Abstract

As a new attractive application of the vortex beams, power coupling of annular vortex beam propagating through a two- Cassegrain-telescope optical system in turbulent atmosphere has been investigated. A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system is established, the general analytical expression of vortex beams with limited apertures and the analytical formulas for the average intensity distribution at the receiver plane are derived. Under the H-V 5/7 turbulence model, the average intensity distribution at the receiver plane and power coupling efficiency of the optical system are numerically calculated, and the influences of the optical topological charge, the laser wavelength, the propagation path and the receiver apertures on the power coupling efficiency are analyzed. These studies reveal that the average intensity distribution at the receiver plane presents a central dark hollow profile, which is suitable for power coupling by the Cassegrain telescope receiver. In the optical system with optimized parameters, power coupling efficiency can keep in high values with the increase of the propagation distance. Under the atmospheric turbulent conditions, great advantages of vortex beam in power coupling of the two-Cassegrain-telescope optical system are shown in comparison with beam without vortex.

©2013 Optical Society of America

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References

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  1. A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
    [Crossref] [PubMed]
  2. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
    [Crossref] [PubMed]
  3. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
    [Crossref] [PubMed]
  4. R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44(7), 2015–2019 (2012).
    [Crossref]
  5. G. Gallatin and B. McMorran, “Propagation of vortex electron wave functions in a magnetic field,” Phys. Rev. A 86(1), 012701 (2012).
    [Crossref]
  6. C. Kamacıoğlu and Y. Baykal, “Generalized expression for optical source fields,” Opt. Laser Technol. 44(6), 1706–1712 (2012).
    [Crossref]
  7. V. P. Lukin, P. A. Konyaev, and V. A. Sennikov, “Beam spreading of vortex beams propagating in turbulent atmosphere,” Appl. Opt. 51(10), C84–C87 (2012).
    [Crossref] [PubMed]
  8. G. Fang, W. Zhu, X. Chen, and J. Pu, “Propagation of partially coherent double-vortex beams in turbulent atmosphere,” Opt. Laser Technol. 44(6), 1780–1785 (2012).
    [Crossref]
  9. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation properties of an optical vortex carried by an Airy beam: experimental implementation,” Opt. Lett. 36(9), 1617–1619 (2011).
    [Crossref] [PubMed]
  10. X. Chu and G. Zhou, “Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path,” Opt. Express 15(12), 7697–7707 (2007).
    [Crossref] [PubMed]
  11. H. Wu, W. Wu, X. Xu, J. Chen, and Y. Zhao, “A new method to improve power efficiencies of optical systems with Cassegrain-telescope receivers,” Opt. Commun. 284(13), 3361–3364 (2011).
    [Crossref]
  12. G. Ren, “Current situation and development trend of high energy laser weapon,” Laser Optoelectronics Prog. 45(9), 62–69 (2008).
    [Crossref]
  13. J. Simpson, “Tactical laser relay mirror demonstration anticipated before 2011,” Inside the Air Force 18, 3-7 (2007).
  14. H. Wu, Z. Sun, and J. Chen, “Improving performances of the optical systems with Cassegrain-telescope receivers by using vortex sources and phase optimizations,” Opt. Laser Technol. 45, 132–136 (2013).
    [Crossref]
  15. H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
    [Crossref]
  16. Z. Mei, D. Zhao, and J. Gu, “Comparison of two approximate methods for hard-edged diffracted flat-topped light beams,” Opt. Commun. 267(1), 58–64 (2006).
    [Crossref]
  17. H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equation of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22(4), 647–653 (2005).
    [Crossref]
  18. G. Zhou, Y. Cai, and X. Chu, “Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere,” Opt. Express 20(9), 9897–9910 (2012).
    [Crossref] [PubMed]
  19. H. T. Eyyuboglu, C. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006).
    [Crossref] [PubMed]
  20. C. Arpali, C. Yazıcıoğlu, H. T. Eyyuboğlu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14(20), 8918–8928 (2006).
    [Crossref] [PubMed]
  21. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
    [Crossref] [PubMed]

2013 (1)

H. Wu, Z. Sun, and J. Chen, “Improving performances of the optical systems with Cassegrain-telescope receivers by using vortex sources and phase optimizations,” Opt. Laser Technol. 45, 132–136 (2013).
[Crossref]

2012 (8)

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
[Crossref]

G. Zhou, Y. Cai, and X. Chu, “Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere,” Opt. Express 20(9), 9897–9910 (2012).
[Crossref] [PubMed]

