Coherent Property Evolution of Partially Coherent Elliptical Vortex Beam Propagation Through Turbulence

Guo Zheng

doi:10.3788/LOP202158.0901001

Journals >Laser & Optoelectronics Progress >Volume 58 >Issue 9 >Page 0901001 > Article

Laser & Optoelectronics Progress
Vol. 58, Issue 9, 0901001 (2021)

Coherent Property Evolution of Partially Coherent Elliptical Vortex Beam Propagation Through Turbulence

Guo Zheng^*

Author Affiliations

Jiangsu Automation Research Institute, Lianyungang , Jiangsu 222000, China

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DOI: 10.3788/LOP202158.0901001 Cite this Article Set citation alerts

Guo Zheng. Coherent Property Evolution of Partially Coherent Elliptical Vortex Beam Propagation Through Turbulence[J]. Laser & Optoelectronics Progress, 2021, 58(9): 0901001 Copy Citation Text

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Abstract

Based on the generalized Huygens-Fresnel principle, the spectral density, degree of coherence, and the coherent vortex of partially coherent elliptical vortex beam propagation through anisotropic non-Kolmogorov turbulence are studied. Among them, we focus on the important properties of coherent vortex beams as partially coherent vortex beams. It is found that the elliptic rate, coherence length, and turbulence parameters all have influences on the conservation distance. Moreover, it is easier to separate the newly generated coherent vortex points of partially coherent elliptical vortex beams than that of partially coherent circular vortex beams.

Keywords

anisotropic non-Kolmogorov turbulence atmospheric optics coherence coherent vortex partially coherent elliptical vortex beam

U (x^{'}, y^{'}) = (α x^{'} + i y^{'})^{n} U_{0} (x^{'}, y^{'})

，（1）

U_{0} (x^{'}, y^{'}) = e x p (- \frac{{x^{'}}^{2} + {y^{'}}^{2}}{{w_{0}}^{2}})

，（2）

W ({ρ_{1}}^{'}, {ρ_{2}}^{'}) = 〈U^{*} ({ρ_{1}}^{'}) U ({ρ_{2}}^{'})〉 = (α x_{1}^{'} - i y_{1}^{'})^{n} (α x_{2}^{'} + i y_{2}^{'})^{n} \times e x p (- \frac{{ρ_{1}}^{'}^{2} + {ρ_{2}}^{'}^{2}}{{w_{0}}^{2}}) e x p [- \frac{({ρ_{1}}^{'} - {ρ_{2}}^{'})^{2}}{2 δ^{2}}]

，（3）

W (ρ_{1}, ρ_{2}, z) = {(\frac{k}{2 π z})}^{2} \iint W (ρ_{1}^{'}, ρ_{2}^{'}, 0) e x p [- \frac{i k}{2 z} (ρ_{1} - ρ_{1}^{'})^{2} + \frac{i k}{2 z} (ρ_{2} - ρ_{2}^{'})^{2}] \times {〈e x p [ψ^{*} (ρ_{1}^{'}, ρ_{1}, z) + ψ (ρ_{2}^{'}, ρ_{2}, z)]〉}_{m} d^{2} ρ_{1}^{'} d^{2} ρ_{2}^{'}

，（4）

{〈e x p [ψ^{*} (ρ_{1}^{'}, ρ_{1}, z) + ψ (ρ_{2}^{'}, ρ_{2}, z)]〉}_{m} = e x p \{- M_{T} [(ρ_{1} - ρ_{2})^{2} + (ρ_{1} - ρ_{2}) (ρ_{1}^{'} - ρ_{2}^{'}) + (ρ_{1}^{'} - ρ_{2}^{'})^{2}]\}

，（5）

M_{T} = \frac{π^{2} k^{2} z}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ

。（6）

Φ_{n} (κ) = A (α^{'}) {\tilde{C}}_{n}^{2} ς^{2} {[{κ_{z}}^{2} + ς^{2} ({κ_{x}}^{2} + {κ_{y}}^{2}) + {κ_{0}}^{2}]}^{- α^{'} / 2} e x p [- \frac{ς^{2} ({κ_{x}}^{2} + {κ_{y}}^{2}) + {κ_{z}}^{2}}{{κ_{m}}^{2}}] (0 \leq κ < \infty, 3 < α^{'} < 4)

，（7）

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

，（8）

A (α^{'}) = \frac{1}{4 π^{2}} Γ (α^{'} - 1) c o s (\frac{α^{'} π}{2})

，（9）

M_{T} = \frac{{\tilde{C}}_{n}^{2} A (α^{'}) π^{2} k^{2} z}{6 ς^{2} (α^{'} - 2)} {κ_{0}}^{4 - α^{'}} \{{(\frac{{κ_{0}}^{2}}{κ_{m}^{2}})}^{2 / α^{'} - 2} (α^{'} - 2 + 2 \frac{{κ_{0}}^{2}}{κ_{m}^{2}}) e x p (\frac{{κ_{0}}^{2}}{κ_{m}^{2}}) \times Γ [2 - \frac{α^{'}}{2}, \frac{{κ_{0}}^{2}}{κ_{m}^{2}}] - 2\}

