abstract

Images reconstructed from binary digital holograms are degraded by errors due to the binary representation of the complex-valued object spectrum and by errors due to computational and plotter limitations. In this paper, representation-related errors are analyzed in terms of false images that appear in the desired reconstruction order and in adjacent diffraction orders. It is shown that the false images are strongly dependent on the manner in which the object spectrum is sampled and on the mapping from spectral sample to binary transmittance. Three categories of digital holograms are distinguished: those that sample the object spectrum at the center of each hologram cell; those that sample at the center of each aperture; and those that sample at each resolvable spot. In going from the first to the third category, the reconstruction is successively less degraded by false images. For the third category, there are no false images in the desired reconstruction order and only one false image in each adjacent order. The two methods in this category differ only in the suppression of these false images.

© 1981 Optical Society of America

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References

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1980 (2)

R. C. Fairchild, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 1 (1980).

J. P. Allebach, Appl. Opt. 19, 2513 (1980).
[CrossRef] [PubMed]

1979 (4)

1978 (7)

1977 (2)

1976 (3)

1975 (5)

1974 (3)

1971 (1)

1970 (4)

1969 (1)

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

1967 (1)

1966 (1)

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 976 (1966).

Allebach, J. P.

Angus, J. C.

Bartelt, H. O.

H. O. Bartelt, K. D. Forster, Opt. Commun. 26, 12 (1978).
[CrossRef]

Bastiaans, M. J.

Becker, H.

H. Becker, W. J. Dallas, Opt. Commun. 15, 50 (1975).
[CrossRef]

Biedermann, K.

Brown, B. R.

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

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 976 (1966).

Bucklew, J.

Burckhardt, C. B.

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]

Coffield, F. E.

Dallas, W. J.

Edwards, R. V.

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 1 (1980).

Fienup, J. R.

R. C. Fairchild, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 1 (1980).

Forster, K. D.

H. O. Bartelt, K. D. Forster, Opt. Commun. 26, 12 (1978).
[CrossRef]

Gabel, R. A.

Gallagher, N. C.

Giriappa, T.

Goodman, J. W.

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

Holmgren, O.

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]

Ih, C. S.

Keegan, J. J.

J. P. Allebach, J. J. Keegan, J. Opt. Soc. Am. 68, 1440A (1978).

Kermisch, D.

Kong, N.

Lee, W. H.

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

W. H. Lee, Appl. Opt. 13, 1677 (1974).
[CrossRef] [PubMed]

W. H. Lee, Appl. Opt. 9, 639 (1970).
[CrossRef] [PubMed]

W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1978), Vol. 16, p. 121.
[CrossRef]

Liu, B.

Lohmann, A. W.

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]

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 976 (1966).

Mann, J. A.

Mitsuhashi, Y.

Morikawa, T.

Naida, P. S.

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

Nakajima, M.

Paris, D. P.

Pearlman, W. A.

W. A. Pearlman, Ph.D. Thesis, Stanford, U., Standard, Calif., (University Microfilms, Ann Arbor, Mich., 1974).

Roetling, P. G.

Sahara, M.

Sawchuk, A. A.

Silvestri, A. M.

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

Strand, T. C.

T. C. Strand, Opt. Eng. 13, 219 (1974).
[CrossRef]

Appl. Opt. (20)

IBM J. Res. Dev. (2)

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

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

J. Opt. Soc. Am. (6)

Opt. Commun. (4)

H. O. Bartelt, K. D. Forster, Opt. Commun. 26, 12 (1978).
[CrossRef]

H. Becker, W. J. Dallas, Opt. Commun. 15, 50 (1975).
[CrossRef]

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

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

Opt. Eng. (1)

T. C. Strand, Opt. Eng. 13, 219 (1974).
[CrossRef]

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

R. C. Fairchild, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 215, 1 (1980).

Other (2)

W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1978), Vol. 16, p. 121.
[CrossRef]

W. A. Pearlman, Ph.D. Thesis, Stanford, U., Standard, Calif., (University Microfilms, Ann Arbor, Mich., 1974).

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Figures (2)

Fig. 1
Fig. 1

General structure of the hologram cell. U × V hologram cell is divided into an M × N array of S × T sample cells. Binary transmittance pattern within each sample cell depends only on the value of the object spectrum at the center of the sample cell and the position of the sample cell in the array.

Fig. 2
Fig. 2

Structure of the hologram cell for the methods that sample the object spectrum at the center of each hologram cell.

