Abstract

A two-step imaging process discovered by Gabor involves photographing the Fresnel diffraction pattern of an object and using this recorded pattern, called a hologram, to construct an image of this object. Here, the process is described from a communication-theory viewpoint. It is shown that construction of the hologram constitutes a sequence of three well-known operations: a modulation, a frequency dispersion, and a square-law detection. In the reconstruction process, the inverse-frequency-dispersion operation is carried out. The process as normally carried out results in a reconstruction in which the signal-to-noise ratio is unity. Techniques which correct this shortcoming are described and experimentally tested. Generalized holograms are discussed, in which the hologram is other than a Fresnel diffraction pattern.

© 1962 Optical Society of America

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References

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  1. D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949).
    [CrossRef]
  2. L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).
  3. A discussion of various similar techniques for eliminating the twin image is given by Lohmann, Optica Acta (Paris) 3, 97 (1956). These are likewise developed by use of a communication theory approach.
    [CrossRef]
  4. See reference 2, p. 391.
  5. See reference 3.
  6. The impulse response of the half-plane filter is derived from Hilbert transform relations. For a discussion of Hilbert transforms, see, for example, W. Brown, Analysis of Linear Time-Invariant SystemsMcGraw-Hill Book Company, Inc., New York, 1962). As given by Brown, the Hilbert transform of a function s(t) is represented in the frequency domain by a filter which phase-shifts the positive frequencies by π, and in the time domain by the convolution integral1/π∫-∞∞s(τ)(t-τ)-1dτ.

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

1956 (1)

A discussion of various similar techniques for eliminating the twin image is given by Lohmann, Optica Acta (Paris) 3, 97 (1956). These are likewise developed by use of a communication theory approach.
[CrossRef]

1948 (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949).
[CrossRef]

Brown, W.

The impulse response of the half-plane filter is derived from Hilbert transform relations. For a discussion of Hilbert transforms, see, for example, W. Brown, Analysis of Linear Time-Invariant SystemsMcGraw-Hill Book Company, Inc., New York, 1962). As given by Brown, the Hilbert transform of a function s(t) is represented in the frequency domain by a filter which phase-shifts the positive frequencies by π, and in the time domain by the convolution integral1/π∫-∞∞s(τ)(t-τ)-1dτ.

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Gabor, D.

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949).
[CrossRef]

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Lohmann,

A discussion of various similar techniques for eliminating the twin image is given by Lohmann, Optica Acta (Paris) 3, 97 (1956). These are likewise developed by use of a communication theory approach.
[CrossRef]

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Nature (1)

D. Gabor, Nature 161, 777 (1948); Proc. Roy. Soc. (London) A197, 454 (1949).
[CrossRef]

Optica Acta (Paris) (1)

A discussion of various similar techniques for eliminating the twin image is given by Lohmann, Optica Acta (Paris) 3, 97 (1956). These are likewise developed by use of a communication theory approach.
[CrossRef]

Trans. IRE (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Other (3)

See reference 2, p. 391.

See reference 3.

The impulse response of the half-plane filter is derived from Hilbert transform relations. For a discussion of Hilbert transforms, see, for example, W. Brown, Analysis of Linear Time-Invariant SystemsMcGraw-Hill Book Company, Inc., New York, 1962). As given by Brown, the Hilbert transform of a function s(t) is represented in the frequency domain by a filter which phase-shifts the positive frequencies by π, and in the time domain by the convolution integral1/π∫-∞∞s(τ)(t-τ)-1dτ.

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Figures (12)

Fig. 1
Fig. 1

Construction of the hologram.

Fig. 2
Fig. 2

Hologram and reconstructed images.

Fig. 3
Fig. 3

Meaning of dispersion.

Fig. 4
Fig. 4

Optics for wavefront reconstruction.

Fig. 5
Fig. 5

System for producing generalized holograms.

Fig. 6
Fig. 6

Wedge for two-beam hologram construction.

Fig. 7
Fig. 7

Spectrum of hologram.

Fig. 8
Fig. 8

Optical system for single-sideband holograms.

Fig. 9
Fig. 9

Zone-plate imaging properties.

Fig. 10
Fig. 10

Results of half-plane filtering: (a) original object, (b) double sideband hologram, (c) reconstruction from (b), (d) single sideband hologram, (e) reconstruction from (d).

Fig. 11
Fig. 11

Modified carrier-frequency system.

