Abstract

An angular spectrum representation is applied for a description of statistical properties of arbitrary beamlike fields propagating through atmospheric turbulence. The Rytov theory is used for the characterization of the perturbation of the field by the atmosphere. In particular, we derive expressions for the cross-spectral density of a coherent and a partially coherent beam of arbitrary type in the case when the power spectrum of atmospheric fluctuations is described by the von Karman model. We illustrate the method by applying it to the propagation of several model beams through the atmosphere.

© 2007 Optical Society of America

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  1. M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
    [CrossRef]
  2. J. C. Leader, "Atmospheric propagation of partially coherent radiation," J. Opt. Soc. Am. 68, 175-185 (1978).
    [CrossRef]
  3. J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  4. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  5. J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
    [CrossRef]
  6. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).
  7. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  8. O. Korotkova, L. C. Andrews, and R. L. Phillips, "A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
    [CrossRef]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  10. Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003).
    [CrossRef]
  11. Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
    [CrossRef]
  12. T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, "Nondiffracting narrow light beam with small atmospheric turbulence-induced propagation," Appl. Opt. 38, 3152-3156 (1999).
    [CrossRef]
  13. K. Wang, L. Zeng, and C. Yin, "Influence of the incident wave-front on intensity distribution of the non-diffracting beam used in large-scale measurement," Opt. Commun. 216, 99-103 (2003).
    [CrossRef]
  14. Y. Zhang, M. Tang, and C. Tao, "Partially coherent vortex beams propagation in a turbulent atmosphere," Chin. Opt. Lett. 3, 559-561 (2005).
  15. C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005).
    [CrossRef] [PubMed]
  16. Z. Bouchal, "Resistance of nondiffracting vortex beam against amplitude and phase perturbations," Opt. Commun. 210, 155-164 (2002).
    [CrossRef]
  17. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  19. G. S. Agarwal and E. Wolf, "Higher-order coherence functions in the space-frequency domain," J. Mod. Opt. 40, 1489-1496 (1993).
    [CrossRef]
  20. E. A. Vitrichenko, V. V. Voitsekhovich, and M. I. Mishchenko, "Effect of atmospheric turbulence on the field of view of adaptive systems," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 20, 758-759 (1984).
  21. V. V. Voitesekovich, V. G. Orlov, S. Cuevas, and R. Avila, "Efficiency of off-axis astronomical adaptive systems: comparison of theoretical and experimental data," Astron. Astrophys., Suppl. Ser. 133, 427-430 (1998).
    [CrossRef]

2005 (2)

Y. Zhang, M. Tang, and C. Tao, "Partially coherent vortex beams propagation in a turbulent atmosphere," Chin. Opt. Lett. 3, 559-561 (2005).

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

2004 (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, "A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

2003 (2)

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003).
[CrossRef]

K. Wang, L. Zeng, and C. Yin, "Influence of the incident wave-front on intensity distribution of the non-diffracting beam used in large-scale measurement," Opt. Commun. 216, 99-103 (2003).
[CrossRef]

2002 (3)

1999 (1)

1998 (2)

V. V. Voitesekovich, V. G. Orlov, S. Cuevas, and R. Avila, "Efficiency of off-axis astronomical adaptive systems: comparison of theoretical and experimental data," Astron. Astrophys., Suppl. Ser. 133, 427-430 (1998).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

1993 (1)

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

1991 (1)

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

1990 (1)

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1984 (1)

E. A. Vitrichenko, V. V. Voitsekhovich, and M. I. Mishchenko, "Effect of atmospheric turbulence on the field of view of adaptive systems," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 20, 758-759 (1984).

1983 (1)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

1978 (1)

1977 (1)

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Agarwal, G. S.

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

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Aruga, T.

Avila, R.

V. V. Voitesekovich, V. G. Orlov, S. Cuevas, and R. Avila, "Efficiency of off-axis astronomical adaptive systems: comparison of theoretical and experimental data," Astron. Astrophys., Suppl. Ser. 133, 427-430 (1998).
[CrossRef]

Banakh, V. A.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

Belenkii, M. S.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Boardman, A. D.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Bouchal, Z.