R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44(7), 2015–2019 (2012).
[Crossref]

G. Gallatin and B. McMorran, “Propagation of vortex electron wave functions in a magnetic field,” Phys. Rev. A 86(1), 012701 (2012).
[Crossref]

C. Kamacıoğlu and Y. Baykal, “Generalized expression for optical source fields,” Opt. Laser Technol. 44(6), 1706–1712 (2012).
[Crossref]

V. P. Lukin, P. A. Konyaev, and V. A. Sennikov, “Beam spreading of vortex beams propagating in turbulent atmosphere,” Appl. Opt. 51(10), C84–C87 (2012).
[Crossref] [PubMed]

G. Fang, W. Zhu, X. Chen, and J. Pu, “Propagation of partially coherent double-vortex beams in turbulent atmosphere,” Opt. Laser Technol. 44(6), 1780–1785 (2012).
[Crossref]

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (2)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

2008 (1)

G. Ren, “Current situation and development trend of high energy laser weapon,” Laser Optoelectronics Prog. 45(9), 62–69 (2008).
[Crossref]

2007 (1)

2006 (3)

2005 (1)

Ambrosio, A.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
[Crossref] [PubMed]

Arpali, C.

Arpali, S. A.

Baykal, Y.

Baykal, Y. K.

Borbone, F.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
[Crossref] [PubMed]

Cai, Y.

Chan, C. T.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Chen, J.

H. Wu, Z. Sun, and J. Chen, “Improving performances of the optical systems with Cassegrain-telescope receivers by using vortex sources and phase optimizations,” Opt. Laser Technol. 45, 132–136 (2013).
[Crossref]

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
[Crossref]

H. Wu, W. Wu, X. Xu, J. Chen, and Y. Zhao, “A new method to improve power efficiencies of optical systems with Cassegrain-telescope receivers,” Opt. Commun. 284(13), 3361–3364 (2011).
[Crossref]

Chen, R.

R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44(7), 2015–2019 (2012).
[Crossref]

Chen, X.

G. Fang, W. Zhu, X. Chen, and J. Pu, “Propagation of partially coherent double-vortex beams in turbulent atmosphere,” Opt. Laser Technol. 44(6), 1780–1785 (2012).
[Crossref]

Chew, K.

R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44(7), 2015–2019 (2012).
[Crossref]

Chu, X.

Dai, H. T.

Eyyuboglu, H. T.

Fang, G.

G. Fang, W. Zhu, X. Chen, and J. Pu, “Propagation of partially coherent double-vortex beams in turbulent atmosphere,” Opt. Laser Technol. 44(6), 1780–1785 (2012).
[Crossref]

Gallatin, G.

G. Gallatin and B. McMorran, “Propagation of vortex electron wave functions in a magnetic field,” Phys. Rev. A 86(1), 012701 (2012).
[Crossref]

Gu, J.

Z. Mei, D. Zhao, and J. Gu, “Comparison of two approximate methods for hard-edged diffracted flat-topped light beams,” Opt. Commun. 267(1), 58–64 (2006).
[Crossref]

Kamacioglu, C.

C. Kamacıoğlu and Y. Baykal, “Generalized expression for optical source fields,” Opt. Laser Technol. 44(6), 1706–1712 (2012).
[Crossref]

Konyaev, P. A.

Lin, Z.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Liu, X.

Liu, Y. J.

Liu, Z.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
[Crossref]

Lukin, V. P.

Luo, D.

Ma, H.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
[Crossref]

Maddalena, P.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
[Crossref] [PubMed]

Mao, H.

Marrucci, L.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
[Crossref] [PubMed]

McMorran, B.

G. Gallatin and B. McMorran, “Propagation of vortex electron wave functions in a magnetic field,” Phys. Rev. A 86(1), 012701 (2012).
[Crossref]

Mei, Z.

Z. Mei, D. Zhao, and J. Gu, “Comparison of two approximate methods for hard-edged diffracted flat-topped light beams,” Opt. Commun. 267(1), 58–64 (2006).
[Crossref]

Ng, J.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Pu, J.

G. Fang, W. Zhu, X. Chen, and J. Pu, “Propagation of partially coherent double-vortex beams in turbulent atmosphere,” Opt. Laser Technol. 44(6), 1780–1785 (2012).
[Crossref]

X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
[Crossref] [PubMed]

Ren, G.

G. Ren, “Current situation and development trend of high energy laser weapon,” Laser Optoelectronics Prog. 45(9), 62–69 (2008).
[Crossref]

Roviello, A.

