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Photonics Research
Vol. 9, Issue 11, 2265 (2021)

Control of phase, polarization, and amplitude based on geometric phase in a racemic helix array

Chao Wu^{1、2、3、*}, Quan Li^1、2、4, Zhihui Zhang^1、2, Song Zhao^1、2, and Hongqiang Li^{1、2、3、5}

Author Affiliations

¹School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

²Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, Tongji University, Shanghai 200092, China

³The Institute of Dongguan-Tongji University, Dongguan 523808, China

⁴College of Electronic and Information Engineering, Tongji University, Shanghai 200092, China

⁵e-mail: hqlee@tongji.edu.cn

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DOI: 10.1364/PRJ.440166 Cite this Article Set citation alerts

Chao Wu, Quan Li, Zhihui Zhang, Song Zhao, Hongqiang Li. Control of phase, polarization, and amplitude based on geometric phase in a racemic helix array[J]. Photonics Research, 2021, 9(11): 2265 Copy Citation Text

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Abstract

The Pancharatnam–Berry geometric phase has attracted great interest due to the elegant phase control strategy via geometric transformation of optical elements. The commonly used geometric phase is associated with circular polarization states. Here, we show that by exploiting the geometric phase associated with the two elliptical eigen-polarization states in a racemic metallic helix array, exotic features including full range phase modulation for linear polarization states, diverse polarization conversion, and full complex amplitude modulation can be obtained with rotation of the helices. As a proof of concept, several devices for implementing polarization conversion, vortex beam generating, and lateral dual focusing are built with a racemic helix array in the microwave regime. The calculated and experimental results validate our proposals, which can stimulate various advanced metadevices.

M_{θ_{r}, θ_{l}} = R (- β) (\begin{matrix} e^{i ϕ_{x, x}} & 0 \\ 0 & e^{i ϕ_{y, y}} \end{matrix}) R (β),

(1)

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ϕ_{x, x} = - 2 \arctan [\tan (ψ + α) \tan χ],

(2)

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ϕ_{y, y} = 2 \arctan [\cot (ψ + α) \tan χ],

(3)

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E = M_{θ_{r}, θ_{l}} \cdot (\begin{matrix} 1 \\ - i \end{matrix}) = e^{i (ϕ_{x} - β)} R (- β) (\begin{matrix} 1 \\ e^{i (Δ ϕ - π / 2)} \end{matrix}) .

(4)

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M = R (- β) (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) R (β) = (\begin{matrix} \cos 2 β & \sin 2 β \\ \sin 2 β & - \cos 2 β \end{matrix}) .

(5)

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r_{x, x} = \cos^{2} β e^{i ϕ_{x, x}} + \sin^{2} β e^{i ϕ_{y, y}} .

(6)

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A (x, y, λ_{0}) = A_{1} e^{i \frac{2 π}{λ_{0}} (\sqrt{{(x - x_{1})}^{2} + {(y - y_{1})}^{2} + f_{1}^{2}} - f_{1})} + A_{2} e^{i \frac{2 π}{λ_{0}} (\sqrt{{(x - x_{2})}^{2} + {(y - y_{2})}^{2} + f_{2}^{2}} - f_{2})},

(7)

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A (x, λ_{0}) = A_{1} e^{i \frac{2 π}{λ_{0}} (\sqrt{{(x - x_{1})}^{2} + f_{1}^{2}} - f_{1})} + A_{2} e^{i \frac{2 π}{λ_{0}} (\sqrt{{(x - x_{2})}^{2} + f_{2}^{2}} - f_{2})} .

(8)

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M_{sub} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} e_{x} \\ e_{y} \end{matrix}) = κ (\begin{matrix} e_{x} \\ e_{y} \end{matrix}),

(A1)

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κ_{1, 2} = \frac{r_{x, x} - r_{y, y} \pm \sqrt{{(r_{x, x} + r_{y, y})}^{2} - 4 r_{x, y} r_{y, x}}}{2} .

(A2)

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e_{1} = (\begin{matrix} 1 \\ R_{1} e^{i η_{1}} \end{matrix}), e_{2} = (\begin{matrix} 1 \\ R_{2} e^{i η_{2}} \end{matrix}),

(A3)

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ψ_{1} = \frac{1}{2} \arctan (\frac{2 R_{1} \cos η_{1}}{1 - R_{1}^{2}}), χ_{1} = \frac{1}{2} \arcsin (\frac{2 R_{1} \sin η_{1}}{1 + R_{1}^{2}}), ψ_{2} = \frac{1}{2} \arctan (\frac{2 R_{2} \cos η_{2}}{1 - R_{2}^{2}}), χ_{2} = \frac{1}{2} \arcsin (\frac{2 R_{2} \sin η_{2}}{1 + R_{2}^{2}}) .

(A4)

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(\begin{matrix} 1 \\ 0 \end{matrix}) = A_{R} \cdot R (- ψ - θ_{r}) (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} \cos χ \\ i \sin χ \end{matrix}) + A_{L} \cdot R (ψ - θ_{l}) (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} \cos χ \\ - i \sin χ \end{matrix}),

(B1)

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r_{x} = A_{R} \cdot R (- ψ - θ_{r}) (\begin{matrix} \cos χ \\ i \sin χ \end{matrix}) + A_{L} \cdot R (ψ - θ_{l}) (\begin{matrix} \cos χ \\ - i \sin χ \end{matrix}) .

(B2)

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r_{x} = (\begin{matrix} \cos^{2} β e^{i ϕ_{x, x}} + \sin^{2} β e^{i ϕ_{y, y}} \\ \sin β \cos β e^{i ϕ_{x, x}} - \sin β \cos β e^{i ϕ_{y, y}} \end{matrix}),

(B3)

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e^{i ϕ_{x, x}} = \frac{\cos (ψ + α) \cos χ - i \sin (ψ + α) \sin χ}{\cos (ψ + α) \cos χ + i \sin (ψ + α) \sin χ}, e^{i ϕ_{y, y}} = \frac{\sin (ψ + α) \cos χ + i \cos (ψ + α) \sin χ}{\sin (ψ + α) \cos χ - i \cos (ψ + α) \sin χ},

(B4)

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r_{y} = (\begin{matrix} \sin β \cos β e^{i ϕ_{x, x}} - \sin β \cos b e^{i ϕ_{y, y}} \\ \sin^{2} β e^{i ϕ_{x, x}} + \cos^{2} β e^{i ϕ_{y, y}} \end{matrix}) .

(B5)

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Chao Wu, Quan Li, Zhihui Zhang, Song Zhao, Hongqiang Li. Control of phase, polarization, and amplitude based on geometric phase in a racemic helix array[J]. Photonics Research, 2021, 9(11): 2265

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