Abstract

An exact theory of free-space radiative energy transfer is given in terms of a generalized specific intensity that is constant along geometrical rays. General and explicit relations are derived for the generalized specific intensity expressed in terms of the field variables. Such relations are also derived for the cross-spectral density function of the field expressed in terms of the generalized specific intensity. For an arbitrary, freely propagated field, the theory is shown to reproduce the exact results of wave theory by transfer equations that are almost identical to the classical ones. The description reduces to the classical theory within a quasi-homogeneous field approximation. Similarly, it reduces to the geometrical-optics energy expressions in that approximation. For two-wave interference, additional ray contributions to the energy transport are found along the interference fringes. These interference rays serve only to describe the effects of the interference on the local energy transport.

© 1991 Optical Society of America

Full Article  |  PDF Article

Errata

Hans M. Pedersen, "Exact geometrical theory of free-space radiative energy transfer: errata," J. Opt. Soc. Am. A 8, 1518-1518 (1991)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-8-9-1518

References

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  1. M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).
  2. S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950).
  3. L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).
  4. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  5. Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969) [Zh. Eksp. Teor. Fiz. 56, 1262–1272 (1969)].
  6. G. V. Rozenberg, “The statistical-electrodynamic content of photometric quantities and the basic concepts of radiation transfer theory,” Opt. Spectrosc. (USSR) 28, 210–213 (1970).
  7. G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherent theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972) [Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
    [CrossRef]
  8. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  9. G. V. Rozenberg, “Coherence, observability and the photometric aspect of beam optics,” Appl. Opt. 12, 2855–2862 (1973).
    [CrossRef] [PubMed]
  10. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,”J. Opt. Soc. Am. 64, 1219–1226 (1974).
    [CrossRef]
  11. E. W. Marchand, E. Wolf, “Walther’s definition of generalized radiance,”J. Opt. Soc. Am. 64, 1273–1274 (1974).
    [CrossRef]
  12. A. Walther, “Reply to Marchand and Wolf,”J. Opt. Soc. Am. 64, 1275 (1974).
    [CrossRef]
  13. W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,”J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [CrossRef]
  14. E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
    [CrossRef]
  15. E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 868–886 (1976).
    [CrossRef]
  16. A. S. Marathay, “Radiometry of partially coherent fields I,” Opt. Acta 23, 785–794 (1976).
    [CrossRef]
  17. A. S. Marathay, “Radiometry of partially coherent fields II,” Opt. Acta 23, 795–798 (1976).
    [CrossRef]
  18. G. V. Rozenberg, “The light ray (contribution to the theory of the light field),” Sov. Phys. Usp. 20, 55–79 (1977) [Usp. Fiz. Nauk 121, 97–138 (1977)].
    [CrossRef]
  19. M. S. Zubairy, E. Wolf, “Exact equations for radiative transfer of energy and momentum in free electromagnetic fields,” Opt. Commun. 20, 321–324 (1977).
    [CrossRef]
  20. E. Collett, J. T. Foley, E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,”J. Opt. Soc. Am. 67, 475–477 (1977).
    [CrossRef]
  21. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  22. E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  23. H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 5.
    [CrossRef]
  24. A. T. Friberg, “On the existence. of a radiance function for a partially coherent planar source,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 459.
  25. A. T. Friberg, “On the question of the existence of nonradiating primary sources of finite extent,”J. Opt. Soc. Am. 68, 1281–1283 (1978).
    [CrossRef]
  26. E. Wolf, “The radiant intensity from planar sources of any state of coherence,”J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [CrossRef]
  27. A. Walther, “Propagation of generalized radiance through lenses,”J. Opt. Soc. Am. 68, 1606–1610 (1978).
    [CrossRef]
  28. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary state of coherence,”J. Opt. Soc. Am. 69, 192–198 (1979).
    [CrossRef]
  29. A. T. Friberg, “Phase-space methods for partially coherent wave-fields,” in Optics in Four Dimensions—1980 (ICO, Ensenada), M. A. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 313–331 (1981).
  30. A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous source,” Opt. Acta 28, 261–277 (1981).
    [CrossRef]
  31. A. T. Friberg, “Effects of coherence in radiometry,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 55–70 (1979).
    [CrossRef]
  32. A. T. Friberg, “Radiation from partially coherent sources,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 71–83 (1979).
    [CrossRef]
  33. M. S. Zubairy, “Radiative energy transfer in a randomly fluctuating medium,” Opt. Commun. 37, 315–320 (1981).
    [CrossRef]
  34. E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
    [CrossRef]
  35. H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
    [CrossRef]
  36. J. T. Foley, E. Wolf, “Radiometry as a short wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
    [CrossRef]
  37. M. Nieto-Vesperinas, “Classical radiometry and radiative transfer theory: a short-wavelength limit of a general mapping of cross-spectral densities in second-order coherence theory,” J. Opt. Soc. Am. A 3, 1354–1359 (1986).
    [CrossRef]
  38. K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).
    [CrossRef]
  39. E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).
  40. E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979).
  41. E. C. G. Sudarshan, “Geometry of wave electromagnetics,” in Optics in Four Dimensions—1980 (ICO, Ensenada), M. A. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 95–105 (1981).
  42. L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [CrossRef]
  43. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, New York, 1975).
  44. In Ref. 35 the radiance function was defined as an angular energy density and not as an angular energy flux density. To correct that mistake, the energy expressions there should be divided by the phase velocity c.
  45. A. J. Devaney, G. C. Sherman, “Plane-wave representation for scalar wavefields,”SIAM Rev. 15, 765–786 (1973).
    [CrossRef]
  46. H. H. Barrett, “The radon transform and its applications,” in Progress In Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 217–286.
    [CrossRef]
  47. H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
    [CrossRef]
  48. E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. 75, 269–280 (1959).
    [CrossRef]

