M. S. Zubairy and E. Wolf, “Exact equations for radiative transfer
of energy and momentum in free electromagnetic
fields,” Opt. Commun. 20, 321–324 (1977).

[CrossRef]

Formally a slightly different but essentially equivalent class of sources has been considered by H. A. Ferwerda and M. G. van Heel “On the coherence properties of
thermionic emission sources,” Optik 47, 357–362 (1977). See also H. A. Ferwerda and M. G. van Heel, “Determination of Coherence Length from
Directionality,” in Proceedings of the
Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

W. H. Carter and E. Wolf, “Coherence and radiometry with
quasi-homogeneous planar sources,” J. Opt.
Soc. Am. 67, 785–796 (1977).

[CrossRef]

The angular distribution of radiant intensity from some other types of model sources is discussed in the following papers: 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); B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence
for finite planar sources,” J. Opt. Soc.
Am. 67, 241–247 (1977); H. P. Baltes, B. Steinle, and G. Antes “Radiometric and correlation properties
of bounded planar sources,” in Proceedings
of the Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press); W. H. Carter and M. Bertolotti, “An analysis of the far-field coherence
and radiant intensity of light scattered from liquid
crystals” (J. Opt. Soc. Am., in press).

[CrossRef]

A somewhat different approach was described by A. S. Marathay, “Radiometry of partially coherent fields
I,” Opt. Acta 23, 785–794 (1976); II, ibid.23, 795–798 (1976).

[CrossRef]

L. Mandel and E. Wolf, “Spectral coherence and the concept of
cross-spectral purity,” J. Opt. Soc.
Am. 66, 529–535 (1976).

[CrossRef]

E. Wolf, “New theory of radiative energy transfer
in free electromagnetic fields,” Phys. Rev.
D 13, 869–886 (1976).

[CrossRef]

For a discussion of Wigner’s theorem and of related researches, see M. D. Srinivas and E. Wolf, “Some nonclassical features of
phase-space representations of quantum mechanics,” Phys. Rev. D 11, 1477–1485 (1975).

[CrossRef]

This relationship appears to have been first considered, for the special case of radiation from large statistically homogeneous sources, by 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).

[CrossRef]

See, for example, F. Scudieri, M. Bertolotti, and R. Bartolino, “Light scattered by a liquid crystal: A
new quasi-thermal source,” Appl.
Opt. 13, 181–185 (1974); M. Bertolotti, F. Scudieri, and S. Verginelli, “Spatial coherence of light scattered by
media with large correlation length of refractive index
fluctuations,” Appl. Opt. 15, 1842–1844 (1976).

[CrossRef]
[PubMed]

E. W. Marchand and E. Wolf, “Radiometry with sources of any state of
coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).

[CrossRef]

E. W. Marchand and E. Wolf, “Walther’s definition of
generalized radiance,” J. Opt. Soc.
Am. 64, 1273–1274 (1974); see also A. Walther, “Reply to Marchand and
Wolf”, J. Opt. Soc. Am. 64, 1275 (1974).

[CrossRef]

(a)See, for example, the following publications and the references quoted therein: Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “Status of the theory of propagation of
waves in randomly inhomogeneous medium,” Sov. Phys.—Usp. 13, 551–575 (1971); (b)V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave
Propagation (U.S. Department of Commerce, National Technical Service, Springfield, Va., 1971), Sec. 63; (c)Yu. A. Kravtsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction
theory,” Sov. Phys.—Usp. 18, 118–130 (1975); (d)Yu. N. Barabanenkov, “Multiple scattering of waves by
ensembles of particles and the theory of radiation
transport,” Sov. Phys.—Usp. 18, 673–689 (1976); (e)A. Ishimaru, “Theory and application of wave
propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).

A general theory of such mappings was formulated by G. S. Agarwal and E. Wolf, “Calculus of functions of noncommuting
operators and general phase-space methods in quantum mechanics. I. Mapping
theorems and ordering of functions of noncommuting
operators,” Phys. Rev. D 2, 2161–2186 (1970); “II. Quantum
mechanics in phase space,” Phys. Rev.
D 2, 2187–2205 (1970); “III. A
generalized Wick theorem and multitime mapping,” Phys. Rev. D 2, 2206–2225 (1970). These papers also contain an extensive bibliography of earlier publications on this subject.

