Abstract

A family of interconnected analogs of the radiance or specific intensity from classical radiometry are defined for coherent monochromatic wave fields. These generalized radiances are explicitly conserved along rays, regardless of wavelength. Unlike several previous definitions, the new forms are suitable for the description of fields propagating in all directions.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 193–199.
  2. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.
  3. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.
  4. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 69–78.
  5. L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).
  6. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  7. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  8. Yu. N. Barabanenkov, “On the spectral theory of radiation transfer equations,” Sov. Phys. JETP 29, 679–684 (1969).
  9. E. W. Marchand, E. Wolf, “Generalized radiometry for radiation from partially coherent sources,” Opt. Commun. 6, 305–308 (1972).
    [CrossRef]
  10. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
    [CrossRef]
  11. 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]
  12. G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
    [CrossRef]
  13. See also Ref. 2, pp. 292–297.
  14. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  15. A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
    [CrossRef]
  16. A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
    [CrossRef]
  17. 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]
  18. 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]
  19. A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
    [CrossRef]
  20. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1980).
    [CrossRef]
  21. E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024–2026 (1994).
    [CrossRef] [PubMed]
  22. A. T. Friberg, S. Yu. Popov, “Radiometric description of intensity and coherence in generalized holographic axicon images,” Appl. Opt. 35, 3039–3046 (1996).
    [CrossRef] [PubMed]
  23. A. S. Marathay, “Radiometry of partially coherent fields. I,” Opt. Acta 23, 785–794 (1976).
    [CrossRef]
  24. H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
    [CrossRef]
  25. R. Martinez-Herrero, P. M. Mejias, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. A 1, 556–558 (1984).
    [CrossRef]
  26. R. Martinez-Herrero, P. M. Mejias, “Radiometric definitions from second-order coherence characteristics of planar sources,” J. Opt. Soc. Am. A 3, 1055–1058 (1986).
    [CrossRef]
  27. G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
    [CrossRef]
  28. H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887.
  29. H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata, 8, 1518 (1991).
    [CrossRef]
  30. H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A Math. Gen. 25, 5263–5278 (1992).
    [CrossRef]
  31. H. M. Pedersen, “Propagation of generalized specific intensity in refracting media,” J. Opt. Soc. Am. A 9, 1623–1625 (1992).
    [CrossRef]
  32. H. M. Pedersen, “Geometric theory of fields radiated from three-dimensional, quasi-homogeneous sources,” J. Opt. Soc. Am. A 9, 1626–1632 (1992).
    [CrossRef]
  33. R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  34. H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]
  35. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
    [CrossRef]
  36. N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
    [CrossRef]
  37. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
    [CrossRef]
  38. E. P. Wigner, “Quantum mechanical distribution functions revisited,” in Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.
  39. The existence of such a plane for all directions u imposes some extra restrictions on the region of interest. For Eq. (2.6) to be valid for at least one plane normal to each direction u, the region must contain the minimum sphere that intersects all rays that carry some amount of light.
  40. This result was obtained during a discussion with Greg W. Forbes.
  41. S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
    [CrossRef]
  42. M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).
    [CrossRef]
  43. See Ref. 2, p. 289.
  44. K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  45. M. A. Alonso, G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A 17, 2391–2402 (2000).
    [CrossRef]
  46. The only reference given here where I is identified with H is Ref. 2, page 302.
  47. A similar result was presented in R. G. Littlejohn, R. Winston, “Generalized radiance and measurement,” J. Opt. Soc. Am. A 12, 2736–2743 (1995). The treatment presented there, however, was based on Walther’s first generalized radiance. It is therefore limited to forward-propagating fields and leads to results that are not invariant under rotations of the reference frame.
    [CrossRef]
  48. This function is the Helmholtz analog of the Husimi function from quantum theory. See Ref. 34.
  49. The localization properties of this function are studied in Ref. 45.

