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互易波导模式耦合理论

陈云天 王经纬 陈伟锦 徐竞

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互易波导模式耦合理论

陈云天, 王经纬, 陈伟锦, 徐竞

Reciprocal waveguide coupled mode theory

Chen Yun-Tian, Wang Jing-Wei, Chen Wei-Jin, Xu Jing
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  • 波导中模式耦合是一种普遍的现象. 在光纤通信中不同导模之间的耦合会引起串扰, 导模和辐射模的耦合会降低导模的功率. 另一方面, 利用模式耦合现象能设计出具有特定功能的耦合器和分束器等光学器件. 模式耦合在光纤通信和光纤传感中也具有广泛应用. 因此, 分析研究波导模式如何耦合具有重要的应用价值. 模式耦合理论是研究波导中模式耦合的常用方法, 不仅提供了一种直观的物理图景来描述光学模式如何杂化, 而且还对相关模式如何杂化给出定量评估. 近年来, 以宇称时间对称性结构为代表的非厄米波导成为研究热点, 但传统模式耦合理论在这种情况下不再适用. 本文简述了模式耦合理论的发展历史, 详细介绍了构造互易波导模式耦合理论的关键概念和方法, 进一步回顾了在波导模式耦合理论方面的一系列代表性工作, 尤其是手征对称模式耦合理论以及广义模式耦合理论, 总结了这些模式耦合理论和传统模式耦合理论之间的联系, 最后简单介绍了它们在宇称时间对称波导及各向异性波导中的应用.
    Mode coupling is a common phenomenon in waveguides. The mode coupling among different guided modes in fiber-optic communication can cause crosstalk, and the mode coupling of guided mode and radiated mode can reduce the power of the guided mode. Application of mode coupling can guide the design of optical devices such as couplers and beam splitters with specific functions, which have been widely used in fiber optic communication and fiber sensing. So it is important to analyze how waveguide modes are coupled. The coupled-mode theory is a common method of studying mode coupling in waveguides. It provides not only an intuitive picture of how the photonic modes are hybridized, but also a quantitative assessment of how the hybridization among those relevant modes evolves. In recent years, non-Hermitian waveguides, represented by parity-time symmetrical structures, have become a research hotspot. However the conventional coupled-mode theory no longer works in this case. In this review, we briefly summarize the development history of coupled-mode theory and introduce the representative work in reciprocal waveguide coupled-mode theory in detail. Then the relationship among several coupled-mode theories is analyzed and their applications are briefly introduced.
      通信作者: 徐竞, jing_xu@hust.edu.cn
    • 基金项目: 国家级-国家自然科学基金(11874026,61775063, 61735006)
      Corresponding author: Xu Jing, jing_xu@hust.edu.cn
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  • 图 1  前向和后向传播模式之间对称关系 (a) 手征对称; (b) 时间反演对称; (c) 宇称对称

    Fig. 1.  Symmetry relations between the forward and backward propagating modes: (a) Chiral symmetry; (b) time reversal symmetry; (c) parity symmetry.

    图 2  有效折射率$n_{\rm{eff}}$实部与$\Delta {{\varepsilon}} $ (a) 纤芯1 相对介电常数为${{\varepsilon}}= {{\varepsilon}}_0 + \Delta {{\varepsilon}} $, 纤芯2 相对介电常数为${{\varepsilon}}= {{\varepsilon}}_0 + \Delta {{\varepsilon}} $; (b) 纤芯1 相对介电常数为${{\varepsilon}}= {{\varepsilon}}_0 + \Delta {{\varepsilon}} $, 纤芯2 相对介电常数为${{\varepsilon}}= {{\varepsilon}}_0 - \Delta {{\varepsilon}} $

    Fig. 2.  Real part of effective mode indices $n_{\rm{eff}}$ versus $\Delta {{\varepsilon}} $: (a) ${{\varepsilon}}= {{\varepsilon}}_0 + \Delta {{\varepsilon}} $ in core layer 1 and ${{\varepsilon}}= {{\varepsilon}}_0 + \Delta {{\varepsilon}} $ in core layer 2; (b) ${{\varepsilon}}= {{\varepsilon}}_0 + \Delta {{\varepsilon}} $ in core layer 1 and ${{\varepsilon}}= {{\varepsilon}}_0 - \Delta {{\varepsilon}} $ in core layer 2.

    图 3  有效折射率$n_{\rm{eff}}$实部与虚部及$\Delta {{\varepsilon}} $ (a), (c) 有效折射率$n_{\rm{eff}}$实部; (b), (d) 有效折射率$n_{\rm{eff}}$虚部

    Fig. 3.  Real part and imaginary part of effective mode indices $n_{\rm{eff}}$ versus $\Delta {{\varepsilon}} $: (a), (c) Real part of effective mode indices $n_{\rm{eff}}$; (b), (d) imaginary part of effective mode indices $n_{\rm{eff}}$.

