Abstract

In the first part of this paper, the concept of the angular correlation function of a stationary optical field is introduced. This function characterizes the correlation that exists between the complex amplitudes of any two plane waves in the angular spectrum description of the statistical ensemble that represents the field. Relations between this function and the more commonly known correlation functions are derived. In particular, it is shown that the angular correlation function is essentially the four-dimensional spatial Fourier transform of the cross-spectral density function of the source. The angular correlation function is shown to characterize completely the second-order coherence properties of the far field. An expression for the intensity distribution in the far zone of a field generated by a source of any state of coherence is deduced. Some generalizations of the far-zone form of the Van Cittert-Zernike theorem are also obtained.

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  1. A. Walther, J. Opt. Soc. Am. 58, 1256 (1968).
  2. 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).
  3. 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.
  4. 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).]
  5. 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).
  6. 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.
  7. 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).
  8. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 625 (1962).
  9. 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).

Beran, M.

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).

Born, M.

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).

Bouwkamp, C. J.

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).

Lumley, J.

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).]

Miyamoto, K.

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 625 (1962).

Parrent, G.

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).

Walther, A.

A. Walther, J. Opt. Soc. Am. 58, 1256 (1968).

Wolf, E.

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).

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 625 (1962).

Other (9)

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).

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