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003).
[CrossRef]

Z. Bouchal, "Resistance of nondiffracting vortex beam against amplitude and phase perturbations," Opt. Commun. 210, 155-164 (2002).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Buldakov, V. M.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Cuevas, S.

V. V. Voitesekovich, V. G. Orlov, S. Cuevas, and R. Avila, "Efficiency of off-axis astronomical adaptive systems: comparison of theoretical and experimental data," Astron. Astrophys., Suppl. Ser. 133, 427-430 (1998).
[CrossRef]

Davidson, F. M.

Gbur, G.

Kon, A. I.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Korotkova, O.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

Leader, J. C.

Li, R.

Li, S. W.

Mandel, L.

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

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Mishchenko, M. I.

E. A. Vitrichenko, V. V. Voitsekhovich, and M. I. Mishchenko, "Effect of atmospheric turbulence on the field of view of adaptive systems," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 20, 758-759 (1984).

Orlov, V. G.

V. V. Voitesekovich, V. G. Orlov, S. Cuevas, and R. Avila, "Efficiency of off-axis astronomical adaptive systems: comparison of theoretical and experimental data," Astron. Astrophys., Suppl. Ser. 133, 427-430 (1998).
[CrossRef]

Paterson, C.

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Ricklin, J. C.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Takabe, M.

Tang, M.

Tao, C.

Vitrichenko, E. A.

E. A. Vitrichenko, V. V. Voitsekhovich, and M. I. Mishchenko, "Effect of atmospheric turbulence on the field of view of adaptive systems," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 20, 758-759 (1984).

Voitesekovich, V. V.

V. V. Voitesekovich, V. G. Orlov, S. Cuevas, and R. Avila, "Efficiency of off-axis astronomical adaptive systems: comparison of theoretical and experimental data," Astron. Astrophys., Suppl. Ser. 133, 427-430 (1998).
[CrossRef]

Voitsekhovich, V. V.

E. A. Vitrichenko, V. V. Voitsekhovich, and M. I. Mishchenko, "Effect of atmospheric turbulence on the field of view of adaptive systems," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 20, 758-759 (1984).

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

Wang, K.

K. Wang, L. Zeng, and C. Yin, "Influence of the incident wave-front on intensity distribution of the non-diffracting beam used in large-scale measurement," Opt. Commun. 216, 99-103 (2003).
[CrossRef]

Wolf, E.

G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

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

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

Wu, J.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

Yin, C.

K. Wang, L. Zeng, and C. Yin, "Influence of the incident wave-front on intensity distribution of the non-diffracting beam used in large-scale measurement," Opt. Commun. 216, 99-103 (2003).
[CrossRef]

Yoshikado, S.

Zeng, L.

K. Wang, L. Zeng, and C. Yin, "Influence of the incident wave-front on intensity distribution of the non-diffracting beam used in large-scale measurement," Opt. Commun. 216, 99-103 (2003).
[CrossRef]

Zhang, Y.

Appl. Opt. (1)

Astron. Astrophys., Suppl. Ser. (1)

V. V. Voitesekovich, V. G. Orlov, S. Cuevas, and R. Avila, "Efficiency of off-axis astronomical adaptive systems: comparison of theoretical and experimental data," Astron. Astrophys., Suppl. Ser. 133, 427-430 (1998).
[CrossRef]

Chin. Opt. Lett. (1)

Czech. J. Phys. (1)

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

E. A. Vitrichenko, V. V. Voitsekhovich, and M. I. Mishchenko, "Effect of atmospheric turbulence on the field of view of adaptive systems," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 20, 758-759 (1984).