1987

1986

1985

J. T. Foley, E. Wolf, “Radiometry as a short wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

1982

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[CrossRef]

1981

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

M. S. Zubairy, “Radiative energy transfer in a randomly fluctuating medium,” Opt. Commun. 37, 315–320 (1981).
[CrossRef]

E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

1979

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).

E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979).

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary state of coherence,”J. Opt. Soc. Am. 69, 192–198 (1979).
[CrossRef]

1978

1977

G. V. Rozenberg, “The light ray (contribution to the theory of the light field),” Sov. Phys. Usp. 20, 55–79 (1977) [Usp. Fiz. Nauk 121, 97–138 (1977)].
[CrossRef]

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

E. Collett, J. T. Foley, E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,”J. Opt. Soc. Am. 67, 475–477 (1977).
[CrossRef]

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

1976

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 868–886 (1976).
[CrossRef]

A. S. Marathay, “Radiometry of partially coherent fields I,” Opt. Acta 23, 785–794 (1976).
[CrossRef]

A. S. Marathay, “Radiometry of partially coherent fields II,” Opt. Acta 23, 795–798 (1976).
[CrossRef]

1975

E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,”J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

1974

1973

1972

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherent theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972) [Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
[CrossRef]

1970

G. V. Rozenberg, “The statistical-electrodynamic content of photometric quantities and the basic concepts of radiation transfer theory,” Opt. Spectrosc. (USSR) 28, 210–213 (1970).

1969

Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969) [Zh. Eksp. Teor. Fiz. 56, 1262–1272 (1969)].

1968

1965

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

1964

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

1959

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. 75, 269–280 (1959).
[CrossRef]

1953

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[CrossRef]

Baltes, H. P.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 5.
[CrossRef]

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969) [Zh. Eksp. Teor. Fiz. 56, 1262–1272 (1969)].

Barrett, H. H.

H. H. Barrett, “The radon transform and its applications,” in Progress In Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. XXI, pp. 217–286.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, New York, 1975).

Carter, W. H.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950).

Collett, E.

E. Collett, J. T. Foley, E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,”J. Opt. Soc. Am. 67, 475–477 (1977).
[CrossRef]

Devaney, A. J.

A. J. Devaney, G. C. Sherman, “Plane-wave representation for scalar wavefields,”SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Dolin, L. S.

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

Foley, J. T.

J. T. Foley, E. Wolf, “Radiometry as a short wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

E. Collett, J. T. Foley, E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,”J. Opt. Soc. Am. 67, 475–477 (1977).
[CrossRef]

Friberg, A. T.

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary state of coherence,”J. Opt. Soc. Am. 69, 192–198 (1979).
[CrossRef]

A. T. Friberg, “On the question of the existence of nonradiating primary sources of finite extent,”J. Opt. Soc. Am. 68, 1281–1283 (1978).
[CrossRef]

A. T. Friberg, “On the existence. of a radiance function for a partially coherent planar source,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 459.

