E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence," Opt. Commun. 13, 205–209 (1975), Eq. (11).
Some extensions of the analysis given in Refs. 13 and 14 were recently carried out by H. P. Baltes, B. Steinle, and G. Antes, "Spectral Coherence and the Radiant Intensity from Statistically Homogeneous and Isotropic Planar Sources," Opt. Commun. 18, 242–246 (1976) and by B. Steinle and H. P. Baltes, "Radiant Intensity and Spatial Coherence for Finite Planar Sources," J. Opt. Soc. Am. 67, 241–247 (1977).
This limit must be interpreted with some caution, since we assumed that the linear dimensions of the source are large compared to the correlation length in the source plane. More precisely the limit must be considered in the sense that as kσ→∞, the linear dimensions of the source (characterized by a length l say) must also become infinite, while l/σ »1.
See, for example, G. W. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., Cambridge, England, 1922), p. 20. Eq. (5) (with an obvious substitution).
I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 688, formula 1 of §6.567, with υ = 0, µ = ½.
The right-hand sides of Eqs. (5.22) and (5.24) may be expressed in terms of the spherical Bessel functions j1 and j0, respectively, and one then obtains the formulas [Equation] We use here the same definitions of the spherical Bessel functions as employed by A. Messiah in Quantum Mechanics, Vol. I, (North-Holland, Amsterdam, 1961), pp. 488–490.
A somewhat more satisfactory generalization of the concept of statistical homogeneity that applies to sources of finite extent and of nonuniform intensity was introduced recently in Ref. 11, under the name "quasi-homogeneity" (see also Ref. 12). However, in order to keep our analysis as simple as possible we will not consider this generalization in the present paper.
The term "localized" implies that Q(r,t) ≡ 0 for all points r outside some finite domain.
For the definition of an analytic signal see, for example, M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.2.
For the definition of wide-sense stationarity see, for example,W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York. 1958), p. 60.
See, for example, C. Kittel, Elementary Statistical Physics (Wiley, New York, 1958), p. 133ff.
See, for example, J. Peřina, Coherence of Light (Van Nostrand, London, 1972), Sec. 4.2.
Throughout this paper a spatial Fourier transform is denoted by a tilde. An elegant formal analogy between formulas of the Wiener- Khintchine type [e.g., Eqs. (2.13)] and some of the formulas derived in the present paper [e.g., Eqs. (6.8) and (6.11)] would have been more clearly brought out had we adopted consistently a circumflex to denote temporal Fourier transforms [as we did in Eqs. (2.12) for the-correlation functions], i.e., had we written V⌃(r, ω) in place of ν(r, ω) and Q⌃(r,ω) in place of ρ(r,ω). We have not adopted this more consistent notation because of the evident problems that it would present to the printers in connection with formulas that involve variables that are spatial, as well as temporal Fourier transforms of the basic variables.
Corresponding formulas for W(∞)υ(r1s1,r2s2,ω) and for J(s,ω) in terms of W¯(0)υ rather than W¯(0)ρ were derived previously elsewhere. These formulas can be shown to follow from the Eqs. (3.5) and (3.10), respectively, with the help of an important relation [Eq. (6.17)] derived below, as is demonstrated at the end of Sec. 6.