[CrossRef]

For a fuller account of optical coherence theory see, for example, L. Mandel and E. Wolf “Coherence properties of optical
fields,” Rev. Mod. Phys. 37, 231–287 (1965) or Ref. 5 quoted below.

[CrossRef]

The formula (7.3) is a generalization to quasi-homogeneous sources of an expression derived for large homogeneous sources by M. Beran and G. Parrent, “The mutual coherence function of
incoherent radiation,” Nuovo
Cimento 27, 1049–1063 (1963), Sec. 8; A. Walther, Ref. 9, Sec. 4; W. H. Carter and E. Wolf, Ref. 24, Sec. II.

[CrossRef]

E. Wigner, “On the quantum correction for
thermodynamic equilibrium,” Phys.
Rev. 40, 749–759 (1932).

[CrossRef]

A general theory of such mappings was formulated by G. S. Agarwal and E. Wolf, “Calculus of functions of noncommuting
operators and general phase-space methods in quantum mechanics. I. Mapping
theorems and ordering of functions of noncommuting
operators,” Phys. Rev. D 2, 2161–2186 (1970); “II. Quantum
mechanics in phase space,” Phys. Rev.
D 2, 2187–2205 (1970); “III. A
generalized Wick theorem and multitime mapping,” Phys. Rev. D 2, 2206–2225 (1970). These papers also contain an extensive bibliography of earlier publications on this subject.

[CrossRef]

See, for example, A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Interscience, New York, 1965), Sec. 2.2, or W. Pauli, “Die Allgemeinen Principien der
Wellenmechanik,” in Handbuch der
Physik, 2 Aufl., Band 24, 1 Teil, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), pp. 92 and 260.

The angular distribution of radiant intensity from some other types of model sources is discussed in the following papers: 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); B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence
for finite planar sources,” J. Opt. Soc.
Am. 67, 241–247 (1977); H. P. Baltes, B. Steinle, and G. Antes “Radiometric and correlation properties
of bounded planar sources,” in Proceedings
of the Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press); W. H. Carter and M. Bertolotti, “An analysis of the far-field coherence
and radiant intensity of light scattered from liquid
crystals” (J. Opt. Soc. Am., in press).

[CrossRef]

The angular distribution of radiant intensity from some other types of model sources is discussed in the following papers: 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); B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence
for finite planar sources,” J. Opt. Soc.
Am. 67, 241–247 (1977); H. P. Baltes, B. Steinle, and G. Antes “Radiometric and correlation properties
of bounded planar sources,” in Proceedings
of the Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press); W. H. Carter and M. Bertolotti, “An analysis of the far-field coherence
and radiant intensity of light scattered from liquid
crystals” (J. Opt. Soc. Am., in press).

[CrossRef]

(a)See, for example, the following publications and the references quoted therein: Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “Status of the theory of propagation of
waves in randomly inhomogeneous medium,” Sov. Phys.—Usp. 13, 551–575 (1971); (b)V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave
Propagation (U.S. Department of Commerce, National Technical Service, Springfield, Va., 1971), Sec. 63; (c)Yu. A. Kravtsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction
theory,” Sov. Phys.—Usp. 18, 118–130 (1975); (d)Yu. N. Barabanenkov, “Multiple scattering of waves by
ensembles of particles and the theory of radiation
transport,” Sov. Phys.—Usp. 18, 673–689 (1976); (e)A. Ishimaru, “Theory and application of wave
propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).

See, for example, F. Scudieri, M. Bertolotti, and R. Bartolino, “Light scattered by a liquid crystal: A
new quasi-thermal source,” Appl.
Opt. 13, 181–185 (1974); M. Bertolotti, F. Scudieri, and S. Verginelli, “Spatial coherence of light scattered by
media with large correlation length of refractive index
fluctuations,” Appl. Opt. 15, 1842–1844 (1976).