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
[Crossref] [PubMed]

Schattschneider, P.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Sennikov, V. A.

Sun, X. W.

Sun, Z.

H. Wu, Z. Sun, and J. Chen, “Improving performances of the optical systems with Cassegrain-telescope receivers by using vortex sources and phase optimizations,” Opt. Laser Technol. 45, 132–136 (2013).
[Crossref]

Tian, H.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Verbeeck, J.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Wu, H.

H. Wu, Z. Sun, and J. Chen, “Improving performances of the optical systems with Cassegrain-telescope receivers by using vortex sources and phase optimizations,” Opt. Laser Technol. 45, 132–136 (2013).
[Crossref]

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
[Crossref]

H. Wu, W. Wu, X. Xu, J. Chen, and Y. Zhao, “A new method to improve power efficiencies of optical systems with Cassegrain-telescope receivers,” Opt. Commun. 284(13), 3361–3364 (2011).
[Crossref]

Wu, Q.

R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44(7), 2015–2019 (2012).
[Crossref]

Wu, W.

H. Wu, W. Wu, X. Xu, J. Chen, and Y. Zhao, “A new method to improve power efficiencies of optical systems with Cassegrain-telescope receivers,” Opt. Commun. 284(13), 3361–3364 (2011).
[Crossref]

Xu, X.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
[Crossref]

H. Wu, W. Wu, X. Xu, J. Chen, and Y. Zhao, “A new method to improve power efficiencies of optical systems with Cassegrain-telescope receivers,” Opt. Commun. 284(13), 3361–3364 (2011).
[Crossref]

Yazicioglu, C.

Zhao, D.

Zhao, Y.

H. Wu, W. Wu, X. Xu, J. Chen, and Y. Zhao, “A new method to improve power efficiencies of optical systems with Cassegrain-telescope receivers,” Opt. Commun. 284(13), 3361–3364 (2011).
[Crossref]

Zhong, L.

R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44(7), 2015–2019 (2012).
[Crossref]

Zhou, G.

Zhu, W.

G. Fang, W. Zhu, X. Chen, and J. Pu, “Propagation of partially coherent double-vortex beams in turbulent atmosphere,” Opt. Laser Technol. 44(6), 1780–1785 (2012).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Laser Optoelectronics Prog. (1)

G. Ren, “Current situation and development trend of high energy laser weapon,” Laser Optoelectronics Prog. 45(9), 62–69 (2008).
[Crossref]

Nat Commun (1)

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat Commun 3, 989–991 (2012).
[Crossref] [PubMed]

Nature (1)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
[Crossref] [PubMed]

Opt. Commun. (3)

H. Wu, W. Wu, X. Xu, J. Chen, and Y. Zhao, “A new method to improve power efficiencies of optical systems with Cassegrain-telescope receivers,” Opt. Commun. 284(13), 3361–3364 (2011).
[Crossref]

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285(6), 859–863 (2012).
[Crossref]

Z. Mei, D. Zhao, and J. Gu, “Comparison of two approximate methods for hard-edged diffracted flat-topped light beams,” Opt. Commun. 267(1), 58–64 (2006).
[Crossref]

Opt. Express (5)

Opt. Laser Technol. (4)

C. Kamacıoğlu and Y. Baykal, “Generalized expression for optical source fields,” Opt. Laser Technol. 44(6), 1706–1712 (2012).
[Crossref]

G. Fang, W. Zhu, X. Chen, and J. Pu, “Propagation of partially coherent double-vortex beams in turbulent atmosphere,” Opt. Laser Technol. 44(6), 1780–1785 (2012).
[Crossref]

R. Chen, L. Zhong, Q. Wu, and K. Chew, “Propagation properties and M2 factors of a vortex Airy beam,” Opt. Laser Technol. 44(7), 2015–2019 (2012).
[Crossref]

H. Wu, Z. Sun, and J. Chen, “Improving performances of the optical systems with Cassegrain-telescope receivers by using vortex sources and phase optimizations,” Opt. Laser Technol. 45, 132–136 (2013).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

G. Gallatin and B. McMorran, “Propagation of vortex electron wave functions in a magnetic field,” Phys. Rev. A 86(1), 012701 (2012).
[Crossref]

Phys. Rev. Lett. (1)

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Other (1)

J. Simpson, “Tactical laser relay mirror demonstration anticipated before 2011,” Inside the Air Force 18, 3-7 (2007).

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Figures (8)

Fig. 1
Fig. 1 A typical model of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere.

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Fig. 2
Fig. 2 The optical field of the annular vortex source: (a) intensity distribution, (b) phase distribution.

