Abstract

In this study we investigate holography by scanning the receiver, the source, the object, or both source and receiver. We show that by scanning source and receiver we obtain increased resolution, an image closer to the hologram and the possibility of undistorted magnification. We present an analysis giving image position, magnification, and aberrations. We also present experimental evidence in support of the theory.

© 1969 Optical Society of America

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References

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  1. K. Preston and J. L. Kreuzer, Appl. Phys. Letters 10, 150 (1967).
    [Crossref]
  2. A. F. Metherell and S. Spinak, Appl. Phys. Letters 13, 22 (1968).
    [Crossref]
  3. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [Crossref]
  4. L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. IEEE 54, 1026 (1966).
    [Crossref]

1968 (1)

A. F. Metherell and S. Spinak, Appl. Phys. Letters 13, 22 (1968).
[Crossref]

1967 (2)

E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
[Crossref]

K. Preston and J. L. Kreuzer, Appl. Phys. Letters 10, 150 (1967).
[Crossref]

1966 (1)

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. IEEE 54, 1026 (1966).
[Crossref]

Champagne, E. B.

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. IEEE 54, 1026 (1966).
[Crossref]

Kreuzer, J. L.

K. Preston and J. L. Kreuzer, Appl. Phys. Letters 10, 150 (1967).
[Crossref]

Leith, E. N.

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. IEEE 54, 1026 (1966).
[Crossref]

Metherell, A. F.

A. F. Metherell and S. Spinak, Appl. Phys. Letters 13, 22 (1968).
[Crossref]

Porcello, L. J.

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. IEEE 54, 1026 (1966).
[Crossref]

Preston, K.

K. Preston and J. L. Kreuzer, Appl. Phys. Letters 10, 150 (1967).
[Crossref]

Spinak, S.

A. F. Metherell and S. Spinak, Appl. Phys. Letters 13, 22 (1968).
[Crossref]

Vivian, W. E.

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. IEEE 54, 1026 (1966).
[Crossref]

Appl. Phys. Letters (2)

K. Preston and J. L. Kreuzer, Appl. Phys. Letters 10, 150 (1967).
[Crossref]

A. F. Metherell and S. Spinak, Appl. Phys. Letters 13, 22 (1968).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. IEEE 54, 1026 (1966).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Geometry and symbolism for scanned holography.

Fig. 2
Fig. 2

Scanned-receiver hologram (p = 0) (a) and reconstructed true image (b).

Fig. 3
Fig. 3

Optical system for hologram reconstruction.

Fig. 4
Fig. 4

Scanned-source and receiver hologram (p = 1) (a) and reconstructed true image (b).

Fig. 5
Fig. 5

Fourier transform of scanned receiver (a) and scanned-source and receiver (b) holograms.

Fig. 6
Fig. 6

Reconstructed true image of scanned-object hologram with point source.

Fig. 7
Fig. 7

Reconstructed true image of in-line scanned-object hologram with plane-source irradiation.

Equations (24)

