Abstract

Partially coherent fields with the electric field parallel to the azimuthal coordinate are analyzed by use of the exact angular spectrum representation. The known results for fully coherent fields are used to find the permitted forms of azimuthally polarized, partially coherent fields. The derived result is then used to show that this class of fields is severely restricted because the azimuthal polarization state is particularly sensitive to the correlation properties of the electric-field components. Two examples of azimuthally polarized fields are briefly examined. The first is a class of nondiffracting fields that retain the polarization state upon propagation, whereas the second is an example in which the azimuthal polarization is broken because the cross-spectral density function is not of the permitted form.

© 2003 Optical Society of America

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).
  3. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1990).
  4. E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
    [CrossRef]
  5. T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [CrossRef]
  6. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  7. R. H. Jordan, D. G. Hall, “Free-space azimuthal wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef] [PubMed]
  8. R. H. Jordan, D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field expansion method,” J. Opt. Soc. Am. A 12, 84–94 (1995).
    [CrossRef]
  9. P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
    [CrossRef]
  10. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [CrossRef] [PubMed]
  11. A. A. Tovar, G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997).
    [CrossRef]
  12. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
    [CrossRef]
  13. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  14. J. Tervo, P. Vahimaa, J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
    [CrossRef]
  15. P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
    [CrossRef] [PubMed]
  16. A. Lapucci, M. Ciofini, “Polarization state modifications in the propagation of high azimuthal order annular beams,” Opt. Express 9, 603–609 (2001).
    [CrossRef] [PubMed]
  17. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  18. S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 17, 780–789 (2000).
    [CrossRef]
  19. S. R. Seshadri, “Spatial coherence of azimuthally symmetric Gaussian electromagnetic beams,” J. Appl. Phys. 88, 6973–6980 (2000).
    [CrossRef]
  20. S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” J. Appl. Phys. 87, 4084–4093 (2000).
    [CrossRef]
  21. P. Östlund, A. T. Friberg, “Radiation efficiency of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 18, 1696–1703 (2001).
    [CrossRef]
  22. 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]
  23. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 2737–2753 (1959).
    [CrossRef]
  24. F. Gori, “Matrix treatment for partially polarized, partially coherent fields,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  25. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  26. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  27. J. Peřina, Coherence of Light (Reidel, Dordrecht, The Netherlands, 1985).
  28. E. W. Marchand, E. Wolf, “Angular correlation and thefar-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
    [CrossRef]
  29. W. H. Carter, “Properties of electromagnetic radiation from partially correlated current distribution,” J. Opt. Soc. Am. 70, 1067–1074 (1980).
    [CrossRef]
  30. J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
    [CrossRef]
  31. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]
  32. L. Raåde, B. Westergren, Mathematics Handbook for Science and Engineering (Studentlitteratur, Lund, Sweden, 1998), p. 336.
  33. G. Gbur, D. James, E. Wolf, “Energy conservation law for randomly fluctuating electromagnetic fields,” Phys. Rev. E 59, 4594–4599 (1999).
    [CrossRef]
  34. T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  35. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  36. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  37. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
    [CrossRef]
  38. J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
    [CrossRef]
  39. Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
    [CrossRef]
  40. J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optic,” J. Opt. Soc. Am. A 8, 282–289 (1991).
    [CrossRef]
  41. A. V. Schegrov, E. Wolf, “Partially coherent conical beams,” Opt. Lett. 25, 141–143 (2000).
    [CrossRef]
  42. A. S. Ostrovsky, G. Martı́nez-Niconoff, J. C. Ramı́rez-San-Juan, “Coherent-mode representation of propagation-invariant fields,” Opt. Commun. 195, 27–34 (2001).
    [CrossRef]
  43. J. Turunen, “Invariant propagation of uniform-intensity Schell-model felds,” J. Mod. Opt. 49, 1795–1799 (2002).
    [CrossRef]
  44. L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
    [CrossRef]
  45. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
    [CrossRef]
  46. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 8.
  47. 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]
  48. 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]
  49. T. Saastamoinen, J. Tervo, J. Turunen, “Radiation from arbitrarily polarized spatially incoherent planar sources,” Opt. Commun. 221, 257–269 (2003).
    [CrossRef]
  50. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
    [CrossRef]

2003 (1)

T. Saastamoinen, J. Tervo, J. Turunen, “Radiation from arbitrarily polarized spatially incoherent planar sources,” Opt. Commun. 221, 257–269 (2003).
[CrossRef]

2002 (6)

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]

