Equation (16) of Ref. 1. There is a misprint in that equation: Jω(f) should be replaced by Jω(S) on the left-hand side of that equation.
The expression (5. 1) for the radiance was proposed by A. Walther in the paper quoted in Ref. 5. Essentially the same expression for the radiance was proposed in a somewhat different context by L. S. Dolin, in Izv. VUZov: Radiophys. 7, 559–563 (1964), whose title, in English translation, is "Description in Weakly Inhomogeneous Wave Fields."
A function g(r) is said to be non-negative definite if for any non-negative integer N, any set of N vectors ri, and any set of N (real or complex) numbers ai, [equation] That this condition is obeyed by our degree of spatial coherence follows at once from the non-negative definiteness of the cross-spectral density function W(r1, r2), established in the Appendix of Ref. 4, and from Eq. (2.7) above if we take into account the fact that I(r)≥0.
For an account of Bochner's theorem see, for example, R. R. Goldberg, Fourier Transforms (Cambridge U. P., Cambridge, England, 1965), Chap. 5.
It has been shown in Ref. 17 that, in general, the radiant emittance of a partially coherent source acquires negative values.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stds. (U.S. GPO, Washington, D. C., 1964), p. 319; also (Dover, New York, 1965); W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 2785 (1931).
The integral in Eq. (7.3) extends only formally over the complete plane z=0, since I(0)(r) vanishes at point r outside the domain occupied by the source.
The limit kσg→∞ must be interpreted with some caution, since for a quasihomogeneous source we must have σg«l, where l represents a typical linear dimension of the source. Hence in proceeding to the limit kσg/l we must allow kl→∞ in such a way that σg/l tends to some number that is much smaller than unity.
The situation here is somewhat similar to that encountered in connection with the Kirchhoff approximation in elementary diffraction theory (cf. Ref. 6, Sec. 8.3.2).
See, for example R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, Now York, 1961), pp. 160–161.
M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.4.2.
In physical optics the quantity I(r) defined by Eq. (2. 1), is usually referred to simply as intensity (at frequency ω). We use here the adjective "optical" to distinguish it clearly from the radiometric concept of radiant intensity, that we will also encounter in the present paper.
V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (reprinted by Dover, New York, 1967), Sec. 1. 3.
A. C. Schell, "The Multiple Plate Antenna," Doctoral dissertation (Massachusetts Institute of Technology, 1961) (unpublished), Sec. 7.5.
See also R. A. Shore, "Partially Coherent Diffraction by a Circular Aperture," in Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, London, 1963), Part 2, pp. 787–795; and A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far-field Diffraction Plane," Nuovo Cimento B 15, 295–307 (1973).
Strictly speaking this condition only ensures that |µ(r1, r2)| has appreciable values when the separation |r1 - r2| is small compared with the linear dimensions of the source. This condition does not ensure that µ(r1, r2) is a function of r1 - r2 only over the whole domain occupied by the secondary source. The size of the domain where µ(r1, r2) is only a function of r1 - r2 depends on the imaging properties of the optical system and must be separately examined in each particular case. In any case µ(r1, r2) cannot be expected to be a function of r1 - r2 only when either of the two points is close to the boundary of the source, whether the source is a secondary or a primary one. This fact is, however, of no great practical consequence if the source is sufficiently large.
Our concept of a quasihomogeneity bears some similarity to that of local homogeneity, well known in the statistical theory of turbulence (cf. Ref. 8, Secs. 5 and 7), but the two concepts are not equivalent. A factorization (of the mutual intensity rather than of the cross-spectral density function) of the form (2. 9) was assumed in a recent paper by S. Wadaka and T. Sato, "Merits of symmetric scanning for detection of coherence function in incoherent imaging system," J. Opt. Soc. Am. 66, 145–147 (1976). However in this work no motivation for this factorization is given, nor are any restrictions stated on the behavior of the two factors.