Abstract

The complex spectral degree of coherence of a general random, statistically stationary electromagnetic field is introduced in a manner similar to the way it is defined for a beamlike field, namely, by means of Young’s interference experiment. Both its modulus and its phase are measurable. We illustrate the definition by applying it to blackbody radiation emerging from a cavity. The results are of particular interest for near-field optics.

© 2004 Optical Society of America

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References

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  1. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
    [CrossRef]
  2. F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. London 61, 158–164 (1948).
    [CrossRef]
  3. E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London, Ser. A 230, 246–265 (1955).
    [CrossRef]
  4. B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
    [CrossRef]
  5. W. H. Carter, E. Wolf, “Far-zone behavior of electromag-netic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
    [CrossRef] [PubMed]
  6. L. Mandel, E. Wolf, “Spectral coherence and concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  7. 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]
  8. E. Wolf, W. H. Carter, “A radiometric generalization of the van Cittert–Zernike theorem for fields generated by sources of arbitrary state of coherence,” Opt. Commun. 16, 297–302 (1976). see also Ref. 9.
    [CrossRef]
  9. M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
    [CrossRef]
  10. E. Wolf, “Unified theory of coherence and polarization of statistical electromagnetic beams,” Phys. Rev. Lett. 312, 263–267 (2003).
    [CrossRef]
  11. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  12. B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: Coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
    [CrossRef]
  13. R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964), pp. 319–320.
  14. M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).
  15. Expression (3.14) was noted previously (Ref. 5, Eq. 6.16) as the degree of coherence in a particular case, namely, in the far field generated by three-dimensional fluctuating charge current distribution in free space.
  16. D. F. V. James, E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1–4 (1998).
    [CrossRef]
  17. V. N. Kumar, D. N. Rao, “Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence,” J. Mod. Opt. 48, 1455–1465 (2001).
  18. An analogous definition of the degree of coherence of a beamlike field in the space–time domain was obtained many years ago by Karczewski in a little-known paper.4
  19. C. L. Mehta, E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328–1334 (1967).
    [CrossRef]

2003

E. Wolf, “Unified theory of coherence and polarization of statistical electromagnetic beams,” Phys. Rev. Lett. 312, 263–267 (2003).
[CrossRef]

2001

V. N. Kumar, D. N. Rao, “Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence,” J. Mod. Opt. 48, 1455–1465 (2001).

1998

D. F. V. James, E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1–4 (1998).
[CrossRef]

1987

W. H. Carter, E. Wolf, “Far-zone behavior of electromag-netic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

1977

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

1976

E. Wolf, W. H. Carter, “A radiometric generalization of the van Cittert–Zernike theorem for fields generated by sources of arbitrary state of coherence,” Opt. Commun. 16, 297–302 (1976). see also Ref. 9.
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (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]

1967

C. L. Mehta, E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328–1334 (1967).
[CrossRef]

1966

1963

B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
[CrossRef]

1955

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London, Ser. A 230, 246–265 (1955).
[CrossRef]

1948

F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. London 61, 158–164 (1948).
[CrossRef]

1938

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

Carter, W. H.

W. H. Carter, E. Wolf, “Far-zone behavior of electromag-netic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

E. Wolf, W. H. Carter, “A radiometric generalization of the van Cittert–Zernike theorem for fields generated by sources of arbitrary state of coherence,” Opt. Commun. 16, 297–302 (1976). see also Ref. 9.
[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]

James, D. F. V.

D. F. V. James, E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1–4 (1998).
[CrossRef]

Karczewski, B.

Kumar, V. N.

V. N. Kumar, D. N. Rao, “Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence,” J. Mod. Opt. 48, 1455–1465 (2001).

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964), pp. 319–320.

Mandel, L.

L. Mandel, E. Wolf, “Spectral coherence and concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

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

Mehta, C. L.

C. L. Mehta, E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328–1334 (1967).
[CrossRef]

Rao, D. N.

V. N. Kumar, D. N. Rao, “Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence,” J. Mod. Opt. 48, 1455–1465 (2001).

Wolf, E.

E. Wolf, “Unified theory of coherence and polarization of statistical electromagnetic beams,” Phys. Rev. Lett. 312, 263–267 (2003).
[CrossRef]

D. F. V. James, E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1–4 (1998).
[CrossRef]

W. H. Carter, E. Wolf, “Far-zone behavior of electromag-netic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

E. Wolf, W. H. Carter, “A radiometric generalization of the van Cittert–Zernike theorem for fields generated by sources of arbitrary state of coherence,” Opt. Commun. 16, 297–302 (1976). see also Ref. 9.
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[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]

C. L. Mehta, E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328–1334 (1967).
[CrossRef]

B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: Coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London, Ser. A 230, 246–265 (1955).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

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

Zernike, F.

F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. London 61, 158–164 (1948).
[CrossRef]

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

J. Mod. Opt.

V. N. Kumar, D. N. Rao, “Two-beam interference experiments in the frequency-domain to measure the complex degree of spectral coherence,” J. Mod. Opt. 48, 1455–1465 (2001).

J. Opt. Soc. Am.

Nuovo Cimento

B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
[CrossRef]

Opt. Acta

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

Opt. Commun.

D. F. V. James, E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1–4 (1998).
[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. Wolf, W. H. Carter, “A radiometric generalization of the van Cittert–Zernike theorem for fields generated by sources of arbitrary state of coherence,” Opt. Commun. 16, 297–302 (1976). see also Ref. 9.
[CrossRef]

Phys. Rev.