A. T. Friberg, “Phase-space methods for partially coherent wave-fields,” in Optics in Four Dimensions—1980 (ICO, Ensenada), M. A. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 313–331 (1981).

A. T. Friberg, “Effects of coherence in radiometry,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 55–70 (1979).
[CrossRef]

A. T. Friberg, “Radiation from partially coherent sources,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 71–83 (1979).
[CrossRef]

Geist, J.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 5.
[CrossRef]

Green, H. S.

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[CrossRef]

Kim, K.

Mandel, L.

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Marathay, A. S.

A. S. Marathay, “Radiometry of partially coherent fields I,” Opt. Acta 23, 785–794 (1976).
[CrossRef]

A. S. Marathay, “Radiometry of partially coherent fields II,” Opt. Acta 23, 795–798 (1976).
[CrossRef]

Marchand, E. W.

Nieto-Vesperinas, M.

Ovchinnikov, G. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherent theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972) [Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
[CrossRef]

Pedersen, H. M.

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[CrossRef]

Planck, M.

M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).

Rozenberg, G. V.

G. V. Rozenberg, “The light ray (contribution to the theory of the light field),” Sov. Phys. Usp. 20, 55–79 (1977) [Usp. Fiz. Nauk 121, 97–138 (1977)].
[CrossRef]

G. V. Rozenberg, “Coherence, observability and the photometric aspect of beam optics,” Appl. Opt. 12, 2855–2862 (1973).
[CrossRef] [PubMed]

G. V. Rozenberg, “The statistical-electrodynamic content of photometric quantities and the basic concepts of radiation transfer theory,” Opt. Spectrosc. (USSR) 28, 210–213 (1970).

Sherman, G. C.

A. J. Devaney, G. C. Sherman, “Plane-wave representation for scalar wavefields,”SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979).

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).

E. C. G. Sudarshan, “Geometry of wave electromagnetics,” in Optics in Four Dimensions—1980 (ICO, Ensenada), M. A. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 95–105 (1981).

Tatarskii, V. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherent theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972) [Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
[CrossRef]

Walther, A.

Wolf, E.

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).
[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

E. Wolf, “The radiant intensity from planar sources of any state of coherence,”J. Opt. Soc. Am. 68, 1597–1605 (1978).
[CrossRef]

E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

E. Collett, J. T. Foley, E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,”J. Opt. Soc. Am. 67, 475–477 (1977).
[CrossRef]

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

M. S. Zubairy, 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, 868–886 (1976).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,”J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
[CrossRef]

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

E. W. Marchand, E. Wolf, “Walther’s definition of generalized radiance,”J. Opt. Soc. Am. 64, 1273–1274 (1974).
[CrossRef]

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. 75, 269–280 (1959).
[CrossRef]

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, New York, 1975).

Zubairy, M. S.

M. S. Zubairy, “Radiative energy transfer in a randomly fluctuating medium,” Opt. Commun. 37, 315–320 (1981).
[CrossRef]

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

Appl. Opt.

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

J. Opt. Soc. Am.

E. Collett, J. T. Foley, E. Wolf, “On an investigation of Tatarskii into the relationship between coherence theory and the theory of radiative transfer,”J. Opt. Soc. Am. 67, 475–477 (1977).
[CrossRef]

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

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

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,”J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

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

E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

A. T. Friberg, “On the question of the existence of nonradiating primary sources of finite extent,”J. Opt. Soc. Am. 68, 1281–1283 (1978).
[CrossRef]

E. Wolf, “The radiant intensity from planar sources of any state of coherence,”J. Opt. Soc. Am. 68, 1597–1605 (1978).
[CrossRef]

A. Walther, “Propagation of generalized radiance through lenses,”J. Opt. Soc. Am. 68, 1606–1610 (1978).
[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary state of coherence,”J. Opt. Soc. Am. 69, 192–198 (1979).
[CrossRef]

E. W. Marchand, E. Wolf, “Walther’s definition of generalized radiance,”J. Opt. Soc. Am. 64, 1273–1274 (1974).
[CrossRef]

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]

J. Opt. Soc. Am. A

Opt. Acta

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[CrossRef]

A. S. Marathay, “Radiometry of partially coherent fields I,” Opt. Acta 23, 785–794 (1976).
[CrossRef]

A. S. Marathay, “Radiometry of partially coherent fields II,” Opt. Acta 23, 795–798 (1976).
[CrossRef]

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

Opt. Commun.