J. Mod. Opt. (3)

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

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998).
[CrossRef]

K. Wang, L. Zeng, and C. Yin, "Influence of the incident wave-front on intensity distribution of the non-diffracting beam used in large-scale measurement," Opt. Commun. 216, 99-103 (2003).
[CrossRef]

Z. Bouchal, "Resistance of nondiffracting vortex beam against amplitude and phase perturbations," Opt. Commun. 210, 155-164 (2002).
[CrossRef]

Opt. Eng. (Bellingham) (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, "A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

Opt. Spektrosk. (1)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

Phys. Rev. Lett. (1)

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (1)

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Other (3)

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

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Figures (5)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Intensity profile of a typical coherent Gaussian beam, with σ = 3 cm , at propagation distances L = 1000 m and L = 1500 m . The solid curve is the angular spectrum result, while the dashed curve is based on a standard method. The dashed–dotted curve shows the profile of the beam in the absence of turbulence.

Fig. 3
Fig. 3

Intensity profile of a typical Gaussian Schell-model beam, with σ I = 3 cm and σ μ = 1 cm , at propagation distances L = 1000 m and L = 1500 m . The solid curve is the angular spectrum result, while the dashed curve is based on a standard method. The dashed–dotted curve shows the profile of the beam in the absence of turbulence.

Fig. 4
Fig. 4

Intensity profile of a nondiffracting beam, with r 0 = 1 cm , for different propagation distances L.

Fig. 5
Fig. 5

Intensity profile of a nondiffracting beam, with r 0 = 3 cm , for different propagation distances L.

Equations (49)

