Abstract

We construct the coherent-mode representation for fluctuating, statistically stationary electromagnetic fields. The modes are shown to be spatially fully coherent in the sense of a recently introduced spectral degree of electromagnetic coherence. We also prove that the electric cross-spectral density tensor can be rigorously expressed as a correlation tensor averaged over an appropriate ensemble of strictly monochromatic vectorial wave functions. The formalism is demonstrated for partially polarized, partially coherent Gaussian Schell-model beams, but the theory applies to arbitrary random electromagnetic fields and can find applications in radiation and propagation and in inverse problems.

© 2004 Optical Society of America

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  2. E. Wolf, “New spectral representation of random sources and the partially coherent fields they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  3. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  4. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [CrossRef]
  5. G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
    [CrossRef]
  6. F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  7. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  8. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  9. E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1984).
    [CrossRef]
  10. K. Kim, D.-Y. Park, “Approximate method in coherent-mode representation,” Opt. Lett. 17, 1043–1045 (1992).
    [CrossRef] [PubMed]
  11. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
    [CrossRef]
  12. K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
    [CrossRef]
  13. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
    [CrossRef]
  14. R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
    [CrossRef]
  15. T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
    [CrossRef]
  16. J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
    [CrossRef]
  17. P. Vahimaa, J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
    [CrossRef]
  18. J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
    [CrossRef] [PubMed]
  19. B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
    [CrossRef]
  20. T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
    [CrossRef]
  21. M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
    [CrossRef]
  22. A number of recent papers consider beams in which the field consists of orthogonal, partially correlated transverse modes. The cross-spectral density then has, besides the “diagonal” elements, nonzero “off-diagonal” elements, and hence the represention is not the coherent-mode decomposition.
  23. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
    [CrossRef]
  24. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
  25. D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, London, UK, 2003).
  26. E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).
  27. T. Setälä, M. Kaivola, A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
    [CrossRef] [PubMed]
  28. T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  29. J. Tervo, T. Setälä, A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [CrossRef] [PubMed]
  30. T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef]
  31. T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).
    [CrossRef]
  32. C. L. Mehta, E. Wolf, A. P. Balachandran, “Some theorems on the unimodular complex degree of optical coherence,” J. Math. Phys. 7, 133–138 (1966).
    [CrossRef]
  33. In Refs. 2and 3, Wolf actually examined the properties of statistically stationary random sources rather than fields, but the results are naturally applicable also to the field-domain analysis.
  34. L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
    [CrossRef]
  35. E. Wolf, “Completeness of coherent-mode eigenfunctions of Schell-model sources,” Opt. Lett. 9, 387–389 (1984).
    [CrossRef] [PubMed]
  36. P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—The basic field equations,” Nuovo Cimento 17, 462–476 (1960).
    [CrossRef]
  37. P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—Conservation laws,” Nuovo Cimento 17, 477–490 (1960).
    [CrossRef]
  38. For notational simplicity we sometimes represent the electric correlation tensors by the corresponding matrices.
  39. C. L. Mehta, E. Wolf, “Correlation theory of quantized electromagnetic fields. II. Stationary fields and their spectral properties,” Phys. Rev. 157, 1188–1197 (1967).
    [CrossRef]
  40. This may be seen at once similar to the scalar case of Ref. 3, i.e., by examining the properties of the Schwartz inequality.
  41. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.
  42. E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton U. Press, Princeton, N.J., 1975), p. 16.
  43. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, San Diego, Calif., 2001).
  44. E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series (Dover, New York, 1957), Vol. 2, p. 326.
  45. J. L. Lumley, Stochastic Tools in Turbulence (Academic, New York, 1970).
  46. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  47. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  48. An alternative definition for the degree of electromagnetic coherence in the space–time domain, based on the visibility of interference fringes in Young’s double-slit experiment, was introduced in 1963 by Karczewski.49,50Several consequences of such a definition have been discussed.29A space–frequency analog of this degree of electromagnetic coherence was later employed in radiation analysis,51,52and in 2003 it was formalized by Wolf for electromagnetic beams in the two-slit interference geometry.53
  49. B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
    [CrossRef]
  50. B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
    [CrossRef]
  51. W. H. Carter, E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
    [CrossRef] [PubMed]
  52. T. Setälä, K. Blomstedt, M. Kaivola, A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
    [CrossRef]
  53. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [CrossRef]
  54. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 484.
  55. P. Holmes, J. L. Lumley, C. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge U. Press, Cambridge, UK, 1996), Chap. 3.
  56. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  57. P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A Pure Appl. Opt. 6, S41–S44 (2004).
    [CrossRef]
  58. M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A 1, 711–715 (1984).
    [CrossRef]
  59. L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
    [CrossRef]