Tables (7)

Tables Icon

Table I Functions P and Q for Methods that Use One Spectral Sample per Hologram Cell

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Table II Weighting p(x,y) and Nonlnearities drs[a exp()] for Methods that Use One Spectral Sample per Hologram Cell

Tables Icon

Table III Normalized Bounds for False Images in the Reconstruction Order with Methods that Use One Spectral Sample per Hologram Cell a

Tables Icon

Table IV Function Qmn[a exp()] for Methods that Use One Spectral Sample per Resolution Cell

Tables Icon

Table V Thresholds for Allebach-Keegan Hologram a

Tables Icon

Table VI Nonlinearities drs[a exp()] for Methods that Use One Spectral Sample per Resolution Cell

Tables Icon

Table VII Normalized Maxima of the Nonlinearities for the Lee (1974) and Allebach-Keegan Holograms with M = N = 16

Equations (47)

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F ( u , v ) = f ( x , y ) exp [ - i 2 π ( u x + v y ) ] d x d y ,
H ( u , v ) = m n O m n [ u - m S , v - n T ; F ( m S , n T ) ] .
O m n [ u , v ; b ] = { 0 or 1 , u < S / 2 and v < T / 2 0 , u S / 2 or v T / 2 ;
O m + k M , n + l N [ u , v ; b ] = O m n [ u , v ; b ] ,
O m n [ u , v ; b ] = P ( u , v ) * Q m n [ u , v , b ] .
O m n [ u , v ; a exp ( i α ) ] = rect [ 2 ( u - U / 2 π ) / U , v / ( a V ) ] , P ( u , v ) = rect ( 2 u / U ) δ ( v ) , Q m n [ u , v ; a exp ( i α ) ] = δ ( u - α U / 2 π ) rect [ v / ( a V ) ] ,
h ( x , y ) = 1 S T p ( x , y ) m n g m n ( x - m / U , y - n / V ) ,
g m n ( x , y ) = k l sinc [ S ( x - k / S ) , T ( y - l / T ) f m + k M , n + l N ( x , y ) ,
f r s ( x , y ) CFT d r s [ F ( u , v ) ] ,
d r s [ b ] = 1 M N k , M - 1 l = 0 N - 1 q k l [ r / U , s / V ; b ] × exp [ i 2 π ( k r / M + l s / N ) ] ,
d r s ¯ [ b ] = ρ b ,
η = 1 S T p ( r ¯ / U , s ¯ / V ) ρ .
max x , y f r s ( x , y ) W 2 max u , v d r s [ F ( u , v ) ] W 2 max b d r s [ b ] ,
g m n ( x , y ) = k l sinc [ U ( x - y / U ) , V ( y - l / V ) ] f m + k , n + l ( x , y ) ;
d r s [ b ] = q [ r / U , s / V ; b ] ,
Q m [ u , v ; a exp ( i α ) ] = δ ( u ) rect [ v / b m U ) ] , M = 0 , 1 , 2 , 3 ,
c m = ( 2 a / 3 ) cos ( α - 2 π m / 3 ) + K / 3 ,
d r s [ b ] = 1 M k = 0 M - 1 q k [ r / U , s / V ; b ] exp [ i 2 π k r / M ] .
g m n ( x , y ) = l sinc [ V ( y - l / V ) ] f m , n + l ( x , y ) .
P ( u , v ) = rect ( u / R , v / R ) ;
Q m n [ u , v ; b ] = Q m n [ b ] δ ( u , v ) .
d r s [ b ] = 1 M N k , M - 1 l = 0 N - 1 Q k l [ b ] exp [ i 2 π ( k r / M + l s / N ) ] ,
g m n ( x , y ) = f m n ( x , y ) .
p ( x , y ) = R 2 sinc ( R x , R y ) .
d 10 [ a exp ( i α ) ] = ρ a ^ exp ( i α ^ ) , ρ 1 / π ,
d ˜ M / 4 , 0 [ b j ] = 4 M N k , M / 4 - 1 l = 0 N - 1 Q 4 k , l [ b j ] = b ^ j ,
d M / 4 , 0 [ a exp ( i α ) ] = 1 4 j = 0 3 exp ( i π j / 2 ) b ^ j
P ( u , v ) = rect ( 2 u U ) δ ( v ) ; Q [ u , v ; a exp ( i α ) ] = δ ( u - α U 2 π ) rect ( v a V ) .
