Abstract

This paper presents an analysis of the number of diffraction-limited resolvable elements and the spatial extent of object space when a hologram of a specific number of independent elements is used. The analysis includes a study of both plane and spherical reference waves, and both plane and solid objects. A generalized spatial offset of the reference wave is used to separate the desired hologram terms. The spatial extent and number of elements that can be defined in the object are reduced by spatial offset. A system is mentioned in which separation is accomplished without spatial offset, by temporal offset of the frequency of the reference beam. Without spatial offset, in the optimum configuration, the number of resolvable elements of a plane object, using the Rayleigh criterion, is equal to the number of independent elements in the hologram. For solid objects, a hologram with n × n independent samples can resolve n3/3 elements in object space, if the optimum configuration is used.

© 1970 Optical Society of America

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References

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  1. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  3. A. Macovski, Ph.D. dissertation, 1968, Stanford University; University Microfilms, Ann Arbor, Mich., Order No. 69-258.
  4. A. Macovski, Appl. Phys. Letters 14, 166 (1969).
    [Crossref]
  5. L. H. Enloe, W. C. Jakes, and C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968).
    [Crossref]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill Book Co., New York, 1968).
  7. Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press, Ltd., London, 1965), 3rd ed.

1969 (1)

A. Macovski, Appl. Phys. Letters 14, 166 (1969).
[Crossref]

1968 (1)

L. H. Enloe, W. C. Jakes, and C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968).
[Crossref]

1962 (1)

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Enloe, L. H.

L. H. Enloe, W. C. Jakes, and C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968).
[Crossref]

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill Book Co., New York, 1968).

Jakes, W. C.

L. H. Enloe, W. C. Jakes, and C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968).
[Crossref]

Leith, E. N.

Macovski, A.

A. Macovski, Appl. Phys. Letters 14, 166 (1969).
[Crossref]

A. Macovski, Ph.D. dissertation, 1968, Stanford University; University Microfilms, Ann Arbor, Mich., Order No. 69-258.

Rubinstein, C. B.

L. H. Enloe, W. C. Jakes, and C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968).
[Crossref]

Upatnieks, J.

Appl. Phys. Letters (1)

A. Macovski, Appl. Phys. Letters 14, 166 (1969).
[Crossref]

Bell System Tech. J. (1)

L. H. Enloe, W. C. Jakes, and C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Other (3)

A. Macovski, Ph.D. dissertation, 1968, Stanford University; University Microfilms, Ann Arbor, Mich., Order No. 69-258.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill Book Co., New York, 1968).

Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press, Ltd., London, 1965), 3rd ed.

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Figures (10)

Fig. 1
Fig. 1

Plane object with point-source reference in the object plane. The extent of the object is X by Y, and the extent of the hologram is W by L. The reference point source is separated from the center of the object by a distance s in the x dimension.

Fig. 2
Fig. 2

Spatial-frequency domain for a point-source-reference hologram. The minimum value of spatial offset necessary to separate the conjugate image terms from the |U|2 term is used.

Fig. 3
Fig. 3

Spatial-frequency domain for a plane-reference-wave hologram. The minimum angular offset necessary to separate the conjugate image terms from the |U|2 term is used.

Fig. 4
Fig. 4

Defined object space for a point-source-reference hologram with which nxλ/W>W/Z and nyλ/L>L/Z. The point source is a distance s from the hologram plane.

Fig. 5
Fig. 5

Defined object space for a point-source-reference hologram with which (nxλ/W) <L/Z.

Fig. 6
Fig. 6

Number of cross-section elements, Nx and Ny, vs distance from the hologram plane, zi, where the point-source reference is a distance z from the hologram plane. For the cases in which (nyλ/L) <L/Z and (nxλ/W)<W/Z, Nx and Ny go to zero with increasing zi, whereas in the opposite case they reach asymptotes of nxW2 z and nyL2z, respectively.

Fig. 7
Fig. 7

Defined object space for a point-source-reference hologram with which nλ/D=D/z. A symmetrical hologram is used, where L=W=D.

Fig. 8
Fig. 8

Defined object space for a plane-reference-wave hologram.