[CrossRef]
[PubMed]

The formula (7.3) is a generalization to quasi-homogeneous sources of an expression derived for large homogeneous sources by M. Beran and G. Parrent, “The mutual coherence function of
incoherent radiation,” Nuovo
Cimento 27, 1049–1063 (1963), Sec. 8; A. Walther, Ref. 9, Sec. 4; W. H. Carter and E. Wolf, Ref. 24, Sec. II.

[CrossRef]

See, for example, A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Interscience, New York, 1965), Sec. 2.2, or W. Pauli, “Die Allgemeinen Principien der
Wellenmechanik,” in Handbuch der
Physik, 2 Aufl., Band 24, 1 Teil, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), pp. 92 and 260.

See, for example, F. Scudieri, M. Bertolotti, and R. Bartolino, “Light scattered by a liquid crystal: A
new quasi-thermal source,” Appl.
Opt. 13, 181–185 (1974); M. Bertolotti, F. Scudieri, and S. Verginelli, “Spatial coherence of light scattered by
media with large correlation length of refractive index
fluctuations,” Appl. Opt. 15, 1842–1844 (1976).

[CrossRef]
[PubMed]

See, for example, M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.2.

W. H. Carter and E. Wolf, “Coherence and radiometry with
quasi-homogeneous planar sources,” J. Opt.
Soc. Am. 67, 785–796 (1977).

[CrossRef]

This relationship appears to have been first considered, for the special case of radiation from large statistically homogeneous sources, by 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).

[CrossRef]

As in the previous sections we describe the coherence properties of a source in terms of correlation functions involving the field distribution in the source plane. Such a description may be employed whether the source is a primary or a secondary one. However, when the source is a primary one, one may characterize its coherence properties in an alternative way, by means of correlation functions involving the true source variable (e.g., the charge-current density distribution). For a primary scalar source this alternative approach is discussed in a forthcoming paper by W. H. Carter and E. Wolf, “Coherence and radiant intensity in
scalar wavefields generated by fluctuating primary planar
sources (submitted to J. Opt. Soc.
Am.).

E. Wolf and W. H. Carter, “On the radiation efficiency of
quasi-homogeneous sources of different degrees of spatial
coherence,” in Proceedings of the Fourth
Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

Formally a slightly different but essentially equivalent class of sources has been considered by H. A. Ferwerda and M. G. van Heel “On the coherence properties of
thermionic emission sources,” Optik 47, 357–362 (1977). See also H. A. Ferwerda and M. G. van Heel, “Determination of Coherence Length from
Directionality,” in Proceedings of the
Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

A. T. Friberg, “On the existence of a radiance function
for a partially coherent planar source,” in Proceedings of the Fourth Rochester Conference on Coherence and
Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

See, for example, E. Hopf, Mathematical Problems of Radiative Equilibrium (Cambridge U. P., Cambridge, 1934), Sec. 2.

(a)See, for example, the following publications and the references quoted therein: Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “Status of the theory of propagation of
waves in randomly inhomogeneous medium,” Sov. Phys.—Usp. 13, 551–575 (1971); (b)V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave
Propagation (U.S. Department of Commerce, National Technical Service, Springfield, Va., 1971), Sec. 63; (c)Yu. A. Kravtsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction
theory,” Sov. Phys.—Usp. 18, 118–130 (1975); (d)Yu. N. Barabanenkov, “Multiple scattering of waves by
ensembles of particles and the theory of radiation
transport,” Sov. Phys.—Usp. 18, 673–689 (1976); (e)A. Ishimaru, “Theory and application of wave
propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).

A somewhat different approach was described by A. S. Marathay, “Radiometry of partially coherent fields
I,” Opt. Acta 23, 785–794 (1976); II, ibid.23, 795–798 (1976).

[CrossRef]

E. W. Marchand and E. Wolf, “Radiometry with sources of any state of
coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).

[CrossRef]

E. W. Marchand and E. Wolf, “Walther’s definition of
generalized radiance,” J. Opt. Soc.
Am. 64, 1273–1274 (1974); see also A. Walther, “Reply to Marchand and
Wolf”, J. Opt. Soc. Am. 64, 1275 (1974).

[CrossRef]

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. (34).