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Fig. 3
Fig. 3 The average intensity distribution at the receiver plane with the parameters ( l=4, λ=3.8um, σ=0.5, N=10, ω 0 =0.2m, L=10km, ξ= 30 ∘ ).

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Fig. 4
Fig. 4 Evolution of the average intensity distribution with the increase of the optical topological charge ( λ=3.8um, σ=0.5, N=10, ω 0 =0.2m, L=10km, ξ= 30 ∘ ): (a) l≤3, (b) l≥4.

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Fig. 5
Fig. 5 Distribution of the average intensity with different laser wavelengths ( l=4, σ=0.5, N=10, ω 0 =0.2m, L=10km, ξ= 30 ∘ ).

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Fig. 6
Fig. 6 Evolution of the average intensity distribution with the increase of the propagation distance ( l=4, λ=3.8um, σ=0.5, N=10, ω 0 =0.2m, ξ= 30 ∘ ).

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Fig. 7
Fig. 7 Evolution of the average intensity distribution with the increase of the zenith angle ( l=4, λ=3.8um, σ=0.5, N=10, ω 0 =0.2m, L=10km ).

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Fig. 8
Fig. 8 Evolution of the relations between power coupling efficiency and the receiver aperture as different parameter increases: (a) the optical topological charge, (b) the laser wavelength, (c) the propagation distance, (d) the zenith angle.

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Tables (2)

Tables Icon

Table 1 Power Coupling Efficiency of the Optical System with Different Receiver Apertures ( l=4, λ=3.8um, σ=0.5, N=10, ω 0 =0.2m, ξ= 30 ∘ , a=0.15m, b=0.4m ).

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Tables Icon

Table 2 ower Coupling Efficiency of the Optical System with Different Sources ( λ=3.8um, σ=0.5, N=10, ω 0 =0.2m, ξ= 30 ∘ , a=0.15m, b=0.4m, a ′ =0.15m, b ′ =0.6m ).

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Equations (26)

