In this connection it is of interest to note that it follows at once from Eq. (3.8) that no finite planar Lambertian source exists, if "finite" is interpreted in the sense that the field distribution v^{(0)} (r′) in the source plane vanishes outside a finite domain. For when the source is Lambertian, J _{ω}(s) = J_{0}cosθ, where J^{(0)} is a constant, and Eq. (3.8) then implies that for such a source we would necessarily have [Equation] Since the right-hand side of this equation becomes singular when θ = π/2, the Fourier transform C¯_{v}(f) would become singular when |f| = k|s_{| = k [cf. Eq. (4.2)] and this contradicts our earlier result that C¯v(f) is an entire analytic function.
}

A. C. Schell, The Multiple Plate Antenna (D.Sc. Thesis, Massachusetts Institute of Technology, 1961), Sec. 7.5. (unpublished).

C. H. Papas: Theory of Electromagnetic Wave Propagation (McGraw Hill, New York, 1965), Chap. 2.

If we substitute from (6.14) into (6.13) and change the variables of integration via the transformation (2.11) we obtain the following expression for C¯_{p} (f): [Equation] where [Equation] On substituting this expression for C¯_{ρ}(f) into (6.16) we obtain the following alternative but equivalent expression for the radiant intensity: [Equation] This expression for the radiant intensity generated by a finite primary planar source was recently derived in a different manner by E. Wolf and W. H. Carter, "Coherence and radiant intensity in scalar wavefields generated by fluctuating primary planar sources," [J. Opt. Soc. Am., 68, 953–964 (1978), Eq. (3.10)].

See for example R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), Sec. 45. A simple derivation of Rayleigh's formula (Al) is sketched out in E. W. Marchand and E. Wolf, "Boundary diffraction wave in the Rayleigh-Kirchhoff diffraction theory," J. Opt. Soc. Am., 52, 761–767 (1962), Sec. III. The Rayleigh formula given in this reference as Eq. (3.5) differs in sign from Eq. (Al) of the present paper. This difference arises from the fact that in the paper by Marchand and Wolf the differentiation is taken with respect to the z coordinate of the boundary point, i.e., with respect to the z coordinate of the point (x′,y′,z′), followed by letting z′ → 0, whereas in (Al) the differentiation is taken with respect to the z coordinate of the field point (x,y,z).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Appendix III.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1971), p. 367.

See, for example, J. Peřina: Coherence of Light (Van Nostrand, London, 1971), Sec. 4.2.

We ignore here certain refinements that are needed in order to define the cross-spectral density function in a mathematically rigorous manner.

If we substitute from (3.6) into (3.4) and change the variables of integration via the transformation (2.11) we find that C¯_{v}(f) may be expressed in the form [Equation] where [Equation] On substituting this expression for C¯_{v} (f) into (3.8) we obtain the following alternative expression for the radiant intensity that was derived previously^{1} in a different manner:[Equation]

See, for example, B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (Amer. Math. Soc., Providence, R.I., 1963), p. 352.

We stress that by a "finite planar source" we mean here a field distribution v^{(0)}(r′) in the plane z = 0 that identically vanishes outside a finite domain a. This type of a source must be distinguished from a finite primary planar source (considered in Sec. VI), for which the true source distribution ρ^{(0)}(r′) vanishes outside a finite domain of the plane in which the source is situated.

Actually, because of the fact that C¯_{v}(f) is an entire analytic function of two complex variables, it is sufficient that it has the same elements throughout any finite two-dimensional domain, however small, of the real f-plane.