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 areJ(x1,x2)=K∫∫J(ξ1,ξ2)exp[i(x2⋅ξ2−x1⋅ξ1)]dξ1dξ2for the propagation of the mutual intensity between planes sufficiently separated to allow the far-field approximation andJ′(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 J(ξ1,ξ2) 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 we use. 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 Γ0(u1−u2) = [sin(u2−u1)∊]/[(u2−u1)∊].

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.

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.

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