Abstract

The radiometric theory of spatial coherence is presented with special attention to the validity of the approximations on which it is based. A new definition of the transverse coherence area is introduced and shown to be in general agreement with earlier definitions. In free-space propagation the product of the transverse coherence area and the intensity is shown to be constant along rectilinear rays, and, for radiation from uniform Lambert sources, a well-known paraxial formula for the transverse coherence area is extended to the extraparaxial domain. A decrease of the spatial coherence in free-space propagation takes place in regions with an increase of the intensity. For imaging systems this occurs in a finite part of image space whenever a real image of a diffusely radiating, extended object is formed at a finite distance.

© 2000 Optical Society of America

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References

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  1. A. J. Devaney, A. T. Friberg, A. T. N. Kumar, E. Wolf, “Decrease in spatial coherence of light propagating in free space,” Opt. Lett. 22, 1672–1673 (1997).
    [CrossRef]
  2. L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559–563 (1964).
  3. Yu. N. Barabanenkov, “On the spectral theory of radiation transfer equations,” Sov. Phys. JETP 29, 679–684 (1969) [Zh. Eksp. Teor. Fiz. 56, 1262–1272 (1969)].
  4. 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)[Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
    [CrossRef]
  5. H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1981).
  6. H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata 8, 1518 (1991).
    [CrossRef]
  7. H. M. Pedersen, “Propagation of generalized specific intensity in refractive media,” J. Opt. Soc. Am. A 9, 1623–1625 (1992).
    [CrossRef]
  8. H. M. Pedersen, “Geometrical theory of fields radiated from three-dimensional, quasi-homogeneous sources,” J. Opt. Soc. Am. A 9, 1626–1632 (1992).
    [CrossRef]
  9. H. M. Pedersen, J. J. Stamnes, “van Cittert–Zernike theorem for quasi-homogeneous wavefields and the modified Debye integral,” Pure Appl. Opt. 1, 13–28 (1992).
    [CrossRef]
  10. H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).
    [CrossRef]
  11. H. M. Pedersen, “Simplified geometrical theory of radiated fields from three-dimensional quasi-homogeneous sources,” Pure Appl. Opt. 2, 179–184 (1993).
    [CrossRef]
  12. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  13. E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
    [CrossRef]
  14. 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]
  15. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  16. 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]
  17. A. T. Friberg, “On the generalized radiance associated with radiation from a quasi-homogeneous source,” Opt. Acta 28, 261–277 (1981).
    [CrossRef]
  18. R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).
  20. H. M. Pedersen, J. J. Stamnes, “Reciprocity principles for focused wavefields and the modified Debye integral,” Opt. Acta 30, 1434–1454 (1983).
    [CrossRef]
  21. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986).
  22. H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
    [CrossRef]

1997 (1)

1993 (2)

H. M. Pedersen, “Simplified geometrical theory of radiated fields from three-dimensional quasi-homogeneous sources,” Pure Appl. Opt. 2, 179–184 (1993).
[CrossRef]

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

1992 (4)

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

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

H. M. Pedersen, J. J. Stamnes, “van Cittert–Zernike theorem for quasi-homogeneous wavefields and the modified Debye integral,” Pure Appl. Opt. 1, 13–28 (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 25, 5263–5278 (1992).
[CrossRef]

1991 (1)

1983 (1)

H. M. Pedersen, J. J. Stamnes, “Reciprocity principles for focused wavefields and the modified Debye integral,” Opt. Acta 30, 1434–1454 (1983).
[CrossRef]

1981 (3)

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1981).

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

1979 (1)

1977 (2)

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 (1)

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

1972 (1)

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)[Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
[CrossRef]

1969 (1)

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

1964 (1)

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

Baltes, H. P.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Barabanenkov, Yu. N.

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

Born, M.

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

Carter, W. H.

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.

Dolin, L. S.

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

Ferwerda, H. A.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Foley, J. T.

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.

Glass, A. S.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Kumar, A. T. N.

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 25, 5263–5278 (1992).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

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)[Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
[CrossRef]

Pedersen, H. M.

H. M. Pedersen, “Simplified geometrical theory of radiated fields from three-dimensional quasi-homogeneous sources,” Pure Appl. Opt. 2, 179–184 (1993).
[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 25, 5263–5278 (1992).
[CrossRef]

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

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

H. M. Pedersen, J. J. Stamnes, “van Cittert–Zernike theorem for quasi-homogeneous wavefields and the modified Debye integral,” Pure Appl. Opt. 1, 13–28 (1992).
[CrossRef]

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

H. M. Pedersen, J. J. Stamnes, “Reciprocity principles for focused wavefields and the modified Debye integral,” Opt. Acta 30, 1434–1454 (1983).
[CrossRef]

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1981).

Stamnes, J. J.

H. M. Pedersen, J. J. Stamnes, “van Cittert–Zernike theorem for quasi-homogeneous wavefields and the modified Debye integral,” Pure Appl. Opt. 1, 13–28 (1992).
[CrossRef]

H. M. Pedersen, J. J. Stamnes, “Reciprocity principles for focused wavefields and the modified Debye integral,” Opt. Acta 30, 1434–1454 (1983).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986).

Steinle, B.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (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)[Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
[CrossRef]

Winston, R.

Wolf, E.

A. J. Devaney, A. T. Friberg, A. T. N. Kumar, E. Wolf, “Decrease in spatial coherence of light propagating in free space,” Opt. Lett. 22, 1672–1673 (1997).
[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]

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

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

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

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

J. Phys. A (1)

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

Opt. Acta (4)

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1981).

