Abstract

Recent researches have revealed that there exists an intimate connection between radiometry and the theory of partial coherence. In this paper a review is presented of some of these developments. After a brief discussion of various models for energy transport in optical fields and of some of the basic concepts of the classical theory of optical coherence, the following topics are discussed: the foundations of radiometry, the coherence properties of Lambertian sources, and the relationship between the state of coherence of a source and the directionality of the light that the source generates. Some very recent work is also described which reveals that certain sources that are spatially highly incoherent in a global sense will generate light that is just as directional as a laser beam.

© 1978 Optical Society of America

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  1. 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.
  2. See, for example, E. Hopf, Mathematical Problems of Radiative Equilibrium (Cambridge U. P., Cambridge, 1934), Sec. 2.
  3. 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.
  4. 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.
  5. See, for example, M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.2.
  6. 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.
  7. 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˙ΔV* + 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).
  8. L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).
  9. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256–1259 (1968).
  10. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 63, 1622–1623 (1973).
  11. E. W. Marchand and E. Wolf, "Radiometry with sources of any state of coherence," J. Opt. Soc. Am. 64, 1219–1226 (1974).
  12. 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).
  13. 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).
  14. 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).
  15. 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)]phys. opt. denotes here the radiant intensity when calculated in this more direct manner.
  16. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).
  17. 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.
  18. 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).
  19. Carets denote operators.
  20. 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.
  21. 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).
  22. 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).
  23. 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.).
  24. W. H. Carter and E. Wolf "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).
  25. E. Wolf and W. H. Carter, "On the radiation efficiency of quasihomogeneous 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).
  26. 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).
  27. 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).
  28. E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27–29 (1978).
  29. 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).
  30. Actually all of them have the same degree of spatial coherence gQ (r1 - r 2). They can only differ by their intensity distributions IQ (r).
  31. 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).
  32. 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].
  33. 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.
  34. Ref. 24, Sec. II, especially Eqs. (31) and (32).
  35. See, for example, the following publications and the references quoted therein: (a) 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).
  36. E. Wolf, "New theory of radiative energy transfer in free electromagnetic fields," Phys. Rev. D 13, 869–886 (1976).
  37. 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).

1978 (1)

1977 (2)

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).

W. H. Carter and E. Wolf "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).

1976 (3)

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

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).

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

1975 (2)

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).

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).

1974 (3)

1973 (1)

1972 (1)

1970 (1)

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.

1968 (1)

1965 (1)

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.

1963 (1)

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.

1932 (1)

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

Agarwal, G. S.

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.

Akhiezer, A. I.

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.

Antes, G.

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).

Baltes, H. P.

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).

Barabanenkov, Yu. N.

See, for example, the following publications and the references quoted therein: (a) 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).

Bartolino, R.

Beran, M.

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.

Berestetskii, V. B.

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.

Bertolotti, M.

Born, M.

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

Carter, W. H.

W. H. Carter and E. Wolf "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).

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).

E. Wolf and W. H. Carter, "On the radiation efficiency of quasihomogeneous 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).

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.).

Collett, E.

Ferwerda, H. A.

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).

Friberg, A. T.

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).

Hopf, E.

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

Kravtsov, Yu. A.

See, for example, the following publications and the references quoted therein: (a) 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).

Mandel, L.

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

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.

Marathay, A. S.

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).

Marchand, E. W.

Parrent, G.

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.

Rytov, S. M.

See, for example, the following publications and the references quoted therein: (a) 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).

Scudieri, F.

Skinner, T. J.

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].

Srinivas, M. D.

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).

Steinle, B.

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).

Tatarskii, V. I.

See, for example, the following publications and the references quoted therein: (a) 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).

van Heel, M. G.

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).

Walther, A.

Wigner, E.

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

Wigner, E. P.

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.

Wolf, E.

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

W. H. Carter and E. Wolf "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).

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).

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

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

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).

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).

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).

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

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).

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.

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.

E. Wolf and W. H. Carter, "On the radiation efficiency of quasihomogeneous 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).

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.).

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

Zubairy, M. S.

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).

Appl. Opt. (1)

J. Opt. Soc. Am. (7)

Nuovo Cimento (1)

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.

Opt. Acta (1)

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).

Opt. Commun. (2)

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).

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).

Opt. Lett. (1)

Phys. Rev. (1)

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

Phys. Rev. D (3)

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).

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

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.

Rev. Mod. Phys. (1)

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.

Other (19)

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˙ΔV* + 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)]phys. 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.

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.

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 quasihomogeneous 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).

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).

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].

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).

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

See, for example, the following publications and the references quoted therein: (a) 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 - r 2). They can only differ by their intensity distributions IQ (r).

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