J. Turunen, “Invariant propagation of uniform-intensity Schell-model felds,” J. Mod. Opt. 49, 1795–1799 (2002).
[CrossRef]

J. Tervo, P. Vahimaa, J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[CrossRef]

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[CrossRef] [PubMed]

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

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

2001 (4)

A. S. Ostrovsky, G. Martı́nez-Niconoff, J. C. Ramı́rez-San-Juan, “Coherent-mode representation of propagation-invariant fields,” Opt. Commun. 195, 27–34 (2001).
[CrossRef]

P. Östlund, A. T. Friberg, “Radiation efficiency of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 18, 1696–1703 (2001).
[CrossRef]

A. Lapucci, M. Ciofini, “Polarization state modifications in the propagation of high azimuthal order annular beams,” Opt. Express 9, 603–609 (2001).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

2000 (4)

S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 17, 780–789 (2000).
[CrossRef]

S. R. Seshadri, “Spatial coherence of azimuthally symmetric Gaussian electromagnetic beams,” J. Appl. Phys. 88, 6973–6980 (2000).
[CrossRef]

S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” J. Appl. Phys. 87, 4084–4093 (2000).
[CrossRef]

A. V. Schegrov, E. Wolf, “Partially coherent conical beams,” Opt. Lett. 25, 141–143 (2000).
[CrossRef]

1999 (2)

G. Gbur, D. James, E. Wolf, “Energy conservation law for randomly fluctuating electromagnetic fields,” Phys. Rev. E 59, 4594–4599 (1999).
[CrossRef]

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
[CrossRef]

1998 (3)

1997 (1)

1996 (2)

1995 (2)

1994 (2)

1993 (1)

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1991 (2)

1990 (1)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

1987 (4)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

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]

1982 (1)

1981 (1)

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

1980 (1)

1972 (2)

1964 (1)

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 2737–2753 (1959).
[CrossRef]

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

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

Bouchal, Z.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Carter, W. H.

Ciofini, M.

Clark, G. H.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Friberg, A. T.

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]

P. Östlund, A. T. Friberg, “Radiation efficiency of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 18, 1696–1703 (2001).
[CrossRef]

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optic,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

Gbur, G.

G. Gbur, D. James, E. Wolf, “Energy conservation law for randomly fluctuating electromagnetic fields,” Phys. Rev. E 59, 4594–4599 (1999).
[CrossRef]

Gori, F.

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

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

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

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Greene, P. L.

Guattari, G.

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

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Hall, D. G.

James, D.

G. Gbur, D. James, E. Wolf, “Energy conservation law for randomly fluctuating electromagnetic fields,” Phys. Rev. E 59, 4594–4599 (1999).
[CrossRef]

James, D. F. V.

Jordan, R. H.

Kaivola, M.

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]

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1990).

Lapucci, A.

Mandel, L.

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

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

Marcatili, E. A. J.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

Marchand, E. W.

Marti´nez-Niconoff, G.

A. S. Ostrovsky, G. Martı́nez-Niconoff, J. C. Ramı́rez-San-Juan, “Coherent-mode representation of propagation-invariant fields,” Opt. Commun. 195, 27–34 (2001).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Mishra, S. R.

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Olivi´k, M.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Östlund, P.

Ostrovsky, A. S.

A. S. Ostrovsky, G. Martı́nez-Niconoff, J. C. Ramı́rez-San-Juan, “Coherent-mode representation of propagation-invariant fields,” Opt. Commun. 195, 27–34 (2001).
[CrossRef]

Pääkkönen, P.

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Perina, J.

J. Peřina, Coherence of Light (Reidel, Dordrecht, The Netherlands, 1985).

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Raåde, L.

L. Raåde, B. Westergren, Mathematics Handbook for Science and Engineering (Studentlitteratur, Lund, Sweden, 1998), p. 336.

Rami´rez-San-Juan, J. C.

A. S. Ostrovsky, G. Martı́nez-Niconoff, J. C. Ramı́rez-San-Juan, “Coherent-mode representation of propagation-invariant fields,” Opt. Commun. 195, 27–34 (2001).
[CrossRef]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Saastamoinen, T.

T. Saastamoinen, J. Tervo, J. Turunen, “Radiation from arbitrarily polarized spatially incoherent planar sources,” Opt. Commun. 221, 257–269 (2003).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

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

Schegrov, A. V.

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

Seshadri, S. R.

