Asymmetric rotating array beams with free movement and revolution

Jia Xu; Zhenglin Liu; Keming Pan; Daomu Zhao

doi:10.3788/COL202220.022602

Journals >Chinese Optics Letters >Volume 20 >Issue 2 >Page 022602 > Article

Chinese Optics Letters
Vol. 20, Issue 2, 022602 (2022)

Asymmetric rotating array beams with free movement and revolution

Jia Xu, Zhenglin Liu, Keming Pan, and Daomu Zhao^*

Author Affiliations

Zhejiang Provincial Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China

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DOI: 10.3788/COL202220.022602 Cite this Article Set citation alerts

Jia Xu, Zhenglin Liu, Keming Pan, Daomu Zhao. Asymmetric rotating array beams with free movement and revolution[J]. Chinese Optics Letters, 2022, 20(2): 022602 Copy Citation Text

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Abstract

We introduce a new class of partially coherent asymmetric array beams. When the beam propagates, the spectral density of each lobe and the corresponding degree of coherence have rotating behavior. Especially, not only can array-like lattices revolve arbitrarily, but also they can move freely by controlling transverse plane shifts. Furthermore, we have generated this kind of beam experimentally, and the experimental phenomena are consistent with the numerical simulation results. Such a rotating beam with free movement and revolution may broaden the way for optical applications. More importantly, it inspires further studies in the field of asymmetric coherence gratings and lattices.

Keywords

asymmetric array beams free movement and revolution partial coherence rotating behavior transverse plane shifts

W^{(0)} (r_{1}, r_{2}; ω) = 〈 U^{*} (r_{1}; ω) U (r_{2}; ω) 〉,

(1)

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W^{(0)} (r_{1}, r_{2}) = \int p (v) H_{0}^{*} (r_{1}, v) H_{0} (r_{2}, v) d^{2} v,

(2)

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p (v_{x}, v_{y}) = \frac{4 π δ_{x} δ_{y}}{C_{x} C_{y}} \sum_{n_{x} = 1}^{N_{x}} \sum_{n_{y} = 1}^{N_{y}} {\exp [- δ_{x}^{2} {(2 π v_{x} + a_{x})}^{2}] \times \exp [- δ_{y}^{2} {(2 π v_{y} + a_{y})}^{2}]},

(3)

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H_{0} (r, v) = τ (r) \exp [- 2 π i v_{x} (x \cos α - y \sin α)] \times \exp [- 2 π i v_{y} (x \sin α + y \cos α)] \times \exp [i u (x \cos θ - y \sin θ) (x \sin θ + y \cos θ)],

(4)

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τ (r) = \exp (- \frac{x^{2}}{4 σ_{x}^{2}}) \exp (- \frac{y^{2}}{4 σ_{y}^{2}}),

(5)

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W^{(0)} (r_{1}, r_{2}) = \frac{1}{C_{x} C_{y}} \sum_{n_{x}}^{N_{x}} \sum_{n_{y} = 1}^{N_{y}} {\exp (- \frac{x_{1}^{2} + x_{2}^{2}}{4 σ_{x}^{2}} - \frac{y_{1}^{2} + y_{2}^{2}}{4 σ_{y}^{2}}) \times \exp [(\frac{\sin 2 α}{4 δ_{x}^{2}} - \frac{\sin 2 α}{4 δ_{y}^{2}}) (x_{2} - x_{1}) (y_{2} - y_{1})] \times \exp [- (\frac{\cos^{2} α}{4 δ_{x}^{2}} + \frac{\sin^{2} α}{4 δ_{y}^{2}}) {(x_{2} - x_{1})}^{2}] \times \exp [- (\frac{\sin^{2} α}{4 δ_{x}^{2}} + \frac{\cos^{2} α}{4 δ_{y}^{2}}) {(y_{2} - y_{1})}^{2}] \times \exp [i (a_{x} \cos α + a_{y} \sin α) (x_{2} - x_{1}) + i (a_{y} \cos α - a_{x} \sin α) (y_{2} - y_{1}) + \frac{i u \sin 2 θ}{2} (x_{2}^{2} - x_{1}^{2} + y_{1}^{2} - y_{2}^{2}) + \frac{i u \cos 2 θ}{2} (x_{2} y_{2} - x_{1} y_{1})]} .

