Abstract

The spatial coherence radius in moderate-to-strong maritime turbulence is derived on the basis of the modified Rytov approximation. Models are developed to simulate the spiral spectrum of Airy beams propagating through moderate-to-strong maritime turbulence. In the moderate-to-strong irradiance fluctuation region, we analyze the effects of maritime turbulence on the spread of the spiral spectrum of Airy beams in a horizontal propagation path. Results indicate that the increment in the inner-scale significantly increases the received power. By contrast, the outer-scale elicits a negligible effect on the received power if the ratio of the inner-scale to the outer-scale is less than 0.01. The outer-scale affects the received power only if the ratio is greater than 0.01. The performance of a light source is essential for the received power of Airy beams carrying orbital angular momentum (OAM) through moderate-to-strong maritime turbulence. Airy beams with longer wavelengths, smaller OAM numbers, larger radii of the main ring, and smaller diameters of the circular aperture are less affected by maritime turbulence. Autofocusing of Airy beams is beneficial for the propagation of the spiral spectrum in a certain propagation distance. These results contribute to the design of optical communication systems with OAM encoding for moderate-to-strong maritime turbulence.

© 2016 Optical Society of America

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References

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2015 (2)

2014 (3)

2013 (1)

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

2012 (2)

2011 (2)

2010 (2)

2009 (2)

2008 (4)

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
[Crossref] [PubMed]

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

2007 (1)

2005 (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

2004 (2)

2001 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1979 (1)

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

1978 (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

1975 (1)

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Anguita, J. A.

Baazs, N. L.

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Barnett, S.

Baumgartl, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bouchal, Z.

Z. Bouchal and R. Celechovský, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Boyd, R. W.

Broky, J.

Celechovský, R.

Z. Bouchal and R. Celechovský, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Champagne, F. H.

Chen, R.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Chen, Z.

Cheng, M.

Christodoulides, D. N.

Chu, X.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Cizmár, T.

Courtial, J.

Cui, Z.

Dan, W.

Dholakia, K.

Dogariu, A.

Dreyer, G. F.

Efremidis, N. K.

Franke-Arnold, S.

Friehe, C. A.

Gao, C.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gao, J.

Gao, M.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gibson, C. H.

Gibson, G.

Grayshan, K.

Grayshan, K. J.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Guo, H.

Guo, L.

He, Y. T.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Hill, R. J.

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

Hu, Z.

Jiang, Y. S.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

La Rue, J. C.

Lavery, M. P.

Li, F.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Liao, J. L.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Liu, X.

Liu, Y.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Luo, B.

Malik, M.

Mazilu, M.

Mills, M. S.

Mirhosseini, M.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Morris, J. E.

Neifeld, M. A.

O’Sullivan, M. N.

Ou, J.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Padgett, M.

Pas’ko, V.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Prakash, J.

Qu, J.

Robertson, D. J.

Rodenburg, B.

Siviloglou, G. A.

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Tang, H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Tyler, G. A.

Vasic, B. V.

Vasnetsov, M.

Vetelino, F. S.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
[Crossref] [PubMed]

Wang, S. H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wu, G.

Xu, H.

Young, C. Y.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
[Crossref] [PubMed]

Yu, S.

Zhang, J. H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Zhang, L.

Zhang, P.

Zhang, Y.

Zhang, Z.

Zhao, F.

Zhou, G.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Zhu, Y.

Am. J. Phys. (1)

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

New J. Phys. (1)

Z. Bouchal and R. Celechovský, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Opt. Commun. (3)

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Opt. Express (8)

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[Crossref] [PubMed]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17(15), 13236–13245 (2009).
[Crossref] [PubMed]

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref] [PubMed]

Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22(18), 22101–22110 (2014).
[Crossref] [PubMed]

Y. Zhu, L. Zhang, Z. Hu, and Y. Zhang, “Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams,” Opt. Express 23(7), 9137–9146 (2015).
[Crossref] [PubMed]

M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
[Crossref] [PubMed]

H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011).
[Crossref] [PubMed]

Opt. Lett. (5)

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Phys. Rev. Lett. (2)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Radio Sci. (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

Waves Random Complex Media (1)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Other (2)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).

A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products (Academic, 2007).

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Figures (4)

Fig. 1
Fig. 1 Received power of the detected OAM states which equals the original azimuthal index m0 of Airy beams through a maritime environment as a function of the propagation distance with different wavelengths.
Fig. 2
Fig. 2 Received power of the detected OAM states of Airy beams through maritime environment as a function of propagation distance with different azimuthal indices: (a) m= m 0 ; (b) m 0 =1, Δm=4 to 4 .
Fig. 3
Fig. 3 Received power of the detected OAM states of Airy beams through maritime environment as a function of propagation distance with different variables: (a) the radius of the main ring r 0 ; (b) diameter of the circular aperture D.
Fig. 4
Fig. 4 Received power of the detected OAM states of Airy beams through maritime environment with different values of l 0 and L 0 : (a) L 0 =1 m and (b) l 0 =1 cm . Spatial coherence radius with different values of l 0 and L 0 : (c) C n 2 = 10 14 m 2/3 and (d) C n 2 = 10 12 m 2/3 .

