See also E. Wolf, "A new description of second-order coherence phenomena in the space-frequency domain," in Optics in Four Dimensions—1980, M. A. Machado and L. M. Narducci, eds., Conference Proceedings #65 (American Institute of Physics, New York, 1981), pp. 42–48; "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).
M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
C. L. Mehta and L. Mandel, "Some properties of higher order coherence functions," in Electromagnetic Wave Theory, Part II, J. Brown, ed. (Pergamon, Oxford, 1967), pp. 1069–1075. See also E. Wolf, "Light fluctuations as a new spectroscopic tool," Jpn. J. Appl. Phys. 4, Suppl. I, Sec. 6, 1–14 (1965).
Our method of proof follows closely an argument given in connection with positive definite functions by S. Bochner in Lectures on Fourier Integrals (Princeton U. Press, Princeton, N.J., 1959), pp. 326–327. A nonrigorous proof of a related nonnegative definiteness condition satisfied by the cross-spectral density was given by L. Mandel and E. Wolf in the appendix of Ref. 15.
For pertinent references and a discussion of this point, see A. M. Yaglom; An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962), especially pp. 35–39.
S. Goldman, Information Theory (Prentice-Hall, New York, 1953), Sec. 8.4, or D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), Sec. 3.2.
W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 107–108.
In classical wave theory the fluctuating function is usually real, say, Q(r)(r, t). It is often convenient to associate with Q(r)(r, t) a certain complex function Q(r, t), known as the analytic signal (see, for example, Ref. 1, Sec. 3.1, or Ref. 17, Sec. 2). However, when Q(r)(r, t) is a member of a stationary ensemble, the transition to the complex analytic signal is mathematically unsatisfactory for reasons similar to those that we discussed in a some-what different context at the beginning of Section 2. This difficulty may be overcome by starting from the real cross-correlation function Γ(r)(r1, r2, τ) = 〈Q(r)(r1, t)Q(r)(r2, t + τ)t and defining ΓQ(r1, r2, τ) not by Eq. (3.1) but rather as the complex analytic signal associated with Γ(r)(r1, r2, τ). This procedure is applicable whenever Γ(r) is absolutely integrable with respect to τ. If the complex cross-correlation function is introduced in this way, all our subsequent results remain valid. The fact that ΓQ is an analytic signal implies that the cross-spectral density WQ(r1, r2, ω) will then vanish identically for ω < 0.
R. R. Goldberg, Fourier Transforms (Cambridge U. Press, Cambridge, 1965), p. 6.
A simple proof of this result is as follows: One readily finds from Eq. (3.1), with the help of Schwarz's inequality, that [equation] Hence [equation] and consequently [equation] Now the first two terms on the right-hand-side of this formula are necessarily finite, as is apparent from Eq. (3.1), and the integral on the right-hand side is finite by assumption [Eq. (3.2)]. Hence the left-hand side must also be finite, implying that ΓQ is square integrable with respect to τ.
More explicitly, this assumption means that there exists a nonnegative function F(τ) of τ alone such that for all r1 Ε D, r2 Ε D: [equation] and [equation].
The corresponding result for real functions of a smaller number of variables is established, for example, in E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier Series (Cambridge U. Press, Cambridge, 1926), Vol. II, p. 326.
Actually we use here an obvious multidimensional generalization of Mercer's theorem to the case in which the kernel depends on two vector variables r1 and r2 rather than on two scalar variables (x1 and x2, say). For a discussion of the usual form of Mercer's theorem see, for example, F. Smithies, Integral Equations (Cambridge U. Press, Cambridge, 1970), p. 128, or F. Riecz and B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955), p. 245.
In two recent papers cited below, Mercer's expansion was used in the treatment of other problems of optical coherence theory. However, the functions that were represented in this way were the equal-time correlation function (the mutual intensity) Γ(r1, r2, 0) and the equal-time degree of coherence γ(r1, r2, 0). Some advantages of basing the theory on the Mercer expansion of the cross-spectral density W(r1, r2, ω), as is done in this paper, will become apparent in Part II of this investigation. The two papers are R. Martínez-Herrero, "Expansion of the complex degree of coherence," Nuovo Cimento 54B, 205–210 (1979); R. Martínez-Herrero and P. M. Majias, "Relation between the expansions of the correlation function at the object and image planes for partially coherent illumination," Opt. Commun. 37, 234–238 (1981). In this connection see also the early pioneering article by H. Gamo, "Matrix treatment of partial cohrence," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 18–336, especially Sec. 4.4.