E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

M. S. Zubairy, “Radiative energy transfer in a randomly fluctuating medium,” Opt. Commun. 37, 315–320 (1981).
[CrossRef]

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

Opt. Spectrosc. (USSR)

G. V. Rozenberg, “The statistical-electrodynamic content of photometric quantities and the basic concepts of radiation transfer theory,” Opt. Spectrosc. (USSR) 28, 210–213 (1970).

Phys. Lett.

E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979).

Phys. Rev. A

E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

Phys. Rev. D

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 868–886 (1976).
[CrossRef]

Physica

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).

Proc. Phys. Soc.

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. 75, 269–280 (1959).
[CrossRef]

Proc. Phys. Soc. A

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[CrossRef]

Radiophys. Quantum Electron.

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Sov. Phys. Usp.

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A. T. Friberg, “Effects of coherence in radiometry,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 55–70 (1979).
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A. T. Friberg, “Radiation from partially coherent sources,” in Applications of Optical Coherence, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.194, 71–83 (1979).
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H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 5.
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A. T. Friberg, “On the existence. of a radiance function for a partially coherent planar source,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 459.

A. T. Friberg, “Phase-space methods for partially coherent wave-fields,” in Optics in Four Dimensions—1980 (ICO, Ensenada), M. A. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 313–331 (1981).

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Equations (95)