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U ( r , ω ) = a ( u , ω ) P u ( r , ω ) d 2 u ,
P u ( r , ω ) = exp [ i k ( u r ) ]
u z = + 1 u 2 .
a ( u , ω ) = 1 ( 2 π ) 2 U 0 ( r , ω ) P u * ( r , ω ) d 2 r ,
U ( r , ω ) = a ( u , ω ) P u T ( r , ω ) d 2 u ,
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) ,
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) = a * ( u 1 , ω ) a ( u 2 , ω ) P u 1 T * ( r 1 , ω ) P u 2 T ( r 2 , ω ) T d 2 u 1 d 2 u 2 .
I ( r ) W ( r , r , ω ) = U * ( r , ω ) U ( r , ω ) .
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) ,
W ( r 1 , r 2 , ω ) = A ( u 1 , u 2 , ω ) P u 1 * ( r 1 , ω ) P u 2 ( r 2 , ω ) d 2 u 1 d 2 u 2 ,
A ( u 1 , u 2 , ω ) = a * ( u 1 , ω ) a ( u 2 , ω )
A ( u 1 , u 2 , ω ) = 1 ( 2 π ) 4 W ( 0 ) ( r 1 , r 2 , ω ) P u 1 * ( r 1 , ω ) P u 2 ( r 2 , ω ) d 2 r 1 d 2 r 2 .
W ( r 1 , r 2 , ω ) = A ( u 1 , u 2 , ω ) P u 1 T * ( r 1 , ω ) P u 2 T ( r 2 , ω ) d 2 u 1 d 2 u 2 .
P u T ( r , ω ) = P u ( r , ω ) exp [ ψ u ( 1 ) ( r , ω ) + ψ u ( 2 ) ( r , ω ) + ] ,
W u 1 , u 2 ( r 1 , r 2 , ω ) = P u 1 * ( r 1 , ω ) P u 2 ( r 2 , ω ) exp [ ψ u 2 ( 1 ) * ( r 1 , ω ) + ψ u 1 ( 2 ) * ( r 1 , ω ) + ψ u 2 ( 1 ) ( r 2 , ω ) + ψ u 2 ( 2 ) ( r 2 , ω ) ] .
exp [ ψ ] exp [ ψ + 1 2 ( ψ 2 ψ 2 ) ] .
W u 1 , u 2 ( r 1 , r 2 , ω ) = P u 1 * ( r 1 , ω ) P u 2 ( r 2 , ω ) exp [ 2 E u 1 , u 2 ( 1 ) ( r 1 , r 2 , ω ) + E u 1 , u 2 ( 2 ) ( r 1 , r 2 , ω ) ] ,
E u 1 , u 2 ( 1 ) ( r 1 , r 2 , ω ) = π k 2 0 L d z d 2 κ Φ n ( z , κ ) ,
E u 1 , u 2 ( 2 ) ( r 1 , r 2 , ω ) = 2 π k 2 0 L d z d 2 κ Φ n ( z , κ ) exp [ i ( L z ) ( u 1 u 2 ) κ ] exp [ i ( r 2 r 1 ) κ ] ,
E u 1 , u 2 ( 1 ) ( r 2 r 1 , ω ) = 2 π 2 k 2 0 L d z 0 κ d κ Φ n ( z , κ ) ,
E u 1 , u 2 ( 2 ) ( r 2 r 1 , ω ) = 4 π 2 k 2 0 L d z 0 κ d κ Φ n ( z , κ ) J 0 [ κ ( r 2 r 1 ) ( L z ) ( u 2 u 1 ) ] ,
E u 1 , u 2 ( 1 ) ( r 2 r 1 , ω ) = 2 π 2 k 2 L 0 κ d κ Φ n ( κ ) .
J 0 [ κ ( r 2 r 1 ) z ( u 2 u 1 ) ] = m J m [ κ r 2 r 1 ] J m [ κ z u 2 u 1 ] e i m ( ϕ r ϕ u ) ,
0 L d z J m [ κ z u 2 u 1 ] = 1 κ u 2 u 1 [ 0 κ L u 2 u 1 J 0 ( t ) d t 2 k = 0 m 1 J 2 k + 1 ( κ L u 2 u 1 ) ]
0 L d z J m [ κ z u 2 u 1 ] = 1 κ u 2 , u 1 [ 1 J 0 ( κ L u 2 u 1 ) 2 k = 0 m J 2 k ( κ L u 2 u 1 ) ]
W ( r 1 , r 2 , ω ) = A ( u 1 , u 2 , ω ) P u 1 * ( r 1 , ω ) P u 2 ( r 2 , ω ) × exp [ 2 E u 1 , u 2 ( 1 ) ( r 1 , r 2 , ω ) + E u 1 , u 2 ( 2 ) ( r 1 , r 2 , ω ) ] d 2 u 1 d 2 u 2 .