C. L. Mehta, E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328–1334 (1967).
[CrossRef]

Phys. Rev. A

W. H. Carter, E. Wolf, “Far-zone behavior of electromag-netic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett.

E. Wolf, “Unified theory of coherence and polarization of statistical electromagnetic beams,” Phys. Rev. Lett. 312, 263–267 (2003).
[CrossRef]

Physica

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Proc. Phys. Soc. London

F. Zernike, “Diffraction and optical image formation,” Proc. Phys. Soc. London 61, 158–164 (1948).
[CrossRef]

Proc. R. Soc. London, Ser. A

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London, Ser. A 230, 246–265 (1955).
[CrossRef]

Other

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

R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964), pp. 319–320.

M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

Expression (3.14) was noted previously (Ref. 5, Eq. 6.16) as the degree of coherence in a particular case, namely, in the far field generated by three-dimensional fluctuating charge current distribution in free space.

An analogous definition of the degree of coherence of a beamlike field in the space–time domain was obtained many years ago by Karczewski in a little-known paper.4

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Figures (1)

Fig. 1
Fig. 1

Illustration of the notation relating to determining the spectral degree of coherence of a three-dimensional electromagnetic field from Young’s interference experiment.

Equations (34)

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W(r1, r2, ω)
[Wαβ(r1, r2, ω)]=[Eα*(r1, ω)Eβ(r2, ω)]
(α=x,y,z; β=x,y,z),
Ex(r, ω)=-12πEx(ρ, ω)Gz(r, ρ)d2ρ,
Ey(r, ω)=-12πEy(ρ, ω)Gz(r, ρ)d2ρ,
Ez(r, ω)=12π[Ex(ρ, ω)Gx(r, ρ)+Ey(ρ, ω)Gy(r, ρ)]d2ρ,
G(r, r)=exp(ik|r-r|)|r-r|,
Gx(r, ρ)=xexp(ik|r-r|)|r-r|r=ρ,
k=ω/c
Ez(ρ, ω)=limz0 Ez(ρ, z, ω)=12πE(ρ, ω)  (ρ-ρ) ik|ρ-ρ|-1|ρ-ρ|3/2×exp(ik|ρ-ρ|)d2ρ.
Eα(r, ω)=-12πA1+A2Eα(ρ, ω)Gz(r, ρ)d2ρ
(α=x,y,z),
Eα(r, ω)=-12π [Eα(ρ1, ω)Gz(r, ρ1)dA1+Eα(ρ2, ω)Gz(r, ρ2)dA2]
(α=x,y,z).
S(r, ω)=Tr W(r, r, ω)=Sx(r, ω)+Sy(r, ω)+Sz(r, ω),
Sα(r, ω)=Eα*(r, ω)Eα(r, ω)(α=x,y,z),
Gz(r, ρ)=zikR-1R2exp(ikR)R,
R=|ρ-r|.
Gz(r, ρ)ikzRexp(ikR)R.
Eα(r, ω)=-iλEα(1)(ρ1, ω)zR1exp(ikR1)R1dA1+Eα(2)(ρ2, ω)zR2exp(ikR2)R2dA2
(α=x,y,z).
Sα(r, ω)=1λ2Sα(ρ1, ω)zR12(dA1)2R12+Sα(ρ2, ω)zR22(dA2)2R22+2 ReWαα(ρ1, ρ2, ω)×exp[ik(R2-R1)]R1R2z2R1R2dA1dA2
(α=x,y,z),
S(r, ω)=1λ2S(ρ1, ω)zR12(dA1)2R12+S(ρ2, ω)zR22(dA2)2R22+2 ReTr W(ρ1, ρ2, ω)×exp[ik(R2-R1)]R1R2z2R1R2dA1dA2.
S(1)(r, ω)=1λ2 S(ρ1, ω)zR12(dA1)2R12,
S(2)(r, ω)=1λ2 S(ρ2, ω)zR22(dA2)2R22,
S(r, ω)=S(1)(r, ω)+S(2)(r, ω)+2S(1)(r, ω)S(2)(r, ω)×Reη(ρ1, ρ2, ω) exp[ik(R2-R1)]R1R2,
η(ρ1, ρ2, ω)=TrW(ρ1, ρ2, ω)S(ρ1, ω)S(ρ2, ω)
TrW(ρ1, ρ2, ω)TrW(ρ1, ρ1, ω)TrW(ρ2, ρ2, ω).
Smax(r, ω)=S(1)(r, ω)+S(2)(r, ω)+2S(1)(r, ω)S(2)(r, ω)|η(ρ1, ρ2, ω)|,
Smin(r, ω)=S(1)(r, ω)+S(2)(r, ω)-2S(1)(r, ω)S(2)(r, ω)|η(ρ1, ρ2, ω)|.
V(r, ω)Smax(r, ω)-Smin(r, ω)Smax(r, ω)+Smin(r, ω)=2S(1)(r, ω)S(2)(r, ω)S(1)(r, ω)+S(2)(r, ω) |η(ρ1, ρ2, ω)|.
V(r, ω)|η(ρ1, ρ2, ω)|;
η(ρ1, ρ2, ω)=sin[k|ρ2-ρ1|]k|ρ2-ρ1|.

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