Abstract

A simple procedure is presented for determining the Fourier-optics-regime object–image relationship for spatially coherent wave fields in a general imaging system. The procedure links straightforward ray-optics methods for imaging system analysis with parameters of a wave-optics-based superposition kernel that represents the Fourier-optics aspects of the system. Standard formulas are presented, and several confirming examples are treated.

© 1994 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  3. A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).
  4. F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983).
  5. S. H. Lee, ed., Optical Information Processing Fundamentals (Springer-Verlag, New York, 1981).
    [CrossRef]
  6. D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), Chap. 2.
  7. As used in this Letter, the term space invariant allows for possible magnification and inversion of the image.
  8. D. N. Sitter, “Space invariant modeling in three-dimensional optical image formation,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

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

O’Shea, D. C.

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), Chap. 2.

Sitter, D. N.

D. N. Sitter, “Space invariant modeling in three-dimensional optical image formation,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

Yu, F. T. S.

F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983).

Other (8)

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

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983).

S. H. Lee, ed., Optical Information Processing Fundamentals (Springer-Verlag, New York, 1981).
[CrossRef]

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), Chap. 2.

As used in this Letter, the term space invariant allows for possible magnification and inversion of the image.

D. N. Sitter, “Space invariant modeling in three-dimensional optical image formation,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).

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Figures (2)

Fig. 1
Fig. 1

Two-lens imaging system to illustrate the key steps of the analysis: (a) determining the location of the aperture stop, (b) locating the entrance pupil by finding the image of aperture stop as seen from object space (the exit pupil is found similarly), (c) locating the system pupil by extending marginal axial rays from object and image planes toward the center of system.

Fig. 2
Fig. 2

Examples for the analysis: (a) two single-lens Fourier-transform lenses in cascade, (b) single-lens system with aperture stop in back focal plane, (c) single lens system with aperture stop in the plane of the lens.

Equations (5)

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U im ( x , y ) = - - U obj ( ξ , η ) h sv ( x , y ; ξ , η ) d ξ d η ,
h sv ( x , y ; ξ , η ) = exp [ j k 2 R 1 ( ξ 2 + η 2 ) ] × 1 λ 2 d o d i P sys ( ξ λ d o ± x λ d i , η λ d o ± y λ d i ) × exp [ j k 2 R 2 ( x 2 + y 2 ) ] .
U im ( x , y ) = exp [ j k 2 R 2 ( x 2 + y 2 ) ] × - - U g ( ξ , η ) h ( x - ξ , y - η ) d ξ d η = exp [ j k 2 R 2 ( x 2 + y 2 ) ] [ U g ( x , y ) * * h ( x , y ) ] ,
U g ( x , y ) = 1 M U ( x M , y M ) ,
h ( x , y ) = ( 1 λ d i ) 2 P sys ( x λ d i , y λ d i ) .

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