[CrossRef]

The formula (7.3) is a generalization to quasi-homogeneous sources of an expression derived for large homogeneous sources by M. Beran and G. Parrent, “The mutual coherence function of
incoherent radiation,” Nuovo
Cimento 27, 1049–1063 (1963), Sec. 8; A. Walther, Ref. 9, Sec. 4; W. H. Carter and E. Wolf, Ref. 24, Sec. II.

[CrossRef]

(a)See, for example, the following publications and the references quoted therein: Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “Status of the theory of propagation of
waves in randomly inhomogeneous medium,” Sov. Phys.—Usp. 13, 551–575 (1971); (b)V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave
Propagation (U.S. Department of Commerce, National Technical Service, Springfield, Va., 1971), Sec. 63; (c)Yu. A. Kravtsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction
theory,” Sov. Phys.—Usp. 18, 118–130 (1975); (d)Yu. N. Barabanenkov, “Multiple scattering of waves by
ensembles of particles and the theory of radiation
transport,” Sov. Phys.—Usp. 18, 673–689 (1976); (e)A. Ishimaru, “Theory and application of wave
propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).

See, for example, F. Scudieri, M. Bertolotti, and R. Bartolino, “Light scattered by a liquid crystal: A
new quasi-thermal source,” Appl.
Opt. 13, 181–185 (1974); M. Bertolotti, F. Scudieri, and S. Verginelli, “Spatial coherence of light scattered by
media with large correlation length of refractive index
fluctuations,” Appl. Opt. 15, 1842–1844 (1976).

[CrossRef]
[PubMed]

It has been shown that a spatially completely incoherent source would give rise to radiant intensity that falls off with θ in proportion to cos2θ rather than cosθ. [T. J. Skinner, Ph.D. Thesis (Boston University, 1965), p. 46; E. W. Marchand and E. Wolf, Ref. 11, Sec. V; W. H. Carter and E. Wolf, Ref. 24, Sec. III].

For a discussion of Wigner’s theorem and of related researches, see M. D. Srinivas and E. Wolf, “Some nonclassical features of
phase-space representations of quantum mechanics,” Phys. Rev. D 11, 1477–1485 (1975).

[CrossRef]

The angular distribution of radiant intensity from some other types of model sources is discussed in the following papers: 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); B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence
for finite planar sources,” J. Opt. Soc.
Am. 67, 241–247 (1977); H. P. Baltes, B. Steinle, and G. Antes “Radiometric and correlation properties
of bounded planar sources,” in Proceedings
of the Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press); W. H. Carter and M. Bertolotti, “An analysis of the far-field coherence
and radiant intensity of light scattered from liquid
crystals” (J. Opt. Soc. Am., in press).

[CrossRef]

(a)See, for example, the following publications and the references quoted therein: Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “Status of the theory of propagation of
waves in randomly inhomogeneous medium,” Sov. Phys.—Usp. 13, 551–575 (1971); (b)V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave
Propagation (U.S. Department of Commerce, National Technical Service, Springfield, Va., 1971), Sec. 63; (c)Yu. A. Kravtsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction
theory,” Sov. Phys.—Usp. 18, 118–130 (1975); (d)Yu. N. Barabanenkov, “Multiple scattering of waves by
ensembles of particles and the theory of radiation
transport,” Sov. Phys.—Usp. 18, 673–689 (1976); (e)A. Ishimaru, “Theory and application of wave
propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).

Formally a slightly different but essentially equivalent class of sources has been considered by H. A. Ferwerda and M. G. van Heel “On the coherence properties of
thermionic emission sources,” Optik 47, 357–362 (1977). See also H. A. Ferwerda and M. G. van Heel, “Determination of Coherence Length from
Directionality,” in Proceedings of the
Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

E. Wigner, “On the quantum correction for
thermodynamic equilibrium,” Phys.
Rev. 40, 749–759 (1932).

[CrossRef]

E. P. Wigner, “Quantum mechanical distribution
functions revisited,” in Perspectives in
Quantum Theory, edited by W. Yourgrau and A. van der Merwe (M.I.T. Press, Cambridge, Mass., 1971), pp. 25–36.

E. Collett and E. Wolf, “Is complete spatial coherence necessary
for the generation of highly directional light
beams?,” Opt. Lett. 2, 27–29 (1978).