S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 17, 780–789 (2000).
[CrossRef]

S. R. Seshadri, “Spatial coherence of azimuthally symmetric Gaussian electromagnetic beams,” J. Appl. Phys. 88, 6973–6980 (2000).
[CrossRef]

S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” J. Appl. Phys. 87, 4084–4093 (2000).
[CrossRef]

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
[CrossRef]

Setälä, T.

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]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 8.

Tervo, J.

T. Saastamoinen, J. Tervo, J. Turunen, “Radiation from arbitrarily polarized spatially incoherent planar sources,” Opt. Commun. 221, 257–269 (2003).
[CrossRef]

J. Tervo, P. Vahimaa, J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[CrossRef]

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[CrossRef] [PubMed]

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

Tovar, A. A.

Turunen, J.

T. Saastamoinen, J. Tervo, J. Turunen, “Radiation from arbitrarily polarized spatially incoherent planar sources,” Opt. Commun. 221, 257–269 (2003).
[CrossRef]

J. Turunen, “Invariant propagation of uniform-intensity Schell-model felds,” J. Mod. Opt. 49, 1795–1799 (2002).
[CrossRef]

J. Tervo, P. Vahimaa, J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[CrossRef]

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[CrossRef] [PubMed]

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optic,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

Vahimaa, P.

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[CrossRef] [PubMed]

J. Tervo, P. Vahimaa, J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[CrossRef]

Vasara, A.

Vicalvi, S.

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

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

Westergren, B.

L. Raåde, B. Westergren, Mathematics Handbook for Science and Engineering (Studentlitteratur, Lund, Sweden, 1998), p. 336.

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Wolf, E.

A. V. Schegrov, E. Wolf, “Partially coherent conical beams,” Opt. Lett. 25, 141–143 (2000).
[CrossRef]

G. Gbur, D. James, E. Wolf, “Energy conservation law for randomly fluctuating electromagnetic fields,” Phys. Rev. E 59, 4594–4599 (1999).
[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 I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

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

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

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 2737–2753 (1959).
[CrossRef]

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

Appl. Phys. Lett. (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[CrossRef]

J. Appl. Phys. (3)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

S. R. Seshadri, “Spatial coherence of azimuthally symmetric Gaussian electromagnetic beams,” J. Appl. Phys. 88, 6973–6980 (2000).
[CrossRef]

S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” J. Appl. Phys. 87, 4084–4093 (2000).
[CrossRef]

J. Mod. Opt. (3)

J. Tervo, P. Vahimaa, J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[CrossRef]

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

J. Turunen, “Invariant propagation of uniform-intensity Schell-model felds,” J. Mod. Opt. 49, 1795–1799 (2002).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (10)

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optic,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

A. A. Tovar, G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997).
[CrossRef]

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
[CrossRef]

P. Östlund, A. T. Friberg, “Radiation efficiency of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 18, 1696–1703 (2001).
[CrossRef]

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
[CrossRef]

S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 17, 780–789 (2000).
[CrossRef]

R. H. Jordan, D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field expansion method,” J. Opt. Soc. Am. A 12, 84–94 (1995).
[CrossRef]

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 2737–2753 (1959).
[CrossRef]

Opt. Commun. (6)

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

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

A. S. Ostrovsky, G. Martı́nez-Niconoff, J. C. Ramı́rez-San-Juan, “Coherent-mode representation of propagation-invariant fields,” Opt. Commun. 195, 27–34 (2001).
[CrossRef]

T. Saastamoinen, J. Tervo, J. Turunen, “Radiation from arbitrarily polarized spatially incoherent planar sources,” Opt. Commun. 221, 257–269 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. A (1)

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 (2)

G. Gbur, D. James, E. Wolf, “Energy conservation law for randomly fluctuating electromagnetic fields,” Phys. Rev. E 59, 4594–4599 (1999).
[CrossRef]

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

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

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]

Pure Appl. Opt. (3)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

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

Other (7)

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

J. Peřina, Coherence of Light (Reidel, Dordrecht, The Netherlands, 1985).

L. Raåde, B. Westergren, Mathematics Handbook for Science and Engineering (Studentlitteratur, Lund, Sweden, 1998), p. 336.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1990).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 8.

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Figures (1)

Fig. 1
Fig. 1

Degree of polarization in the far zone of an azimuthally polarized GSM source with w=1000λ, σ0=1λ (dashed curve), σ0=20λ (solid curve with dots), and σ0=100λ (solid curve).

