Researching | Broadband meta-converters for multiple Laguerre-Gaussian modes

Journals >Photonics Research >Volume 9 >Issue 9 >Page 1689 > Article

Photonics Research
Vol. 9, Issue 9, 1689 (2021)

Broadband meta-converters for multiple Laguerre-Gaussian modes

Huade Mao^1、†, Yu-Xuan Ren^2、†, Yue Yu^3、†, Zejie Yu³, Xiankai Sun^3、6, Shuang Zhang^1、4, and Kenneth K. Y. Wong^1、5、*

Author Affiliations

¹Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China

²Institute for Translational Brain Research, Shanghai Medical School, Fudan University, Shanghai 200032, China

³Department of Electronic Engineering, The Chinese University of Hong Kong, Hong Kong, China

⁴Department of Physics, The University of Hong Kong, Hong Kong, China

⁵Advanced Biomedical Instrumentation Centre, Hong Kong Science Park, Hong Kong, China

⁶e-mail: xksun@cuhk.edu.hk

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DOI: 10.1364/PRJ.423344 Cite this Article Set citation alerts

Huade Mao, Yu-Xuan Ren, Yue Yu, Zejie Yu, Xiankai Sun, Shuang Zhang, Kenneth K. Y. Wong. Broadband meta-converters for multiple Laguerre-Gaussian modes[J]. Photonics Research, 2021, 9(9): 1689 Copy Citation Text

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Abstract

Metasurface provides miniaturized devices for integrated optics. Here, we design and realize a meta-converter to transform a plane-wave beam into multiple Laguerre-Gaussian (LG) modes of different orders at various diffraction angles. The metasurface is fabricated with Au nano-antennas, which vary in length and orientation angle for modulation of both the phase and the amplitude of a scattered wave, on a silica substrate. Our error analysis suggests that the metasurface design is robust over a 400 nm wavelength range. This work presents the manipulation of LG beams through controlling both radial and azimuthal orders, which paves the way in expanding the communication channels by one more dimension (i.e., radial order) and demultiplexing different modes.

S_{out} = ⟨ R | Γ (- α) \hat{Q} Γ (α) | L ⟩,

(1)

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{LG}_{p}^{l} = \frac{\sqrt{2} r^{| l |}}{w_{0}^{| l | + 1}} \cdot \exp (- \frac{r^{2}}{w_{0}^{2}}) \cdot L P_{p}^{| l |} (\frac{2 r^{2}}{w_{0}^{2}}) \cdot \exp (i l ϕ),

(2)

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M = \sum_{s} σ_{s} {({LG}_{p}^{l})}_{s} \cdot \exp (i \cdot \sin θ \cdot \vec{g_{s}} \cdot \vec{ρ}) = A_{M} \exp (i ϕ_{M}),

(3)

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C_{p}^{l} = \frac{⟨ {LG}_{p}^{l}, U ⟩}{⟨ {LG}_{p}^{l}, {LG}_{p}^{l} ⟩},

(4)

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p (Z; μ, Σ) = \frac{1}{{(2 π)}^{\frac{d_{i}}{2}} {| Σ |}^{\frac{1}{2}}} \cdot \exp [- \frac{1}{2} {(Z - μ)}^{T} Σ^{- 1} (Z - μ)],

(5)

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ϵ (Z) = \frac{| C (w_{x}, w_{y}; α_{0}) - C (μ_{x}, μ_{y}; α_{0}) |}{0.464},

(6)

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ξ = \int_{μ_{x} - 10}^{μ_{x} + 10} \int_{μ_{y} - 10}^{μ_{y} + 10} ϵ (Z) \cdot p (Z; μ, Σ) d w_{x} d w_{y},

(7)

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H (x, y) = \sum_{m, n} M_{m n} \cdot \frac{\exp [i k R_{m n} (x, y)]}{R_{m n} (x, y)},

(8)

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E_{x} = \vec{e_{x}} \cdot \exp [i (k z - ω t)],

(A1)

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E_{y} = \vec{e_{y}} \cdot \exp [i (k z - ω t)],

(A2)

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| x ⟩ = [\begin{matrix} 1 \\ 0 \end{matrix}],

(A3)

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| y ⟩ = [\begin{matrix} 0 \\ 1 \end{matrix}],

(A4)

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| L ⟩ = \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ i \end{matrix}],

(A5)

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| R ⟩ = \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ - i \end{matrix}],

(A6)

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| Ψ ⟩ = R (- α) \hat{Q} R (α) | L ⟩,

(A7)

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R (α) = (\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}),

(A8)

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\hat{Q} = (\begin{matrix} A_{o} e^{i ϕ_{o} (z)} & 0 \\ 0 & A_{e} e^{i ϕ_{e} (z)} \end{matrix}),

(A9)

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E_{out} = ⟨ R | Ψ (z) ⟩ = \frac{1}{\sqrt{2}} {(1 - i)}^{*} \cdot | Ψ (z) ⟩ = i \cdot \sin [\frac{k d (n_{o} - n_{e})}{2}] e^{i [\frac{k d (n_{o} + n_{e})}{2} + 2 α]},

(A10)

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A_{out} = abs (E_{out}) = \sin [\frac{k d (n_{o} - n_{e})}{2}],

(A11)

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ϕ_{out} = angle (E_{out}) = \frac{k d (n_{o} + n_{e})}{2} + 2 α + \frac{π}{2} .