[CrossRef]
[PubMed]

W. H. Carter and E. Wolf, “Coherence and radiometry with
quasi-homogeneous planar sources,” J. Opt.
Soc. Am. 67, 785–796 (1977).

[CrossRef]

M. S. Zubairy and E. Wolf, “Exact equations for radiative transfer
of energy and momentum in free electromagnetic
fields,” Opt. Commun. 20, 321–324 (1977).

[CrossRef]

E. Wolf, “New theory of radiative energy transfer
in free electromagnetic fields,” Phys. Rev.
D 13, 869–886 (1976).

[CrossRef]

L. Mandel and E. Wolf, “Spectral coherence and the concept of
cross-spectral purity,” J. Opt. Soc.
Am. 66, 529–535 (1976).

[CrossRef]

This relationship appears to have been first considered, for the special case of radiation from large statistically homogeneous sources, by 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).

[CrossRef]

For a discussion of Wigner’s theorem and of related researches, see M. D. Srinivas and E. Wolf, “Some nonclassical features of
phase-space representations of quantum mechanics,” Phys. Rev. D 11, 1477–1485 (1975).

[CrossRef]

E. W. Marchand and E. Wolf, “Radiometry with sources of any state of
coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).

[CrossRef]

E. W. Marchand and E. Wolf, “Walther’s definition of
generalized radiance,” J. Opt. Soc.
Am. 64, 1273–1274 (1974); see also A. Walther, “Reply to Marchand and
Wolf”, J. Opt. Soc. Am. 64, 1275 (1974).

[CrossRef]

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. (34).

[CrossRef]

A general theory of such mappings was formulated by G. S. Agarwal and E. Wolf, “Calculus of functions of noncommuting
operators and general phase-space methods in quantum mechanics. I. Mapping
theorems and ordering of functions of noncommuting
operators,” Phys. Rev. D 2, 2161–2186 (1970); “II. Quantum
mechanics in phase space,” Phys. Rev.
D 2, 2187–2205 (1970); “III. A
generalized Wick theorem and multitime mapping,” Phys. Rev. D 2, 2206–2225 (1970). These papers also contain an extensive bibliography of earlier publications on this subject.

[CrossRef]

For a fuller account of optical coherence theory see, for example, L. Mandel and E. Wolf “Coherence properties of optical
fields,” Rev. Mod. Phys. 37, 231–287 (1965) or Ref. 5 quoted below.

[CrossRef]

See, for example, M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.2.

As in the previous sections we describe the coherence properties of a source in terms of correlation functions involving the field distribution in the source plane. Such a description may be employed whether the source is a primary or a secondary one. However, when the source is a primary one, one may characterize its coherence properties in an alternative way, by means of correlation functions involving the true source variable (e.g., the charge-current density distribution). For a primary scalar source this alternative approach is discussed in a forthcoming paper by W. H. Carter and E. Wolf, “Coherence and radiant intensity in
scalar wavefields generated by fluctuating primary planar
sources (submitted to J. Opt. Soc.
Am.).

E. Wolf and W. H. Carter, “On the radiation efficiency of
quasi-homogeneous sources of different degrees of spatial
coherence,” in Proceedings of the Fourth
Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

M. S. Zubairy and E. Wolf, “Exact equations for radiative transfer
of energy and momentum in free electromagnetic
fields,” Opt. Commun. 20, 321–324 (1977).

[CrossRef]

See, for example, F. Scudieri, M. Bertolotti, and R. Bartolino, “Light scattered by a liquid crystal: A
new quasi-thermal source,” Appl.
Opt. 13, 181–185 (1974); M. Bertolotti, F. Scudieri, and S. Verginelli, “Spatial coherence of light scattered by
media with large correlation length of refractive index
fluctuations,” Appl. Opt. 15, 1842–1844 (1976).

[CrossRef]
[PubMed]

A. Walther, “Radiometry and
coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).

[CrossRef]

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. (34).

[CrossRef]

E. W. Marchand and E. Wolf, “Radiometry with sources of any state of
coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).

[CrossRef]

L. Mandel and E. Wolf, “Spectral coherence and the concept of
cross-spectral purity,” J. Opt. Soc.
Am. 66, 529–535 (1976).

