Abstract

It is shown that the several specific intensities and generalized radiances, given by a mapping of the cross-spectral density in the statistical theory of wave fields, tend, respectively, in the limit of large wave numbers to quantities that have the properties of specific intensities and radiances of the classical theory of radiative energy transfer and radiometry. An example for quasi-homogeneous sources is given.

© 1986 Optical Society of America

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References

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  1. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  2. A. T. Friberg, “On the existence of the radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
    [CrossRef]
  3. E. C. G. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. 23, 2802–2809 (1981).
    [CrossRef]
  4. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  5. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  6. A. Walther, “Propagation of the generalized radiance through lenses,” J. Opt. Soc. Am. 68, 1606–1610 (1978).
    [CrossRef]
  7. V. I. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), Sec. 63. [See also L. Dolin, “Beam description of weakly inhomogeneous fields,” Izv. Vusov (Radiofizika) 7, 559 (1964)].
  8. G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer theory,” Radiophys. Quantum Electron. 15, 1087 (1972).
    [CrossRef]
  9. E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1979).
    [CrossRef]
  10. A. T. Friberg, “On the generalized radiance associated with radiation from a quasi-homogeneous plasma source,” Opt. Acta 28, 261–277 (1981).
    [CrossRef]
  11. 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]
  12. L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781–786 (1966).
    [CrossRef]
  13. E. Wolf, J. Opt. Soc. Am. 72, 343– (1982).
    [CrossRef]
  14. M. S. Zubairy, Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978), Chap. 2.
  15. E. Wigner, “Quantum mechanical distribution functions revisited,” in Perspectives in Quantum Mechanics, W. Yourgrau, A. Van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.
  16. E. Collett, J. T. Foley, E. Wolf, J. Opt. Soc. Am. 67, 465 (1977).
    [CrossRef]
  17. 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]
  18. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  19. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  20. E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
    [CrossRef]

1985 (1)

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 (1)

1981 (2)

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

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

1979 (2)

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

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1979).
[CrossRef]

1978 (2)

1977 (3)

E. Collett, J. T. Foley, E. Wolf, J. Opt. Soc. Am. 67, 465 (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]

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

1973 (1)

1972 (2)

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

E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
[CrossRef]

1968 (1)

1966 (1)

L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781–786 (1966).
[CrossRef]

1932 (1)

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

Carter, W. H.

Cohen, L.

L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781–786 (1966).
[CrossRef]

Collett, E.

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, J. Opt. Soc. Am. 67, 465 (1977).
[CrossRef]

Friberg, A. T.

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

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

Marchand, E. W.

Ovchinnikov, G. I.

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

Sudarshan, E. C. G.

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

Tatarskii, V. I.

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

V. I. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), Sec. 63. [See also L. Dolin, “Beam description of weakly inhomogeneous fields,” Izv. Vusov (Radiofizika) 7, 559 (1964)].

Walther, A.

Wigner, E.

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

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

Wolf, E.

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, J. Opt. Soc. Am. 72, 343– (1982).
[CrossRef]

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1979).
[CrossRef]

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

E. Collett, J. T. Foley, E. Wolf, J. Opt. Soc. Am. 67, 465 (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]

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

E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
[CrossRef]

Zubairy, M. S.

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]

M. S. Zubairy, Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978), Chap. 2.

J. Math. Phys. (1)

L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781–786 (1966).
[CrossRef]

J. Opt. Soc. Am. (9)

Opt. Acta (1)

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

Opt. Commun. (2)

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

Phys. Rev. (2)

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

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

Phys. Rev. D (1)

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1979).
[CrossRef]

Radiophys. Quantum Electron. (1)

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

Other (3)

V. I. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), Sec. 63. [See also L. Dolin, “Beam description of weakly inhomogeneous fields,” Izv. Vusov (Radiofizika) 7, 559 (1964)].

M. S. Zubairy, Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978), Chap. 2.

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

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Figures (1)

Fig. 1
Fig. 1

Source geometry corresponding to Eq. (1).