2004

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A Pure Appl. Opt. 6, S41–S44 (2004).
[CrossRef]

T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef]

2003

2002

T. Setälä, M. Kaivola, A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

1999

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

1998

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

1997

P. Vahimaa, J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
[CrossRef]

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

1996

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

1995

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

1993

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

1992

1989

1987

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

1986

1984

1983

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1982

1981

E. Wolf, “New spectral representation of random sources and the partially coherent fields they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

1980

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1967

C. L. Mehta, E. Wolf, “Correlation theory of quantized electromagnetic fields. II. Stationary fields and their spectral properties,” Phys. Rev. 157, 1188–1197 (1967).
[CrossRef]

1966

C. L. Mehta, E. Wolf, A. P. Balachandran, “Some theorems on the unimodular complex degree of optical coherence,” J. Math. Phys. 7, 133–138 (1966).
[CrossRef]

1963

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

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

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
[CrossRef]

1960

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—The basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—Conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, San Diego, Calif., 2001).

Balachandran, A. P.

C. L. Mehta, E. Wolf, A. P. Balachandran, “Some theorems on the unimodular complex degree of optical coherence,” J. Math. Phys. 7, 133–138 (1966).
[CrossRef]

Bastiaans, M. J.

Berkooz, C.

P. Holmes, J. L. Lumley, C. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge U. Press, Cambridge, UK, 1996), Chap. 3.

Blomstedt, K.

T. Setälä, K. Blomstedt, M. Kaivola, A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Born, M.

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

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

Cai, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Carter, W. H.

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

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Collett, E.

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).

Courjon, D.

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, London, UK, 2003).

Dogariu, A.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 484.

Friberg, A. T.

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef]

J. Tervo, T. Setälä, A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[CrossRef] [PubMed]

T. Setälä, K. Blomstedt, M. Kaivola, A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

T. Setälä, M. Kaivola, A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef] [PubMed]

Gori, F.

Guattari, G.

Habashy, T.

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

Hobson, E. W.

E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series (Dover, New York, 1957), Vol. 2, p. 326.

Holmes, P.

P. Holmes, J. L. Lumley, C. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge U. Press, Cambridge, UK, 1996), Chap. 3.

Huttunen, J.

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Kaivola, M.

T. Setälä, K. Blomstedt, M. Kaivola, A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

T. Setälä, M. Kaivola, A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Karczewski, B.

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

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

Kim, K.

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.

Lü, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Lumley, J. L.

P. Holmes, J. L. Lumley, C. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge U. Press, Cambridge, UK, 1996), Chap. 3.

J. L. Lumley, Stochastic Tools in Turbulence (Academic, New York, 1970).

Mandel, L.

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
[CrossRef]

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

Mehta, C. L.

C. L. Mehta, E. Wolf, “Correlation theory of quantized electromagnetic fields. II. Stationary fields and their spectral properties,” Phys. Rev. 157, 1188–1197 (1967).
[CrossRef]

C. L. Mehta, E. Wolf, A. P. Balachandran, “Some theorems on the unimodular complex degree of optical coherence,” J. Math. Phys. 7, 133–138 (1966).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 484.

Mukunda, N.

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Park, D.-Y.

Piquero, G.

Roman, P.

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—The basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—Conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Setälä, T.

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef]

J. Tervo, T. Setälä, A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[CrossRef] [PubMed]

T. Setälä, K. Blomstedt, M. Kaivola, A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

T. Setälä, M. Kaivola, A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shirai, T.

Simon, R.

Starikov, A.

Stein, E. M.

E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton U. Press, Princeton, N.J., 1975), p. 16.

Sundar, K.

Tervo, J.

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef]

P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A Pure Appl. Opt. 6, S41–S44 (2004).
[CrossRef]

J. Tervo, T. Setälä, A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[CrossRef] [PubMed]

Tervonen, E.

