Researching | Superimposed orbital angular momentum mode of multiple Hankel–Bessel beam propagation in anisotropic non-Kolmogorov turbulence

Journals >Chinese Optics Letters >Volume 14 >Issue 8 >Page 080102 > Article

Chinese Optics Letters
Vol. 14, Issue 8, 080102 (2016)

Superimposed orbital angular momentum mode of multiple Hankel–Bessel beam propagation in anisotropic non-Kolmogorov turbulence

Guanghui Wu^1、*, Chuangming Tong^1、2, Mingjian Cheng³, and Peng Peng¹

Author Affiliations

¹Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China

²State Key Laboratory of Millimeter Waves, Nanjing 210096, China

³School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China

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DOI: 10.3788/COL201614.080102 Cite this Article Set citation alerts

Guanghui Wu, Chuangming Tong, Mingjian Cheng, Peng Peng. Superimposed orbital angular momentum mode of multiple Hankel–Bessel beam propagation in anisotropic non-Kolmogorov turbulence[J]. Chinese Optics Letters, 2016, 14(8): 080102 Copy Citation Text

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Abstract

Mathematical models for the superimposed orbital angular momentum (OAM) mode of multiple Hankel–Bessel (HB) beams in anisotropic non-Kolmogorov turbulence are developed. The effects of anisotropic turbulence and source parameters on the mode detection spectrum of the superimposed OAM mode are analyzed. Anisotropic characteristics of the turbulence in the free atmosphere can enhance the performance of OAM-based communication. The HB beam is a good source for mitigating the turbulence effects due to its nondiffraction and self-focusing properties. Turbulence effects on the superimposed OAM mode can be effectively reduced by the appropriate allocation of OAM modes at the transmitter based on the reciprocal features of the mode cross talk.

U_{free} (ρ, φ, z) = \sum_{m_{0}} t_{m_{0}} E_{free}^{m_{0}} (ρ, φ, z),

(1)

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E_{free}^{m_{0}} (ρ, φ, z) = i^{3 | m_{0} | + 1} | m_{0} |! A_{0} \sqrt{\frac{π}{2 k z}} J_{| m_{0} | / 2} (\frac{k ρ^{2}}{4 z}) \times \exp [i (k z - π | m_{0} | / 4 - π / 4)] \exp (i m_{0} φ),

(2)

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t_{m_{0}} (z) = {(2 π)}^{- 1 / 2} \int_{0}^{2 π} U_{free} (ρ, φ, z) \exp (- i m_{0} φ) ρ d φ .

(3)

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U_{free}^{m_{0}, l_{0}} (ρ, φ, z) = t_{m_{0}} E_{free}^{m_{0}} (ρ, φ, z) + t_{l_{0}} E_{free}^{l_{0}} (ρ, φ, z) .

(4)

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U_{at}^{m_{0}, l_{0}} (ρ, φ, z) = U_{free}^{m_{0}, l_{0}} (ρ, φ, z) \exp [ψ (ρ, φ, z)],

(5)

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U_{at}^{m_{0}, l_{0}} (ρ, φ, z) = \sum_{m = - \infty}^{\infty} t_{m}^{'} (z) \exp (i m φ) .

(6)

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γ_{m}^{'} (z) = {(2 π)}^{- 1} \iint \exp [- i m (φ - φ^{'})] U_{at}^{m_{0}, l_{0}} (ρ, φ, z) U_{at}^{m_{0}, l_{0}, *} (ρ^{'}, φ^{'}, z) d φ d φ^{'} .