2000

1999

1996

1995

1994

1993

1992

1991

1987

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]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

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]

1984

R. Martinez-Herrero, P. M. Mejias, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. A 1, 556–558 (1984).
[CrossRef]

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

1982

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

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
[CrossRef]

1981

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

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

1980

1979

1978

1976

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

1974

1973

1972

E. W. Marchand, E. Wolf, “Generalized radiometry for radiation from partially coherent sources,” Opt. Commun. 6, 305–308 (1972).
[CrossRef]

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
[CrossRef]

1969

Yu. N. Barabanenkov, “On the spectral theory of radiation transfer equations,” Sov. Phys. JETP 29, 679–684 (1969).

1968

1964

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

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
[CrossRef]

Agarwal, G. S.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

Alonso, M. A.

Balasz, N. L.

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, “On the spectral theory of radiation transfer equations,” Sov. Phys. JETP 29, 679–684 (1969).

Bastiaans, M. J.

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 193–199.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.

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.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (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]

Forbes, G. W.

Friberg, A. T.

Jennings, B. K.

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Kim, K.

Lee, H.-W.

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Littlejohn, R. G.

Løkberg, O. J.

H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A Math. Gen. 25, 5263–5278 (1992).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.

Marathay, A. S.

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

Marchand, E. W.

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, “Generalized radiometry for radiation from partially coherent sources,” Opt. Commun. 6, 305–308 (1972).
[CrossRef]

Martinez-Herrero, R.

Mejias, P. M.

Nieto-Vesperinas, M.

Ovchinnikov, G. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
[CrossRef]

Pedersen, H. M.

H. M. Pedersen, “Propagation of generalized specific intensity in refracting media,” J. Opt. Soc. Am. A 9, 1623–1625 (1992).
[CrossRef]

H. M. Pedersen, “Geometric theory of fields radiated from three-dimensional, quasi-homogeneous sources,” J. Opt. Soc. Am. A 9, 1626–1632 (1992).
[CrossRef]

H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A Math. Gen. 25, 5263–5278 (1992).
[CrossRef]

H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata, 8, 1518 (1991).
[CrossRef]

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

H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887.

Popov, S. Yu.

Steinberg, S.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

Tatarskii, V. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
[CrossRef]

Walther, A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
[CrossRef]

Wigner, E. P.

E. P. Wigner, “Quantum mechanical distribution functions revisited,” in Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.

Winston, R.

Wolf, E.

E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024–2026 (1994).
[CrossRef] [PubMed]

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

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]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (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, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[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, “Generalized radiometry for radiation from partially coherent sources,” Opt. Commun. 6, 305–308 (1972).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 193–199.

Wolf, K. B.

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[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. Math. Phys.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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]

R. Martinez-Herrero, P. M. Mejias, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. A 1, 556–558 (1984).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, “Radiometric definitions from second-order coherence characteristics of planar sources,” J. Opt. Soc. Am. A 3, 1055–1058 (1986).
[CrossRef]

H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata, 8, 1518 (1991).
[CrossRef]

H. M. Pedersen, “Propagation of generalized specific intensity in refracting media,” J. Opt. Soc. Am. A 9, 1623–1625 (1992).
[CrossRef]

H. M. Pedersen, “Geometric theory of fields radiated from three-dimensional, quasi-homogeneous sources,” J. Opt. Soc. Am. A 9, 1626–1632 (1992).
[CrossRef]

R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
[CrossRef]

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]

M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).
[CrossRef]

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

M. A. Alonso, G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A 17, 2391–2402 (2000).
[CrossRef]

A similar result was presented in R. G. Littlejohn, R. Winston, “Generalized radiance and measurement,” J. Opt. Soc. Am. A 12, 2736–2743 (1995). The treatment presented there, however, was based on Walther’s first generalized radiance. It is therefore limited to forward-propagating fields and leads to results that are not invariant under rotations of the reference frame.
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. A Math. Gen.