Turunen, J.

Vahimaa, P.

P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A Pure Appl. Opt. 6, S41–S44 (2004).
[CrossRef]

P. Vahimaa, J. Turunen, “Bragg diffraction of spatially partially coherent fields,” J. Opt. Soc. Am. A 14, 54–59 (1997).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, San Diego, Calif., 2001).

Weiss, G.

E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton U. Press, Princeton, N.J., 1975), p. 16.

Wolf, E.

T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[CrossRef]

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

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

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

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[CrossRef]

E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1984).
[CrossRef]

E. Wolf, “Completeness of coherent-mode eigenfunctions of Schell-model sources,” Opt. Lett. 9, 387–389 (1984).
[CrossRef] [PubMed]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

E. Wolf, “New spectral representation of random sources and the partially coherent fields they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

C. L. Mehta, E. Wolf, “Correlation theory of quantized electromagnetic fields. II. Stationary fields and their spectral properties,” Phys. Rev. 157, 1188–1197 (1967).
[CrossRef]

C. L. Mehta, E. Wolf, A. P. Balachandran, “Some theorems on the unimodular complex degree of optical coherence,” J. Math. Phys. 7, 133–138 (1966).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—Conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—The basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

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

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

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B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Zhang, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

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IEEE J. Quantum Electron.

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[CrossRef]

Inverse Probl.

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

J. Math. Phys.

C. L. Mehta, E. Wolf, A. P. Balachandran, “Some theorems on the unimodular complex degree of optical coherence,” J. Math. Phys. 7, 133–138 (1966).
[CrossRef]

J. Mod. Opt.

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

J. Opt. A Pure Appl. Opt.

P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A Pure Appl. Opt. 6, S41–S44 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nuovo Cimento

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

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—The basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—Conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

Opt. Commun.

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

E. Wolf, “New spectral representation of random sources and the partially coherent fields they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett.

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

Phys. Lett. A

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

Phys. Rev.

C. L. Mehta, E. Wolf, “Correlation theory of quantized electromagnetic fields. II. Stationary fields and their spectral properties,” Phys. Rev. 157, 1188–1197 (1967).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. E

T. Setälä, K. Blomstedt, M. Kaivola, A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Phys. Rev. Lett.

T. Setälä, M. Kaivola, A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Proc. Phys. Soc. London

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
[CrossRef]

Other

A number of recent papers consider beams in which the field consists of orthogonal, partially correlated transverse modes. The cross-spectral density then has, besides the “diagonal” elements, nonzero “off-diagonal” elements, and hence the represention is not the coherent-mode decomposition.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, London, UK, 2003).

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).

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

This may be seen at once similar to the scalar case of Ref. 3, i.e., by examining the properties of the Schwartz inequality.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.

E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton U. Press, Princeton, N.J., 1975), p. 16.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, San Diego, Calif., 2001).

E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series (Dover, New York, 1957), Vol. 2, p. 326.

J. L. Lumley, Stochastic Tools in Turbulence (Academic, New York, 1970).

For notational simplicity we sometimes represent the electric correlation tensors by the corresponding matrices.

In Refs. 2and 3, Wolf actually examined the properties of statistically stationary random sources rather than fields, but the results are naturally applicable also to the field-domain analysis.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 484.

P. Holmes, J. L. Lumley, C. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge U. Press, Cambridge, UK, 1996), Chap. 3.

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

An alternative definition for the degree of electromagnetic coherence in the space–time domain, based on the visibility of interference fringes in Young’s double-slit experiment, was introduced in 1963 by Karczewski.49,50Several consequences of such a definition have been discussed.29A space–frequency analog of this degree of electromagnetic coherence was later employed in radiation analysis,51,52and in 2003 it was formalized by Wolf for electromagnetic beams in the two-slit interference geometry.53

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Figures (1)

Fig. 1
Fig. 1

Behavior of the ratio of the eigenvalues Λn(j)/Λ0(1) as a function of n for three values of the global degree of coherence q. Solid curves, j=1 [Eq. (81)]; dashed curves, j=2 [Eq. (83)]. The degree of polarization is P(ω)=0.6.

Equations (99)

Equations on this page are rendered with MathJax. Learn more.