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E ( r ) = 1 v g I 0 ( r , n ) d Ω ( n ) ,
F ( r ) = I 0 ( r , n ) n d Ω ( n ) .
n · I 0 ( r , n ) = 0 ,
I 0 ( r , n ) 0.
· F ( r ) = 0
J ( n ) = n · n 0 I 0 ( r , n ) δ ( r · n 0 ) d 3 r = B 0 ( r , n ) δ ( r · n 0 ) d 3 r ,
E ( r ) = 1 c ( 1 + 1 4 k 2 2 ) W ( r , 0 ) ,
F ( r ) = 1 i k [ ξ W ( r , ξ ) ] ξ = 0 .
W ( r , ξ ) = 1 2 π - Γ ( r - ξ / 2 , r + ξ / 2 , τ ) exp ( i ω τ ) d τ ,
Γ ( r 1 , r 2 , τ ) = V * ( r 1 , t - τ ) V ( r 2 , t )
ξ 2 W ( r , ξ ) + k 2 W ( r , ξ ) + 1 4 2 W ( r , ξ ) = 0 ,
ξ · W ( r , ξ ) = 0.
2 U ( r ) + k 2 U ( r ) = 0.
W ( r , ξ ) = U * [ ( r - ξ ) / 2 ] U [ ( r + ξ ) / 2 ] .
W ( r , ξ ) = U * [ ( r - ξ ) / 2 ] U [ ( r + ξ / 2 ) ] ,
W ( r , ξ ) = ( 1 ( 2 π ) 2 A ( n , s ) δ ( n · s ) × exp { i [ s · r + k 2 - s 2 / 4 n · ξ ] } d 3 s ) d Ω ( n ) .
W ( r , ξ ) = ( k 2 π ) 2 C ( n 1 , n 2 ) × exp [ i k ( n 2 · r 2 - n 1 · r 1 ) ] d Ω 1 d Ω 2 = ( k 2 π ) 2 C ( n 1 , n 2 ) exp { i k [ ( n 2 - n 1 ) · r + ½ ( n 1 + n 2 ) · ξ ] } d Ω 1 d Ω 2 .
W ( r , ξ ) = { ( k 2 π ) 2 δ ( n · q ) S ( 1 - 1 4 q 2 n , q ) × exp [ i k ( q · r ) + 1 - q 2 / 4 n · ξ ) ] d 3 q } d Ω ( n ) ,
A ( n , s ) = S [ 1 - ¼ ( s / k ) 2 n , s / k ] .
A ( n , 0 ) = S ( n , 0 ) = C ( n , n )
A ( n , s ) = A * ( n , - s ) .
F ( r ) = [ ( 1 2 π ) 2 A ( n , s ) δ ( n · s ) 1 - s 2 4 k 2 × exp ( i s · r ) d 3 s ] n d Ω ( n ) ,
F ( r ) = I ( r , n ) n d Ω ( n ) ,
I ( r , n ) = ( 1 2 π ) 2 A ( n , s ) δ ( n · s ) 1 - s 2 4 k 2 exp ( i s · r ) d 3 s
n · I ( r , n ) = 0.
W ( r , 0 ) = [ ( 1 2 π ) 2 A ( n , s ) δ ( n · s ) exp ( i s · r ) d 3 s ] d Ω ( n ) .
E ( r ) = 1 + 1 4 k 2 2 1 c I ( r , n ) d Ω ( n ) ,
E ( r ) = ( 1 + 1 8 k 2 2 - ) 1 c I ( r , n ) d Ω ( n ) .
J ( n ) = n · n 0 I ( r , n ) δ ( r · n 0 ) d 3 r .
J ( n ) = ( 1 2 π ) 2 A ( n , s ) n · n 0 δ ( n · s ) 1 - s 2 4 k 2 × [ δ ( r · n 0 ) exp ( i s · r ) d 3 r ] d 3 s = A ( n , s ) n · n 0 δ ( n · s ) 1 - s 2 4 k 2 δ 2 ( s ) d 3 s ,
J ( n ) = - A ( n , s n 0 ) n · n 0 δ ( n · s n 0 ) 1 - s 2 4 k 2 d s = A ( n , 0 ) .
W ( r , ξ ) = [ δ ( n · r ) I ( r - r , n ) G ( r , n · ξ ) d 3 r ] d Ω ( n ) ,
G ( r , z ) = 1 ( 2 π ) 2 δ ( n · s ) × { exp [ i ( s · r + k 2 - s 2 / 4 z ] / 1 - s 2 4 k 2 } d 3 s = k π exp ( 2 i k r 2 + z 2 / 4 ) / r 2 + z 2 / 4 .
W ( r , ξ ) = { exp [ i ( k 2 + 2 / 4 n · ξ ) ] / 1 + 2 4 k 2 } × I ( r , n ) d Ω ( n ) .
