Abstract

A series representation developed from holography theory is used to determine the image distances and relative image intensities for three types of zone plates. The effect of nonlinearity in film reproduction on the hologram of a point object and the images produced by this hologram are discussed.

© 1967 Optical Society of America

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References

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  1. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  2. A. W. Lohmann, Appl. Opt. 4, 1667 (1965).
    [CrossRef]
  3. L. Mertz, J. Opt. Soc. Am. 51, No. 1 (1961).
  4. N. O. Young, Sky and Telescope 26, 8 (1963).
  5. D. Gabor, Proc. Soc. Roy. (London) A197, 454 (1966).
  6. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill Book Co., Inc., New York, 1957), 3rd ed., p. 361.
  7. M. H. Horman, unpublished. Details of this work are available in Boeing Document D2-90157, June1962.
  8. G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).

1966 (1)

D. Gabor, Proc. Soc. Roy. (London) A197, 454 (1966).

1965 (1)

1963 (1)

N. O. Young, Sky and Telescope 26, 8 (1963).

1962 (1)

1961 (1)

L. Mertz, J. Opt. Soc. Am. 51, No. 1 (1961).

1952 (1)

G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).

Gabor, D.

D. Gabor, Proc. Soc. Roy. (London) A197, 454 (1966).

Horman, M. H.

M. H. Horman, unpublished. Details of this work are available in Boeing Document D2-90157, June1962.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill Book Co., Inc., New York, 1957), 3rd ed., p. 361.

Leith, E. N.

Lohmann, A. W.

Mertz, L.

L. Mertz, J. Opt. Soc. Am. 51, No. 1 (1961).

Rogers, G. L.

G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).

Upatnieks, J.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill Book Co., Inc., New York, 1957), 3rd ed., p. 361.

Young, N. O.

N. O. Young, Sky and Telescope 26, 8 (1963).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Proc. Roy. Soc. (Edinburgh) (1)

G. L. Rogers, Proc. Roy. Soc. (Edinburgh) A63, 193 (1952).

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

D. Gabor, Proc. Soc. Roy. (London) A197, 454 (1966).

Sky and Telescope (1)

N. O. Young, Sky and Telescope 26, 8 (1963).

Other (2)

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill Book Co., Inc., New York, 1957), 3rd ed., p. 361.

M. H. Horman, unpublished. Details of this work are available in Boeing Document D2-90157, June1962.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of a Gabor zone plate production.

Fig. 2
Fig. 2

Geometry for the Gabor zone plate derivation.

Fig. 3
Fig. 3

Schematic diagram showing the effect of film nonlinearity.

Fig. 4
Fig. 4

Comparison of a binary zone plate and a Fresnel zone plate.

Fig. 5
Fig. 5

Schematic diagram showing the first four real image points of a binary zone plate.

Fig. 6
Fig. 6

Photographs taken at the points specified in Fig. 5.

Fig. 7
Fig. 7

Schematic diagram of an off-axis zone plate production.

Fig. 8
Fig. 8

An overexposed hologram for an off-axis zone plate.

Fig. 9
Fig. 9

Schematic diagram of the wavefront reconstruction of an off-axis overexposed hologram showing the position of the first four images.

Fig. 10
Fig. 10

A series of multiple exposed pictures taken at the positions indicated on Fig. 9.

Equations (17)

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x 2 = 2 R b λ + b 2 λ 2 2 R b λ ,
A ( x ) = [ A 0 2 + A 1 2 / R 2 + 2 A 0 ( A 1 / R ) cos ( 2 π x 2 / 2 R λ ] ½ .
T ( x ) = K [ A 0 2 + A 1 2 / R 2 + 2 A 0 ( A 1 / R ) cos π x 2 / R λ ] ,
T n ( x ) = K [ A 0 2 + ( n A n R ) 2 + 2 A 0 n A n R cos ( n π x 2 R λ ) ] .
T n ( θ ) = I 0 + I n cos ( n θ ) .
T ˜ ( θ ) = [ I ¯ 0 + I ¯ 1 cos θ + I ¯ n cos n θ ] ,
I ¯ 0 = K [ A 0 2 + i = 1 n ( i A i R ) 2 ] I ¯ 1 = 2 K [ A 0 A 1 R + i = 1 n - 1 ( i ( i + 1 ) A i A i + 1 R 2 ) ] I ¯ 2 = 2 K [ A 0 ( 2 A 2 ) R + i = 1 n - 2 i ( i + 2 ) A i A i + 2 R 2 ] : I n = 2 K A 0 ( n A n / R )
θ = π x 2 / R λ .
F ( θ ) = { 0 , - π < θ < - π / 2 K A 0 2 , - π / 2 θ < π / 2 0 , π / 2 θ π ,
F ( θ ) = K A 0 2 [ 1 2 + 2 π j = 1 ( - 1 ) j - 1 cos [ ( 2 j - 1 ) θ ] ( 2 j - 1 ) ] .
A 0 2 K 2 2 K 1 = A 0 2 + i = 1 ( i A i R ) 2 = A 0 2 + j = 1 [ ( 2 j - 1 ) A ( 2 j - 1 ) R ] 2
2 A 0 2 K 2 π K 1 = 2 A 0 A 1 R + 2 i = 1 i ( i + 1 ) A i A i + 1 R 2 = 2 A 0 A 1 R
0 = 2 A 0 2 A 2 R + 2 i = 1 i ( i + 2 ) A i A i + 2 R 2 = 2 j = 1 ( 2 j - 1 ) ( 2 j + 1 ) A 2 j - 1 A 2 i + 1 R 2
- 2 A 0 2 K 2 3 π K 1 = 2 A 0 3 A 3 R + 2 i = 1 i ( i + 3 ) A i A i + 3 R 2 = 2 A 0 3 A 3 R
0 = 2 A 0 4 A 4 R + 2 i = 1 i ( i + 4 ) A i A i + 4 R 2 = 2 j = 1 ( 2 j - 1 ) ( 2 j + 3 ) A 2 j - 1 A 2 j + 3 R 2 .
H ( θ ) = { 0 , - π < θ < - K A 0 2 , - θ 0 , < θ π .
H ( θ ) = K A 0 2 [ π - 2 π n = 1 sin ( n ) n cos ( n θ ) ] ] .

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