Abstract

We present a general theory of electromagnetic diffraction-free beams composed of uncorrelated Bessel modes. Our approach is based on the direct application of the nonnegativity constraint to the cross-spectral density tensor describing the electromagnetic field distribution. The field correlation properties are most conveniently derived in the spatial frequency domain, where the angular spectrum takes on the form of an infinitely thin ring. We also present several examples, including a vector generalization of the recently introduced dark and antidark diffraction-free beams.

© 2009 Optical Society of America

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  1. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microwaves Opt. Acoust. 2, 105-112 (1978).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1502 (1987).
    [CrossRef] [PubMed]
  3. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150-152 (1989).
    [CrossRef]
  4. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75-87 (1993).
    [CrossRef]
  5. H. Sonajalg, M. Ratsep, and P. Saari, “Demonstrations of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310-312 (1997).
    [CrossRef] [PubMed]
  6. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364-1366 (2001).
    [CrossRef]
  7. D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X-waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29, 1446-1448 (2004).
    [CrossRef] [PubMed]
  8. S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1-4 (2006).
    [CrossRef]
  9. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable coherence optics,” J. Opt. Soc. Am. A 8, 282-289 (1991).
    [CrossRef]
  10. M. W. Kowarz and G. S. Agarwal, “Bessel-beam respresentation for partially coherent fields,” J. Opt. Soc. Am. A 12, 1324-1330 (1995).
    [CrossRef]
  11. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508-2510 (2007).
    [CrossRef] [PubMed]
  12. H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605-611 (2008).
    [CrossRef]
  13. Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
    [CrossRef]
  14. D. G. Hall, “Vector-beam solutions of Maxwell's wave equation,” Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
  15. J. Tervo, P. Vahimaa, and J. Turunen, “On propagation invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537-1543 (2002).
    [CrossRef]
  16. J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A 20, 1974-1980 (2003).
    [CrossRef]
  17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  18. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).
  19. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part 1. Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343-351 (1982).
    [CrossRef]
  20. F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301-305 (1980).
    [CrossRef]
  21. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311-316 (1987).
    [CrossRef]
  22. R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal mode spectrum,” J. Opt. Soc. Am. A 10, 2008-2016 (1993).
    [CrossRef]
  23. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150-156 (2001).
    [CrossRef]
  24. F. Gori, M. Santarsiero, R. Borghi, and Chun-Fang Li, “Partially correlated thin annular sources: the scalar case,” J. Opt. Soc. Am. A 25, 2826-2832 (2008).
    [CrossRef]
  25. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241-243 (1998).
    [CrossRef]
  26. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarisation matrix,” J. Opt. A, Pure Appl. Opt. 7, 941-951 (1998).
  27. 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]
  28. E. Wolf, “Unified theory of coherence and polarization,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  29. C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
  30. F. Gori, M. Santarsiero, R. Simon, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78-84 (2003).
    [CrossRef]
  31. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).
  32. E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642-644 (2008).
    [CrossRef] [PubMed]
  33. F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239-243 (1988).
    [CrossRef]
  34. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
    [CrossRef]

2008 (3)

2007 (1)

2006 (1)

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1-4 (2006).
[CrossRef]

2004 (2)

2003 (3)

2002 (1)

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

2001 (2)

1998 (3)

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

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

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

1994 (1)

1993 (2)

1991 (1)

1989 (1)

1988 (1)

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239-243 (1988).
[CrossRef]

1987 (2)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1502 (1987).
[CrossRef] [PubMed]

1982 (1)

1980 (1)

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

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microwaves Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

Agarwal, G. S.

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1-4 (2006).
[CrossRef]

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[CrossRef]

Baykal, Y.

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605-611 (2008).
[CrossRef]

Besieris, I. M.

Borghi, R.

Born, M.

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

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Cada, M.

Cai, Y.

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605-611 (2008).
[CrossRef]

Christodoulides, D. N.

Di Trapani, P.

Durnin, J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1502 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1502 (1987).
[CrossRef] [PubMed]

Efremidis, N. K.

Eyyuboglu, H. T.

