Abstract

When a hologram is desired from an object which does not exist physically but is known in mathematical terms, one can compute the hologram. An automatic plotter will make a drawing at a large scale which is then reduced photographically. Since the drawing can contain only black and white areas, we have developed a theory for binary holograms. They are equivalent in terms of image reconstruction with ordinary holograms. This has been proven theoretically and verified experimentally.

© 1967 Optical Society of America

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References

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  1. B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [Crossref] [PubMed]
  2. D. Hauk, A. Lohmann, Optik 15, 275 (1958).
  3. C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
    [Crossref]
  4. A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
    [Crossref]
  5. J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
    [Crossref]
  6. M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
    [Crossref]
  7. A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).
  8. J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
    [Crossref]
  9. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  10. B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).
  11. A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).
  12. A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

1966 (5)

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

1965 (2)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

1964 (1)

1961 (1)

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

1958 (1)

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

1957 (1)

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

Brown, B. R.

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Hauk, D.

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Kelly, D. L.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

Kozma, A.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

Leith, E. N.

Lipson, H.

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

Lohmann, A.

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Lohmann, A. W.

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

Marquet, M.

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Paris, D. P.

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

Taylor, C. A.

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

Tsujiuchi, J.

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Upatnieks, J.

Waters, J. P.

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Letters (1)

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

IEEE J. Quantum Electron. (1)

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

J. Opt. Soc. Am. (3)

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

Math. Comp. (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Opt. Acta (2)

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Optik (1)

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Other (1)

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

Cited By

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Figures (6)

Fig. 1
Fig. 1 Setup for the construction of an image IMG from a binary hologram HOLO, illuminated by a point source x0 in SOU.

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Fig. 2
Fig. 2 Structure of the (n,m) cell in the binary hologram. The rectangular opening has a width cδν, a height Wnmδν, and it is displaced from the center of the cell at (nδν,mδν) by Pnmδν.

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Fig. 3
Fig. 3 Binary Fraunhofer holography: (a) the hologram; (b) the image.

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Fig. 4
Fig. 4 Simulation of a groundglass, i.e., object with random phase: (a) the hologram; (b) the image.

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Fig. 5
Fig. 5 A holographic image with defects owing to zero-order approximation in the calculation.

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Fig. 6
Fig. 6 Images from binary holograms of different orders: (a) M = 1; (b) M = 3.

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Equations (65)

