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Photonics Research
Vol. 4, Issue 6, 233 (2016)

Theory of high-density low-cross-talk waveguide superlattices

Nan Yang^1、2、3, Huashan Yang^1、2、3, Hengrun Hu^1、2、3, Rui Zhu^1、2、3, Shining Chen^1、4, Hongguo Zhang^1、2、3, and Wei Jiang^{1、2、3、*}

Author Affiliations

¹National Laboratory of Solid State Microstructures, Nanjing 210093, China

²College of Engineering and Applied Sciences, Nanjing University, Nanjing 210093, China

³Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

⁴College of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China

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DOI: 10.1364/PRJ.4.000233 Cite this Article Set citation alerts

Nan Yang, Huashan Yang, Hengrun Hu, Rui Zhu, Shining Chen, Hongguo Zhang, Wei Jiang. Theory of high-density low-cross-talk waveguide superlattices[J]. Photonics Research, 2016, 4(6): 233 Copy Citation Text

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Abstract

Waveguide superlattices, a special type of waveguide arrays, can be designed to achieve very low cross talk at submicrometer/subwavelength pitches. The theoretical framework and design rationales for such waveguide superlattices will be presented in depth. Waveguide sidewall roughness can help to deter the coherent coupling between identical waveguides in nearby supercells, but it also induces random fluctuation of transmission. Statistical behavior of the transmission due to roughness in a waveguide superlattice is systematically treated. Complex transmission characteristics due to spectral oscillation and random roughness will be presented, and their evolution with the superlattice length will be analyzed.Institutions.

Keywords

Integrated optics Multiplexing Optical interconnects Waveguides

\hat{A} | ψ ⟩ = - i \frac{\partial}{\partial z} \hat{B} | ψ ⟩,

(1)

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| ψ ⟩ \equiv (\begin{matrix} E_{t} (x, y, z) \\ H_{t} (x, y, z) \end{matrix}) e^{- i ω t}, E_{t} \equiv (\begin{matrix} E_{x} \\ E_{y} \end{matrix}), H_{t} \equiv (\begin{matrix} H_{x} \\ H_{y} \end{matrix}), \hat{A} = (\begin{matrix} ω ϵ - ω^{- 1} \nabla_{t} \times μ^{- 1} \nabla_{t} \times & 0 \\ 0 & ω μ - ω^{- 1} \nabla_{t} \times ϵ^{- 1} \nabla_{t} \times \end{matrix}), \hat{B} = (\begin{matrix} 0 & - \hat{z} \times \\ \hat{z} \times & 0 \end{matrix}),

(2)

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⟨ ψ | \hat{B} | ψ^{'} ⟩ = \hat{z} \cdot \int E_{t}^{*} \times H_{t}^{'} + E_{t}^{'} \times H_{t}^{*'} d x d y .

(3)

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| ψ ⟩ = \sum_{n} c_{n} | ψ_{n} ⟩ = \sum_{n} c_{n} \exp (i β_{n} z) | n ⟩ .

(4)

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0 \approx \sum_{n} c_{n} \exp (i β_{n} z) ⟨ m | (\hat{A} - {\hat{A}}_{n}) | n ⟩ + \sum_{n} i \frac{\partial c_{n}}{\partial z} \exp (i β_{n} z) ⟨ m | \hat{B} | n ⟩ .

(5)

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{[Δ A]}_{m n} = ⟨ m | (\hat{A} - {\hat{A}}_{n}) | n ⟩ .

(6)

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[B] [e^{i β_{n} z} (- i \partial c_{n} / \partial z)] \approx [Δ A] [e^{i β_{n} z} c_{n}] .

(7)

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u_{m} (z) = ⟨ m | B | ψ (z) ⟩ \approx \sum_{n} ⟨ m | B | n ⟩ c_{n} \exp (i β_{m} z) .

(8)

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{\tilde{c}}_{n} = e^{i Δ β_{n} z} c_{n},

(9)

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- i (\partial / \partial z) [{\tilde{c}}_{n}] \approx [K] [{\tilde{c}}_{n}],

(10)

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| Λ_{n} - K_{n n} | \leq \sum_{k \neq n} | K_{n k} | .

(11)

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| Λ_{n} - Λ_{m} | \geq | K_{n n} - K_{m m} | - | Λ_{n} - K_{n n} | - | Λ_{m} - K_{m m} | .

(12)

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| Λ_{n} - Λ_{m} | \geq | K_{n n} - K_{m m} | - [\sum_{k \neq m} | K_{m k} | + \sum_{k \neq n} | K_{n k} |] .

(13)

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| K_{m m} - K_{n n} | ≫ \sum_{k \neq m} | K_{m k} | + \sum_{k \neq n} | K_{n k} |,

(14)

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[B] [e^{i β_{n} z} (- i \partial c_{n} / \partial z)] \approx ([Δ A] + [R]) [e^{i β_{n} z} c_{n}],

(15)

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- i (\partial / \partial z) [{\tilde{c}}_{n}] \approx [K] [{\tilde{c}}_{n}] + [\tilde{R}] [{\tilde{c}}_{n}],

(16)

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[{\tilde{c}}_{n}] = \exp (i [K] z + i \int_{0}^{z} [\tilde{R}] d z^{'}) [{\tilde{c}}_{n} (0)],

(17)

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Nan Yang, Huashan Yang, Hengrun Hu, Rui Zhu, Shining Chen, Hongguo Zhang, Wei Jiang. Theory of high-density low-cross-talk waveguide superlattices[J]. Photonics Research, 2016, 4(6): 233

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