Abstract

The opportunity to process signals in domains other than the time or frequency domains arises naturally in coherently illuminated optical systems that produce Fourier transforms. It is well known that N samples are sufficient to represent the information content in the object, image, and Fourier planes. We extend these results to show that we can accurately represent the intensity signal in any Fresnel plane of a coherently illuminated optical system with exactly N samples, provided that we use a specified nonuniform sampling technique.

© 1990 Optical Society of America

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References

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  1. Special Issue, Proc. IEEE 69 (1981).
  2. G. Toraldo di Francia, “Resolving Power and Information,” J. Opt. Soc. Am. 45, 497–501 (1955); “Capacity of an Optical Channel in the Presence of Noise,” Opt. Acta 2, 5–8 (1955).
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  4. E. L. O’Neill, A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
    [CrossRef]
  5. A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  6. R. Barakat, “Application of the Sampling Theorem to Optical Diffraction Theory,” J. Opt. Soc. Am. 54, 920–930 (1964).
    [CrossRef]
  7. A. VanderLugt, “Design Relationships for Holographic Memories,” Appl. Opt. 12, 1675–1685 (1973).
    [CrossRef]
  8. A. VanderLugt, “Packing Density in Holographic Systems,” Appl. Opt. 14, 1081–1087 (1975).
    [CrossRef]
  9. R. J. Marks, J. F. Walkup, M. O. Hagler, “A Sampling Theorem for Space-Variant Systems,” J. Opt. Soc. Am. 66, 918–921 (1976).
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  10. B. D. Guenther, Modern Optics (Wiley, New York, 1990).
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  12. A. VanderLugt, “Operational Notation for the Analysis and Synthesis of Optical Data Processing Systems,” Proc. IEEE 54, 1055–1063 (1966).
    [CrossRef]

1985

1981

Special Issue, Proc. IEEE 69 (1981).

1976

1975

1973

1966

A. VanderLugt, “Operational Notation for the Analysis and Synthesis of Optical Data Processing Systems,” Proc. IEEE 54, 1055–1063 (1966).
[CrossRef]

1964

1963

E. L. O’Neill, A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

1955

Barakat, R.

Guenther, B. D.

B. D. Guenther, Modern Optics (Wiley, New York, 1990).

Hagler, M. O.

Linfoot, E. H.

Marks, R. J.

O’Neill, E. L.

E. L. O’Neill, A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

Toraldo di Francia, G.

VanderLugt, A.

Walkup, J. F.

Walther, A.

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

E. L. O’Neill, A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Fourier transform system.

Fig. 2
Fig. 2

Impulse of a sample of the object.

Fig. 3
Fig. 3

Sample spacing to produce (a) maximum frequency but minimum overlap and (b) minimum frequency but maximum overlap.

Fig. 4
Fig. 4

Optimum sampling spacing for the Fresnel transform: (a) the chirp function representation of the spatial frequency distribution; (b) the sample spacing at the lens plane.

Fig. 5
Fig. 5

Optimum sampling for Fresnel transforms. The height gives the value of the maximum spatial frequency as a function of position.

Fig. 6
Fig. 6

High bandwidth imaging system: (a) the scissors diagram; (b) the space/frequency distribution.

Fig. 7
Fig. 7

General trapezoidal region.

Fig. 8
Fig. 8

Low bandwidth imaging system: (a) the scissors diagram; (b) the space/frequency distribution.

Fig. 9
Fig. 9

Maximum frequency as a function of position.

Equations (25)

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g ( u ) = - f ( x ) exp [ - j π λ F ( u - x ) 2 ] d x ,
h ( u ) = g ( u ) exp ( j π λ F u 2 ) .
F ( ξ ) = - h ( u ) exp [ - j π λ F ( ξ - u 2 ) ] d u .
F ( ξ ) = - - f ( x ) exp [ - j π λ F ( u - x ) 2 ] × exp ( j π λ F u 2 ) exp [ - j π λ F ( ξ - u ) 2 ] d x d u = - f ( x ) exp ( j 2 π λ F ξ x ) d x .
r ( u ) = - d ( x ) exp [ - j π λ D ( u - x ) 2 ] d x ,
r ( u ) = { exp ( - j π λ D u 2 ) ; u θ co D , 0 ; else .
f ( x ) = d ( x - n d 0 / 2 ) + d ( x + n d 0 / 2 ) ,
I ( u ) = | - f ( x ) exp [ - j π λ D ( u - x ) 2 ] d x | 2 = | - [ d ( x - n d 0 / 2 ) + d ( x + n d 0 / 2 ) ] × exp [ - j π λ D ( u - x ) 2 ] d x | 2 .
I ( u ) = | exp [ - j π λ D ( u - n d 0 / 2 ) 2 ] + exp [ - j π λ D ( u + n d 0 / 2 ) 2 ] | 2 = 2 [ 1 + cos ( 2 π n d 0 u / λ D ) ] .
α f = n d 0 λ D .
α f ( u ) = L λ F ( 1 - u L ) .
c ( u ) = 1 + cos [ π λ F ( L - u ) 2 ] ,
π λ F ( L - u ) 2 = π L 2 λ F - π
u = L - L 2 - λ F ,
u = L - L 1 - λ F L 2 = L ( 1 - 1 - 1 L α co ) = L ( 1 - 1 - 2 N ) L N d 0 .
π λ F ( L - u ) 2 = π
u = L - λ F .
N f = L - 2 θ co D d 0 + 4 θ co D 2 d 0 = L d 0 = N ,
N f = 2 ( - L / 2 × θ co D ) d f + 2 L 2 d f = 2 θ co D d f ,
d F = 2 ξ co N = 2 λ F α co N = λ F L ,
α f = L λ D ,
α f = - M L λ d ,
D = - M L L F - 2 θ co M ,
α co M L ( 1 + M ) λ F .
α co - M 2 L ( M - 1 ) λ F .

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