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

E 0 (r,θ,0)=E(r,θ,0)t(r,a,b),
E(r,θ,0)=A(r)exp( ilθ ),
t(r,a,b)={ 1 a≤r≤b 0 else ,
E(r,θ,0)= ∑ n=1 N (−1) n−1 N ( N n ) [ exp( − n r 2 ω 0 2 )−exp( − n r 2 σ ω 0 2 ) ]exp( ilθ ),
t(r,a,b)= ∑ w=1 M B w [ exp( − C w b 2 r 2 )−exp( − C w a 2 r 2 ) ],
E 0 (r,θ,0)= ∑ n=1 N ∑ w=1 M (−1) n−1 B w N ( N n ) [ exp( − C w b 2 r 2 )−exp( − C w a 2 r 2 ) ] ×[ exp( − n r 2 ω 0 2 )−exp( − n r 2 σ ω 0 2 ) ]exp( ilθ ).
<I(R,φ,L)>= k 2 (2πL) 2 ∫ 0 ∞ ∫ 0 ∞ ∫ 0 2π ∫ 0 2π E 0 ( r 1 , θ 1 ,0)exp{ ik 2L [ R 2 + r 1 2 −2R r 1 cos(φ− θ 1 ) ] } × { E 0 ( r 2 , θ 2 ,0)exp{ ik 2L [ R 2 + r 2 2 −2R r 2 cos(φ− θ 2 ) ] } } ∗ ×<exp[ ψ(R,φ, r 1 , θ 1 )+ ψ ∗ (R,φ, r 2 , θ 2 ) ]> r 1 r 2 d r 1 d r 2 d θ 1 d θ 2 ,
<exp[ ψ(R,φ, r 1 , θ 1 )+ ψ ∗ (R,φ, r 2 , θ 2 ) ]>=exp[ −0.5 D ψ ( r 1 , r 2 , θ 1 , θ 2 ) ] =exp{ − 1 ρ 0 2 [ r 1 2 + r 2 2 −2 r 1 r 2 cos( θ 1 − θ 2 ) ] },
<I(R,φ,L)>= k 2 (2πL) 2 ∑ n=1 N ∑ m=1 N ∑ w=1 M ∑ v=1 M (−1) n+m N 2 ( N n )( N m ) B w B v ∗ ∫ 0 ∞ ∫ 0 ∞ ∫ 0 2π ∫ 0 2π exp( − n r 1 2 ω 0 2 )exp( − m r 2 2 ω 0 2 ) ×exp( − C w a 2 r 1 2 )exp( − C v ∗ a 2 r 2 2 )exp{ ik 2L [ r 1 2 −2R r 1 cos(φ− θ 1 )− r 2 2 +2R r 2 cos(φ− θ 2 ) ] } ×exp[ il( θ 1 − θ 2 ) ]exp{ − 1 ρ 0 2 [ r 1 2 −2 r 1 r 2 cos( θ 1 − θ 2 )+ r 2 2 ] } r 1 r 2 d r 1 d r 2 d θ 1 d θ 2 .
exp[ ikRr α cos(φ−θ) ]= ∑ l=−∞ ∞ i l J l ( kRr α )exp[ il( φ−θ ) ],
∫ 0 2π exp[ −ilθ+2αRrcos(θ−φ) ]dθ =2πexp( −ilφ ) I l ( 2αRr ),
I l ( x )= ∑ m=0 ∞ ( x/2 ) 2m+l m!Γ(m+l) ,
J l (x)= ( −i ) l 2π ∫ 0 2π exp( ixcosθ+ilθ ) dθ,
∫ 0 ∞ t m−1 exp( −α t 2 ) J l (βt)dt= α −m/2 2l! ( β 2 4α ) l/2 exp( − β 2 4α )Γ( l+m 2 ) F 1 ( −m+l+2 2 ;l+1; β 2 4α ),
<I(R,L) >= k 2 (2πL) 2 ∑ n=1 N ∑ m=1 N ∑ w=1 M ∑ v=1 M (−1) n+m N 2 ( N n )( N m ) B w B v ∗ [ ( T 1 − T 2 + T 3 − T 4 )−( T ′ 1 − T ′ 2 + T ′ 3 − T ′ 4 ) ],
T 1 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 1,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 1,1 L 2 ) ( H 1,1 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 1,1 L 2 ) ( H ′ 1,1 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,2 L 2 ) ( H ′ 2,2 ) − P 1 ],
T 2 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 1,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 1,1 L 2 ) ( H 1,1 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 1,2 L 2 ) ( H ′ 1,2 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,1 L 2 ) ( H ′ 2,1 ) − P 1 ],
T 3 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 1,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 1,2 L 2 ) ( H 1,2 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 1,2 L 2 ) ( H ′ 1,2 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,1 L 2 ) ( H ′ 2,1 ) − P 1 ],
T 4 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 1,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 1,2 L 2 ) ( H 1,2 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 1,1 L 2 ) ( H ′ 1,1 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,2 L 2 ) ( H ′ 2,2 ) − P 1 ],
T ′ 1 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 2,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 2,1 L 2 ) ( H 2,1 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 1,1 L 2 ) ( H ′ 1,1 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,2 L 2 ) ( H ′ 2,2 ) − P 1 ],
T ′ 2 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 2,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 2,1 L 2 ) ( H 2,1 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 1,2 L 2 ) ( H ′ 1,2 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,1 L 2 ) ( H ′ 2,1 ) − P 1 ],
T ′ 3 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 2,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 2,2 L 2 ) ( H 2,2 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,2 L 2 ) F 1 ( P 2 1 ;s+1; k 2 R 2 4 H ′ 1,2 L 2 ) ( H ′ 1,2 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,1 L 2 ) ( H ′ 2,1 ) − P 1 ],
T ′ 4 = ∑ s=−∞ ∞ ∑ t=0 ∞ [ Γ( P 1 ) ] 2 γ 2t+s+l ( s! ) 2 t!Γ( s+t+l ) ( k 2 R 2 4 L 2 ) s [ exp( − k 2 R 2 4 H 2,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H 2,2 L 2 ) ( H 2,2 ) − P 1 ]× [ exp( − k 2 R 2 4 H ′ 1,1 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 1,1 L 2 ) ( H ′ 1,1 ) − P 1 +exp( − k 2 R 2 4 H ′ 2,2 L 2 ) F 1 ( P 2 ;s+1; k 2 R 2 4 H ′ 2,2 L 2 ) ( H ′ 2,2 ) − P 1 ],
η= 2π ∫ a ′ b ′ <I(R,L) >RdR ∫ a b ∫ 0 2π E 0 (r,θ,0) E 0 ∗ (r,θ,0)rdrdθ ,
C n 2 ( h )=8.2× 10 −56 V (h) 2 h 10 exp( −h / 1000 )+2.7× 10 −16 exp( −h / 1500 )+ C 0 exp( −h/100 ),
V(h)=5+30exp{ [ −( h−9400 ) / 4800 ] 2 },

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