Equations (58)

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W(r1; r2; ω)=[Wij(r1; r2; ω)]=[Ei*(r1; ω)Ej(r2; ω)],
[Wij(r1; r2; ω)]*=Wji(r2; r1; ω),
i,jfi*(r1)fj(r2)Wij(r1; r2; ω)d3r1d3r20,
Wij(r1; r2; ω)=-Aij(k1; k2; ω)×exp[i(k2xx2-k1xx1+k2yy2-k1yy1)]exp[i(k2zz2-k1z*z1)]dk1xdk1ydk2xdk2y,
kz=(k2-kx2-ky2)1/2,kx2+ky2<k2i(kx2+ky2-k2)1/2,kx2+ky2k2,
Aij(k1; k2; ω)=1(2π)4-Wij×(x1, y1, 0; x2, y2, 0; ω)×exp[i(k1xx1-k2xx2)]×exp[i(k1yy1-k2yy2)]×dx1dy1dx2dy2
Eρ(r; ω)=cos ϕEx(r; ω)+sin ϕEy(r; ω),
Eϕ(r; ω)=-sin ϕEx(r; ω)+cos ϕEy(r; ω),
Eϕ(r; ω)=0f(α; ω)J1(αρ)exp(ikzz)αdα,
Wϕϕ(r1; r2; ω)=Eϕ*(r1; ω)Eϕ(r2; ω)=sin ϕ1 sin ϕ2Wxx(r1; r2; ω)-sin ϕ1 cos ϕ2Wxy(r1; r2; ω)-cos ϕ1 sin ϕ2Wyx(r1; r2; ω)+cos ϕ1 cos ϕ2Wyy(r1; r2; ω).
Wϕϕ(r1; r2; ω)=Wϕϕ(ρ1, z1; ρ2, z2; ω)=0f*(α; ω)J1(αρ1)×exp(-ikz*z1)αdα×0f(α; ω)J1(αρ2)exp(ikzz2)αdα=0F(α1; α2; ω)J1(α1ρ1)J1(α2ρ2)×exp[i(k2zz2-k1z*z1)]α1α2dα1dα2,
F(α1; α2; ω)=f*(α1; ω)f(α2; ω).
F(α1; α2; ω)=0Wϕϕ(ρ1, 0;ρ2, 0;ω)×J1(α1ρ1)J1(α2ρ2)ρ1ρ2dρ1dρ2.
Wxx(r1; r2; ω)=sin ϕ1 sin ϕ2Wϕϕ(ρ1, z1; ρ2, z2; ω),
Wxy(r1; r2; ω)=-sin ϕ1 cos ϕ2Wϕϕ(ρ1, z1; ρ2, z2; ω),
Wyx(r1; r2; ω)=-cos ϕ1 sin ϕ2Wϕϕ(ρ1, z1; ρ2, z2; ω),
Wyy(r1; r2; ω)=cos ϕ1 cos ϕ2Wϕϕ(ρ1, z1; ρ2, z2; ω).
μij(r1; r2; ω)=Wij(r1; r2; ω)[Wii(r1; r1; ω)Wjj(r2; r2; ω)]1/2.
μii(r1; r2; ω)=μii(ρ1; ρ2; ω)=Wϕϕ(ρ1; ρ2; ω)[Wϕϕ(ρ1; ρ1; ω)Wϕϕ(ρ2; ρ2; ω)]1/2,
μij(r1; r2; ω)=μij(ρ1; ρ2; ω)=-Wϕϕ(ρ1; ρ2; ω)[Wϕϕ(ρ1; ρ1; ω)Wϕϕ(ρ2; ρ2; ω)]1/2,
Wij(ρ1, ϕ1, z; ρ2, ϕ2, z; ω)=Wij(ρ1, ϕ1, 0;ρ2, ϕ2, 0; ω)
F(α1; α2; ω)=F(α1; ω)δ(α1-α2),
Wϕϕ(ρ1, z1; ρ2, z2; ω)=0F(α; ω)J1(αρ1)J1(αρ2)exp[ikz(z2-z1)]α2dα.
S(ρ; ω)=0F(α; ω)J12(αρ)α2dα,
F(α; ω)=F0(ω)δ(α-α0).
Wϕϕ(ρ1, z1; ρ2, z2; ω)=α02F0(ω)J1(α0ρ1)exp(-ikz0z1)J1(α0ρ2)exp(ikz0z2),
Wxx(x1, y1, 0, x2, y2, 0)=Ky1y2g(x1, x2, y1, y2),
Wxy(x1, y1, 0, x2, y2, 0)=-Kx1y2g(x1, x2, y1, y2),