P u 1 * T ( r 1 ) P u 2 T ( r 1 ) P u 3 * T ( r 2 ) P u 4 T ( r 2 ) = P u 1 * ( r 1 ) P u 2 ( r 2 ) P u 3 * ( r 3 ) P u 4 ( r 4 ) exp [ 4 E ( 1 ) + E 1 , 2 ( 2 ) ( 1 , 1 ) + E 1 , 4 ( 2 ) ( 1 , 2 ) + E 3 , 2 ( 2 ) ( 2 , 1 ) + E 3 , 4 ( 2 ) ( 2 , 2 ) ] + [ E 1 , 3 ( 3 ) ( 1 , 2 ) + E 2 , 4 ( 3 ) * ( 1 , 2 ) ] ,
E a , b ( 3 ) ( 1 , 2 ) = 2 π k 2 0 L d z d 2 κ Φ n ( z , κ ) e i z κ ( u a u b ) e i κ ( r 1 r 2 ) e i z κ 2 k ,
Φ n ( κ ) = 0.33 C n 2 exp [ κ 2 κ m 2 ] ( κ 2 + κ 0 2 ) 11 6 ,
U 0 ( r , ω ) = U 0 exp ( r 2 2 σ 2 ) ,
a ( u , ω ) = σ 2 2 π U 0 exp ( k 2 u 2 σ 2 2 ) .
W 0 ( r 1 , r 2 , ω ) = I 0 exp [ r 1 2 + r 2 2 2 σ I 2 ] exp [ ( r 2 + r 1 ) 2 2 σ μ 2 ] ,
A ( u 1 , u 2 , ω ) = 1 8 π 2 I 0 σ I 2 σ 2 exp [ k 2 ( u 2 + u 1 ) 2 σ I 2 4 ] exp [ k 2 ( u 2 2 u 1 2 ) σ 2 8 ] ,
σ 2 = 1 2 ( 1 4 σ I 2 + 1 2 σ μ 2 ) 1 .
U 0 ( r , ω ) = A 0 J 0 ( k α r ) ,
a ( u , ω ) = A 0 δ ( k u k α ) ,
ϕ u ( 1 ) ( r , ω ) = i k 0 L d z d 2 κ g n ( z , κ ) exp [ i ( L z ) u κ ] exp [ i κ r ] exp [ i ( L z ) 2 k κ 2 ] ,
g n ( z , κ ) = 1 4 π 2 n 1 ( r , z ) exp [ i κ r ] .
ϕ u ( 1 ) ( r , ω ) = 0 .
exp [ ψ u 1 ( 1 ) * ( r 1 , ω ) + ψ u 1 ( 2 ) * ( r 1 , ω ) + ψ u 2 ( 1 ) ( r 2 , ω ) + ψ u 2 ( 2 ) ( r 2 , ω ) ] = exp [ ψ u 1 ( 2 ) * ( r 1 , ω ) + ψ u 2 ( 2 ) ( r 2 , ω ) + 1 2 ( ψ u 1 ( 1 ) * ( r 1 , ω ) + ψ u 2 ( 1 ) ( r 2 , ω ) ) 2 ] ,
exp [ ψ u 1 ( 1 ) * ( r 1 , ω ) + ψ u 1 ( 2 ) * ( r 1 , ω ) + ψ u 2 ( 1 ) ( r 2 , ω ) + ψ u 2 ( 2 ) ( r 2 , ω ) ] = exp [ ψ u 1 ( 2 ) * ( r 1 , ω ) + 1 2 ( ψ u 1 ( 1 ) * ( r 1 , ω ) ) 2 + ψ u 2 ( 2 ) ( r 2 , ω ) + 1 2 ( ψ u 2 ( 1 ) ( r 2 , ω ) ) 2 ] exp [ ψ u 1 ( 1 ) * ( r 1 , ω ) ψ u 2 ( 1 ) ( r 2 , ω ) ] .
2 π Φ n ( z , κ ) δ ( z z ) δ ( 2 ) ( κ κ ) = g n * ( z , κ ) g n ( z , κ ) ,
ψ u 1 ( 1 ) * ( r 1 , ω ) ψ u 2 ( 1 ) ( r 2 , ω ) = 4 π k 2 0 L d z d 2 κ Φ n ( z , κ ) exp [ i z ( u 1 u 2 ) κ + i κ ( r 2 r 1 ) ] .
ψ u ( 1 ) ( r , ω ) ϕ u ( 1 ) ( r , ω ) ,
ψ u ( 2 ) ( r , ω ) ϕ u ( 2 ) ( r , ω ) 1 2 ϕ u ( 1 ) 2 ( r , ω ) .
ϕ u ( 2 ) ( r , ω ) = ψ u ( 2 ) ( r , ω ) + 1 2 ψ u ( 1 ) 2 ( r , ω ) ,
ϕ u ( 2 ) ( r , ω ) = i k 3 2 π 0 L d z 0 z d z d 2 s d 2 κ d 2 κ exp [ i k ( L z ) ] exp [ i k s r 2 ( L z ) ] exp [ i u z ( z L ) ] exp [ i k u ( s r ) ] exp [ i ( κ + κ ) s ] g n ( z , κ ) g n ( z , κ ) exp [ i ( z z ) u κ ] exp [ i ( z z ) 2 k κ 2 ] .
2 π Φ n ( z , κ ) δ ( z z ) δ ( 2 ) ( κ + κ ) = g n ( z , κ ) g n ( z , κ ) ,
ϕ u ( 2 ) ( r , ω ) = 4 k 2 0 L d z 0 κ d κ Φ n ( z , k ) .

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