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ϕ 1 ( x , y , z ) = ± k 1 ( r 1 + r 0 - r 2 ) - k 2 r a ,
ϕ 2 ( x , y , z ) = - k r b .
ϕ 1 ( x , y , z ) = ± k 1 { r 1 + ( x / m x ) ( x / m x - 2 x 1 ) 2 r 1 - ( x / m x ) 2 ( x / m x - 2 x 1 ) 2 8 r 1 3 + ( y / m y ) ( y / m y - 2 y 1 ) 2 r 1 - ( y / m y ) 2 ( y / m y - 2 y 1 ) 2 8 r 1 3 + ( z / m z ) ( z / m z - 2 z 1 ) 2 r 1 - ( z / m z ) 2 ( z / m z - 2 z 1 ) 2 8 r 1 3 + - ( x 2 / m x 2 - 2 x 1 x / m x ) ( y 2 / m y 2 - 2 y 1 y / m y ) 4 r 1 3 - ( y 2 / m y 2 - 2 y 1 y / m y ) ( z 2 / m z 2 - 2 z 1 z / m z ) 4 r 1 3 - ( z 2 / m z 2 - 2 z 1 z / m z ) ( x 2 / m x 2 - 2 x 1 x / m x ) 4 r 1 3 + r 0 + μ [ μ - 2 ( x 1 - x 0 ) ] 2 r 0 - μ 2 [ μ - 2 ( x 1 - x 0 ) ] 2 8 r 0 3 + ξ [ ξ - 2 ( y 1 - y 0 ) ] 2 r 0 - ξ 2 [ ξ - 2 ( y 1 - y 0 ) ] 2 8 r 0 3 + η [ η - 2 ( Z 1 - z 0 ) ] 2 r 0 - η 2 [ η - 2 ( z 1 - z 0 ) ] 2 8 r 0 3 - [ μ 2 - 2 μ ( x 1 - x 0 ) ] [ ξ 2 - 2 ξ ( y 1 - y 0 ) ] 4 r 0 3 - [ μ 2 - 2 μ ( x 1 - x 0 ) ] [ η 2 - 2 η ( z 1 - z 0 ) ] 4 r 0 3 - [ ξ 2 - 2 ξ ( y 1 - z 0 ) ] [ η 2 - 2 η ( z 1 - z 0 ) ] 4 r 0 3 + - r 2 - ( x / m x ) ( x / m x - 2 x 2 ) 2 r 2 + ( x / m x ) 2 ( x / m x - 2 x 2 ) 2 8 r 2 3 - ( y / m y ) ( y / m y - 2 y 2 ) 2 r 2 + ( y / m y ) 2 ( y / m y - 2 y 2 ) 2 8 r 2 3 - ( z / m z ) ( z / m z - 2 z 2 ) 2 r 2 + ( z / m z ) 2 ( z / m z - 2 z 2 ) 2 8 r 2 3 + [ ( x / m x ) 2 - 2 x 2 ( x / m x ) ] [ ( y / m y ) 2 - 2 y 2 ( y / m y ) ] 4 r 2 3 + [ ( x / m x ) 2 - 2 x 2 ( x / m x ) ] [ ( z / m z ) 2 - 2 z 2 ( z / m z ) ] 4 r 2 3 + [ ( y / m y ) 2 - 2 y 2 ( y / m y ) ] [ ( z / m z ) 2 - 2 z 2 ( z / m z ) ] 4 r 2 3 + } - k 2 { r a + x ( x - 2 x a ) 2 r a - x 2 ( x - 2 x a ) 2 8 r a 3 + y ( y - 2 y a ) 2 r a - y 2 ( y - 2 y a ) 2 8 r a 3 + z ( z - 2 z a ) 2 r a - z 2 ( z - 2 z a ) 2 8 r a 3 - ( x 2 - 2 x a x ) ( y 2 - 2 y a y ) 4 r a 3 - ( x 2 - 2 x a x ) ( z 2 - 2 z a z ) 4 r a 3 - ( y 2 - 2 y a y ) ( z 2 - 2 z a z ) 4 r a 3 + }
ϕ 2 ( x , y , z ) = - k 2 { r b + x ( x - 2 x b ) 2 r b - x 2 ( x - 2 x b ) 2 8 r b 3 + y ( y - 2 y b ) 2 r b - y 2 ( y - 2 y b ) 2 8 r b 3 + z ( z - 2 z b ) 2 r b - z 2 ( z - 2 z b ) 2 8 r b 3 - ( x 2 - 2 x b x ) ( y 2 - 2 y b y ) 4 r b 3 - ( x 2 - 2 x b x ) ( z 2 - 2 z b z ) 4 r b 3 - ( y 2 - 2 y b y ) ( z 2 - 2 z b z ) 4 r b 3 + } .