(7)

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〈 γ_{m}^{'} (z) 〉 = {(2 π)}^{- 1} \iint \exp [- i m (φ - φ^{'})] U_{free}^{m_{0}, l_{0}} (ρ, φ, z) U_{free}^{m_{0}, l_{0}, *} (ρ^{'}, φ^{'}, z) \times {〈 \exp [ψ_{1} (ρ, φ, z) + ψ_{1}^{*} (ρ^{'}, φ^{'}, z)] 〉}_{at} d φ d φ^{'},

(8)

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{〈 \exp [ψ_{1} (ρ, φ, z) + ψ_{1}^{*} (ρ^{'}, φ^{'}, z)] 〉}_{at} \approx \exp [- (ρ^{2} + ρ^{' 2} - 2 ρ ρ^{'} \cos (φ - φ^{'})) / ρ_{0}^{2}],

(9)

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ρ_{0} = {[π^{2} k^{2} z / 3 \int_{0}^{\infty} κ^{3} ϕ_{n} (κ) d κ]}^{- 1 / 2},

(10)

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ϕ_{n} (κ) = A (α) C_{n}^{2} μ^{2} \frac{\exp {- [μ^{2} (κ_{x}^{2} + κ_{y}^{2}) + κ_{z}^{2}] / κ_{l}^{2}}}{{[μ^{2} (κ_{x}^{2} + κ_{y}^{2}) + κ_{z}^{2} + κ_{0}^{2}]}^{α / 2}},

(11)

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A (α) = Γ (α - 1) \cos (π α / 2) / 4 π^{2},

(12)

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c (α) = {π A (α) Γ (3 / 2 - α / 2) [(3 - α) / 3]}^{1 / (α - 5)},

(13)

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ρ_{0} = {μ^{2 - α} \frac{π^{2} k^{2} z A (α)}{6 (α - 2)} C_{n}^{2} [{\tilde{κ}}_{l}^{2 - α} γ \exp (\frac{κ_{0}^{2}}{κ_{l}^{2}}) Γ (2 - \frac{α}{2}, \frac{κ_{0}^{2}}{κ_{l}^{2}}) - 2 {\tilde{κ}}_{0}^{4 - α}]}^{- 1 / 2},

(14)

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M_{m} (z) = \iint 〈 γ_{m}^{'} (z) 〉 ρ^{'} ρ d ρ^{'} d ρ .

(15)

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\int_{0}^{2 π} \exp [- i n φ_{1} + η \cos (φ_{1} - φ_{2})] d φ_{1} = 2 π \exp (- i n φ_{2}) I_{n} (η),

(16)

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\int_{0}^{\infty} J_{n} (ρ, z) J_{m} (ρ^{'}, z) ρ^{'} d ρ^{'} = {\begin{cases} {| J_{n} (ρ, z) |}^{2}, & if m = n \\ 0, & if m \neq n \end{cases},

(17)

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M_{m} (z) = \int \frac{π}{2 k z} A_{0}^{2} \exp (- \frac{2 ρ^{2}}{ρ_{0}^{2}}) [γ_{m_{0}} (z) {(| m_{0} |!)}^{2} {| J_{| m_{0} | / 2} (\frac{k ρ^{2}}{4 z}) |}^{2} I_{m - m_{0}} (\frac{2 ρ^{2}}{ρ_{0}^{2}}) + γ_{l_{0}} (z) {(| l_{0} |!)}^{2} {| J_{| l_{0} | / 2} (\frac{k ρ^{2}}{4 z}) |}^{2} I_{m - l_{0}} (\frac{2 ρ^{2}}{ρ_{0}^{2}})] ρ d ρ .

(18)

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p_{m} (z) = M_{m} (z) / I,

(19)

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I = \int_{0}^{\infty} \int_{0}^{2 π} {| U_{free}^{m_{0}, l_{0}} (ρ, φ, z) |}^{2} ρ d ρ d ϕ = \sum_{m} M_{m} (z) .

(20)

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References (30)

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Guanghui Wu, Chuangming Tong, Mingjian Cheng, Peng Peng. Superimposed orbital angular momentum mode of multiple Hankel–Bessel beam propagation in anisotropic non-Kolmogorov turbulence[J]. Chinese Optics Letters, 2016, 14(8): 080102

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