Abstract

The cross-talk among different orbital angular momentum (OAM) modes induced by the turbulent atmosphere is a challenging effect commonly occurring in OAM-based free-space optical (FSO) communication. The aim of this study is to propose a simple method to reduce the crosstalk and demonstrate its effect by analytical derivation and numerical simulation. It is found that the crosstalk is largely reduced by using a focusing mirror. Our results will be useful in free-space optical communication.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

X. Yan, L. Guo, M. Cheng, and J. Li, “Controlling abruptly autofocusing vortex beams to mitigate crosstalk and vortex splitting in free-space optical communication,” Opt. Express 26, 12605–12619 (2018).
[Crossref] [PubMed]

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

2017 (2)

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25, 15286–15298 (2017).
[Crossref] [PubMed]

2016 (3)

J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4, B14–B28 (2016).
[Crossref]

Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel–Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33, 2277–2283 (2016).
[Crossref]

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4, B1–B4 (2016).
[Crossref]

2014 (3)

2013 (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

2010 (1)

2009 (1)

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[Crossref] [PubMed]

2008 (1)

2005 (1)

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

2004 (1)

2002 (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, 4 (2002).

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

2000 (1)

1995 (1)

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Ahmed, N.

Anguita, J. A.

Bao, C.

Barnett, S.

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Cai, Y.

Campos, J.

Chen, M.

Cheng, M.

Chu, X.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

Cottrell, D. M.

Courtial, J.

Dan, W.

Davis, J. A.

Dolinar, S.

Erkmen, B. I.

Fan, C.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

Fedoruk, M. P.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[Crossref] [PubMed]

Feng, X.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

Fickler, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Fink, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Frankearnold, S.

Friese, M.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Fu, S.

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4, B1–B4 (2016).
[Crossref]

Gao, C.

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4, B1–B4 (2016).
[Crossref]

Gao, J.

Gao, S.

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Gibson, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed. (Academic, 2000).

Guo, L.

Handsteiner, J.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

He, H.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Heckenberg, N.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Hu, Z.

Huang, H.

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, and M. Tur, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Huang, Q.

Krenn, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Lavery, M. P. J.

Lei, T.

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Li, J.

Li, N.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

Li, Y.

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel–Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33, 2277–2283 (2016).
[Crossref]

Li, Z.

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Malik, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

McNamara, D. E.

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, 4 (2002).

Neifeld, M. A.

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, 153901 (2005).
[Crossref] [PubMed]

Qiao, C.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Ren, Y.

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, and M. Tur, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Rubenchik, A. M.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[Crossref] [PubMed]

Rubinsztein-Dunlop, H.

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed. (Academic, 2000).

Scheidl, T.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Schmidt, J. D.

J. D. Schmidt, Numerical simulation of optical wave propagation with examples in matlab,(SPIEBellingham, Washington, USA, 2010).
[Crossref]

Sun, R.

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, 4 (2002).

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, 4 (2002).

Tur, M.

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I. Erkmen, S. Dolinar, and M. Tur, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Turitsyn, S. K.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[Crossref] [PubMed]

Ursin, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Vasic, B. V.

Vasnetsov, M.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Wang, F.

Wang, J.

J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4, B14–B28 (2016).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Xie, G.

Xie, Z.

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Yan, X.

Yan, Y.

Yuan, X.

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Yuan, Y.

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Zeilinger, A.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Zhang, P.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

Zhang, Y.

Zhao, F.

Zhu, Y.

Appl. Opt. (1)

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

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

New J. Phys. (1)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Opt. & Laser Technol. (1)

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. & Laser Technol. 98, 7–11 (2018).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Photonics Res. (2)

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4, B1–B4 (2016).
[Crossref]

J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4, B14–B28 (2016).
[Crossref]

Phys. Rev. Lett. (4)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 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, 4 (2002).

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[Crossref] [PubMed]

H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Sci. Reports (1)

Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Reports 7, 42276 (2017).
[Crossref]

Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Other (2)

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed. (Academic, 2000).