Equations (52)

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d E = I ( r , s ) cos θ d σ d Ω .
W ( r 1 , r 2 ) = V * ( r 1 ) V ( r 2 ) .
U ( r ) = 1 c W ( r , 0 ) ,
P ( r ) = 1 i k ρ W ( r , ρ ) | ρ = 0 ,
U ( r ) = 1 c I ( r , s ) d Ω ,
P ( r ) = R ( r , s ) d Ω
I ( r , s ) d Ω = W ( r , 0 ) ,
R ( r , s ) d Ω = 1 i k ρ W ( r , ρ ) | ρ = 0 .
I ( r , s ) = d 3 ρ d 3 v K ( r , s , ρ , v ) W ( v , ρ ) ,
R ( r , s ) = d 3 ρ d 3 v K ( r , s , ρ , v ) W ( v , ρ ) ,
K ( r , s , ρ , v ) = 1 ( 2 π ) 6 0 d K K 2 d 3 u e i K · ρ e i u · r e + i u · v g ( u , ρ ) K = K s , s = ( s , s z ) , s z = cos θ ,
g ( u , 0 ) = g ( 0 , ρ ) = 1.
I ( r , s ) = 0 d K K 2 f ( r , K ) ,
R ( r , s ) = 1 i k 0 d K K 2 h ( r , K ) ,
f ( r , K ) = 1 ( 2 π ) 6 d 3 d 3 d 3 υ e i K · ρ × e i u · r e i u · v g ( u , ρ ) W ( v , ρ ) ,
h ( r , K ) = 1 ( 2 π ) 6 d ρ 3 d u 3 d υ 3 e i K · ρ × e i u · r e i u · v g ( u , ρ ) ρ W ( v , ρ ) .
P ( r ) = s I ( r , s ) d Ω ;
1 ( 2 π ) 6 0 d K K 2 d 3 r V ( r ) ( ) ( k 2 π ) 2 cos θ ( 2 π ) 2 d 3 R V ( R ) ( ) .
B ( R , s ) = ( k 2 π ) 2 cos θ ( 2 π ) 2 d 2 ρ d 2 u d 2 υ exp ( i k s · ρ ) × e i u · R e i u · v g ( u , ρ ) W ( v , ρ ) ,
( v 2 + 4 ρ 2 + 4 k 2 ) W ( v , ρ ) = 0 ,
( v · ρ ) W ( v , ρ ) = 0.
r 2 I ( r , s ) + 4 0 d K K 2 ( k 2 K 2 ) × d 3 ρ d 3 u d 3 υ e i u · r e i k · ρ e i u · v g ( u , ρ ) W ( v , ρ ) 4 0 d K K 2 d 3 ρ d 3 u d 3 υ e i K · ρ e i u · r e i u · v ρ 2 g ( u , ρ ) W ( v , ρ ) 8 0 d K K 2 d 3 ρ d 3 u d 3 υ ρ g ( u , ρ ) · ρ W ( v , ρ ) = 0.
r . 0 d K K 2 d 3 ρ d 3 u d 3 υ e i K · ρ e i u · r × e + i u · v g ( u , ρ ) ρ W ( v , ρ ) = 0.
R ( r , s ) d Ω = s I ( r , s ) d Ω ,
d 3 k [ h ( r , K ) i k s f ( r , K ) ] = 0.
e i u · r u g ( u ) d 3 u = 1 i r d 3 u e i u · r g ( u )
h ( r , K ) = g ( 1 i r , 1 i K ) i K g 1 ( 1 i r , 1 i K ) f ( r , K ) ,
d Ω d K K 2 [ g ( 1 i r , 1 i K ) K × g 1 ( 1 i r , 1 i K ) k s ] f ( r , K ) = 0.
lim k g ( u , ρ / k ) = 1.
f ( r , K ) = I ( r , s ) δ ( K k ) k 2 ,
d Ω [ g ( 1 i r , 1 i k s ) s × g 1 ( 1 i r , 1 i k s ) s ] I ( r , s ) = 0 ,
