Abstract

We model the effects of weak fluctuations on the probability densities and normalized powers of vortex models for the Bessel–Gauss photon beam with fractional topological charge in the paraxial non-Kolmogorov turbulence channel. We find that probability density of signal vortex models is a function of deviation from the center of the photon beam, and the farther away from the beam center it is, the smaller the probability density is. For fractional topological charge, the average probability densities of signal/crosstalk vortex modes oscillate along the beam radius except the half-integer order. As the beam waist of the photon source grows, the average probability density of signal and crosstalk vortex modes grow together. Moreover, the peak of the average probability density of crosstalk vortex modes shifts outward from the beam center as the beam waist gets larger. The results also show that the smaller index of non-Kolmogorov turbulence and the smaller generalized refractive-index structure parameter may lead to the higher average probability densities of signal vortex modes and lower average probability densities of crosstalk vortex modes. Lower-coherence radius or beam waist can give rise to less reduction of the normalized powers of the signal vortex modes, which is opposite to the normalized powers of crosstalk vortex modes.

© 2016 Chinese Laser Press

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References

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  1. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
    [Crossref]
  2. W. N. Plick, M. Krenn, R. Fickler, and S. Ramelow, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013).
    [Crossref]
  3. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
    [Crossref]
  4. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15, 6300–6313 (2007).
    [Crossref]
  5. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, and S. M. Barnett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
    [Crossref]
  6. S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21, 1192–1197 (2004).
    [Crossref]
  7. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
    [Crossref]
  8. E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Processes Geophys. 13, 297–301 (2006).
    [Crossref]
  9. J. C. Gutirrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
    [Crossref]
  10. F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type,” Opt. Lett. 36, 606–608 (2011).
    [Crossref]
  11. F. G. Mitri, “High-order Bessel nonvortex beam of fractional type,” Phys. Rev. A 85, 025801 (2012).
    [Crossref]
  12. F. G. Mitri, “High-order Bessel non-vortex beam of fractional type: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67, 135 (2013).
    [Crossref]
  13. J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
    [Crossref]
  14. J. Gao, Y. Zhang, W. Dan, and Z. Hu, “Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams,” Opt. Express 23, 17024–17034 (2015).
    [Crossref]
  15. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2th ed. (SPIE, 2005).
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    [Crossref]
  17. Y. Jiang, S. Wang, J. Zhang, and J. Ou, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
    [Crossref]
  18. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
    [Crossref]
  19. X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
    [Crossref]
  20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).
  21. 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, 2414–2429 (2008).
    [Crossref]

2015 (1)

2013 (3)

W. N. Plick, M. Krenn, R. Fickler, and S. Ramelow, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67, 135 (2013).
[Crossref]

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

2012 (2)

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
[Crossref]

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type,” Phys. Rev. A 85, 025801 (2012).
[Crossref]

2011 (1)

2008 (5)

2007 (2)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15, 6300–6313 (2007).
[Crossref]

2006 (1)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Processes Geophys. 13, 297–301 (2006).
[Crossref]

2004 (3)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21, 1192–1197 (2004).
[Crossref]

2000 (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2th ed. (SPIE, 2005).

Anguita, J. A.

Barnett, S. M.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, and S. M. Barnett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

Branover, H.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Processes Geophys. 13, 297–301 (2006).
[Crossref]

Burnham, D.

Dan, W.

Fickler, R.

W. N. Plick, M. Krenn, R. Fickler, and S. Ramelow, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013).
[Crossref]

Flossmann, F.

Franke-Arnold, S.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, and S. M. Barnett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Gao, J.

Golbraikh, E.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Processes Geophys. 13, 297–301 (2006).
[Crossref]

Götte, J. B.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, and S. M. Barnett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15, 6300–6313 (2007).
[Crossref]

Gutirrez-Vega, J. C.

J. C. Gutirrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

Hu, Z.

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[Crossref]

Jiang, Y.

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

Kopeika, N. S.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Processes Geophys. 13, 297–301 (2006).
[Crossref]

Krenn, M.

W. N. Plick, M. Krenn, R. Fickler, and S. Ramelow, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013).
[Crossref]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[Crossref]

Lopez-Mariscal, C.

J. C. Gutirrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

López-Mariscal, C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

C. López-Mariscal, D. Burnham, D. Rudd, and D. McGloin, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16, 11411–11422 (2008).
[Crossref]

McGloin, D.

Mitri, F. G.

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67, 135 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type,” Phys. Rev. A 85, 025801 (2012).
[Crossref]

F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type,” Opt. Lett. 36, 606–608 (2011).
[Crossref]

Neifeld, M. A.

O’Holleran, K.

Ou, J.

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

Padgett, M. J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2th ed. (SPIE, 2005).

Plick, W. N.

W. N. Plick, M. Krenn, R. Fickler, and S. Ramelow, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013).
[Crossref]

Preece, D.

Ramelow, S.

W. N. Plick, M. Krenn, R. Fickler, and S. Ramelow, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013).
[Crossref]

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[Crossref]

Rudd, D.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Sheng, X.

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
[Crossref]

Tao, S. H.

Vasic, B. V.

Wang, S.

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

Wang, X.

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
[Crossref]

Wang, Z.