J. D. Schmidt, Numerical simulation of optical wave propagation with examples in matlab,(SPIEBellingham, Washington, USA, 2010).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic of the OAM state disturbed by turbulent atmosphere after passing through a focussing mirror.
Fig. 2
Fig. 2 The probability density of OAM mode for different values of C0 with F = (a) 1 km,(b) infinity and (c) 1 km, respectively.
Fig. 3
Fig. 3 The received OAM spectrum for different quantum number. The first row and second row denote there exists a focusing mirror and does not exist a focusing mirror, respectively.
Fig. 4
Fig. 4 The received OAM spectrum varying with different valued of R0 for C 0 = 0 (the first row) and C 0 0 (the second row).
Fig. 5
Fig. 5 The varying of the normalized energy weight of the central OAM mode ( l = l 0 = 3) as a function of R0 for C 0 = 0 and C 0 0.
Fig. 6
Fig. 6 The received intensity and phase (one realization) of the Gaussian vortex beam in turbulent atmosphere for C 0 = 0 and C 0 0.
Fig. 7
Fig. 7 The received OAM spectrum (averaged with 200 realizations) of the Gaussian vortex beam in turbulent atmosphere for C 0 = 0 and C 0 0.

Equations (16)

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E ( r , θ , 0 ) = exp  [ ( 1 + i C 0 ) 2 R 0 2 r 2 + i l 0 θ ] ,
E f r e e ( ρ , φ , z ) = i λ z exp   ( i k z ) × E ( r , θ , 0 ) exp   { i k 2 z [ r 2 + ρ 2 2 r ρ cos   ( φ θ ) ] } r d r d θ ,
0 2 π exp   [ i n φ 1 i x cos   ( φ 2 φ 1 ) ] d φ 1 = 2 π ( i ) n J n ( x ) exp   [ i n φ 2 ] ,
E f r e e ( ρ , φ , z ) = 2 π ( i ) l 0 ( i λ z ) exp   ( i k z ) exp   ( i l 0 φ ) exp   ( i k 2 z ρ 2 ) × π k ρ 8 z α 3 / 2 exp   ( k 2 ρ 2 8 α z 2 ) [ I 0.5 l 0 0.5 ( k 2 ρ 2 8 α z 2 ) I 0.5 l 0 + 0.5 ( k 2 ρ 2 8 α z 2 ) ] ,
E ( ρ , φ , z ) = E f r e e ( ρ , φ , z ) exp  [ ψ ( ρ , φ , z ) ] ,
E ( ρ , φ , z ) = 1 2 π l = + a l ( ρ , z ) exp  ( i l φ ) ,
a l ( ρ , z ) = 1 2 π 0 2 π E ( ρ , φ , z ) exp  ( i l φ ) d φ ,
| a l ( ρ , z ) | 2 = 1 2 π E f r e e * ( ρ , φ 1 , z ) E f r e e ( ρ , φ 2 , z ) × exp [ ψ * ( ρ , φ 1 , z ) + ψ ( ρ , φ 2 , z ) ]   exp [ i l ( φ 2 φ 1 ) ] d φ 1 d φ 2 ,
exp   [ ψ * ( ρ , φ 1 , z ) + ψ ( ρ , φ 2 , z ) ] = exp   [ 2 ρ 2 2 ρ 2 cos   ( φ 2 φ 1 ) ρ 0 2 ]
ρ 0 = [ π 2 3 k 2 z 0 κ 3 ϕ n ( κ ) d κ ] 1 / 2 .
ϕ n ( κ ) = 0.033 C n 2 κ 11 / 3 exp  ( κ 2 / κ m 2 ) ,
ρ 0 = ( 0.5466 k 2 z C n 2 l i 1 / 3 ) 1 / 2 .
| a l ( ρ , z ) | 2 = π 4 k 2 8 z 2 | α 1.5 | 2 ( 1 λ z ) 2 ρ 2 exp   [ k 2 ρ 2 8 α * z 2 k 2 ρ 2 8 α z 2 ] exp   [ ( 2 ρ 0 2 ) ρ 2 ] × I l 0 l [ ( 2 ρ 0 2 ) ρ 2 ] | [ I 0.5 l 0 0.5 ( k 2 ρ 2 8 α z 2 ) I 0.5 l 0 + 0.5 ( k 2 ρ 2 8 α z 2 ) ] | 2 .
ω l = 0 R | a l ( ρ , z ) | 2 ρ d ρ ,
ω l = π 4 k 2 8 λ 2 z 4 | α 1.5 | 2 0 R ρ 3 exp   [ q ρ 2 ] exp   [ t ρ 2 ] × I l 0 l [ t ρ 2 ] | [ I 0.5 l 0 0.5 ( p ρ 2 ) I 0.5 l 0 + 0.5 ( p ρ 2 ) ] | 2 d ρ ,
P l = ω l l = ω l ,

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