μ = i = 0 a i x i ξ = i = 0 b i y i η = i = 0 c i z i .
μ = a 0 + a 1 x ξ = b 0 + b 1 y η = c 0 + c 1 z .
1 r b = ± k 1 m x 2 k 2 [ 1 r 1 - 1 r 2 + a 1 2 r 0 ] - 1 r a = ± k 1 m y 2 k 2 [ 1 r 1 - 1 r 2 + b 1 2 r 0 ] - 1 r a = ± k 1 m z 2 k 2 [ 1 r 1 - 1 r 2 + c 1 2 r 0 ] - 1 r a
x b r b = ± k 1 m x k 2 { x 1 r 1 - x 2 r 2 + a 1 r 0 ( x 1 - x 0 ) } - x a r a y b r b = ± k 1 m y k 2 { y 1 r 1 - y 2 r 2 + b 1 r 0 ( y 1 - y 0 ) } - y a r a z b r b = ± k 1 m z k 2 { z 1 r 1 - z 2 r 2 + c 1 r 0 ( z 1 - z 0 ) } - z a r a .
1 r b = ± k 1 m 2 k 2 [ 1 r 1 + p 2 r 0 - 1 r 2 ] - 1 r a x b r b = ± k 1 m k 2 [ x 1 ( 1 r 1 + p r 0 ) - x 2 r 2 - p r 0 x 0 ] - x a r a y b r b = ± k 1 m k 2 [ y 1 ( 1 r 1 + p r 0 ) - y 2 r 2 - p r 0 y 0 ] - y a r a .
radial M R = r b r 1 = ± k 1 m 2 k 2 ( r b r 1 ) 2 × { 1 + p 2 ( r 1 r 0 ) 3 [ 1 - ( x 0 x 1 + y 0 y 1 + z 0 z 1 r 1 2 ) ] } angular M α = α b α 1 = ± k 1 m k 2 ( sin α 1 sin α b ) × { 1 + p r 1 r 0 [ 1 - D 0 ( x 1 + x 0 ) r 0 2 sin ( A 0 + α 1 ) sin α 1 ] } ,
lateral M L = ( r b / r 1 ) ( α b / α 1 ) = ( r b / r 1 ) M α .
M R = k 2 k 1 ( sin α b sin α 1 ) 2 1 + p 2 ( 1 + p ) 2 · M L 2 .
M R = ( k 2 / k 1 ) [ ( sin α b / sin α 1 ) 2 ] M L 2 .
ϕ 0 ( x , y , z ) = ± k 1 ( r 1 + r 0 - r 2 ) .
Δ ϕ 0 ( x , y , z ) = [ δ ϕ 0 ( x , y , z ) / δ x 1 ] Δ x 1 = k 1 [ ( x 1 - x ) r 1 + ( x 1 - p x - x 0 ) r 0 ] Δ x 1 .
Δ x 1 λ 1 / 2 x max ( 1 r 1 + p r 0 ) ,
( λ 1 R / 2 X ) Δ x 1 ,
1 / R = ( 1 / R 1 ) + 1 / R 0
R 1 = r 1 when x = X R 0 = r 0 when x 0 = p X .
( λ 1 R / 2 X ) Δ x 1 ( λ 1 R 0 / 2 X ) .
Δ x 1 = λ 1 R 1 / 2 X .
S x = ± k 1 m 4 k 2 [ 1 r 1 3 + p 4 r 0 3 - 1 r 2 3 ] - 1 r a 3 + 1 r b 3
C x = ± k 1 m 3 k 2 [ x 1 r 1 3 + p 3 x 0 r 0 3 - x 2 r 2 3 ] + x a r a 3 - x b r b 3 C y = ± k 1 m 3 k 2 [ y 1 r 1 3 + p 3 y 0 r 0 3 - y 2 r 2 3 ] + y a r a 3 - y b r b 3
A x = ± k 1 m 2 k 2 [ x 1 2 r 1 3 + p 2 x 0 2 r 0 3 - x 2 2 r 2 3 ] + x a 2 r a 3 - x b 2 r b 3 A y = ± k 1 m 2 k 2 [ y 1 2 r 1 3 + p 2 y 0 2 r 0 3 - y 2 2 r 2 3 ] + y a 2 r a 3 - y b 2 r b 3 A x y = ± k 1 m 2 k 2 [ x 1 y 1 r 1 3 + p 2 x 0 y 0 r 0 3 - x 2 y 2 r 2 3 ] + x a y a r a 3 - x b y b r b 3 .