，（10）

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = {(\frac{k}{2 π z})}^{2} e x p [- \frac{i k}{2 z} ({ρ_{1}}^{2} - {ρ_{2}}^{2})] e x p [- M_{T} (ρ_{1} - ρ_{2})^{2}] \times \\ \iint d^{2} u d^{2} v [α^{2} ({u_{x}}^{2} - \frac{{v_{x}}^{2}}{4}) + ({u_{y}}^{2} - \frac{{v_{y}}^{2}}{4}) - i α (u_{x} v_{y} - u_{y} v_{x})] e x p (- \frac{2 u^{2}}{{w_{0}}^{2}}) \times \\ e x p [\frac{i k}{z} (ρ_{1} - ρ_{2}) u] e x p (- Q v^{2} - \frac{i k}{z} u v) \times e x p \{v [\frac{i k}{2 z} (ρ_{1} + ρ_{2}) - M_{T} (ρ_{1} - ρ_{2})]\} \end{array}

。（11）

Q = \frac{1}{2 {w_{0}}^{2}} + \frac{1}{2 δ^{2}} + M_{T}

。（12）

\int_{- \infty}^{\infty} x^{n} e x p [- p x^{2} + 2 q x] d x = n! e x p (\frac{q^{2}}{p}) \sqrt[]{\frac{π}{p}} {(\frac{q}{p})}^{n} \sum_{m = 0}^{[n / 2]} \frac{1}{m! (n - 2 m)!} {(\frac{p}{4 q^{2}})}^{m}

，（13）

W (ρ_{1}, ρ_{2}, z) = {(\frac{k}{2 π z})}^{2} e x p [- \frac{i k}{2 z} ({ρ_{1}}^{2} - {ρ_{2}}^{2})] e x p [- M_{T} (ρ_{1} - ρ_{2})^{2}] \times [α^{2} I_{1} + I_{2} - α (I_{3} - I_{4})]

。（14）

\begin{array}{l} I_{1} = \frac{π^{2}}{Q A} C_{x} C_{y} e x p (\frac{{D_{x}}^{2} + {D_{y}}^{2}}{A}) (\frac{1}{2 A} + \frac{{D_{x}}^{2}}{A^{2}}) - \frac{π^{2} {w_{0}}^{2}}{8 B} e x p [- \frac{k^{2} {w_{0}}^{2}}{8 z^{2}} (ρ_{1} - ρ_{2})^{2}] \times \\ e x p (\frac{{E_{x}}^{2} + {E_{y}}^{2}}{B}) (\frac{{E_{x}}^{2}}{B^{2}} + \frac{1}{2 B}) \end{array}

，（15）

\begin{array}{l} I_{2} = \frac{π^{2}}{Q A} C_{x} C_{y} e x p (\frac{{D_{x}}^{2} + {D_{y}}^{2}}{A}) (\frac{1}{2 A} + \frac{{D_{y}}^{2}}{A^{2}}) - \frac{π^{2} {w_{0}}^{2}}{8 B} e x p [- \frac{k^{2} {w_{0}}^{2}}{8 z^{2}} (ρ_{1} - ρ_{2})^{2}] \times \\ e x p (\frac{{E_{x}}^{2} + {E_{y}}^{2}}{B}) (\frac{{E_{y}}^{2}}{B^{2}} + \frac{1}{2 B}) \end{array}

，（16）

I_{3} = \frac{i π^{2} w_{0}}{\sqrt[]{2 Q A B}} C_{x} \frac{D_{x} E_{y}}{A B} e x p [- \frac{k^{2} {w_{0}}^{2}}{8 z^{2}} (y_{1} - y_{2})^{2}] e x p (\frac{{D_{x}}^{2}}{A} + \frac{{E_{y}}^{2}}{B})

，（17）

I_{4} = \frac{i π^{2} w_{0}}{\sqrt[]{2 Q A B}} C_{y} \frac{D_{y} E_{x}}{A B} e x p [- \frac{k^{2} {w_{0}}^{2}}{8 z^{2}} (x_{1} - x_{2})^{2}] e x p (\frac{{D_{y}}^{2}}{A} + \frac{{E_{x}}^{2}}{B})

，（18）

A = \frac{2}{{w_{0}}^{2}} + \frac{k^{2}}{4 Q z^{2}}

，（19）

B = Q + \frac{k^{2} {w_{0}}^{2}}{8 z^{2}}

，（20）

C_{x} = e x p \{\frac{1}{4 Q} [\frac{i k}{2 z} (x_{1} + x_{2}) - M_{T} (x_{1} - x_{2} {)]}^{2}\}

，（21）

D_{x} = \frac{1}{2} [\frac{i k}{z} (x_{1} - x_{2}) + \frac{k^{2}}{4 Q z^{2}} (x_{1} + x_{2}) + \frac{i k}{2 Q z} M_{T} (x_{1} - x_{2})]

，（22）

E_{x} = \frac{1}{2} [\frac{i k}{2 z} (x_{1} + x_{2}) - M_{T} (x_{1} - x_{2}) + \frac{k^{2} {w_{0}}^{2}}{4 z^{2}} (x_{1} - x_{2})]

，（23）

S (ρ, z) = W (ρ, ρ, z)

，（24）

μ (ρ_{1}, ρ_{2}, z) = \frac{W (ρ_{1}, ρ_{2}, z)}{\sqrt[]{W (ρ_{1}, ρ_{1}, z) W (ρ_{2}, ρ_{2}, z)}}

。（25）

R e [μ (ρ_{1}, ρ_{2}, z)] = 0

，（26）

I m [μ (ρ_{1}, ρ_{2}, z)] = 0

。（27）

Guo Zheng. Coherent Property Evolution of Partially Coherent Elliptical Vortex Beam Propagation Through Turbulence[J]. Laser & Optoelectronics Progress, 2021, 58(9): 0901001

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