Abstract

Radiometric properties of the generalized radiance function as defined in terms of the Wigner function in six-dimensional phase space are considered in connection with the traditional definition of classical radiance and the space–time transport equation of classical radiometry in free space. In the asymptotic limit of short wavelength and sufficient incoherence, we show that the main properties of this generalized radiance function can be attributed to classical radiance in a nonstationary field. In addition, a correction to the transport equation in which the wavelength is finite is derived, and errors in this equation and the energy-conservation law of classical radiometry, are estimated.

© 1998 Optical Society of America

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References

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  1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  3. E. W. Marchand, E. Wolf, “Radiometry with source of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
    [CrossRef]
  4. E. W. Marchand, E. Wolf, “Walther’s definitions of generalized radiance,” J. Opt. Soc. Am. 64, 1273–1274 (1974).
    [CrossRef]
  5. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  6. L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
    [CrossRef]
  7. A. T. Friberg, “Summary of research,” in Selected Papers on Coherence and Radiometry, A. T. Friberg, B. J. Thompson, eds., SPIE Milestone Series, Vol. MS69 (SPIE, Bellingham, Wash., 1993), pp. xv–xxxi.
  8. R. G. LittleJohn, R. Winston, “Correction to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  9. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), p. 1.
  10. K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997).
    [CrossRef]
  11. E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  12. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
    [CrossRef]
  13. K. Yoshimori, K. Itoh, “On the generalized radiance function for a polychromatic field,” J. Opt. Soc. Am. 15, 2786–2787 (1998).
    [CrossRef]
  14. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), p. 185.
  15. E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
    [CrossRef]
  16. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  17. See, for example, Ref. 14, p. 224.
  18. 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]
  19. 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]

1998 (1)

K. Yoshimori, K. Itoh, “On the generalized radiance function for a polychromatic field,” J. Opt. Soc. Am. 15, 2786–2787 (1998).
[CrossRef]

1997 (1)

1993 (1)

1992 (1)

1987 (1)

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]

1984 (1)

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

1981 (1)

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

1978 (1)

1977 (1)

1974 (2)

1973 (1)

1968 (1)

1949 (1)

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (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]

Apresyan, L. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Carter, W. H.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), p. 1.

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]

Friberg, A. 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]

A. T. Friberg, “Summary of research,” in Selected Papers on Coherence and Radiometry, A. T. Friberg, B. J. Thompson, eds., SPIE Milestone Series, Vol. MS69 (SPIE, Bellingham, Wash., 1993), pp. xv–xxxi.

Itoh, K.

K. Yoshimori, K. Itoh, “On the generalized radiance function for a polychromatic field,” J. Opt. Soc. Am. 15, 2786–2787 (1998).
[CrossRef]

K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997).
[CrossRef]

Kravtsov, Yu. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

LittleJohn, R. G.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), p. 185.

Marchand, E. W.

Moyal, J. E.

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Sudarshan, E. C. G.

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

Walther, A.

Wigner, E.

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

Winston, R.

Wolf, E.

Yoshimori, K.

K. Yoshimori, K. Itoh, “On the generalized radiance function for a polychromatic field,” J. Opt. Soc. Am. 15, 2786–2787 (1998).
[CrossRef]

K. Yoshimori, K. Itoh, “Interferometry and radiometry,” J. Opt. Soc. Am. A 14, 3379–3387 (1997).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

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]

Phys. Rev. (1)

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

Phys. Rev. A (1)

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

Proc. Cambridge Philos. Soc. (1)

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Usp. Fiz. Nauk. (1)

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Other (4)

A. T. Friberg, “Summary of research,” in Selected Papers on Coherence and Radiometry, A. T. Friberg, B. J. Thompson, eds., SPIE Milestone Series, Vol. MS69 (SPIE, Bellingham, Wash., 1993), pp. xv–xxxi.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), p. 1.

See, for example, Ref. 14, p. 224.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), p. 185.

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Equations (39)

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2-1c22t2V(r, t)=0,
V(r, t)= d3k(2π)3exp(ik·r)V˜(k, t).
d3r|V(r, t)|2=d3k|V˜(k, t)|2.
V˜(k, t)=V˜0(k)exp(-iωt),
Bω(r, s, t)=k2(2π)3cK(r, k; t).
K(r, k; t)=d3ρ exp(-ik·ρ)×V*(r-ρ/2, t)V(r+ρ/2, t)=d3q exp(iq·r)×V˜*(k-q/2, t)V˜(k+q/2, t),
 d3k(2π)3K(r, k; t)=|V(r, t)|2,
 d3r(2π)3K(r, k; t)=|V˜(k, t)|2.
 d3rd3k(2π)3K(r, k; t)=d3r|V(r, t)|2.
dFω=(n·s)Bω(r, s, t)dσdΩdωdt,
Bω(r, s, t)t=k2(2π)3cicd3q exp(iq·r)(κ--κ+)×V˜*(k-q/2, t)V˜(k+q/2, t),
κ±=[(k±q/2)2]1/2.
κ--κ+=-s·q+18k2[(s·q)q2-(s·q)3]+O(q5).
k2(2π)3cicd3q exp(iq·r)(-s·q)×V˜*(k-q/2, t)V˜(k+q/2, t)
=-cs·Bω(r, s, t),
Bω(r, s, t)t+cs·Bω(r, s, t)=0.
k2(2π)3cic8k2d3q exp(iq·r)[(s·q)q2-(s·q)3]×V˜*(k-q/2, t)V˜(k+q/2, t)
=c8k2s2(s·)Bω(r, s, t),
Bω(r, s, t)t+cs·Bω(r, s, t)
=c8k2s2(s·)Bω(r, s, t).
Bωt+cs·Bω(λ/L)2cs·Bω,
uω(r, t)=1cdΩBω(r, s, t).
u(r, t)=0dωuω(r, t)=1c d3k(2π)3K(r, k; t)=|V(r, t)|2c,
Φω(r, t)=dΩsBω(r, s, t).
Φ(r, t)=0dωΦω(r, t)= d3k(2π)3sK(r, k; t).
Φ(r, t)= d3kd3ρ(2π)3iρkexp(-ik·ρ)×V*(r-ρ/2, t)V(r+ρ/2, t)= d3kd3ρ(2π)3exp(-ik·ρ)×1kImV*(r-ρ/2, t)V(r+ρ/2, t).
D(ρ)=i(2π)3 d3kkexp(-ik·ρ).
D(ρ)=12πρ[δ(+)(ρ)-δ(-)(ρ)],
δ(±)(ρ)=12π0dk exp(±ikρ)
δ(±)(ρ)=±i2πP 1ρ+12δ(ρ),
Φ(r, t)=-id3ρD(ρ)×ImV*(r-ρ/2, t)V(r+ρ/2, t).
div Φ(r, t)+u(r, t)t=0.
div Φ(r, t)+u(r, t)t
=0dωdiv Φω(r, t)+uω(r, t)t.
div Φω(r, t)+uω(r, t)t
=1cdΩcs·Bω(r, s, t)+Bω(r, s, t)t.
div Φω+uωt(λ/L)2 div Φω.
-ddtVd3ru(r, t)
=0dωσdσdΩ(n·s)Bω(r, s, t).

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