(A12)

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ξ = \frac{| η_{P} e^{i \times (- 1.129)} - η_{ac} e^{i \times ϕ_{ac}} |}{0.464},

(A13)

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{LG}_{p}^{l} = A_{pl} (x, y) \cdot \exp [i ϕ_{pl} (x, y)],

(B1)

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U = \sum_{p = 0}^{\infty} \sum_{l = - \infty}^{\infty} σ_{p l} \cdot A_{p l} (x, y) \cdot \exp [i ϕ_{p l} (x, y)],

(B2)

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\sum_{p = 0}^{\infty} \sum_{l = - \infty}^{\infty} \sum_{p^{'} = 0}^{\infty} \sum_{l^{'} = - \infty}^{\infty} σ_{p l} σ_{p^{'} l^{'}} A_{p l} A_{p^{'} l^{'}} \cdot \exp {i [ϕ_{p l} (x, y) - ϕ_{p^{'} l^{'}} (x, y)]} = {| C |}^{2} .

(B3)

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U = \sum_{p, l} C_{p}^{l} \cdot L G_{p}^{l},

(C1)

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C_{p}^{l} = \frac{⟨ {LG}_{p}^{l}, U ⟩}{⟨ {LG}_{p}^{l}, {LG}_{p}^{l} ⟩} = \frac{\int {LG}_{p}^{l *} \cdot U d S}{\int {LG}_{p}^{l *} \cdot {LG}_{p}^{l} d S},

(C2)

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\frac{\int {LG}_{p}^{l} * \cdot U d S}{\int {LG}_{p}^{l} * \cdot {LG}_{p}^{l} d S} = {\begin{matrix} 1, & p = p^{'} and l = l^{'} \\ 0, & otherwise \end{matrix} .

(C3)

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{(C_{p}^{l})}_{k} = \frac{\int {({LG}_{p}^{l})}_{k}^{*} \cdot U d S}{\int {({LG}_{p}^{l})}_{k}^{*} \cdot {({LG}_{p}^{l})}_{k} d S} = σ_{k},

(C4)

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{(I_{p}^{l})}_{k} \frac{σ_{k}^{2}}{⟨ U, U ⟩} = \frac{σ_{k}^{2}}{\sum_{i} ⟨ σ_{i} {(L G_{p}^{l})}_{i}, σ_{i} {(L G_{p}^{l})}_{i} ⟩} = \frac{σ_{k}^{2}}{\sum_{i} σ_{i}^{2}},

(C5)

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U = A (x, y) \cdot \exp [i ϕ (x, y)] = A_{U} \cdot \exp (i ϕ_{U}) .

(C6)

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U_{fab} = [A (x, y) + t] \cdot \exp [i ϕ (x, y)],

(C7)

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{(C_{p}^{l})}_{fab} = \frac{⟨ {LG}_{p}^{l}, U_{fab} ⟩}{⟨ {LG}_{p}^{l}, {LG}_{p}^{l} ⟩} \to \int {LG}_{p}^{l} \cdot U_{fab} d S, = \int A_{p}^{l} (A_{U} + t) \cdot \exp [i (ϕ_{U} - ϕ_{p}^{l})] d S = \int A_{p}^{l} A_{U} \cdot \exp [i (ϕ_{U} - ϕ_{p}^{l})] d S + \int t \cdot A_{p}^{l} \cdot \exp [i (ϕ_{U} - ϕ_{p}^{l})] d S .

(C8)

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{(C_{p}^{l})}_{fab} \leftarrow E [{(C_{p}^{l})}_{fab}] = \int A_{p}^{l} A_{U} \cdot \exp [i (ϕ_{U} - ϕ_{p}^{l})] d S + \int t \cdot N (μ_{a}, σ_{a}^{2}) d t \int A_{p}^{l} \cdot \exp [i (ϕ_{U} - ϕ_{p}^{l})] d S = \int A_{p}^{l} A_{U} \cdot \exp [i (ϕ_{U} - ϕ_{p}^{l})] d S + μ_{a} \cdot \int A_{p}^{l} \cdot \exp [i (ϕ_{U} - ϕ_{p}^{l})] d S = C_{p}^{l} + μ_{a} {(C_{p}^{l})}_{phase},

(C9)

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ϵ = \sum_{i} | \frac{ω_{i}}{ω_{0}} - \frac{σ_{i}}{σ_{0}} | .

(C10)

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γ = \sum_{i} σ_{i} .

(C11)

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Huade Mao, Yu-Xuan Ren, Yue Yu, Zejie Yu, Xiankai Sun, Shuang Zhang, Kenneth K. Y. Wong. Broadband meta-converters for multiple Laguerre-Gaussian modes[J]. Photonics Research, 2021, 9(9): 1689

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