[CrossRef]

W. H. Carter and E. Wolf, “Coherence and radiometry with
quasi-homogeneous planar sources,” J. Opt.
Soc. Am. 67, 785–796 (1977).

[CrossRef]

E. W. Marchand and E. Wolf, “Walther’s definition of
generalized radiance,” J. Opt. Soc.
Am. 64, 1273–1274 (1974); see also A. Walther, “Reply to Marchand and
Wolf”, J. Opt. Soc. Am. 64, 1275 (1974).

[CrossRef]

A. Walther, “Radiometry and
coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).

[CrossRef]

The formula (7.3) is a generalization to quasi-homogeneous sources of an expression derived for large homogeneous sources by M. Beran and G. Parrent, “The mutual coherence function of
incoherent radiation,” Nuovo
Cimento 27, 1049–1063 (1963), Sec. 8; A. Walther, Ref. 9, Sec. 4; W. H. Carter and E. Wolf, Ref. 24, Sec. II.

[CrossRef]

A somewhat different approach was described by A. S. Marathay, “Radiometry of partially coherent fields
I,” Opt. Acta 23, 785–794 (1976); II, ibid.23, 795–798 (1976).

[CrossRef]

This relationship appears to have been first considered, for the special case of radiation from large statistically homogeneous sources, by 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).

[CrossRef]

The angular distribution of radiant intensity from some other types of model sources is discussed in the following papers: 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); B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence
for finite planar sources,” J. Opt. Soc.
Am. 67, 241–247 (1977); H. P. Baltes, B. Steinle, and G. Antes “Radiometric and correlation properties
of bounded planar sources,” in Proceedings
of the Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press); W. H. Carter and M. Bertolotti, “An analysis of the far-field coherence
and radiant intensity of light scattered from liquid
crystals” (J. Opt. Soc. Am., in press).

[CrossRef]

M. S. Zubairy and E. Wolf, “Exact equations for radiative transfer
of energy and momentum in free electromagnetic
fields,” Opt. Commun. 20, 321–324 (1977).

[CrossRef]

Formally a slightly different but essentially equivalent class of sources has been considered by H. A. Ferwerda and M. G. van Heel “On the coherence properties of
thermionic emission sources,” Optik 47, 357–362 (1977). See also H. A. Ferwerda and M. G. van Heel, “Determination of Coherence Length from
Directionality,” in Proceedings of the
Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

E. Wigner, “On the quantum correction for
thermodynamic equilibrium,” Phys.
Rev. 40, 749–759 (1932).

[CrossRef]

For a discussion of Wigner’s theorem and of related researches, see M. D. Srinivas and E. Wolf, “Some nonclassical features of
phase-space representations of quantum mechanics,” Phys. Rev. D 11, 1477–1485 (1975).

[CrossRef]

A general theory of such mappings was formulated by G. S. Agarwal and E. Wolf, “Calculus of functions of noncommuting
operators and general phase-space methods in quantum mechanics. I. Mapping
theorems and ordering of functions of noncommuting
operators,” Phys. Rev. D 2, 2161–2186 (1970); “II. Quantum
mechanics in phase space,” Phys. Rev.
D 2, 2187–2205 (1970); “III. A
generalized Wick theorem and multitime mapping,” Phys. Rev. D 2, 2206–2225 (1970). These papers also contain an extensive bibliography of earlier publications on this subject.

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E. Wolf, “New theory of radiative energy transfer
in free electromagnetic fields,” Phys. Rev.
D 13, 869–886 (1976).

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For a fuller account of optical coherence theory see, for example, L. Mandel and E. Wolf “Coherence properties of optical
fields,” Rev. Mod. Phys. 37, 231–287 (1965) or Ref. 5 quoted below.

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(a)See, for example, the following publications and the references quoted therein: Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “Status of the theory of propagation of
waves in randomly inhomogeneous medium,” Sov. Phys.—Usp. 13, 551–575 (1971); (b)V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave
Propagation (U.S. Department of Commerce, National Technical Service, Springfield, Va., 1971), Sec. 63; (c)Yu. A. Kravtsov, C. M. Rytov, and V. I. Tatarskii, “Statistical problems in diffraction
theory,” Sov. Phys.—Usp. 18, 118–130 (1975); (d)Yu. N. Barabanenkov, “Multiple scattering of waves by
ensembles of particles and the theory of radiation
transport,” Sov. Phys.—Usp. 18, 673–689 (1976); (e)A. Ishimaru, “Theory and application of wave
propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).