W(r1, r2, ω)=nλn(ω)ψn*(r1, ω)ψn(r2, ω).
DW(r1, r2, ω)ψn(r1, ω)d3r1=λn(ω)ψn(r2, ω),
Dψm*(r, ω)ψn(r, ω)d3r=δmn,
W(r1, r2, ω)=U*(r1, ω)U(r2, ω),
E(r1, r2, τ)=E*(r1, t)E(r2, t+τ).
E(r2, r1, -τ)=E(r1, r2, τ),
f*(r1, t1)E(r1, r2, t2-t1)f(r2, t2)d3r1dt1d3r2dt20,
-|Eij(r1, r2, τ)|dτ<,
-|Eij(r1, r2, τ)|2dτ<,
-E(r1, r2, τ)F2dτ<.
AF2=tr(AA)=i=1mj=1m|Aij|2,
Wij(r1, r2, ω)=12π-Eij(r1, r2, τ)exp(iωτ)dτ,
Eij(r1, r2, τ)=0Wij(r1, r2, ω)exp(-iωτ)dω,
0|Wij(r1, r2, ω)|2dω<,
DD|Wij(r1, r2, ω)|2d3r1d3r2<.
DDW(r1, r2, ω)F2d3r1d3r2<,
W(r2, r1, ω)=W(r1, r2, ω).
g*(r1)W(r1, r2, ω)g(r2)d3r1d3r20,
Wjj(r1, r2, ω)=nλj,n(ω)ψj,n*(r1, ω)ψj,n(r2, ω).
{a(r), b(r)}=Da*(r)b(r)d3r.
W(r1, r2, ω)=nλn(ω)φn*(r1, ω)φn(r2, ω),
Dφn(r1, ω)W(r1, r2, ω)d3r1=λn(ω)φn(r2, ω).
{φm(r, ω), φn(r, ω)}=δmn,
Wn(r1, r2, ω)=λn(ω)φn*(r1, ω)φn(r2, ω).
μ(r1, r2, ω)=W(r1, r2, ω)F[S(r1, ω)S(r2, ω)]1/2.
k2E(r1, r2, τ)=1c22τ2 E(r1, r2, τ)
ii1Eij(r1, r2, τ)=jj2Eij(r1, r2, τ)=0,
-ddτ f(τ)exp(iωτ)dτ=-iω-f(τ)exp(iωτ)dτ,
k2W(r1, r2, ω)=-(ω/c)2W(r1, r2, ω),
ii1Wij(r1, r2, ω)=jj2Wij(r1, r2, ω)=0.
2φn(r, ω)+(ω/c)2φn(r, ω)=0,
φn(r, ω)=0.
k2Wn(r1, r2, ω)=-(ω/c)2Wn(r1, r2, ω),
ii1Wij(n)(r1, r2, ω)=jj2Wij(n)(r1, r2, ω)=0.
W(r1, r2, ω)F2=n,mλn(ω)λm(ω)φn*(r1, ω)φm(r1, ω)φm*(r2, ω)φn(r2, ω).
DDW(r1, r2, ω)F2d3r1d3r2=nλn2(ω),
S(r, ω)=nλ(ω)φn*(r, ω)φn(r, ω).
DS(r, ω)d3r=nλn(ω),
DDS(r1, ω)S(r2, ω)μ2(r1, r2, ω)d3r1d3r2DDS(r1, ω)S(r2, ω)d3r1d3r2
=nλn2(ω)nλn(ω)2,
F(r, ω)=nan(ω)φn(r, ω),
am*(ω)an(ω)=λn(ω)δmn,
n|an(ω)|2<.
F*(r1, ω)F(r2, ω)=nλn(ω)φn*(r1, ω)φn(r2, ω).
W(r1, r2, ω)=F*(r1, ω)F(r2, ω),
an(ω)=λn(ω)exp[iαn(ω)],
2F(r, ω)+(ω/c)2F(r, ω)=0,
F(r, ω)=0.
W(x1, x2, ω)=J(ω)exp-x12+x224w02(ω)×exp-(x1-x2)22σ02(ω),
S(x, ω)=tr J(ω)exp-x22w02(ω),
μ(x1-x2, ω)=1+P2(ω)21/2exp-(x1-x2)22σ02(ω),