H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A Math. Gen. 25, 5263–5278 (1992).
[CrossRef]

Opt. Acta

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

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

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

Opt. Commun.

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]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

E. W. Marchand, E. Wolf, “Generalized radiometry for radiation from partially coherent sources,” Opt. Commun. 6, 305–308 (1972).
[CrossRef]

Opt. Eng.

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
[CrossRef]

Opt. Lett.

Phys. Rep.

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Phys. Rev.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
[CrossRef]

Radiophys. Quantum Electron.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
[CrossRef]

Sov. Phys. JETP

Yu. N. Barabanenkov, “On the spectral theory of radiation transfer equations,” Sov. Phys. JETP 29, 679–684 (1969).

Other

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 193–199.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 69–78.

See also Ref. 2, pp. 292–297.

H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887.

See Ref. 2, p. 289.

E. P. Wigner, “Quantum mechanical distribution functions revisited,” in Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.

The existence of such a plane for all directions u imposes some extra restrictions on the region of interest. For Eq. (2.6) to be valid for at least one plane normal to each direction u, the region must contain the minimum sphere that intersects all rays that carry some amount of light.

This result was obtained during a discussion with Greg W. Forbes.

This function is the Helmholtz analog of the Husimi function from quantum theory. See Ref. 34.

The localization properties of this function are studied in Ref. 45.

The only reference given here where I is identified with H is Ref. 2, page 302.

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Figures (3)

Fig. 1
Fig. 1

Radiance is given by the flux element through an area element dσ centered at r and with unit normal n, for light traveling in a direction within a solid angle dΩ around the unit vector u, divided by n·udσdΩ.

Fig. 2
Fig. 2

When there is a source, sink, or inhomogeneity along the path of a ray, within an exterior concavity of the region of interest, the amount of light carried by this ray might change after this obstacle.

Fig. 3
Fig. 3

Plots of M(n)/(k|φ0|2) as functions of kl for n=0, 1, and 2, for a perfect cylindrical wave converging to the origin. Note that different numeric scale factors were used for each curve to show the relative reduction of the negative regions for increasing n. The area under the three unscaled curves is the same, and the central peak is higher for the curves corresponding smaller n, compensating for the more significant negative regions. These curves are even in l and are therefore shown for l0 only.

Equations (64)

Equations on this page are rendered with MathJax. Learn more.

