A. Walther, J. Opt. Soc. Am. 58, 1256 (1968).
For a more complete review of coherence theory see, for example, M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford and New York, 1970), Ch. X, or L. Mandel and E.Wolf, Rev. Mod. Phys. 37, 231 (1965).
Traditionally, the mutual coherence function has been defined as the time average of the expression in the angular brackets in Eq. (1). For the purpose of the present investigation it is, however, somewhat more convenient to employ the ensemble rather than the time average. The two averages are equal, of course, if the ensemble is ergodic, as is usually the case.
For a stationary field, with which we are dealing here, the Fourier-integral representation does not exist in the ordinary sense and must be interpreted in the sense of the theory of generalized functions. [See, for example, J. Lumley, Stochastic Tools in Turbulence Theory (Academic, New York, 1970).]
For a discussion of the angular-spectrum representation, see, for example, C. J. Bouwkamp, in Rept. Progr. Phys. 17, 41 (1954); J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968); J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968). A generalization of the angular spectrum employing the theory of generalized functions has been discussed by G. C. Sherman and H. J. Bremmermann, J. Opt. Soc. Am. 59, 146 (1969).
By a wave propagated in a complex direction (p,q,m) we mean a wave in the angular-spectrum representation for which the parameter m is pure imaginary, in accordance with Eq. (8b). This is an evanescent wave, whose surfaces of constant phase are propagated at right angle to the z axis in the direction making angles cos-1[p/(p2+q2)½] and cos-1[q/(p2+q2)½] with the x and y axes, respectively. The amplitude of such a wave decreases exponentially with distance from the plane z =0, with decay factor exp(-k|m|z) =exp[-k(p2+q2-1)½z], with p2+q2>1.
This identity follows directly from Eq. (4) if we take the four-dimensional Fourier inverse of both sides of that equation with respect to the variables x1y1 x2, y2 and use Eq. (21a).
K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 625 (1962).
This customary representation of the cross-spectral density of a spatially incoherent source in terms of the Dirac delta function is not entirely satisfactory. For a discussion of this point and of a more refined mathematical procedure, see M. Beran and G. Parrent, Nuovo Cimento 27, 1049 (1963).