Abstract

The theoretical irradiance is presented for the images of sharp-edged objects as a function of the coherence of the illumination of the object. These distributions are presented for optical systems possessing both square apertures and circular apertures. The irradiance for degraded edged objects is also determined for optical systems with square apertures. The influence of the degree of coherence upon contrast and edge gradient is considered in detail. Experimental results are given which show good agreement with the theoretical irradiance. In all cases the systems are assumed free from aberration and the quasimonochromatic assumption is employed.

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  1. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.
  3. D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).
  4. M. De and S. C. Som, Opt. Acta 4, 17 (1962).
  5. W. N. Charman, J. Opt. Soc. Am. 53, 410 (1963).
  6. W. T. Welford, Optics in Meteorology (Pergamon Press, Inc., New York, 1960), pp. 85–91.
  7. H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
  8. The mutual intensity is found by setting τ equal to zero in the expression for the mutual-coherence function which is useful when investigating the imaging process employing quasimonochromatic light.
  9. One of the restrictions imposed by Hopkins, namely that the mutual-intensity function in the object plane is spatially stationary, can be removed by employing other expressions for the propagation of the mutual-intensity function. These are J(x1,x2)=K∫∫J12) exp[i(x2·ξ-x1·ξ1)]dξ1dξ2 for the propagation of the mutual intensity between planes sufficiently separated to allow the far-field approximation and J′(x1,x2)=J(x1,x2)T(x1)T*(x2) for propagation through diffracting objects or apertures having a transmittance function T(x). We note in the first expression that the "far-field approximation may be valid" in the near field as shown by the case when J12) represents an incoherent source and the first expression reduces to the Van Cittert–Zernike theorem, which Hopkins uses to describe the effective source.
  10. D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which wefuse. However, the normalization criteria are different.
  11. In this case the normalized mutual-intensity function or complex degree of coherence measured perpendicular to the edged object is given by Г0(u1-u2)=[sin(u2-u1)ε]/[u2-u1)ε]
  12. By coherence interval, we mean the minimum separation between points in the object plane (spatial separation) for which the mutual-intensity function vanishes.
  13. See Fig. 8 of Ref. 5. Private communication with W. N. Charman revealed that a typographical error resulted in the mislabeling of the "s" value at the low-intensity region of the curves in this figure.
  14. That is, the width of the image of the square effective source in the plane of the entrance aperture normalized by the width of the square entrance aperture.

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.

Canals-Frau, D.

D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which wefuse. However, the normalization criteria are different.

D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).

Charman, W. N.

W. N. Charman, J. Opt. Soc. Am. 53, 410 (1963).

De, M.

M. De and S. C. Som, Opt. Acta 4, 17 (1962).

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Rousseau, M.

D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which wefuse. However, the normalization criteria are different.

D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).

Som, S. C.

M. De and S. C. Som, Opt. Acta 4, 17 (1962).

Welford, W. T.

W. T. Welford, Optics in Meteorology (Pergamon Press, Inc., New York, 1960), pp. 85–91.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.

Other

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Chap. X.

D. Canals-Frau and M. Rousseau, Opt. Acta 5, 15 (1958).

M. De and S. C. Som, Opt. Acta 4, 17 (1962).

W. N. Charman, J. Opt. Soc. Am. 53, 410 (1963).

W. T. Welford, Optics in Meteorology (Pergamon Press, Inc., New York, 1960), pp. 85–91.

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

The mutual intensity is found by setting τ equal to zero in the expression for the mutual-coherence function which is useful when investigating the imaging process employing quasimonochromatic light.

One of the restrictions imposed by Hopkins, namely that the mutual-intensity function in the object plane is spatially stationary, can be removed by employing other expressions for the propagation of the mutual-intensity function. These are J(x1,x2)=K∫∫J12) exp[i(x2·ξ-x1·ξ1)]dξ1dξ2 for the propagation of the mutual intensity between planes sufficiently separated to allow the far-field approximation and J′(x1,x2)=J(x1,x2)T(x1)T*(x2) for propagation through diffracting objects or apertures having a transmittance function T(x). We note in the first expression that the "far-field approximation may be valid" in the near field as shown by the case when J12) represents an incoherent source and the first expression reduces to the Van Cittert–Zernike theorem, which Hopkins uses to describe the effective source.

D. Canals-Frau and M. Rousseau, Ref. 3, report a similar analysis employing the Fourier transforms of the functions which wefuse. However, the normalization criteria are different.

In this case the normalized mutual-intensity function or complex degree of coherence measured perpendicular to the edged object is given by Г0(u1-u2)=[sin(u2-u1)ε]/[u2-u1)ε]

By coherence interval, we mean the minimum separation between points in the object plane (spatial separation) for which the mutual-intensity function vanishes.

See Fig. 8 of Ref. 5. Private communication with W. N. Charman revealed that a typographical error resulted in the mislabeling of the "s" value at the low-intensity region of the curves in this figure.

That is, the width of the image of the square effective source in the plane of the entrance aperture normalized by the width of the square entrance aperture.

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