P ( u , v ) = rect ( 4 u U ) δ ( v ) , Q [ u , v ; a exp ( i α ) ] = k = 0 3 δ [ u - ( 2 k - 3 ) U 8 ] rect ( v b k V ) , a exp ( i α ) = b 0 + i b 1 - b 2 - i b 3 , b k = { a cos ( α - l π / 2 ) , k = l α sin ( α - l π / 2 ) , k = ( l + 1 ) mod 4             l = [ α / ( π / 2 ) ] 0 , elsewhere .
P ( u , v ) = rect ( 3 u U ) δ ( v ) , Q [ u , v ; a exp ( i α ) ] = k = 0 2 δ [ u - ( k - 1 ) U 3 ] rect ( v c k V ) , a exp ( i α ) = c 0 + exp ( i 2 π / 3 ) c 1 + exp ( i 4 π / 3 ) c 2 , c k = { a 2 3 cos ( α - π / 6 - l 2 π / 3 ) ,             k = l α 2 3 sin ( α - l 2 π / 3 ) ,             k = ( l + 1 ) mod 3 , l = [ α / ( 2 π / 3 ) ] 0 , else .
P ( u , v ) = rect ( 2 u U , 2 N v V ) , Q [ u , v ; a exp ( i α ) ] = S ( u - α 1 U 2 π , v - V 4 N ) + S ( u - α 2 U 2 π , v + V 4 N ) , S ( u , v ) = k = 0 N - 1 δ [ u , v - ( 2 k - N + 1 ) V 2 N ] , a exp ( i α ) = ½ [ exp ( i α 1 ) + exp ( i α 2 ) ] , α 1 = α + ψ             α 2 = α - ψ             ψ = cos - 1 ( a ) .
p ( x , y ) = U 2 sinc ( U 2 x ) , d r s [ a exp ( i α ) ] = V exp ( i r α ) a sinc ( s a ) .
p ( x , y ) = U 4 sinc ( U 4 x ) , d r s [ a exp ( i α ) ] = V exp ( - i 3 π r / 4 ) k = 0 3 exp ( i π k r / 2 ) b k sinc ( s b k ) .
p ( x , y ) = U 3 sinc ( U 3 x ) , d r s [ a exp ( i α ) ] = V exp ( - i 2 π r / 3 ) k = 0 2 exp ( i 2 π k r / 3 ) c k sinc ( s c k ) .
p ( x , y ) = U V 4 N sinc ( U 2 x , V 2 N y ) , d r s [ a exp ( i α ) ] = 2 N exp { i [ r α - π ( N - 1 ) s / N ] } cos ( r ψ + π s 2 N ) δ s mod N .
Q m n [ a exp ( i α ) ] = { 1 , m α - M 4 m m α + M 4 - n a n n a 0 , else , m α = [ M α 2 π ] , - π α < π , f n a = [ a N 2 ] , 0 a 1.
Q m n [ a exp ( i α ) ] = { 1 , b m mod 4 t m / 4 , n 0 , else , b m , m = 0 , 1 , 2 , 3 given in Table I , t m n             given in Table V .
d r s [ a exp ( i α ) ] = 1 M N exp ( i r α ^ ) sin [ π ( 1 2 + 1 M ) r ] sin ( π r M ) sin ( π s a ^ ) sin ( π s / N ) α ^ = 2 π m α M             a ^ = 2 n a + 1 N ;
d r s [ a exp ( i α ) ] = 1 4 j = 0 3 exp ( i 2 π r j / M ) d ˜ r s [ b j ] , d ˜ r s [ b j ] = 4 M N k , M / 4 - 1 l = 0 N - 1 Q 4 k , l [ b j ] exp [ i 2 π ( 4 k r / M + l s / N ) ] .
Q [ u , v ; b ] = k , M - 1 l = 0 N - 1 Q k l [ u - k S , u - l T ; b ] ,
Q ˜ [ u , v ; b ] = m n Q [ u - m U , v - n V ; b ] ,
F ˜ ( u , v ) = r s F ( r S , s T ) rect [ ( u - r S ) / S , ( v - s T ) / T ] .
H ( u , v ) = P ( u , v ) * Q ˜ [ u , v ; F ˜ ( u , v ) ] ,
h ( x , y ) = p ( x , y ) r s f ˜ r s ( x - r / U , y - s / V ) ,
f ˜ r s ( x , y ) CFT 1 U V q [ r / U , s / V ; F ˜ ( u , v ) ] ,
q [ r / U , s / V ; F ˜ ( u , v ) ] = k l q [ r / U , s / V ; F ( k S , l T ) ] × rect [ ( u - k S ) / S , ( v - l T ) / T ] .
f ˜ r s ( x , y ) = 1 U V sinc ( S x , T y ) k l f r s ( x - k / S , y - l / T ) .

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