P2(ω)=1-4 det W(x, x, ω)tr2 W(x, x, ω)=1-4 det J(ω)tr2 J(ω),
W(x1, x2, ω)=J(ω)tr J(ω)21+P2(ω)1/2[S(x1, ω)]1/2×[S(x2, ω)]1/2μ(x1-x2, ω).
W(x1, x2, ω)=J(ω)n=0λn(ω)ϕn*(x1, ω)ϕn(x2, ω),
ϕn(x, ω)=2cπ1/412nn! Hn(x2c)exp(-cx2),
λn(ω)=πa+b+c1/2ba+b+cn.
a=14w02(ω),b=12σ02(ω),c=a2+2ab.
{ϕn(x, ω), ϕm(x, ω)}=δnm.
J(ω)=JxxJxyJyxJyy=tr J(ω)s^*(ω)sˆ(ω),
sˆ(ω)=1tr J(ω) {Jxxu^x+Jyy   exp[i arg(Jxy)]u^y}.
W(x1, x2, ω)=n=0Λn(ω)φn*(x1, ω)φn(x2, ω),
Λn(ω)=tr J(ω)λn(ω),
φn(x, ω)=ϕn(x, ω)sˆ(ω).
J(ω)=J1(ω)+J2(ω),
J1(ω)=ABB*C,J2(ω)=DC-B-B*A,
A,C,D0,AC=|B|2.
D=tr J2(ω)tr J1(ω).
A=Jxx-DJyy1-D2,B=Jxy1-D,C=Jyy-DJxx1-D2,
D(D+1)2=det J(ω)tr2 J(ω),
D1=1-P(ω)1+P(ω),D2=1+P(ω)1-P(ω).
D=1-P(ω)1+P(ω),0P(ω)1.
J1(ω)=tr J1(ω)s^1*(ω)s^1(ω),
J2(ω)=tr J2(ω)s^2*(ω)s^2(ω),
s^1(ω)=1tr J1(ω) {Au^x+C   exp[i arg(B)]u^y},
s^2(ω)=Dtr J2(ω) {Cu^x-A exp[i arg(B)]u^y}.
W(x1, x2, ω)=n=0Λn(1)φn(1)*(x1, ω)φn(1)(x2, ω)+n=0Λn(2)φn(2)*(x1, ω)φn(2)(x2, ω),
Λn(1)(ω)=tr J1(ω)λn(ω),
φn(1)(x, ω)=ϕn(x, ω)s^1(ω),
Λn(2)(ω)=tr J2(ω)λn(ω),
φn(2)(x, ω)=ϕn(x, ω)s^2(ω),
Λn(j)(ω)Λ0(j)(ω)=ba+b+cn,j=(1, 2),
Λn(j)(ω)Λ0(j)(ω)=1(q/2)2+1+q[(q/2)2+1]1/2n,
j=(1, 2),
Λn(2)(ω)Λn(1)(ω)=1-P(ω)1+P(ω).
Λn(2)(ω)Λ0(1)(ω)=1-P(ω)1+P(ω)   Λn(1)(ω)Λ0(1)(ω),
W(ρ1, ρ2, ω)=J(ω)exp-x12+x22+y12+y224w02×exp-(x1-x2)2+(y1-y2)22σ02,
S(ρ, ω)=tr J(ω)exp-ρ22w02(ω),
μ(ρ1-ρ2, ω)=exp-(ρ1-ρ2)22σ02(ω).
W(ρ1, ρ2, ω)=J(ω)tr J(ω) [S(ρ1, ω)]1/2[S(ρ2, ω)]1/2×μ(ρ1-ρ2, ω),
W(ρ1, ρ2, ω)=m=0n=0Λmn(ω)φmn*(ρ1, ω)φmn(ρ2, ω),
Λmn(ω)=tr J(ω)λm(ω)λn(ω),
φmn(ρ, ω)=ϕm(x, ω)ϕn(y, ω)sˆ(ω),
{φmn(ρ1, ω), φmn(ρ2, ω)}z=0=δmmδnn.
-f*(r1, t1)W(r1, r2, ω)f(r2, t2)
×exp[-iω(t2-t1)]dωd3r1dt1d3r2dt20.
H(ω)=-h(t)exp(-iωt)dt
E(ω)=g*(r1)W(r1, r2, ω)g(r2)d3r1d3r2,
-|H(ω)|2E(ω)dω0.
g*(r1)W(r1, r2, ω)g(r2)d3r1d3r20,

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