Fig. 9
Fig. 9

Spatial-frequency domain for a plane-reference-wave hologram. A symmetrical hologram recorder is used, where L=W=D and nx=ny=n.

Fig. 10
Fig. 10

Spatial-frequency domain of a hologram in which |U|2 is suppressed. The configuration of Fig. 1 is used with sufficient spatial offset to separate the conjugate image terms from each other.

Equations (63)

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I = R e j ω t + U = R 2 + U 2 + R * U e - j ω t + U * R e j ω t ,
U = i a i exp j ( k / 2 z i ) [ ( x - x i ) 2 + ( y - y i ) 2 ] ,
R = exp j ( k / 2 z ) [ ( x + s ) 2 + y 2 ] ,
and             n x = 2 W F x , n y = 2 L F y .
R * U = i a i exp - j k z [ ( x i + s ) x + y i y ] .
f x = ( 2 π ) - 1 ( θ / x ) = ( x i + s ) / λ z , f y = ( 2 π ) - 1 ( θ / y ) = y i / λ z .
f x max = ( X / 2 ) + s λ z ,             f y max = Y / 2 λ z .
X max = 2 ( λ z F x - s ) , Y max = 2 λ z F y .
X max = ( λ z / W ) n x - 2 s , Y max = ( λ z / L ) n y .
h ( x , y ) = sinc [ ( W / λ z ) x ] sinc [ ( L / λ z ) y ] .
Δ x = ( λ / W ) z ;             Δ y = ( λ / L ) z .
N x = X max Δ x = n x - 2 s W λ z , N y = Y max Δ y = n y .
U 2 = i a i 2 + i         j i j a i a j * exp - j k z × [ x ( x i - x j ) + y ( y i - y j ) ] .
f x max = X / λ z ,             f y max = Y / λ z .
N x = n x - 2 s W / λ z = n x / 4.
I = 1 + U 2 + i a i exp - j k 2 z [ ( x - x i ) 2 + ( y - y i + z β ) 2 ] + i a i exp j k 2 z [ ( x - x i ) 2 + ( y - y i + z β ) 2 ] .
f x = ( 2 π ) - 1 ( θ / x ) = ( λ z ) - 1 ( x - x i ) , f y = ( 2 π ) - 1 ( θ / y ) = ( λ z ) - 1 ( y - y i + z β ) .
and             F x = n x / 2 W = ( λ z ) - 1 ( W / 2 + X / 2 ) F y = n y / 2 L = ( λ z ) - 1 [ ( L / 2 ) + ( Y / 2 ) + z β ] .
and             N x = n x - W 2 λ z = n x 1 + W / X N y = n y - L 2 λ z - 2 β L λ = n y - 2 β L / λ 1 + L / Y .
β = ( L + 3 Y ) / 2 z .
N y = n y - 2 β L / λ 1 + L / Y = n y 2 ( 2 + L / Y ) .
I = K 2 + U 2 + 2 K i a i cos k z i × [ ( x i + s ) x + y i y + 1 2 ( z i - z z ) ( x 2 + y 2 ) ] .
f x = 1 2 π θ x = 1 λ z i [ x i + s + x ( z i - z z ) ] f y = 1 2 π θ y = 1 λ z i [ y i + y ( z i - z z ) ] .
f x max = 1 λ z i [ X 2 + s + W 2 ( z i - z z ) ] , f y max = 1 λ z i [ Y 2 + L 2 ( z i - z z ) ] ,
f x max = 1 λ z i [ X 2 + s + W 2 ( z - z i z ) ] , f y max = 1 λ z i [ Y 2 + L 2 ( z - z i z ) ] .
X 0 = 2 F x λ z - 2 s = z n x λ / W - 2 s , Y 0 = 2 F y λ z = z n y λ / L .
X max = ( z i - z ) ( n x λ W - W z ) + X 0 Y max = ( z i - z ) ( n y λ L - L z ) + Y 0 ,
X max = ( z i - z ) ( n x λ W + W z ) + X 0 Y max = ( z i - z ) ( n y λ L + L z ) + Y 0 .
