Abstract

A CRT display called a scanning halftone plotter has been developed. This device can display continuous-tone pictures on a cathode ray tube and serves as the device for displaying pictorial data in digital information processing by a computer. A significant feature of this plotter is the ability to draw computer-generated holograms economically. A very simple and straightforward treatment of the theory of the computer-generated hologram is demonstrated. The simplicity in mathematical treatment is achieved mainly by use of the halftone plotter. A new algorithm for the calculation of the light propagation from object planes composing a three-dimensional object is described. Holograms consisting of nonexistent three-dimensional objects have been synthesized according to this algorithm. By use of this algorithm, the so-called hidden line problem can be solved automatically. A simple method for improving the reconstructed image quality is also described. A halftone object has been successfully reconstructed by this method. Some applications of computer-generated hologram are demonstrated.

© 1971 Optical Society of America

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References

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  1. Y. Ichioka, M. Izumi, T. Suzuki, Appl. Opt. 8, 2461 (1969).
    [CrossRef] [PubMed]
  2. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  3. B. R. Brown, A. W. Lohmann, IBM J. Res. Develop. 13, 160 (1969).
    [CrossRef]
  4. L. I. Goldfisher, J. Opt. Soc. Amer. 55, 247 (1965).
    [CrossRef]
  5. H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, Appl. Opt. 7, 2301 (1968).
    [CrossRef] [PubMed]
  6. K. Miyamoto, J. Opt. Soc. Amer. 51, 17 (1961).
    [CrossRef]

1969 (2)

Y. Ichioka, M. Izumi, T. Suzuki, Appl. Opt. 8, 2461 (1969).
[CrossRef] [PubMed]

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

1968 (1)

1967 (1)

1965 (1)

L. I. Goldfisher, J. Opt. Soc. Amer. 55, 247 (1965).
[CrossRef]

1961 (1)

K. Miyamoto, J. Opt. Soc. Amer. 51, 17 (1961).
[CrossRef]

Appl. Opt. (3)

IBM J. Res. Develop. (1)

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

J. Opt. Soc. Amer. (2)

L. I. Goldfisher, J. Opt. Soc. Amer. 55, 247 (1965).
[CrossRef]

K. Miyamoto, J. Opt. Soc. Amer. 51, 17 (1961).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Carrier and modulated waves in an ordinary hologram and a computer-generated hologram.

Fig. 2
Fig. 2

Virtual filtering system to limit the first-order diffraction beam.

Fig. 3
Fig. 3

Block diagram of the scanning halftone plotter.

Fig. 4
Fig. 4

Block diagram of modulating circuit of spot brightness.

Fig. 5
Fig. 5

Schematic diagram of connection of binary counters with corresponding digital–analog converter and illustration of minimum allowable phase shift and sampling numbers.

Fig. 6
Fig. 6

Waveforms of converted voltages corresponding to X coordinates: (a) waveforms in the case of pictorial pattern and (b) hologram display.

Fig. 7
Fig. 7

Examples of pictorial pattern and continuous-tone hologram plotted by the scanning halftone plotter.

Fig. 8
Fig. 8

Optical system for constructing three-dimensional hologram.

Fig. 9
Fig. 9

Computer-generated hologram of three-dimensional object; (a) hologram with 64 × 64 samples; (b) three-dimensional image focused on letter O; (c) same as (b) but for letter S; (d) same as (b) but for letter A.

Fig. 10
Fig. 10

Illustrating scheme for an opaque plane object O2 illuminated by the diffracted wave from another plane object O1. The two objects are separated.

Fig. 11
Fig. 11

Simulation of eclipse demonstrating the capability of reconstructing a three-dimensional image; (a) and (c) off-axis views (partial eclipse) showing parallax; (b) on-axis view (annular eclipse).

Fig. 12
Fig. 12

Reduction of speckle noise in reconstructed image; (a) reconstructed image from one hologram with 64 × 64 samples; (b) that from three holograms of the same size; (c) that from six holograms.

Fig. 13
Fig. 13

(a) Computer-generated Fraunhofer hologram of a real nonnegative object; (b) that of a synthesized object; (c) reconstructed image from the hologram shown in (b).

Fig. 14
Fig. 14

(a) Computer-generated hologram that can generate spherical wave; (b) hologram that can generate a reference spherical wave; (c) interference pattern of wavefronts generated by two holograms shown in (a) and (b).

Equations (23)

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H ( x , y ) = n m I n m g [ x - ( n + P n m ) δ x , y - m δ x , a δ x ] ,
g ( x , y , z ) = 1 / 2 π z 2 exp [ - ( x 2 + y 2 ) / 2 z 2 ] ,
h ( ν x , ν y ) = H ( x , y ) E ( x ν x + y ν y ) d x d y = const f ( ν x , ν y , a ) n m I n m E { δ x [ ( n + P n m ) ν x + m ν y ] } ,
E ( x ) = exp ( 2 π i x )
f ( ν x , ν y , a ) = exp [ - ( 2 π a δ x ) 2 ( ν x 2 + ν y 2 ) / 2 ] .
H ( x , y ) = - Δ ν / 2 Δ ν / 2 ν 0 - Δ ν / 2 ν 0 + Δ ν / 2 h ( ν x , ν y ) E [ - ( ν x x + ν y y ) ] d ν x d ν y .
U k l = H k l = H ( k δ x , l δ x ) = n m I n m E ( P n m ) - Δ ν / 2 Δ ν / 2 - Δ ν / 2 Δ ν / 2 E ( ) d ν x d ν y ,
= ( n + P n m - k ) ν x δ x + ( m - l ) ν y δ x ,
f ( ν x , ν y , a ) const .
U k l = n I n l E ( P n l ) S a ( n + P n l - k ) .
U k l = I k l E ( P k l ) S a ( P k l ) + n k I n l E ( P n l ) S a ( n + P n l - k ) .
S a ( i + P n m ) = { 1 for i = 0 , 0 otherwise ,
I n m ( 0 ) = A n m , P n m ( 0 ) = ϕ n m / 2 π ,
U k l = I k l ( i ) E ( P k l ( i ) ) S a ( P k l ( i - 1 ) ) + n k I n l ( i - 1 ) E ( P n l ( i - 1 ) ) S a ( n + P n l ( i - 1 ) - k ) , i 1 ,
E ( P n m ν x δ x ) 1 ,
{ X n m = n + P n m , Y n m = m , Z n m = A n m , n , m = 1 , 2 , 3 , 4 , , N ,
F ( x , y ) = O ( x 1 , y 1 ) exp { - i π [ ( x - x 1 ) 2 + ( y - y 1 ) 2 ] / λ g } d x 1 d y 1 ,
f ( ν x , ν y ) = o ( ν x , ν y ) exp [ i π λ g ( ν x 2 + ν y 2 ) ] ,
h ( ν x , ν y ) f ( ν x , ν y ) = o ( ν x , ν y ) S g 1 ( ν x , ν y ) ,
S g 1 ( ν x , ν y ) = exp [ i π λ g 1 ( ν x 2 + ν y 2 ) ] .
h ( ν x , ν y ) = i O i ( ν x , ν y ) S g i ( ν x , ν y ) .
O ˜ 1 ( x 2 , x 2 ) = F - 1 { F [ O 1 ( x 1 , y 1 ) ] S g i ( ν x , ν y ) } ,
h ( ν x , ν y ) = F { M [ O ˜ 1 ( x 2 , y 2 ) ] } S g 2 ( ν x , ν y ) ,

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