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u ˜ ( ν x , ν y ) = ∬ u ( x , y ) E ( - x ν x - y ν y ) d x d y .
const   u ˜ ( ν x , ν y ) = H ( ν x , ν y ) E ( x 0 ν x ) .
u ( x , y ) = 0   outside   of   ∣ x ∣   ≤ Δ x / 2 ;   ∣ y ∣   ≤ Δ y / 2 ; u ˜ ( ν x , ν y ) ≈ 0   outside   of   ∣ ν x ∣   ≤ Δ ν / 2 ;   ∣ v y ∣   ≤ Δ ν ν 2.
h ( x , y ) = rect ( x / Δ x )   rect ( y / Δ x ) ∬ H ( ν x , ν y ) E [ ( x + x 0 ) ν x + y ν y ] d ν x d ν ; .
rect ( z ) = { 1 ;   in   ∣ z ∣   ≤ 1 2 0 ;   otherwise
h ( x , y ) = const u ( x , y ) .
N 2 = ( Δ x Δ y ) / ( δ x δ y ) = ( Δ x / δ x ) 2 = ( Δ x Δ ν ) 2 = S W .
( Δ ν / δ ν ) 2 ≥ ( Δ x / δ x ) 2 = ( Δ x Δ ν ) 2 ;     → δ ν ≤ 1 / Δ x .
u ˜ ( ν x , ν y ) = Σ Σ ( n , m ) u ˜ ( n / Δ x , m / Δ x ) × sinc ( ν x Δ x - n )   sinc ( ν y Δ x - m ) .
∣ ν x - ( n + P n m ) δ ν ∣   ≤ c δ ν / 2 ; ∣ ν y - m δ ν ∣   ≤ W n m δ ν / 2.
∫ ( m - 1 2 ) δ ν ( m + 1 2 ) δ ν H ( ν x , ν y ) d ν y = H m ( ν x ) ;   H m [ ( n + P n m ) δ ν + μ ] = H m [ ( n + P n m ) δ ν - μ ] ,
H ( ν x , ν y ) = Σ Σ ( n , m )   rect [ ν x - ( n + P n m ) δ ν c δ ν ]   rect [ ν y - m δ ν W n m δ ν ] .
∬ H ( ν x , ν y ) E [ ( x + x 0 ) ν x + y ν y ] d ν x d ν y = c ( δ ν ) 2   sinc [ c δ ν ( x + x 0 ) ] Σ Σ ( n , m ) W n m sinc ( y W n m δ ν ) E { δ ν [ ( x + x 0 ) ( n + P n m ) + y m ] } .
rect ( x / Δ x )   rect ( y / Δ x ) ∬ H E d ν x d ν y = h ( x , y ) ;   h ( x , y ) = const u ( x , y ) .
u ( x , y ) = ∬ u ˜ ( ν x , ν y ) E ( x ν x + y ν y ) d ν x d ν y = rect ( x / Δ x )   rect ( y / Δ x ) ∑ ( n , m ) ∑ u ˜ ( n δ ν ) , m δ ν ) E [ δ ν ( x n + y m ) ] .
c ( δ ν ) 2 W n m E [ x 0 δ ν ( n + P n m ) ] ≈ const   u ˜ ( n δ ν , m δ ν ) ; const u ˜ ( n δ ν , m δ ν ) = c ( δ ν ) 2 A n m E ( φ n m / 2 π ) ; W n m ≈ A n m ;   P n m + n ≈ φ n m / 2 π x 0 δ ν .
P n m ≈ φ n m / 2 π M .
E [ x 0 δ ν ( n + P n m ) ] = E ( M P n m ) ;   φ n m = 2 π M P n m ;   φ ¯ = 2 π M P ¯ .
( a )   sinc [ c δ ν ( x + x 0 ) ] ≈ const ,   in   ∣ x ∣   ≤ Δ x / 2 ; ( b )   sinc ( y W n m δ ν ) ≈ 1   in   ∣ y ∣   ≤ Δ x / 2 ; ( c )   E ( x P n m δ ν ) ≈ 1   in   ∣ x ∣   ≤ Δ x / 2.
∣ φ ∣   ≤ π ;   φ = 2 π M P ;   ∣ P ∣   ≤ 1 2 M .
sinc [ c δ ν ( x + x 0 ) ] ≈ const ,   in   ∣ x ∣   ≤ Δ x / 2.
sinc ( y W δ ν ) ≈ 1 ,
E ( x P n m δ ν ) ≈ 1 x   in   ∣ x ∣   ≤ Δ x / 2.
u ( x , y ) ≠ 0   only   in   ∣ x ∣ ,   ∣ y ∣ ≤ ξ Δ x / 2 ;   ξ ≤ 1.
u ( x , y ) = v ( x , y )   sinc [ c δ ν ( x + x 0 ) ] .
h ( x , y ) / sinc [ c δ ν ( x + x 0 ) ] = const v ( x , y )
v ( x , y ) = u ( x , y ) / sinc [ c δ ν ( x + x 0 ) ] = rect ( x / Δ x ) rect ( y / Δ x ) ∑ ( n , m ) ∑ v ˜ n m E [ δ ν ( x n + y m ) ] ;
v ˜ n m = ( 1 / Δ x ) 2 ∬ u / sinc   E [ - δ ν ( x n + y m ) ] d x d y ;
h ( x , y ) / sinc [ c δ ν ( x + x 0 ) ] = rect ( x / Δ x )   rect ( y / Δ x ) c ( δ ν ) 2 Σ Σ ( n , m ) W n m   sinc ( y W n m δ ν ) E [ M P n m + ( x n + y m ) δ ν + x P n m δ ν ] .