z i max = min { W - 2 s W / z - n x λ / W , L L / z - n y λ / L } .
z i min = max { W + 2 s n x λ / W + W / z , L n y λ / L + L / z } .
Δ z = 2 λ ( z i / D ) 2 ,
N x = X max Δ x = n x - 2 s W λ z i - W 2 λ ( 1 z - 1 z i ) , N y = Y max Δ y = n y - L 2 λ ( 1 z - 1 z i )
N x = n x - 2 s W λ z i - W 2 λ ( 1 z i - 1 z ) , N y = n y - L 2 λ ( 1 z i - 1 z ) .
N x N y Δ z = { [ n - 2 s D λ z i - D 2 λ ( 1 z - 1 z i ) ] × [ n - D 2 λ ( 1 z - 1 z i ) ] / 2 λ ( z i / D ) 2 }
N x N y Δ z = { [ n - 2 s D λ z i - D 2 λ ( 1 z i - 1 z ) ] × [ n - D 2 λ ( 1 z i - 1 z ) ] / 2 λ ( z i / D ) 2 }
N = z i min z i max N x N y Δ z d z i .
N = z i min z D 2 2 λ z i 2 [ n - D 2 λ ( 1 z i - 1 z ) ] 2 d z i + z z i max D 2 2 λ z i 2 [ n - D 2 λ ( 1 z - 1 z i ) ] d z i .
N = n 3 / 3.
N = n 3 / 3 + 1 6 ( D 2 / λ z - n ) 3 .
U 2 = i a i 2 + i         j i j a i a j cos k 2 [ 2 x ( x i z i - x j z j ) + 2 y ( y i z i - y j z j ) + ( x 2 + y 2 ) ( 1 z i - 1 z j ) ] .
f x = 1 2 π θ x = 1 λ [ x i + x z i - x j + x z j ] , f y = 1 2 π θ y = 1 λ [ y i + y z i - y j + y z j ] .
f x max = 1 2 λ z [ X 0 + x ] + 1 2 λ z [ X 0 - x ] = X 0 λ z .
X 0 / 2 + s λ z = X 0 λ z ,
s = 3 X 0 / 2.
s = 3 8 D .
N = 7 / 8 z z n 3 2 [ 4 z z i 2 - 11 2 z 2 z i 3 + 7 4 z 3 z i 4 ] d z i + z n 3 8 z 3 z i 4 d z i = 9 / 196 n 3 .
X = n x λ z i / W - W , Y = n y λ z i / L - L - 2 β z i .
z i min = max { W 2 n x λ , L 2 n y λ - 2 β L } .
N x N y Δ z ( = n - D 2 λ z i n - D 2 λ z i - 2 β D λ ) / ( 2 λ z i D ) 2 .
N = D 2 / n λ D 2 2 λ z i 2 ( n - D 2 2 λ z i ) d z i = n 3 / 6.
f y max = 1 2 λ ( Y i z i + Y j z j ) + D 2 λ ( 1 z i - 1 z j ) .
f y max = 1 λ [ n λ D - 2 β - D z j ] .
n D - 2 β λ = 1 λ z i [ - Y 2 - D 2 + z i β ] , β = 3 8 ( n λ / D ) .
N = 4 D 2 / n λ [ D 2 2 λ z i 2 n - D 2 λ z i n 4 - D 2 λ z i ] d z i = 11 768 n 3 .
I = R 2 + U 2 + R * U + U * R R 2 [ 1 + R * U R 2 + U * R R 2 ] .
s = X / 2 ,             N x = n x / 2
β = L + Y 2 Z ,             N y = n y 2 ( 1 + L / Y ) .
s = X 0 / 2 ,             z i min = 3 4 z ,             N = 13 108 n 3
β = n λ / 4 D ,             N = ( 5 / 96 ) n 3 .
Point-source reference , including U 2 term , N = n 2 / 16 , suppressing the U 2 term , N = n 2 / 4.
Plane reference wave , including U 2 term , N = n 2 4 ( 2 + D / Y ) 2 , suppressing the U 2 term , N = n 2 4 ( 1 + D / Y ) 2 .
Point-source reference , including U 2 term , N = n 3 / 84 , suppressing the U 2 term , N = n 3 / 18.
Plane reference wave , including U 2 term , N = n 3 / 384 , suppressing the U 2 term , N = n 3 / 48.