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
[Crossref]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Yuan, X.

Zambrini, R.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Zhang, J.

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

Zhang, Y.

J. Gao, Y. Zhang, W. Dan, and Z. Hu, “Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams,” Opt. Express 23, 17024–17034 (2015).
[Crossref]

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
[Crossref]

Zhu, Y.

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
[Crossref]

Zilberman, A.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Processes Geophys. 13, 297–301 (2006).
[Crossref]

Appl. Opt. (1)

Eur. Phys. J. D (1)

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67, 135 (2013).
[Crossref]

J. Mod. Opt. (2)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[Crossref]

J. Opt. A (3)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

J. C. Gutirrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

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

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Nonlinear Processes Geophys. (1)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Processes Geophys. 13, 297–301 (2006).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (4)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43, 121–127 (2012).
[Crossref]

Phys. Rev. A (2)

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type,” Phys. Rev. A 85, 025801 (2012).
[Crossref]

W. N. Plick, M. Krenn, R. Fickler, and S. Ramelow, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013).
[Crossref]

Other (2)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2th ed. (SPIE, 2005).

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Figures (8)

Fig. 1.
Fig. 1. Distribution of vortex models with fractional topological charge γ when l0 changes. Every subplot has a peak at the nearest integer to γ. When γ=3.5 (half-integer), this results in two peaks of equal height at the two neighboring integers. The spread in the distribution is determined by the fractional value γ.
Fig. 2.
Fig. 2. Average mode probability Dl(r,z) of FoBG beam along the direction of the beam radius r for different values of Δl.
Fig. 3.
Fig. 3. Average probability densities Dl(r,z) of vortex modes for fractional FoBG beam along the direction of the beam radius r with different values of γ. (a) Δl=0, mode probability; (b) Δl=1, crosstalk probability.
Fig. 4.
Fig. 4. Average probability densities Dl(r,z) of vortex modes for fractional FoBG beam along the direction of the beam radius r with different values of α. (a) Δl=0, mode probability; (b) Δl=1, crosstalk probability.
Fig. 5.
Fig. 5. Average probability densities Dl(r,z) of vortex modes for fractional FoBG beam along the direction of the beam radius r with different values of Cn2. (a) Δl=0, mode probability; (b) Δl=1, crosstalk probability.
Fig. 6.
Fig. 6. Average probability densities Dl(r,z) of vortex modes for fractional FoBG beam along the direction of the beam radius r with different values of beam waist w0. (a) Δl=0, mode probability; (b) Δl=1, crosstalk probability.
Fig. 7.
Fig. 7. Normalized powers Lz(l) of fractional vortex modes for FoBG beam along the direction of the transmission distance z with different values of Δl.
Fig. 8.
Fig. 8. Normalized powers Lz(l) of fractional vortex modes for FoBG beam along coherence radius r with different values of beam waist w0. (a) Δl=0, signal normalized powers; (b) Δl=1, crosstalk normalized powers.

Equations (14)

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

FoBG(r,ϕ,z)=FoBGfree(r,ϕ,z)exp[ψ(r,ϕ,z)],
FoBGfree(r,ϕ,z)=1μ(z)(±i)γsin(±πγ)π×exp[ikr2z2kμ(z)r2μ(z)w02]×l0=i|l0|±γl0J|l0|[krrμ(z)]exp(il0ϕ),
FoBG(r,ϕ,z)=|l0|=βl(r,z)exp(ilϕ),
βl(r,z)=12π02πFoBG(r,ϕ,z)exp(ilϕ)dϕ.
Dl(r,z)=|βl(r,z)|2at=(12π)202π02πFoBGfree(r,ϕ,z)×FoBGfree*(r,ϕ,z)×exp[ψ(r,ϕ,z)+ψ*(r,ϕ,z)]at×exp[il(ϕϕ)]dϕdϕ.
exp[ψ(r,ϕ,z)+ψ*(r,ϕ,z)]at=exp[4π2k2z01dξ0κΦn(κ)(1J0(κ|ξ(rr)|))dκ],
Φn(κ)=A(α)Cn2exp(κ2/κm2)(κ2+κ02)α/2,3<α<4,
exp[ψ(r,ϕ,z)+ψ*(r,ϕ,z)]atexp[|rr|2ρ02]=exp[r2+r22rrcos(ϕϕ)ρ02],
ρ0={2Γ[(3α)/2](α1)π1/2k2Γ(1α/2)Cn2z}1/(α2),3<α<4,
02πexp[inϕ1+ηcos(ϕ1ϕ2)]dϕ1=2πexp(inϕ2)In(η),
m=n=Jm(x)Jn(x)={m=|Jm(x)|2,m=n0,mn,
Dl(r,z)=sin2(πγ)w02π2w2(z)l0=1(±γl0)2|J|l0|(krrμ(z))|2×exp[(1w2(z)+1ρ02)2r2]Ill0(2r2ρ02),
Lz(l)=l=lBPll=BPl,
Pl=0Dl(r,z)rdr=sin2(πγ)w02π2w2(z)l0=1(±γl0)2×0|J|l0|(krrμ(z))|2exp[(1w2(z)1ρ02)2r2]×Ill0(2r2ρ02)rdr.

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