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605-611 (2008).
[CrossRef]

Friberg, A. T.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and Chun-Fang Li, “Partially correlated thin annular sources: the scalar case,” J. Opt. Soc. Am. A 25, 2826-2832 (2008).
[CrossRef]

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

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

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

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239-243 (1988).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

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

Guattari, G.

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

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

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239-243 (1988).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

Hall, D. G.

Huang, W.

Indebetouw, G.

James, D. F. V.

Kivshar, Yu. S.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
[CrossRef]

Korotkova, O.

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605-611 (2008).
[CrossRef]

Kowarz, M. W.

Li, Chun-Fang

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
[CrossRef]

Malomed, B. A.

Mandel, L.

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

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1502 (1987).
[CrossRef] [PubMed]

Mukunda, N.

Padovani, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239-243 (1988).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

Palma, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239-243 (1988).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508-2510 (2007).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1-4 (2006).
[CrossRef]

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[CrossRef]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150-156 (2001).
[CrossRef]

Porras, M. A.

Ratsep, M.

Saari, P.

Santarsiero, M.

Sermutlu, E.

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605-611 (2008).
[CrossRef]

Shaarawi, A. M.

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microwaves Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

Simon, R.

Sonajalg, H.

Sundar, K.

Tervo, J.

J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A 20, 1974-1980 (2003).
[CrossRef]

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

Turunen, J.

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

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

Vahimaa, P.

J. Tervo, P. Vahimaa, and 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, and G. Guattari, “Beam coherence-polarisation matrix,” J. Opt. A, Pure Appl. Opt. 7, 941-951 (1998).

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microwaves Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

Wolf, E.

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642-644 (2008).
[CrossRef] [PubMed]

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

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part 1. Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343-351 (1982).
[CrossRef]

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

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

Ziolkowski, R. W.

Appl. Phys. B (1)

H. T. Eyyuboglu, Y. Baykal, E. Sermutlu, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93, 605-611 (2008).
[CrossRef]

IEEE J. Microwaves Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microwaves Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. A, Pure Appl. Opt. (1)

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

J. Opt. Soc. Am. (1)

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

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

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

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150-156 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and Chun-Fang Li, “Partially correlated thin annular sources: the scalar case,” J. Opt. Soc. Am. A 25, 2826-2832 (2008).
[CrossRef]

J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A 20, 1974-1980 (2003).
[CrossRef]

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

M. W. Kowarz and G. S. Agarwal, “Bessel-beam respresentation for partially coherent fields,” J. Opt. Soc. Am. A 12, 1324-1330 (1995).
[CrossRef]

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150-152 (1989).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75-87 (1993).
[CrossRef]

Opt. Commun. (4)

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1-4 (2006).
[CrossRef]

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

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311-316 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239-243 (1988).
[CrossRef]

Opt. Lett. (7)

Phys. Lett. A (1)

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

Phys. Rep. (1)

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
[CrossRef]

Phys. Rev. E (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1502 (1987).
[CrossRef] [PubMed]

Other (4)

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

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

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

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Equations (71)