W ( r , ξ ) = exp ( i k n · ξ ) [ 1 + 1 8 ( i k n · ξ - 1 ) 2 k 2 + 1 128 [ 3 - 3 i k n · ξ - ( k n · ξ ) 2 ] 4 k 4 + ] I ( r , n ) d Ω ( n ) .
W ( r , ξ ) = ( 1 2 π ) 3 W ( r , κ ) exp ( i κ · ξ ) d 3 κ ,
W ( r , κ ) = W ( r , ξ ) exp ( - i κ · ξ ) d 3 ξ ,
F ( r ) = ( 1 2 π ) 2 W ( r , κ ) ( κ / k ) d 3 κ = [ ( 1 2 π ) 3 0 W ( r , κ n ) ( κ 3 / k ) d κ ] n d Ω ( n ) ,
I ( r , n ) = ( 1 2 π ) 3 0 W ( r , κ n ) ( κ 3 / k ) d κ .
R ( r , n , p ) = W ( r , ξ ) δ ( p - n · ξ ) d 3 ξ ,
I ( r , n ) = - 1 i k ( 2 π ) 2 [ 3 p 3 R ( r , n , p ) ] p = 0 ,
R ( r , n , p ) = 1 2 [ R ( r , n , p ) + i π - R ( r , n , p - t ) d t t ]
I ( r , n ) = - 1 i k ( 2 π ) 2 [ 3 p 3 R ( r , n , p ) ] p = 0 = - 1 i k ( 2 π ) 2 δ ( n · ξ ) ( n · ξ ) 3 W ( r , ξ ) d 3 ξ .
I ( r , n ) = ( k 2 π ) 2 ( 1 + 1 4 k 2 2 ) 3 / 2 n z W ( r , ξ ) × exp [ - i k 2 + 2 / 4 ( n x ξ x + n y ξ y ) ] d ξ x d ξ y ,
I ( r , n ) = ( k 2 π ) 2 ( 1 + 1 4 k 2 2 ) 3 / 2 δ ( n · ξ ) W ( r , ξ ) d 3 ξ .
W ( r , κ ) = W ( r , ξ ) exp ( - i κ · ξ ) d 3 ξ = { ( 1 2 π ) 2 A ( n , s ) δ ( n · s ) ( 2 π ) 3 × δ 3 [ κ - k 2 - s 2 / 4 n ] d 3 s } d Ω ( n ) .
δ 3 ( r n - r 0 n 0 ) = 1 r 0 2 δ ( r - r 0 ) δ 2 ( n - n 0 )
I ( r , n ) = [ ( 1 2 π ) 2 A ( n , s ) δ ( n · s ) 1 - s 2 4 k 2 × exp ( i s · r ) d 3 s ] δ 2 ( n - n ) d Ω ( n ) .
I ( r , n ) = ( k 2 π ) 2 C ( n 1 , n 2 ) exp [ i k ( n 2 - n 1 ) · r ] × | n 1 + n 2 2 | δ 2 ( n - n 1 + n 2 n 1 + n 2 ) d Ω 1 d Ω 2 .
I ( r , n ) = ( k 2 π ) 2 δ ( n · q ) S ( 1 - 1 4 q 2 n , q ) × 1 - 1 4 q 2 exp ( i k q · r ) d 3 q .
E ( r ) = 1 c I ( r , n ) d Ω ( n ) ,
J ( n ) = δ ( n · r ) I ( r , n ) d 3 r ,
( 1 / Δ Ω ) Δ Ω I ( r , n ) d Ω ( n ) = 1 Δ Ω ( k 2 π ) 2 C ( n 1 , n 2 ) × exp [ i k ( n 2 - n 1 ) · r ] | n 1 + n 2 2 | d Ω 1 d Ω 2 ,
( 1 / Δ Ω ) Δ Ω I ( r , n ) d Ω ( n ) = 1 Δ Ω ( k 2 π ) 2 Δ Ω Δ Ω C ( n 1 - n 2 ) × exp [ i k ( n 2 - n 1 ) · r ] d Ω 1 d Ω 2 .
I ( r , n ) = ( k 2 π ) 2 W ( r , ξ ) δ ( n · ξ ) d 3 ξ .
I ( r , n ) = I 0 ( r , n ) .
W ( r , ξ ) = I 0 ( r , n ) exp ( i k n · ξ ) d Ω ( n )
I 0 ( r , n ) = ( k 2 π ) 2 δ ( n · q ) S ( n , q ) exp ( i k q · r ) d 3 q .
W ( r , κ ) = I 0 ( r , n ) ( 2 π ) 3 δ 3 ( κ - k n ) d Ω ( n ) = ( 2 π ) 3 k 2 I 0 ( r , κ / k ) δ ( κ - k ) .
I ( r , n ) = ( 1 2 π ) 3 0 W ( r , κ n ) ( k 3 / k ) d κ = I 0 ( r , n ) .
I 0 ( r , n ) = ( k 2 π ) 2 n z W ( r , ξ ) exp [ - i k ( n x ξ x + n y ξ y ) ] d ξ x d ξ y ,
U ( r ) = a ( r ) exp [ i ϕ ( r ) ] ,
