Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence

Jing Wang; Shijun Zhu; Zhenhua Li

doi:10.3788/COL201614.080101

Journals >Chinese Optics Letters >Volume 14 >Issue 8 >Page 080101 > Article

Chinese Optics Letters
Vol. 14, Issue 8, 080101 (2016)

Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence

Jing Wang, Shijun Zhu^*, and Zhenhua Li

Author Affiliations

Department of Information Physics and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

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DOI: 10.3788/COL201614.080101 Cite this Article Set citation alerts

Jing Wang, Shijun Zhu, Zhenhua Li. Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence[J]. Chinese Optics Letters, 2016, 14(8): 080101 Copy Citation Text

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Abstract

Analytical formulas for a class of tunable random electromagnetic beams propagating in a turbulent atmosphere through a complex optical system are derived with the help of a tensor method. One finds that the far field intensity distribution is tunable by modulating the source correlation structure function. The on-axis spectral degree of polarization monotonically increases to the same value for different values of order

M

in free space while it returns to the initial value after propagating a sufficient distance in turbulence. Furthermore, it is revealed that the state of polarization is closely determined by the initial correlation structure rather than by the turbulence parameters.

E_{α} (ρ; ω) = \frac{i k \exp (i k z)}{2 π z} \iint \exp [- \frac{i k {(r - ρ)}^{2}}{2 z}] \times E_{α} (r; ω) \exp [Ψ_{α} (r, ρ; ω)] d^{2} r, (α = x, y),

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W_{α β} (\tilde{ρ}; ω) = \frac{k^{2}}{4 π^{2} {[\det (\tilde{B})]}^{1 / 2}} \iint \iint W_{α β} (\tilde{r}; ω) \times \exp [- \frac{i k}{2} ({\tilde{r}}^{T} {\tilde{B}}^{- 1} \tilde{r} - 2 {\tilde{r}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ})] \times K_{α β} (\tilde{r}, \tilde{ρ}; ω) d^{4} \tilde{r},

(2)

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K_{α β} (\tilde{r}, \tilde{ρ}; ω) \approx \exp [- \frac{i k}{2} {\tilde{r}}^{T} {\tilde{Q}}_{α β} \tilde{r}],

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\tilde{B} = (\begin{matrix} z \cdot I & 0 \cdot I \\ 0 \cdot I & - z \cdot I \end{matrix}), {\tilde{Q}}_{α β} = \frac{2}{i k ρ_{0 α β}^{2}} (\begin{matrix} I & - I \\ - I & I \end{matrix}),

(4)

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ρ_{0 α β} = {[\frac{ξ {\tilde{C}}_{n}^{2} k^{2} Γ (ξ - 1) Cos (π ξ / 2)}{2^{ξ} {(ξ - 1) [\det ({\tilde{B}}^{(r)})]}^{- 1 / 4}}]}^{\frac{- 1}{ξ - 2}},

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W_{α β} (\tilde{r}; ω) = \frac{A_{α} A_{β} B_{α β}}{C_{0}} \sum_{m = 1}^{M} \sum_{n = 1}^{N} \frac{{(- 1)}^{m + n - 2}}{\sqrt{m n}} \times (\begin{array}{l} M \\ m \end{array}) (\begin{array}{l} N \\ n \end{array}) \exp [- \frac{i k}{2} {\tilde{r}}^{T} M_{0 α β}^{- 1} \tilde{r}],

(6)

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M_{0 α β}^{- 1} = (\begin{matrix} \frac{{(σ_{α}^{2})}^{- 1}}{2 i k} + \frac{{(δ_{α β}^{2})}^{- 1}}{i k} & \frac{i {(δ_{α β}^{2})}^{- 1}}{k} \\ \frac{i {(δ_{α β}^{2})}^{- 1}}{k} & \frac{{(σ_{β}^{2})}^{- 1}}{2 i k} + \frac{{(δ_{α β}^{2})}^{- 1}}{i k} \end{matrix}),

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\begin{matrix} {(σ_{α}^{2})}^{- 1} = σ_{α}^{- 2} \cdot I, \\ {(σ_{β}^{2})}^{- 1} = σ_{β}^{- 2} \cdot I, \end{matrix} {(δ_{α β}^{2})}^{- 1} = {(\begin{matrix} m δ_{α β x}^{2} & 0 \\ 0 & n δ_{α β y}^{2} \end{matrix})}^{- 1},

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W_{α β} (\tilde{ρ}; ω) = \frac{A_{α} A_{β} B_{α β}}{C_{0}} \sum_{m = 1}^{M} \sum_{n = 1}^{N} \frac{{(- 1)}^{m + n - 2}}{\sqrt{m n}} \times (\begin{array}{l} M \\ m \end{array}) (\begin{array}{l} N \\ n \end{array}) {[\det (\tilde{I} + \tilde{B} M_{0 α β}^{- 1} + \tilde{B} \tilde{Q})]}^{- 1 / 2} \times \exp {- \frac{i k}{2} {\tilde{ρ}}^{T} {[{(M_{0 α β}^{- 1} + \tilde{Q})}^{- 1} + \tilde{B}]}^{- 1} \tilde{ρ}},

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I (ρ; ω) = W_{x x} (ρ, ρ; ω) + W_{y y} (ρ, ρ; ω),

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P (ρ; ω) = \sqrt{1 - \frac{4 Det [\overset{\leftrightarrow}{W} (ρ, ρ; ω)]}{{Tr [\overset{\leftrightarrow}{W} (ρ, ρ; ω)]}^{2}}} .

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θ (ρ; ω) = \frac{1}{2} \arctan [\frac{2 Re W_{x y} (ρ, ρ; ω)}{W_{x x} (ρ, ρ; ω) - W_{y y} (ρ, ρ; ω)}], (- π / 2 \leq θ \leq π / 2) .

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A_{\pm} (ρ; ω) = \frac{1}{\sqrt{2}} {\sqrt{{(W_{x x} - W_{y y})}^{2} + 4 {| W_{x y} |}^{2}} {\pm \sqrt{{(W_{x x} - W_{y y})}^{2} + 4 {[Re W_{x y}]}^{2}}}}^{1 / 2} .

(13)

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ε = A_{-} (ρ, ρ; ω) / A_{+} (ρ, ρ; ω), 0 \leq ε \leq 1 .

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Jing Wang, Shijun Zhu, Zhenhua Li. Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence[J]. Chinese Optics Letters, 2016, 14(8): 080101

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