Abstract

A free-space solution is found for the field correlation tensors as a Green’s function integral over the correlation tensor for the source currents. This solution is used to formulate a vector theory of partial coherence in an angular spectrum representation. The theory is used to analyze the angular distribution of radiation into the far field for several special cases of source polarization and coherence. It is found that the radiation pattern from unpolarized, quasihomogeneous sources depends on the coherence properties of the source in the same manner as described in scalar theory. However, for sources that are not unpolarized this is not generally the case.

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  1. A. S. Marathay and G. B. Parrent, Jr., "Use of scalar theory in optics," J. Opt. Soc. Am. 60, 243–245 (1970).
  2. M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice-Hall, Englewood Clifts, N.J., 1964).
  3. J. Peřna, Coherence of Light (Van Nostrand, New York, 1971), Chap. 8.
  4. E. Wolf, "Optics in Terms of Observable Quantities," Nuovo Cimento 12, 884–888 (1954).
  5. P. Roman and E. Wolf, "Correlation Theory of Stationary Electro-magnetic Fields, Part I.—The Basic Field Equation," Nuovo Cimento 17, 462–476 (1960); "Part II.—Conservation Laws," 17, 477–490 (1960).
  6. P. Roman, "Correlation Theory of Stationary Electromagnetic Fields, Part III.—The Presence of Random Sources," Nuovo Cimento 20, 759–772 (1961).
  7. M. J. Beran and G. B. Parrent, Jr., "On The Equations Governing the Coherence of the Radiation Emitted by Random Charge and Current Distributions," J. Opt. Soc. Am. 52, 98–99 (1962).
  8. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1967).
  9. C. T. Tai, Dyadic Green's Function's in Electromagnetic Theory (Intext, San Francisco, 1971).
  10. L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965).
  11. A. Ban¯os, Jr., Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Eq. (2.19).
  12. E. W. Marchand and E. Wolf, "Angular correlation and the far-zone behavior of partially coherent fields," J. Opt. Soc. Am. 62, 379–385 (1972).
  13. A. Walther, "Radiometry and Coherence," J. Opt. Soc. Am. 58, 1256–1259 (1968).
  14. W. H. Carter and E. Wolf, "Correlation Theory of Wavefields Generated by Fluctuating, Three-Dimensional, Primary, Scalar Sources, Part I: General Theory," Opt. Acta (to be published).
  15. E. W. Marchand and E. Wolf, "Generalized Radiomietry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972).
  16. E. W. Marchand and E. Wolf, "Radiometry with Sources of any State of Coherence," J. Opt. Soc. Am. 64, 1219–1226 (1974).
  17. W. H. Carter and E. Wolf, "Coherence and Radiometry with Quasihomogeneous Planar Sources," J. Opt. Soc. Am. 67, 785–796 (1977).
  18. Representing a correlation function by a Dirac delta function in the incoherent limit clearly violates the normalization condition g(0) = 1, but is nevertheless a useful approximation [viz., Ref. 2, Eq. (4.31)].
  19. E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence, "Opt. Commun. 13, 205–209 (1975); also see Ref. 17, Eq. (4.19a).
  20. Because J(s) is defined by Eq. (70) only over directions s from the origin into the z ≥ 0 half space, we assume J(s) = 0 for all other s. This limits the integration in Eq. (72) to the domain p2+ + q2+ ≤ 1.
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (NBS Series 55, Washington, D.C., 1964).
  22. E. Wolf and W. H. Carter, "Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources," J. Opt. Soc. Am. 68, 953–964 (1978).
  23. Lamberts law restricts only the portion of the source correlation function with spatial frequencies less than 1/λ, see W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).
  24. W. H. Carter, "Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-Spectral Density," Opt. Commun. 26, 1–4 (1978).
  25. J. Carl Leader, "The Generalized Partial Coherence of a Radiation Source and Its Far Field," Opt. Acta 25, 395–413 (1978).
  26. H. Lass, Vector and Tensor Analysis (McGraw-Hill, New York, 1950).
  27. A. J. McConnel, Applications of Tensor Analysis (Dover, New York, 1957).

1978

W. H. Carter, "Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-Spectral Density," Opt. Commun. 26, 1–4 (1978).

J. Carl Leader, "The Generalized Partial Coherence of a Radiation Source and Its Far Field," Opt. Acta 25, 395–413 (1978).

E. Wolf and W. H. Carter, "Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources," J. Opt. Soc. Am. 68, 953–964 (1978).

1977

1975

Lamberts law restricts only the portion of the source correlation function with spatial frequencies less than 1/λ, see W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).

E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence, "Opt. Commun. 13, 205–209 (1975); also see Ref. 17, Eq. (4.19a).

1974

1972

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

E. W. Marchand and E. Wolf, "Generalized Radiomietry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972).

1970

1968

1965

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965).

1962

1961

P. Roman, "Correlation Theory of Stationary Electromagnetic Fields, Part III.—The Presence of Random Sources," Nuovo Cimento 20, 759–772 (1961).

1960

P. Roman and E. Wolf, "Correlation Theory of Stationary Electro-magnetic Fields, Part I.—The Basic Field Equation," Nuovo Cimento 17, 462–476 (1960); "Part II.—Conservation Laws," 17, 477–490 (1960).

1954

E. Wolf, "Optics in Terms of Observable Quantities," Nuovo Cimento 12, 884–888 (1954).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (NBS Series 55, Washington, D.C., 1964).

Ban¯os, A.

A. Ban¯os, Jr., Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Eq. (2.19).

Beran, M. J.

Carter, W. H.

E. Wolf and W. H. Carter, "Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources," J. Opt. Soc. Am. 68, 953–964 (1978).