(6)

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W (ρ_{1}, ρ_{2}, z) = \frac{k^{2}}{4 B^{2} C_{x} C_{y}} \sqrt{\frac{1}{a_{x s} a_{y s} T_{1} ε}} \times \sum_{n_{x} = 1}^{N_{x}} \sum_{n_{y} = 1}^{N_{y}} {\exp [- \frac{i k D}{2 B} (ρ_{1}^{2} - ρ_{2}^{2})] \times \exp [\frac{- k^{2} {(ρ_{1 x} - ρ_{2 x})}^{2}}{4 a_{x s} B^{2}} + \frac{- k^{2} {(ρ_{1 y} - ρ_{2 y})}^{2}}{4 a_{y s} B^{2}}] \times \exp (- \frac{Λ_{x}^{2}}{4 T_{1}} + \frac{Ω_{1}^{2}}{4 T_{1}} + \frac{i Λ_{x} Ω_{1}}{2 T_{1}}) \times \exp [\frac{1}{4 ε} {(Ω_{2} + i Λ_{y} + \frac{i Λ_{x} G}{2 T_{1}} + \frac{Ω_{1} G}{2 T_{1}})}^{2}]},

(7)

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a_{x s} = \frac{1}{2 σ_{x}^{2}}; a_{y s} = \frac{1}{2 σ_{y}^{2}}; γ = u \cos 2 θ; β_{1} = \frac{k A}{B} + u \sin 2 θ; β_{2} = \frac{k A}{B} - u \sin 2 θ; a_{n x} = a_{x} \cos α + a_{y} \sin α; a_{n y} = - a_{x} \sin α + a_{y} \cos α; T_{1} = \frac{\cos^{2} α}{4 δ_{x}^{2}} + \frac{\sin^{2} α}{4 δ_{y}^{2}} + \frac{1}{8 σ_{x}^{2}} + \frac{β_{1}^{2}}{4 a_{x s}} + \frac{γ^{2}}{4 a_{y s}}; T_{2} = \frac{\sin^{2} α}{4 δ_{x}^{2}} + \frac{\cos^{2} α}{4 δ_{y}^{2}} + \frac{1}{8 σ_{y}^{2}} + \frac{β_{2}^{2}}{4 a_{y s}} + \frac{γ^{2}}{4 a_{x s}}; G = - (\frac{β_{1}}{a_{x s}} + \frac{β_{2}}{a_{y s}}) \frac{γ}{2} + (\frac{1}{4 δ_{x}^{2}} - \frac{1}{4 δ_{y}^{2}}) \sin 2 α; ε = T_{2} - \frac{G^{2}}{4 T_{1}}; Ω_{1} = \frac{k β_{1} (ρ_{2 x} - ρ_{1 x})}{2 a_{x s} B} + \frac{k γ (ρ_{2 y} - ρ_{1 y})}{2 a_{y s} B}; Ω_{2} = \frac{k γ (ρ_{2 x} - ρ_{1 x})}{2 a_{x s} B} + \frac{k β_{2} (ρ_{2 y} - ρ_{1 y})}{2 a_{y s} B}; Λ_{x} = a_{n x} - \frac{k (ρ_{1 x} + ρ_{2 x})}{2 B}; Λ_{y} = a_{n y} - \frac{k (ρ_{1 y} + ρ_{2 y})}{2 B} .

(8)

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[\begin{array}{l} A & B \\ C & D \end{array}] = [\begin{array}{l} 1 & z \\ 0 & 1 \end{array}] [\begin{matrix} 1 & 0 \\ - \frac{1}{f} & 1 \end{matrix}] = [\begin{matrix} 1 - \frac{z}{f} & z \\ - \frac{1}{f} & 1 \end{matrix}],

(9)

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Jia Xu, Zhenglin Liu, Keming Pan, Daomu Zhao. Asymmetric rotating array beams with free movement and revolution[J]. Chinese Optics Letters, 2022, 20(2): 022602

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