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

W ( ρ 1 , ρ 2 , ω ) = V * ( ρ 1 , ω ) V ( ρ 2 , ω ) ,
Q = n , m W ( ρ n , ρ m ) a n * a m ,
Q = d 2 ρ 1 d 2 ρ 2 W ( ρ 1 , ρ 2 ) f * ( ρ 1 ) f ( ρ 2 ) ,
d 2 ρ 1 W ( ρ 1 , ρ 2 ) u ( ρ 1 ) = λ u ( ρ 2 ) ,
W ( ρ 1 , ρ 2 ) = n λ n u n * ( ρ 1 ) u n ( ρ 2 ) .
W ̂ ( ρ 1 , ρ 2 ) = [ W x x ( ρ 1 , ρ 2 ) W x y ( ρ 1 , ρ 2 ) W y x ( ρ 1 , ρ 2 ) W y y ( ρ 1 , ρ 2 ) ] ,
W α β ( ρ 1 , ρ 2 ) = E α * ( ρ 1 ) E β ( ρ 2 ) .
P ̂ ( ρ ) W ̂ ( ρ , ρ ) = [ W x x ( ρ , ρ ) W x y ( ρ , ρ ) W y x ( ρ , ρ ) W y y ( ρ , ρ ) ] .
Q = α , β n , m W α β ( ρ 1 n , ρ 2 m ) a n * b m .
Q = α , β d 2 ρ 1 d 2 ρ 2 W α β ( ρ 1 , ρ 2 ) f α * ( ρ 1 ) f β ( ρ 2 ) ,
β d 2 ρ 1 W α β ( ρ 1 , ρ 2 ) u β ( ρ 1 ) = λ u α ( ρ 2 ) .
W ̂ ( ρ 1 , ρ 2 ) = n λ n u n ( ρ 1 ) u n ( ρ 2 ) ,
s ̃ ( p 1 , p 2 ) = d 2 ρ 1 d 2 ρ 2 s ( ρ 1 , ρ 2 ) exp [ 2 π i ( p 1 ρ 1 + p 2 ρ 2 ) ] ,
Q = α , β d 2 p 1 d 2 p 2 W ̃ α β ( p 1 , p 2 ) f ̃ α * ( p 1 ) f ̃ β ( p 2 ) .
Φ n ( ρ ) = J n ( κ ρ ) exp ( i n ϕ ) ,
V ( ρ ) = n a n Φ n ( ρ )
Φ ̃ n ( p , θ ) = ( i ) n δ [ p κ ( 2 π ) ] exp ( i n θ ) ,
W ( ρ 1 , ρ 2 ) = J 0 ( κ | ρ 1 ρ 2 | ) + χ J 0 ( κ | ρ 1 + ρ 2 | ) ,
W df ( ρ 1 , ρ 2 ) = n = λ n Φ n * ( ρ 1 ) Φ n ( ρ 2 ) ,
λ n = 1 + ( 1 ) n χ .
Q = d 2 ρ 1 d 2 ρ 2 { W d f ( ρ 1 , ρ 2 ) [ f * ( ρ 1 ) f ( ρ 2 ) + g * ( ρ 1 ) g ( ρ 2 ) ] + 2 R [ W x y ( ρ 1 , ρ 2 ) f * ( ρ 1 ) g ( ρ 2 ) ] } ,
Q d f = n λ n ( | d 2 ρ Φ n ( ρ ) f * ( ρ ) | 2 + | d 2 ρ Φ n ( ρ ) g * ( ρ ) | 2 ) .
d 2 ρ Φ n ( ρ ) h * ( ρ ) = d 2 p Φ ̃ n ( p ) h ̃ * ( p ) ,
Q = Q x y = 2 R { d 2 p 1 d 2 p 2 W ̃ x y ( p 1 , p 2 ) f ̃ * ( p 1 ) g ̃ ( p 2 ) } .
Q = Q x y = 2 R { a b * W ̃ x y ( p 0 , p 0 ) } ,
W ̃ x y ( p 1 , p 2 ) = δ ( p 1 κ 2 π ) δ ( p 2 κ 2 π ) F ( θ 1 , θ 2 ) ,
Q d f = κ 2 n λ n ( | f n | 2 + | g n | 2 ) ,
h n = 1 2 π 0 2 π h ̃ ( κ 2 π , θ ) exp ( i n θ ) d θ ,
Q x y = 2 κ 2 R { n , m f n * g m F n m } ,
F n m = 0 2 π 0 2 π d θ 2 d θ 1 4 π 2 F ( θ 1 , θ 2 ) exp ( i n θ 1 i m θ 2 ) .
n λ n ( | f n | 2 + | g n | 2 ) + 2 n , m R { f n * g m F n m } 0 ,
F ( θ 1 , θ 2 ) = n a n exp [ i n ( θ 2 θ 1 ) ] ,