Equations (23)

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ϕ n ( κ )=0.033 C n 2 κ 11/3 ,0κ<,
ϕ n, eff ( κ )= ϕ n ( κ )[ f( κ l 0 )g( κ L 0 ) G x ( κ )+ G y ( κ ) ] = ϕ n ( κ )[ G x ( κ, l 0 , L 0 )+ G y ( κ ) ],
f( κ l 0 )=exp( κ 2 κ H 2 )[ 10.061 κ κ H +2.836 ( κ κ H ) 7/6 ], κ H = 3.41 l 0 ,
g( κ L 0 )=1exp( κ 2 κ 0 2 ),
G x ( κ )=exp( κ 2 κ x 2 ),
G y ( κ )= κ 11 /3 ( κ 2 + κ y 2 ) 11 /6 ,
η x 2.61 Q 0 / [ 2.61+ Q 0 ( 1+0.65 d 2 +0.45 σ R 2 Q H 1/6 ) ] ,
η y =3 ( σ R σ P ) 12/5 (1+0.69 σ P 12/5 ),
σ P 2 =3.86 σ 1 2 { ( 1+ 1 Q H 2 ) 11/12 [ sin( 11 6 tan 1 Q H ) 0.051sin( 4 3 tan 1 Q H ) ( 1+ Q H 2 ) 1/4 + 3.052sin( 5 4 tan 1 Q H ) ( 1+ Q H 2 ) 7/ 24 ] 5.581 Q H 5/6 }.
ϕ n,eff ( κ )=0.033 C n 2 { 1 [ κ 2 + κ y 2 ] 11/6 + κ 11/3 [ 10.061 κ κ H +2.836 ( κ κ H ) 7/6 ][ exp( κ 2 κ xH 2 )exp( κ 2 κ x0H 2 ) ] },
A i 0 ( r,φ,z=0 )=Ai[ ±( r 0 r ω 0 ) ]exp[ ±a( r 0 r ω 0 ) ]exp( i m 0 φ ),
A i 0 ( r,φ,z )= ik z ω 0 ( r 0 ω 0 a 2 ) J m 0 ( kr r 0 z )exp( ik r 2 2z + a 3 3 i m 0 φ ),
Ai( r,φ,z )=A i 0 ( r,φ,z )exp[ ψ x ( r,φ,z )+ ψ y ( r,φ,z ) ],
Ai( r,φ,z )= 1 2π m β m ( r,z )exp( imφ ) ,
β m ( r,z )= 1 2π 0 2π Ai( r,φ,z ) exp( imφ )dφ.
| β m (r,z) | 2 = 1 2π 0 2π 0 2π A i 0 ( r,φ,z )A i 0 ( r, φ ,z ) exp[ im( φ φ ) ] × exp[ ψ x ( r,φ,z )+ ψ x ( r, φ ,z )+ ψ y ( r,φ,z )+ ψ y ( r, φ ,z ) ] dφd φ .
exp[ ψ x ( r,φ,z )+ ψ x ( r, φ ,z )+ ψ y ( r,φ,z )+ ψ y ( r, φ ,z ) ] =exp{ π 2 k 2 z 3 [ 2 r 2 2 r 2 cos(φ φ ) ] 0 κ 3 ϕ n,eff ( κ )dκ } =exp{ [ 2 r 2 2 r 2 cos(φ φ ) ]( 1 ρ 0x 2 + 1 ρ 0y 2 ) },
ρ 0x 2 = π 2 k 2 z 3 0.033 C n 2 0 κ 2/3 G x ( κ, l 0 , L 0 )dκ =0.054 k 2 C n 2 z{ Γ( 1 6 )[ ( κ xH 2 ) 1/6 ( κ x0H 2 ) 1/6 ] 0.061 κ H Γ( 2 3 )[ ( κ xH 2 ) 2/3 ( κ x0H 2 ) 2/3 ] + 2.836 κ H 7/6 Γ( 3 4 )[ ( κ xH 2 ) 3/4 ( κ x0H 2 ) 3/4 ] } , ρ 0y 2 = π 2 k 2 z 3 0.033 C n 2 0 κ H κ 2/3 G y ( κ )dκ =0.027 k 2 z C n 2 κ y 11 /3 κ H 4 F 2 1 ( 11 6 ;2;3; κ H 2 κ y 2 ).
exp[ ψ x ( r,φ,z )+ ψ x ( r, φ ,z )+ ψ y ( r,φ,z )+ ψ y ( r, φ ,z ) ] =exp[ 2 r 2 2 r 2 cos( φ φ ) ρ 0xy 2 ],
ρ 0xy = ( 0.054 C n 2 k 2 z ) 1/2 ×{ Γ( 1 6 )[ ( κ xH 2 ) 1/6 ( κ x0H 2 ) 1/6 ] 0.061 κ H Γ( 2 3 )[ ( κ xH 2 ) 2/3 ( κ x0H 2 ) 2/3 ] + 2.836 κ H 7/6 Γ( 3 4 )[ ( κ xH 2 ) 3/4 ( κ x0H 2 ) 3/4 ]+0.5 κ y 11 /3 κ H 4 F 2 1 ( 11 6 ;2;3; κ H 2 κ y 2 ) } 1/2 .
0 2π exp[in φ 1 +ηcos( φ 1 φ 2 )] d φ 1 =2πexp(in φ 2 ) I n (η),
| β m (r,z) | 2 = k 2 z 2 ω 0 2 ( r 0 ω 0 a 2 ) 2 exp( 2 a 3 3 ) | J m 0 ( kr r 0 z ) | 2 exp( 2 r 2 ρ 0xy 2 ) I m m 0 ( 2 r 2 ρ 0xy 2 ).
p m = 0 | β m (r,z) | 2 rdr m= 0 | β m (r,z) | 2 rdr .

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