Actually all of them have the same degree of spatial coherence gQ(r1 − r2). They can only differ by their intensity distributions IQ(r).

It has been shown that a spatially completely incoherent source would give rise to radiant intensity that falls off with θ in proportion to cos2θ rather than cosθ. [T. J. Skinner, Ph.D. Thesis (Boston University, 1965), p. 46; E. W. Marchand and E. Wolf, Ref. 11, Sec. V; W. H. Carter and E. Wolf, Ref. 24, Sec. III].

Ref. 24, Sec. II, especially Eqs. (31) and (32).

See, for example, M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.2.

The mutual coherence function is independent of t because of our earlier assumption (cf. Footnote 1) that the radiation is steady in the macroscopic sense. In the language of the theory of random processes this assumption is tantamount to the statement that the fluctuations can be described as a stationary random process.

We mentioned earlier that within the framework of Maxwell’s electromagnetic theory the optical intensity is usually identified with the magnitude of the energy flux vector (the Poynting vector). It would therefore seem more appropriate to identify the optical intensity in a complex scalar wavefield V(r, t) with the magnitude of the flux vector F=α(V˙∇V*+V˙*∇V) associated with that field. (Here V˙ = ∂V/∂t and α is a constant, depending on the choice of units.) However, under experimental conditions frequently encountered in practice (e.g., when measurements are made in the far zone of a radiating system and the field is quasi-monochromatic),⌨ |F|〉 may be shown to be proportional to 〈VV*〉 (at least to a high degree of accuracy).

A. T. Friberg, “On the existence of a radiance function
for a partially coherent planar source,” in Proceedings of the Fourth Rochester Conference on Coherence and
Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

The radiometric definition of the radiant intensity J(s), via the formula (4.4), implies that J(s) represents the average radiated power per unit solid angle around the s direction. One can calculate this radiated power directly from physical optics, without introducing any hypothetical radiance function B(r, s), as will be discussed in Sec. 5. [J(s)]phy.opt. denotes here the radiant intensity when calculated in this more direct manner.

Throughout this talk we shall only be concerned with sources that generate radiation which is steady in the macroscopic sense. Such sources need not be, however, in thermal equilibrium with its surroundings.

See, for example, E. Hopf, Mathematical Problems of Radiative Equilibrium (Cambridge U. P., Cambridge, 1934), Sec. 2.

See, for example, A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Interscience, New York, 1965), Sec. 2.2, or W. Pauli, “Die Allgemeinen Principien der
Wellenmechanik,” in Handbuch der
Physik, 2 Aufl., Band 24, 1 Teil, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), pp. 92 and 260.

The two Helmholtz equations (5.3) for the cross-spectral density function may be obtained, for example, by taking the Fourier transform of the two wave equations that the mutual coherence function is known to satisfy (cf. Ref. 5, Sec. 10.7.1).

As in the previous sections we describe the coherence properties of a source in terms of correlation functions involving the field distribution in the source plane. Such a description may be employed whether the source is a primary or a secondary one. However, when the source is a primary one, one may characterize its coherence properties in an alternative way, by means of correlation functions involving the true source variable (e.g., the charge-current density distribution). For a primary scalar source this alternative approach is discussed in a forthcoming paper by W. H. Carter and E. Wolf, “Coherence and radiant intensity in
scalar wavefields generated by fluctuating primary planar
sources (submitted to J. Opt. Soc.
Am.).

E. Wolf and W. H. Carter, “On the radiation efficiency of
quasi-homogeneous sources of different degrees of spatial
coherence,” in Proceedings of the Fourth
Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press).

Carets denote operators.

E. P. Wigner, “Quantum mechanical distribution
functions revisited,” in Perspectives in
Quantum Theory, edited by W. Yourgrau and A. van der Merwe (M.I.T. Press, Cambridge, Mass., 1971), pp. 25–36.