Fig. 12
Fig. 12

Carrier-frequency hologram and reconstruction: (a) hologram of a straight wire, (b) reconstruction.

Equations (29)

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s e i ω t = [ s b + s r ( x , y ) ] e i ω t .
χ ( x , y ) = i ( λ z ) - 1 [ s b + s r ( α , β ) ] × exp { - i 2 π λ - 1 [ z 2 + ( x - α ) 2 + ( y - β ) 2 ] 1 2 } d α d β ,
[ z 2 + ( x - α ) 2 + ( y - β ) 2 ] 1 2 z + 1 2 ( x - α ) 2 z - 1 + 1 2 ( y - β ) 2 z - 1 .
χ ( x , y ) = i ( λ z ) - 1 [ s b + s r ( α , β ) ] × exp { - i k [ ( x - α ) 2 + ( y - β ) 2 ] } d α d β ,
χ ( x , y ) = s b + i ( λ z ) - 1 s r ( α , β ) × exp { - i k [ ( x - α ) 2 + ( y - β ) 2 ] } d α d β = s b + s r ( x , y ) f ( x , y ) ,
i ( λ z ) - 1 exp [ - i k ( x 2 + y 2 ) ]             and             exp [ i ( 4 k ) - 1 ( ξ 2 + η 2 ) ]
χ ( x , y ) = s b + F { S r ( ξ , η ) exp [ i ( 4 k ) - 1 ( ξ 2 + η 2 ) ] } = s b + F [ S r ( ξ , η ) F ( ξ , η ) ] ,
F ( ξ , η ) = exp [ i ( 4 k ) - 1 ( ξ 2 + η 2 ) ] .
d ϕ / d ξ = ( λ z / 2 π ) ξ             d ϕ / d η = ( λ z / 2 π ) η ,
s e i ω t = s b e i ω t + 1 2 σ e i ( ω t + A x ) + 1 2 σ e i ( ω t - A x ) .
- z sin θ - z θ = - A λ z / ( 2 π ) .
exp [ i A ( x + 1 2 A λ z / π ) ] exp ( - i 1 4 A 2 λ z / π ) = exp i ( A x + ϕ A ) ,
χ A e i ω t = s b e i ω t + 1 2 σ e i ( ω t + A x + ϕ A ) + 1 2 σ e i ( ω t - A x + ϕ A ) .
χ χ * = [ ( s b + s r ) f ] [ ( s b + s r ) f ] * ,
χ χ * = s b 2 + s b ( s r f ) + s b ( s r f ) * + s r f 2
χ A χ A * = s b 2 + 1 2 σ 2 + s b σ cos ( A x + ϕ A ) + s b σ cos ( A x - ϕ A ) + 1 2 σ 2 cos 2 A x = s b 2 + 1 2 σ 2 + 2 s b σ cos A x cos ϕ A + 1 2 σ 2 cos 2 A x .
s b ( s r f ) * f = F [ s b S r ( - ξ , - η ) F * ( - ξ , - η ) F ( ξ , η ) ] ,
a ( α , β ) = i ( λ F 1 ) - 1 s ( x , y ) × exp [ i 2 π ( λ F 1 ) - 1 ( x α + y β ) ] d x d y .
S ( ξ , η ) exp [ - i λ F 1 2 ( ξ 2 + η 2 ) ( 4 π F z ) - 1 ] ,
χ = s c e i ξ c x + s b + s r f
χ χ * = s c 2 + s b 2 + 2 s b s c cos ξ c x + s b [ s r f + ( s r f ) * ] + s c [ ( s r f ) e - i ξ c x + ( s r f ) * e i ξ c x ] + s r f 2 .
s b s c e i ξ c x + s c ( s r f ) * e i ξ c x
s b s c e - i ξ c x + s c ( s r f ) e - i ξ c x ,
χ = s b + ( s r f g )
g = 1 2 δ ( x ) + i / ( π x ) ,
χ χ * = s b 2 + s b ( s r f g ) + s b ( s r f g ) * + s r f g 2 .
( χ χ * ) f = s b 2 f + s b ( s r f g ) * f + s r f g 2 f g * .
( χ χ * ) f = s b 2 + 1 2 s b s r - 1 2 s b s r i / ( π x ) + s r f g 2 f g * .
1/π-s(τ)(t-τ)-1dτ.