const v ˜ j k = c ( δ ν / Δ x ) 2 ∑ ( n , m ) ∑ W n m × ∫ ∫ ( - Δ x / 2 ) ( + Δ x / 2 ) sinc ( y W n m δ ν ) E [ ∊ ] d x d y ;
∊   = M P n m + ( n - j ) x δ ν + ( m - k ) y δ ν + x P n m δ ν .
const v ˜ j k = c ( δ ν ) 2 ∑ ( m , n ) ∑ W n m   E ( M P n m ) σ ( n , j ;   m , k ) ;
σ ( n , j ;   m , k ) = σ x ( n , j ;   m ) σ y ( m , k ;   n ) ;
σ x = ( 1 / Δ x ) ∫ - Δ x / 2 + Δ x / 2 E [ x δ ν ( n - j + P n m ) ] d x = sinc ( n - j + P n m ) ;
σ y = ( 1 / Δ x ) ∫ - Δ x / 2 + Δ x / 2 sinc ( y W n m δ ν ) E [ y δ ν ( m - k ) ] d y = [ Si ( m - k + W n m / 2 ) - Si ( m - k - W n m / 2 ) / W n m .
Si ( z ) = ∫ 0 z sinc ( z ′ ) d z ′ .
sinc ( y W δ ν ) = ( 1 / W δ ν ) ∫ - ∞ + ∞ rect ( μ / W δ ν ) E ( μ y ) d μ .
const v ˜ j k / c ( δ ν ) 2 = A j k E ( φ j k / 2 π ) ;
∑ ( n , m ) ∑ W n m E ( M P n m ) σ ( n , j ;   m , k ) = W j k E ( M P j k ) σ ( j , j ;   k , k ) + R j k ( W n m , P n m ) .
A j k E ( φ j k / 2 π ) = W j k E ( M P j k ) σ ( j , j ;   k , k ) + R j k ;
σ ( j , j ;   k , k ) = sinc ( P j k ) Si ( W j k / 2 ) / ( W j k / 2 ) .
R j k = 0 ;   σ ( j , j ;   k , k ) = 1.
A j k E ( φ j k / 2 π ) = W j k ( 0 ) E ( M P j k ( 0 ) ) ; W j k ( 0 ) = A j k ;   P j k ( 0 ) = φ j k / 2 π M .
A j k E ( φ j k / 2 π ) = W j k ( 1 ) E ( M P j k ( 1 ) ) σ ( W j k ( 0 ) , P j k ( 0 ) ) ;
P j k ( 1 ) = φ j k / 2 π M = P j k ( 0 ) ;
W j k ( 1 ) = A j k / σ ( W j k ( 0 ) , P j k ( 0 ) ) = W j k ( 0 ) / σ ( W j k ( 0 ) , P j k ( 0 ) ) .
A j k E ( φ j k / 2 π ) = W j k ( 2 ) E ( M P j k ( 2 ) ) σ j k ( W j k ( 1 ) , P j k ( 1 ) ) + R j k ( W n m ( 1 ) , P n m ( 1 ) ) ;
A j k E ( φ j k / 2 π ) - R j k ( W n m ( 1 ) , P n m ( 1 ) ) = B j k E ( β j k / 2 π ) ;
W j k ( 2 ) = B j k / σ j k ( W j k ( 1 ) , P j k ( 1 ) ) ;   P j k ( 2 ) = β j k / 2 π M .
σ ( n , j ; m , k ) = σ x ( n , j ; m ) σ y ( m , k ; n ) ;   σ x ( n ≠ j ; m m ) ≈ ( - 1 ) n - j   sin ( π P n m ) / π ( n - j ) ;
σ y ( m ≠ k , n ) ≈ ( - 1 ) m - k - 1 [ W n m / 2 ( m - k ) ] 2 / 3.
u ˜ ( ν x , ν y ) ≈ 0   for   ∣ ν x ∣ ,   ∣ ν y ∣ ≤ Δ ν / 2 = 1 / 2 δ x ; Δ x H = Δ y H = λ f Δ ν .
u ( n δ x , m δ x ) = u n m .
v n m = u n m / sinc [ c δ ν ( x 0 + n δ x ) ] = u n m / sinc [ c ( M + n / N ) ] ; δ ν = 1 / Δ x ;   x 0 δ ν = M .
v n m → v ˜ ( j δ ν , k δ ν ) = ∣ v ˜ j k ∣ E ( φ j k / 2 π )
const ∣ v ˜ ∣ max = c ( δ ν ) 2 W max .
const ∣ ν j k ∣ / c ( δ ν ) 2 = A j k ;   A j k ≤ W max .
∣ n - j ∣ ( m - j ) 2 ≤ n 0 3
u ( x ) → u ( x ) Σ ( n ) δ ( x - n δ x ) = v ( x ) .
u ˜ ( ν ) = ∫ u ( x ) E ( - v x ) d x ;
v ˜ ( ν ) = Σ ( n ) ∫ u ( x ) δ ( x - n δ x ) E ( - ν x ) d x = Σ ( n ) u ( n δ x ) E ( - ν n δ x ) .
Σ ( n ) δ ( x - n δ x ) = Σ ( m ) E ( m x / δ x ) ;
v ˜ ( ν ) = Σ ( m ) ∫ u ( x ) E [ x ( - ν + m / δ x ) ] d x = Σ ( m ) u ˜ ( ν - m / δ x ) .
δ x ≤ 1 / Δ ν 0 .
v ˜ ( ν x , ν y ) = Σ Σ ( n , m ) u ˜ ( ν x - n / δ x , ν y - m / δ x ) .

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