Wyx(x1, y1, 0, x2, y2, 0)=-Ky1x2g(x1, x2, y1, y2),
Wyy(x1, y1, 0, x2, y2, 0)=Kx1x2g(x1, x2, y1, y2),
g(x1, x2, y1, y2)=exp-x12+y12+x22+y22w02×exp-(x1-x2)2+(y1-y2)22σ0,
sx=sin θ cos ϕ,sy=sin θ sin ϕ,sz=cos θ.
Wij(rsˆ, rsˆ)=(2πk/r)2cos2 θAij(ks, ks),
Wxz(rsˆ, rsˆ)=Wzx*(rsˆ, rsˆ)=-(2πk/r)2sin θ cos θ[cos ϕAxx(ks, ks)+sin ϕAxy(ks, ks)],
Wyz(rsˆ, rsˆ)=Wzy*(rsˆ, rsˆ)=-(2πk/r)2sin θ cos θ[cos ϕAyx(ks, ks)+sin ϕAyy(ks, ks)],
Wzz(rsˆ, rsˆ)=(2πk/r)2sin2 θ{cos2 ϕAxx(ks, ks)+sin2 ϕAyy(ks, ks)+cos ϕ sin ϕ[Axy(ks, ks)+Ayx(ks, ks)]}.
Axx(k, k)=(w02+σ02+σ04ky2)h(kx, ky),
Ayy(k, k)=(w02+σ02+σ04kx2)h(kx, ky),
Axy(k, k)=Ayx(k, k)=-σ04kxkyh(kx, ky),
h(kx, ky)=Kσ02w0816(σ02+w02)3exp-(kx2+ky2)w02σ022(σ02+w02).
Wxx(rsˆ, rsˆ)=(2πk/r)2 cos2 θ(w02+σ02+σ04k2 sin2 θ sin2 ϕ)H(θ),
Wxy(rsˆ, rsˆ)=Wyx(rsˆ, rsˆ)=-(2π/r)2k4 cos2 θ sin2 θ sin ϕ×cos ϕσ04H(θ),
Wxz(rsˆ, rsˆ)=Wzx(rsˆ, rsˆ)=-(2πk/r)2sin θ cos θ cos ϕ(w02+σ02)H(θ),
Wyy(rsˆ, rsˆ)=(2πk/r)2 cos2 θ(w02+σ02+σ04k2 sin2 θ cos2 ϕ)H(θ),
Wyz(rsˆ, rsˆ)=Wzy(rsˆ, rsˆ)=-(2πk/r)2 sin θ cos θ sin ϕ(w02+σ02)H(θ),
Wzz(rsˆ, rsˆ)=(2πk/r)2sin2 θ(w02+σ02)H(θ),
H(θ)=h(k sin θ cos ϕ, k sin θ sin ϕ)=Kσ02w0816(σ02+w02)3exp-k2w02σ02 sin2 θ2(σ02+w02).
Er(rsˆ)=Ex(rsˆ)sin θ cos ϕ+Ey(rsˆ)sin θ sin ϕ+Ez(rsˆ)cos θ=Eρ(rsˆ)sin θ+Ez(rsˆ)cos θ,
Eθ(rsˆ)=Ex(rsˆ)cos θ cos ϕ+Ey(rsˆ)cos θ sin ϕ-Ez(rsˆ)sin θ=Eρ(rsˆ)cos θ-Ez(rsˆ)sin θ,
Eϕ(rsˆ)=-Ex(rsˆ)sin ϕ+Ey(rsˆ)cos ϕ.
Wθθ(rsˆ, rsˆ)=(2πk/r)2(w02+σ02)H(θ),
Wϕϕ(rsˆ, rsˆ)=(2πk/r)2 cos2 θ(w02+σ02+k2σ04 sin2 θ)H(θ).
P2(rsˆ)=2 tr Φ2(rsˆ, rsˆ)tr2 Φ(rsˆ, rsˆ)-1,
Φ(rsˆ)=Wθθ(rsˆ, rsˆ)Wθϕ(rsˆ, rsˆ)Wϕθ(rsˆ, rsˆ)Wϕϕ(rsˆ, rsˆ)
P32(θ)=14[3P2(θ)+1],
P(θ)=4(σ02+w02)(1-cos2 θ)-k2σ04 sin2(2θ)4(σ02+w02)(1+cos2 θ)+k2σ04 sin2(2θ).
P(θ)=1-cos2 θ1+cos2 θ.
cos2 θ=σ02+w02k2σ04.

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