W x y ( ρ 1 , ρ 2 ) = n a n Φ n * ( ρ 1 ) Φ n ( ρ 2 ) ,
Q = n λ n ( | f n | 2 + | g n | 2 ) + 2 R { a n f n * g n } .
| a n | λ n
W ̂ ( ρ 1 , ρ 2 ) = n P ̂ n Φ n * ( ρ 1 ) Φ n ( ρ 2 ) ,
P ̂ n = [ λ n a n a n * λ n ] .
U n ( ± ) = 1 2 ( ± exp ( i ɛ n ) 1 ) ,
W ̂ ( ρ 1 , ρ 2 ) = n l = ± μ n ( l ) Ψ n ( l ) ( ρ 1 ) [ Ψ n ( l ) ( ρ 2 ) ] ,
Ψ n ( ± ) ( ρ ) = Φ n ( ρ ) U n ( ± ) .
W ̂ ( p ) ( ρ 1 , ρ 2 ) = J 0 ( κ | ρ 1 ρ 2 | ) [ | a | a a * | a | ] ,
W α α ( u ) ( ρ 1 , ρ 2 ) = ( 1 | a | ) J 0 ( κ | ρ 1 ρ 2 | ) + χ J 0 ( κ | ρ 1 + ρ 2 | ) .
F ( θ 1 , θ 2 ) = G ( θ 1 + θ 2 ) ,
G ( θ ) = n η n exp ( i n θ ) ,
W ̃ df ( p 1 , p 2 ) = δ ( p 1 κ 2 π ) δ ( p 2 κ 2 π ) F d ( θ 2 θ 1 ) ,
F d ( θ ) = n λ n exp ( i n θ ) .
W ̂ ( p 1 , p 2 ) = δ ( p 1 κ 2 π ) δ ( p 2 κ 2 π ) [ F d ( θ 2 θ 1 ) G ( θ 2 + θ 1 ) G ( θ 2 + θ 1 ) F d ( θ 2 θ 1 ) ] .
W ̂ ( p 1 , p 2 ) = δ ( p 1 κ 2 π ) δ ( p 2 κ 2 π ) [ K + ( θ 1 , θ 2 ) 0 0 K ( θ 1 , θ 2 ) ] ,
K ± ( θ 1 , θ 2 ) = F d ( θ 2 θ 1 ) ± G ( θ 2 + θ 1 ) .
Q = d θ 1 d θ 2 [ F d ( θ 2 θ 1 ) ± G ( θ 2 + θ 1 ) ] q * ( θ 1 ) q ( θ 2 )
Q d = d θ 1 d θ 2 F d ( θ 2 θ 1 ) q * ( θ 1 ) q ( θ 2 ) ,
Q a = d θ 1 d θ 2 G ( θ 1 + θ 2 ) q * ( θ 1 ) q ( θ 2 ) ,
Q d | Q a | ,
Q d = n λ n | q n | 2 ,
Q a = n η n q n * q n ,
q n = 0 π d θ q ( θ ) exp ( i n θ )
Q d ± Q a = ( λ 0 ± η 0 ) | q 0 | 2 + n = 1 λ n | q n | 2 + λ n | q n | 2 ± 2 R [ η n q n * q n ] ,
| η 0 | λ 0 ,
| η n | λ n λ n , n 1 .
0 2 π K ± ( θ 1 , θ 2 ) Ψ ( ± ) ( θ 2 ) d θ 2 = 2 π μ ( ± ) Ψ ( ± ) ( θ 1 ) ,
λ n ψ n ± η n ψ n = μ ψ n , n = 0 , ± 1 , ± 2 , ,
ψ n = 1 2 π 0 2 π Ψ ( θ ) exp ( i n θ ) d θ
( λ 0 + γ 0 ) ψ 0 = μ ψ 0 ,
{ λ n ψ n + γ n ψ n = μ ψ n , λ n ψ n + γ n * ψ n = μ ψ n , } n 1 ,
μ 0 ( ± ) = λ 0 ± η 0 ,
μ n , 1 ( ± ) = λ n + λ n 2 + ( λ n λ n 2 ) 2 + | η n | 2 , ( n 1 ) ,
μ n , 2 ( ± ) = λ n + λ n 2 ( λ n λ n 2 ) 2 + | η n | 2 , ( n 1 ) ,
Ψ 0 ( ± ) = 1 ,
Ψ n , j ( ± ) = ψ n , j ( ± ) exp ( i n θ ) + ψ n , j ( ± ) exp ( i n θ ) ,
ψ n , j ( ± ) = ± η n ( μ n , j ( ± ) λ n ) 2 + | η n | 2 ,
ψ n , j ( ± ) = μ n , j ( ± ) λ n ( μ n , j ( ± ) λ n ) 2 + | η n | 2 ,

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