W. H. Carter, "Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-Spectral Density," Opt. Commun. 26, 1–4 (1978).

W. H. Carter and E. Wolf, "Coherence and Radiometry with Quasihomogeneous Planar Sources," J. Opt. Soc. Am. 67, 785–796 (1977).

E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence, "Opt. Commun. 13, 205–209 (1975); also see Ref. 17, Eq. (4.19a).

Lamberts law restricts only the portion of the source correlation function with spatial frequencies less than 1/λ, see W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).

W. H. Carter and E. Wolf, "Correlation Theory of Wavefields Generated by Fluctuating, Three-Dimensional, Primary, Scalar Sources, Part I: General Theory," Opt. Acta (to be published).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1967).

Lass, H.

H. Lass, Vector and Tensor Analysis (McGraw-Hill, New York, 1950).

Leader, J.Carl

J. Carl Leader, "The Generalized Partial Coherence of a Radiation Source and Its Far Field," Opt. Acta 25, 395–413 (1978).

Mandel, L.

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965).

Marathay, A. S.

Marchand, E. W.

McConnel, A. J.

A. J. McConnel, Applications of Tensor Analysis (Dover, New York, 1957).

Parrent, Jr., G. B.

Perina, J.

J. Peřna, Coherence of Light (Van Nostrand, New York, 1971), Chap. 8.

Roman, P.

P. Roman, "Correlation Theory of Stationary Electromagnetic Fields, Part III.—The Presence of Random Sources," Nuovo Cimento 20, 759–772 (1961).

P. Roman and E. Wolf, "Correlation Theory of Stationary Electro-magnetic Fields, Part I.—The Basic Field Equation," Nuovo Cimento 17, 462–476 (1960); "Part II.—Conservation Laws," 17, 477–490 (1960).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (NBS Series 55, Washington, D.C., 1964).

Tai, C. T.

C. T. Tai, Dyadic Green's Function's in Electromagnetic Theory (Intext, San Francisco, 1971).

Walther, A.

Wolf, E.

E. Wolf and W. H. Carter, "Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources," J. Opt. Soc. Am. 68, 953–964 (1978).

W. H. Carter and E. Wolf, "Coherence and Radiometry with Quasihomogeneous Planar Sources," J. Opt. Soc. Am. 67, 785–796 (1977).

E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence, "Opt. Commun. 13, 205–209 (1975); also see Ref. 17, Eq. (4.19a).

Lamberts law restricts only the portion of the source correlation function with spatial frequencies less than 1/λ, see W. H. Carter and E. Wolf, "Coherence properties of Lambertian and non-Lambertian sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).

E. W. Marchand and E. Wolf, "Radiometry with Sources of any State of Coherence," J. Opt. Soc. Am. 64, 1219–1226 (1974).

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

E. W. Marchand and E. Wolf, "Generalized Radiomietry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972).

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965).

P. Roman and E. Wolf, "Correlation Theory of Stationary Electro-magnetic Fields, Part I.—The Basic Field Equation," Nuovo Cimento 17, 462–476 (1960); "Part II.—Conservation Laws," 17, 477–490 (1960).

E. Wolf, "Optics in Terms of Observable Quantities," Nuovo Cimento 12, 884–888 (1954).

W. H. Carter and E. Wolf, "Correlation Theory of Wavefields Generated by Fluctuating, Three-Dimensional, Primary, Scalar Sources, Part I: General Theory," Opt. Acta (to be published).

J. Opt. Soc. Am.

Nuovo Cimento

E. Wolf, "Optics in Terms of Observable Quantities," Nuovo Cimento 12, 884–888 (1954).

P. Roman and E. Wolf, "Correlation Theory of Stationary Electro-magnetic Fields, Part I.—The Basic Field Equation," Nuovo Cimento 17, 462–476 (1960); "Part II.—Conservation Laws," 17, 477–490 (1960).

P. Roman, "Correlation Theory of Stationary Electromagnetic Fields, Part III.—The Presence of Random Sources," Nuovo Cimento 20, 759–772 (1961).

Opt. Acta

J. Carl Leader, "The Generalized Partial Coherence of a Radiation Source and Its Far Field," Opt. Acta 25, 395–413 (1978).

Opt. Commun.

W. H. Carter, "Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-Spectral Density," Opt. Commun. 26, 1–4 (1978).

E. W. Marchand and E. Wolf, "Generalized Radiomietry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972).

E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence, "Opt. Commun. 13, 205–209 (1975); also see Ref. 17, Eq. (4.19a).

Rev. Mod. Phys.

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965).

Other

A. Ban¯os, Jr., Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Eq. (2.19).

W. H. Carter and E. Wolf, "Correlation Theory of Wavefields Generated by Fluctuating, Three-Dimensional, Primary, Scalar Sources, Part I: General Theory," Opt. Acta (to be published).

M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice-Hall, Englewood Clifts, N.J., 1964).

J. Peřna, Coherence of Light (Van Nostrand, New York, 1971), Chap. 8.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1967).

C. T. Tai, Dyadic Green's Function's in Electromagnetic Theory (Intext, San Francisco, 1971).

Because J(s) is defined by Eq. (70) only over directions s from the origin into the z ≥ 0 half space, we assume J(s) = 0 for all other s. This limits the integration in Eq. (72) to the domain p2+ + q2+ ≤ 1.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (NBS Series 55, Washington, D.C., 1964).

Representing a correlation function by a Dirac delta function in the incoherent limit clearly violates the normalization condition g(0) = 1, but is nevertheless a useful approximation [viz., Ref. 2, Eq. (4.31)].

H. Lass, Vector and Tensor Analysis (McGraw-Hill, New York, 1950).

A. J. McConnel, Applications of Tensor Analysis (Dover, New York, 1957).

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