W ( r , ξ ) = a ( r - ξ / 2 ) a ( r + ξ / 2 ) × exp { i [ ϕ ( r + ξ / 2 ) - ϕ ( r - ξ / 2 ) ] } = a 2 ( r ) exp [ i ϕ ( r ) · ξ ] .
W ( r , κ ) = ( 2 π ) 3 a 2 ( r ) δ 3 [ κ - ϕ ( r ) ] .
I ( r , n ) = a 2 ( r ) 0 δ 3 [ κ n - ϕ ( r ) ] ( κ 3 / k ) d κ = a 2 ( r ) ϕ ( r ) k δ 2 [ n - ϕ ( r ) ϕ ( r ) ] ,
E ( r ) = a 2 ( r ) / c ,
F ( r ) = a 2 ( r ) ϕ ( r ) / k .
[ n a 2 ϕ δ 2 ( n - ϕ / ϕ ) / k ] = 0.
· F ( r ) = · [ a 2 ( r ) ϕ ( r ) / k ] = 0 ,
2 a ( r ) · ϕ ( r ) + a ( r ) 2 ϕ ( r ) = 0.
E ( r ) = [ 1 + 1 4 k 2 2 ] a 2 ( r ) / c
F ( r ) = a 2 ( r ) ϕ ( r ) / k ,
2 a · ϕ + a 2 ϕ = 0
[ ϕ ] 2 = k 2 + ( 1 / a ) 2 a .
U = a 1 exp ( i ϕ 1 ) + a 2 exp ( i ϕ 2 ) .
W ( r , ξ ) = a 1 2 exp ( i ϕ 1 · ξ ) + a 2 2 exp ( i ϕ 2 · ξ ) + 2 a 1 a 2 cos ( ϕ 2 - ϕ 1 ) exp [ i ( ϕ 1 + ϕ 2 2 ) · ξ ] .
I ( r , n ) = a 1 2 ( ϕ 1 / k ) δ 2 ( n - ϕ 1 ϕ 1 ) + a 2 2 ( ϕ 2 / k ) δ 2 ( n - ϕ 2 ϕ 2 ) + a 1 a 2 cos ( ϕ 2 - ϕ 1 ) ( ϕ 1 + ϕ 2 / k ) δ 2 ( n - ϕ 1 + ϕ 2 ϕ 1 + ϕ 2 ) .
ϕ 1 = k n 1 ,             ϕ 2 = k n 2 ,
ϕ 2 - ϕ 1 = ψ 0 + k ( n 2 - n 1 ) · r = ψ 0 + k Δ n · r ,
I ( r , n ) = a 1 2 δ 2 ( n - n 1 ) + a 2 2 δ 2 ( n - n 2 ) + a 1 a 2 cos ( ψ 0 + k Δ n · r ) n 1 + n 2 δ 2 ( n - n 1 + n 2 n 1 + n 2 ) .
E ( r ) = 1 c [ a 1 2 + a 2 2 + 2 a 1 a 2 | n 1 + n 2 2 | 2 cos ( ψ 0 + k Δ n · r ) ] ,
F ( r ) = a 1 n 2 1 + a 2 n 2 2 + a 1 a 2 ( n 1 + n 2 ) cos ( ψ 0 + k Δ n · r ) .
a ( r ) = [ a 1 2 + a 2 2 + 2 a 1 a 2 cos ( ϕ 2 - ϕ 1 ) ] 1 / 2
ϕ ( r ) = arctan [ a 1 sin ϕ 1 + a 2 sin ϕ 2 a 1 cos ϕ 1 + a 2 cos ϕ 2 ]
I ( r , n ) = a 2 [ δ 2 ( n - n 1 ) + δ 2 ( n - n 2 ) + cos ( ψ 0 + k Δ n · r ) n 1 + n 2 δ 2 ( n - n 1 + n 2 n 1 + n 2 ) ] ,
E ( r ) = 2 a 2 c [ 1 + | n 1 + n 2 2 | 2 cos ( ψ 0 + k Δ n · r ) ] ,
F ( r ) = a 2 [ 1 + cos ( ψ 0 + k Δ n · r ) ] ( n 1 + n 2 ) .
F dark = 0 ,             E dark = a 2 2 c Δ n 2 .
E ( r ) = 1 2 ρ k 2 [ 1 + 1 4 k 2 2 ] W ϕ ( r , 0 ) ,
F ( r ) = - ½ ρ i ω [ ξ W ϕ ( r , ξ ) ] ξ = 0 .
W ( r , ξ ) = ω 2 W ϕ ( r , ξ ) .
W ( r , ξ ) = ( k 2 π ) 2 C ( Q - q / 2 , Q + q / 2 ) × exp { i k [ q · r + Q · ξ ] } d Ω 1 d Ω 2 ,
J = ( n 1 x , n 1 y , n 2 x , n 2 y ) / ( n x , n y , q x , q y ) = Q 2 .
d Ω 1 d Ω 2 = d q x d q y Q 2 d Ω ( n ) / ( n 1 z n 2 z ) = d q x d q y Q 2 d Ω ( n ) / ( Q z 2 - q z 2 / 4 ) = d q x d q y d Ω ( n ) ,
δ 3 ( r n - r 0 n 0 ) = δ ( r n x ) δ ( r n y ) δ ( r n z - r 0 ) = 1 r 2 n z δ ( n x ) δ ( n y ) δ ( r - r 0 n z ) = 1 r 0 2 δ ( n x ) δ ( n y ) δ ( r - r 0 ) ,

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