B(r, u)u·ndσdΩ=d2Φ,
B(r, u)0.
u·B(r, u)=0.
I(r)=4πB(r, u)dΩ.
F(r)=4πB(r, u)udΩ.
J(u)=AuB(r, u)dσ,
AnB(r, u)dσ=J(u)u·n.
B(r, u)=f(r×u, u).
B(r, u)=u×Lf(r×u, u),
L=r×u.
u(θ)=(sin θ, cos θ).
B(r, u)u·ndsdθ=d2Φ,
B[r, u(θ)]=f[r·u(θ), θ],
u(θ)=(cos θ, -sin θ).
I(r)=2πB[r, u(θ)]dθ,
F(r)=2πB[r, u(θ)]u(θ)dθ,
J[u(θ)]=u(θ)·u(α)RB[su(α),u(θ)]ds=cos(θ-α)RB[su(α),u(θ)]ds,
J[u(θ)]=Rf(l, θ)dl.
(2+k2)U(r)=0,
U(r)=k2π1/22πφ(θ)exp[ikr·u(θ)]dθ,
limR12π2RCR|U(r)|2dσ=2πφ*(θ)φ(θ)dθ=U|U,
U1|U2=2πφ1*(θ)φ2(θ)dθ.
S(r)=|U(r)|2.
H(r)=12[U*(r)U(r)+k-2U*(r)·U(r)].
F(r)=12ik[U*(r)U(r)-U(r)U*(r)].
J[u(θ)]=lim rr121-ikrU[ru(θ)]2=|φ(θ)|2.
M(l, θ)=k2π-ππφθ+α2φ*θ-α2×exp2ikl sinα2dα.
M(n)(l, θ) k2π-ππφθ+α2φ*θ-α2×exp2ikl sinα2cosnα2dα
M(n+1)(l, θ)=RM(n)(l-l, θ)J1(2kl)2ldl.
|U1|U2|2=2πRM1(0)(l, θ)M2(1)(l, θ)d ldθ=2πRM1(1)(l, θ)M2(0)(l, θ)dldθ,
φ(θ)=φ0.
M(0)(l, θ)=k|φ0|2J0(2kl),
M(1)(l, θ)=|φ0|2sin(2kl)πl,
M(2)(l, θ)=|φ0|2J1(2kl)2l.
B(n)[r, u(θ)]=M(n)[r·u(θ), θ].
2πB(n)[r, u(θ)]dθ=k2π2π-ππφθ+α2φ*θ-α2×exp2ikr·u(θ)sinα2×cosnα2dαdθ.
θ=θ+α2,
θ=θ-α2.
2 cos θ sinα2=sin θ-sin θ,
2 sin θ sinα2=cos θ-cos θ,
2πB(n)[r, u(θ)]dθ=k2π2π,|θ-θ|πφ(θ)φ*(θ)×exp{2ikr·[u(θ)-u(θ)]}cosnθ-θ2dθdθ.
2πB(0)[r, u(θ)]dθ=|U(r)|2=S(r).
cos2θ-θ2=1+cos(θ-θ)2=1+u(θ)·u(θ)2.
2πB(2)[r, u(θ)]dθ
=12|U(r)|2+k4π×2πφ(θ)u(θ)exp[2ikr·u(θ)]dθ2
=12[|U(r)|2+k-2|U(r)|2]=H(r),
2πB(n)[r, u(θ)]u(θ)dθ=k2π2π,|θ-θ|πφ(θ)φ*(θ)×sinθ+θ2, cosθ+θ2×exp{2ikr·[u(θ)-u(θ)]}cosθ-θ2dθdθ.
cosθ-θ2sinθ+θ2=sin θ+sin θ2,
cosθ-θ2cosθ+θ2=cos θ+cos θ2,
2πB(1)[r, u(θ)]u(θ)dθ=k2π2πφ(θ)φ*(θ)u(θ)-u(θ)2×exp[2ikr·[u(θ)-u(θ)]]dθdθ=12ik[U*(r)U(r)-U(r)U*(r)]=F(r).
RM(n)(l, θ)dl=k2π-ππφθ+α2φ*θ-α2×Rexp2ikl sinα2dl×cosnα2dα=|φ(θ)|2=J[u(θ)].
2πRM(n)(l, θ)dldθ=U|U.
μ(θ)=12πm=-exp-kwm22+im θ,
μ(θ; r, θ)=μ0(θ-θ)exp[ikr·u(θ)].
Bμ[r, u(θ)]=2πφ*(θ)μ0(θ-θ)exp[ikr·u(θ)]dθ2.
φ(θ)=12πm=-cm exp(imθ).
U[ru(α)]=k2πm=-cm×2πexp[imθ+ikr cos(θ-α)]dθ=km=-cmJm(kr)exp[im(α+π/2)].
CR|U|2dσ=0Rr2π|U|2dαdr=km=-m=-cmcm*0RrJm(kr)Jm(kr)dr×2πexp[i(m-m)α]dα=2πkm=-|cm|20RrJm2(kr)dr.
CR|U|2dσ=2πkR2m=-|cm|201τJm2(kRτ)dτ.
Jm(kRτ)2πkRτ1/2coskRτ+m+12π2.
CR|U|2dσ4π2Rm=-|cm|2×01cos2kRτ+m+12π2dτ.
limR1RCR|U|2dσ=2π2m=-|cm|2.
m=-|cm|2=2πφ*(θ)φ(θ)dθ,
limR1RCR|U|2dσ=2π22πφ*(θ)φ(θ)dθ.

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