I ( r , s ) = I ( r , s ) | 1 / k = 0 + n = 1 [ n I ( r , s ) ( 1 / k ) n ] | 1 / k = 0 ( 1 / k ) n .
I 0 ( r , s ) I ( r , s ) | 1 / k = 0 = lim k I ( r , s )
0 d K K 2 g ( 1 i r , 1 i K ) K · g 1 ( 1 i r , 1 i K ) ] r f ( r , K ) = 0.
g ( 1 i r , 1 i k s ) s · g 1 ( 1 i r , 1 i k s ) ] r I ( r , s ) = 0 ,
s · r I 0 ( r , s ) = 0 ,
r 2 I ( r , s ) = 0
r 2 I ( r , s ) 2 i k s · r I ( r , s ) = 0 ,
ρ 2 g ( u , ρ ) W ˜ ( u , ρ ) + 2 ρ g ( u , ρ ) · ρ W ˜ ( u , ρ ) = 0 ,
B 0 ( R , s , z = 0 ) = B 0 ( R , s , z = R z s z s ) .
W ( R , ρ ) = I ( 0 ) ( R ) g ( 0 ) ( ρ ) ,
B ( R , s , z = 0 ) = ( k 2 π ) 2 cos θ ( 2 π ) 2 g ( 1 i R , 1 i k s ) × d 2 ρ d 2 u exp ( i k s · ρ ) exp ( i u · R ) I ˜ ( 0 ) ( u ) g ( 0 ) ( ρ ) ,
B ( R , s , z = 0 ) = k 2 cos θ g ( 1 i R , 1 i k s ) I ( 0 ) ( R ) g ˜ ( 0 ) ( k s ) ,
B 0 ( R , s , z = 0 ) = k 2 cos θ I ( 0 ) ( R ) g ˜ ( 0 ) ( k s ) ,
W ( R , ρ , z > 0 ) = k 4 d 2 Q d 2 q exp ( i k q · R ) exp ( i k Q · ρ ) × exp { i k [ s z ( Q , q ) s z * ( Q , q ) ] z } I ˜ ( 0 ) ( k q ) g ˜ ( 0 ) ( k Q ) ,
s z ( Q , q ) = ( 1 | Q + 1 2 q | 2 ) 1 / 2 if | Q + 1 2 q | 1 = i ( | Q + 1 2 q | 2 1 ) 1 / 2 if | Q + 1 2 q | > 1 ,
s z ( Q , q ) = ( 1 | Q 1 2 q | 2 ) 1 / 2 if | Q 1 2 q | 1 = i ( | Q 1 2 q | 2 1 ) 1 / 2 if | Q 1 2 q | > 1.
B ( R , s , z > 0 ) = ( k 2 π ) 2 cos θ ( 2 π ) 2 d 2 ρ d 2 u d 2 υ exp ( i k s . ρ ) × exp ( i u · R ) exp ( i u · v ) g ( u , ρ ) W ( v , ρ , z > 0 ) .
B ( R , s , z > 0 ) = k 4 ( 2 π ) 2 cos θ g ( 1 i r , 1 i k s ) g ˜ ( 0 ) ( k s ) × d 2 u d 2 υ exp ( i u · R ) exp ( i u · v ) d 2 q exp ( i k q · v ) × exp { i k [ s z ( s , q ) s z * ( s , q ) ] z } I ˜ ( k q ) .
k [ s z ( s , q ) s z * ( s , q ) ] = k [ ( 1 | s + 1 2 q | 2 ) 1 / 2 ( 1 | s 1 2 q | 2 ) 1 / 2 ] k s · q s z ( for large k ) .
B 0 ( R , s , z > 0 ) = k 2 ( 2 π ) 2 cos θ g ˜ ( 0 ) ( k s ) d 2 u d 2 υ × exp ( i u · R ) × exp ( i u · v ) I ( v s s z z ) ( for large k )
B 0 ( R , s , z > 0 ) = k 2 cos θ I ( R s s z z ) g ˜ ( 0 ) ( k s ) .

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