Abstract

In the literature on coherence theory one almost invariably specifies any correlations that may exist in the source region by means of a correlation function of the field variable. However, when the source is a primary one, it seems more appropriate to specify the correlations by means of a correlation function of the source variable. In the present paper some basic formulas are derived, relating to coherence and radiant intensity in fields generated by primary sources in terms of a source correlation function. A number of results are obtained that exhibit the connection between the "field description" and the "source description." Several illustrative examples are given relating to Gaussian-correlated sources and to Lambertian sources.

© 1978 Optical Society of America

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  1. The term "localized" implies that Q(r,t) ≡ 0 for all points r outside some finite domain.
  2. 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.
  3. 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.
  4. See, for example, E. W. Marchand and E. Wolf, "Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields," J. Opt. Soc. Am. 62, 379–385 (1972), Eq. (4).
  5. See, for example, C. Kittel, Elementary Statistical Physics (Wiley, New York, 1958), p. 133ff.
  6. See, for example, J. Peřina, Coherence of Light (Van Nostrand, London, 1972), Sec. 4.2.
  7. 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.
  8. Corresponding formulas for W(∞)υ(r1s1,r2s2,ω) and for J(s,ω) in terms of (0)υ rather than (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.
  9. E. W. Marchand and E. Wolf, "Radiometry with Sources of any State of Coherence," J. Opt. Soc. Am. 64, 1219–1226 (1974).
  10. 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.
  11. W. H. Carter and E. Wolf, "Coherence and Radiometry with Quasihomogeneous Planar Sources," J. Opt. Soc. Am. 67, 785–796 (1977). On the right-hand side of formula (6.7) of that reference, the factor (2/πx) should be replaced by (2/πx)½.
  12. A slightly different generalization was recently discussed in connection with electron sources by H. A. Ferwerda and M. G. van Heel, "On the Coherence Properties of Thermionic Emission Sources," Optik 47, 357–362 (1977).
  13. 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).
  14. W. H. Carter and E. Wolf, "Coherence Properties of Lambertian and non-Lambertian Sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).
  15. 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).
  16. 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.
  17. 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).
  18. 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.
  19. 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.

1977 (2)

A slightly different generalization was recently discussed in connection with electron sources by H. A. Ferwerda and M. G. van Heel, "On the Coherence Properties of Thermionic Emission Sources," Optik 47, 357–362 (1977).

W. H. Carter and E. Wolf, "Coherence and Radiometry with Quasihomogeneous Planar Sources," J. Opt. Soc. Am. 67, 785–796 (1977). On the right-hand side of formula (6.7) of that reference, the factor (2/πx) should be replaced by (2/πx)½.

1976 (1)

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).

1975 (2)

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).

W. H. Carter and E. Wolf, "Coherence Properties of Lambertian and non-Lambertian Sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).

1974 (1)

1972 (1)

Antes, G.

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).

Baltes, H. P.

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).

Born, M.

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.

Carter, W. H.

Davenport, W. B.

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.

Ferwerda, H. A.

A slightly different generalization was recently discussed in connection with electron sources by H. A. Ferwerda and M. G. van Heel, "On the Coherence Properties of Thermionic Emission Sources," Optik 47, 357–362 (1977).

Gradsteyn, I. S.

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.

Kittel, C.

See, for example, C. Kittel, Elementary Statistical Physics (Wiley, New York, 1958), p. 133ff.

Marchand, E. W.

Messiah, A.

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.

Perina, J.

See, for example, J. Peřina, Coherence of Light (Van Nostrand, London, 1972), Sec. 4.2.

Root, W. L.

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.

Ryzhik, I. M.

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.

Steinle, B.

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).

van Heel, M. G.

A slightly different generalization was recently discussed in connection with electron sources by H. A. Ferwerda and M. G. van Heel, "On the Coherence Properties of Thermionic Emission Sources," Optik 47, 357–362 (1977).

Watson, G. W.

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).

Wolf, E.

J. Opt. Soc. Am. (4)

Opt. Commun. (2)

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).

Optik (1)

A slightly different generalization was recently discussed in connection with electron sources by H. A. Ferwerda and M. G. van Heel, "On the Coherence Properties of Thermionic Emission Sources," Optik 47, 357–362 (1977).

Other (12)

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.

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 (0)υ rather than (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.

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.

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