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

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

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

1961

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

1958

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

1957

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.

Appl. Phys. Letters

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

IEEE J. Quantum Electron.

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

J. Opt. Soc. Am.

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.

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

Opt. Acta

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

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

Other

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

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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.

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.

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

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

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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