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

H. M. Pedersen, J. J. Stamnes, “Reciprocity principles for focused wavefields and the modified Debye integral,” Opt. Acta 30, 1434–1454 (1983).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. D (1)

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

Pure Appl. Opt. (2)

H. M. Pedersen, “Simplified geometrical theory of radiated fields from three-dimensional quasi-homogeneous sources,” Pure Appl. Opt. 2, 179–184 (1993).
[CrossRef]

H. M. Pedersen, J. J. Stamnes, “van Cittert–Zernike theorem for quasi-homogeneous wavefields and the modified Debye integral,” Pure Appl. Opt. 1, 13–28 (1992).
[CrossRef]

Radiophys. Quantum Electron. (1)

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)[Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1419–1421 (1972)].
[CrossRef]

Sov. Phys. JETP (1)

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

Other (3)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986).

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

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Figures (1)

Fig. 1
Fig. 1

Imaging geometry. The geometrical shadow is shown shaded.

Equations (50)

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W(r, ξ)=L(r, s)exp(iksξ)dΩ.
ξW(r, ξ)=0,
ξ2W(r, ξ)+k2W(r, ξ)+142W(r, ξ)=0,
sL(r, s)=0,
ξ2W(r, ξ)+k2W(r, ξ)=0.
W(r, ξ)+14k2 2W(r, ξ)W(r, ξ).
S(r)=1ik [ξW(r, ξ)]ξ=0,
w(r)=12cI(r)+14k2 2I(r)-12ck2 [ξ2W(r, ξ)]ξ=0=1cI(r)+14k2 2I(r)=-1ck2 [ξ2W(r, ξ)]ξ=0.
I(r)+14k2 2I(r)I(r).
S(r)=1ik [ξW(r, ξ)]ξ=0=L(r, s)sdΩ,
w(r)=I(r)c=-1ck2 [ξ2W(r, ξ)]ξ=0=1cL(r, s)dΩ,
I(r)=W(r, 0)=L(r, s)dΩ.
μ(r, ξ)=W(r, ξ)/[I(r-12ξ)I(r+12ξ)]1/2.
Ac(r, s)=μ(r, ξ)δ(ξs)d3ξ,
μ(r, ξ)=W(r, ξ)/I(r)=W(r, ξ)/W(r, 0).
I(r)Ac(r, s)=W(r, ξ)δ(ξs)d3ξ.
I(r)Ac(r, s)=λ2[L(r, s)+L(r,-s)],
I(r)Ac(r, s)=λ2L(r, s).
L(r, s)=L0for sΔΩ(r)0for sΔΩ(r) ,
W(r, ξ)=L0ΔΩ(r)exp(iksξ)dΩ,
I(r)=W(r, 0)=L0ΔΩ(r).
μ(r, ξ)=W(r, ξ)I(r)=1ΔΩ(r)ΔΩ(r)exp(iksξ)dΩ,
Ac(r)ΔΩ(r)=λ2.
14k2 2W(r, ξ)=-L0z2ΔΩ(r)ξnzsz2ik+14 ξn2sz2×exp(iksξ)dΩ,
14k2 2W(r, ξ)-1z2ξnzsz2ik+14 ξn2sz2W(r, ξ)=-1ikzξnz2r+ξn2r2W(r, ξ),
kz[ξn/(2r)]sin θ|ξnz|/(2r),
ξn/(2r)1,
lθr=λScos θ1.
U(r)=Q(r)G(r-r)d3r;
G(r)=-14πrexp(ikr), d3r=dxdydz.
W(r, ξ)=WQ(r-r, ξ)×G*[r-12(ξ-ξ)]G[r+12(ξ-ξ)]d3ξd3r.
 G*(r-12ξ)G(r+12ξ)=exp[ik(|r+12ξ|-|r-12ξ|)](4π)2|r-12ξ||r+12ξ|1(4πr)2expik rr ξ.
L(r, s)=1(4π)20ΨQ(r-rs, ks)dr,
ΨQ(r, q)=WQ(r, ξ)exp(-iqξ)d3ξ
sL(r, s)=ΨQ(r, ks)/(4π)2,
(A2S)=0,
(S)2=1+2A/(k2A).
(S)2=1,
A(r)+(1/k2)2A(r)A(r).
I(r)Ac(r, s)=L(r, s)exp(iksξ)δ(ξs)d3ξdΩ.
I(r)Ac(r, s)=L(r, s)--exp[ik(sxξx+syξy)]dξxdξydΩ=(2π/k)2L(r, s)δ(sx)δ(sy)dΩ.
I(r)Ac(r, s)=λ2L(r, s)δ(sx)δ(sy)dΩ=λ2[L(r, s)+L(r,-s)].
μ(r, ξ)=μ(ξ)=sin(kξ)/(kξ).
W(r, ξ)=L0S f (r-x, ξ)d2x,
f(r, ξ)=zr3expik rr ξ,
s=r-x|r-x|,dΩ=z|r-x|3d2x.
2W(r, ξ)=L0Sg(r-x, ξ)d2x,
g(r, ξ)=2f(r, ξ)=2ikξnzr4-(kξn)2zr5expik rr ξ=2ikξnzrz-(kξn)2r2 f (r, ξ).
2zr3=-z21r=4π z δ3(r).
14k2 2W(r, ξ)=-L0z2ΔΩ(r)ξnzsz2ik+14 ξn2sz2×exp(iksξ)dΩ,

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