    图 4  各向异性波导 (a) 椭圆波导示意图; (b) 使用有限元法进行全波计算得到的有效折射率$n_{\rm{eff}}$实部(红线)和虚部(黑线)与${{\varepsilon}}_{\rm{r}}$函数关系; (c)有效折射率$n_{\rm{eff}}$ 实部; (d) 有效折射率$n_{\rm{eff}}$ 虚部; (e)—(h) 图(b)中黑色实心圆标记点前向/后向传播模式$\phi$$E_x$分量的实部/虚部

    Fig. 4.  Anisotropic waveguide: (a) The schematic of elliptical waveguide; (b) the real (red line) and imaginary (black line) part of effective modal indices, calculated from fullwave simulation using finite element method, as a function of ${{\varepsilon}}_{\rm{r}}$; (c) real part of effective mode indices $n_{\rm{eff}}$; (d) imaginary part of effective mode indices $n_{\rm{eff}}$; (e)−(h) the real/imaginary part of $E_x$ obtained from fullwave simulation is shown for the modes $\phi$

    表 1  $\beta_i > 0$时, 互易波导中原始场和伴随场之间的对称关系

    Table 1.  Symmetric relation of original field and adjoint field in the reciprocal waveguides with $\beta_i > 0$

    $\beta_i$对应的模式$-\beta_i$对应的模式
    $({\bar{{L}}}, {\bar{{B}}})$$\left[\beta_i, {{\phi}}_i\right]$$\left[-\beta_i, {{\psi}}_i\right]$
    $({\bar{{L}}}^{\rm{a}}, {\bar{{B}}}^{\rm{a}})$$\left[\beta_i, {{\psi}}_i\right]$$\left[-\beta_i, {{\phi}}_i\right]$
    下载: 导出CSV

    表 2  互易波导中原始场和伴随场之间的对称关系

    Table 2.  Symmetry relations of original field and adjoint field in the reciprocal waveguides.

    对称关系算符对称性关系约束条件
    手征对称${\sigma}$${{{\psi}}}_i({{r}}) = {\bar{\sigma}}{{{\phi}}}_i({{r}})$${{{\varepsilon}}}_{\rm r}^{zt} = {{{\varepsilon}}}_{\rm r}^{tz} = 0$, ${{{\mu}}}_{\rm r}^{zt} = {{{\mu}}}_{\rm r}^{tz} = 0$和${\bar{ \chi}} = 0$
    时间反演对称${\cal{T}}$${{{\psi}}}_i({{r}}) = {\bar{\sigma}}({{{\phi}}}_i({{r}}))^*$${\bar{{{\varepsilon}}}}_{\rm r}$, ${\bar{{{\mu}}}}_{\rm r}$和${\bar{ \chi}}$是实数
    宇称对称${\cal{P}}$${{{\psi}}}_i({{r}}) = {\bar{\sigma}}{{{\phi}}}_i(-{{r}})$${\bar{{{\varepsilon}}}}_{\rm r}({{r}}) = {\bar{{{\varepsilon}}}}_{\rm r}(-{{r}})$, ${\bar{{{\mu}}}}_{\rm r}({{r}}) = {\bar{{{\mu}}}}_{\rm r}(-{{r}})$和 ${\bar{ \chi}}({{r}}) = -{\bar{ \chi}}(-{{r}})$
    下载: 导出CSV

    表 3  CCMT, GCMT和GCMF之间比较

    Table 3.  Comparison between CCMT, GCMT and GCMF.

    模式耦合理论传统模式耦合理论(CCMT)手征对称模式耦合理论(GCMT)广义模式耦合理论(GCMF)
    耦合模式展开式形式$\varPhi =\displaystyle \sum a_i\phi _i$$\varPhi = \displaystyle\sum a_i\phi _i$$\varPhi = \displaystyle\sum a_i\phi _i ^+ +b_i \psi _i ^-$
    守恒量光功率守恒
    $\nabla \left({{{{E}}} _1 \times {{{H}}} _2 ^{\ast}} + {{{E}}} _2 ^{\ast} \times {{{H}}}_1\right) = 0$
    作用量守恒
    $\nabla \left({{{{E}}} _1 \times {{{H}}} _2 } + {{{E}}} _2 \times {{{H}}}_1\right) = 0$
    作用量守恒
    $\nabla \left({{{{E}}} _1 \times {{{H}}} _2 } + {{{E}}} _2 \times {{{H}}}_1\right) = 0$
    测试函数$\phi _j ^{\ast} $$\sigma \phi _j$$ \psi _j ^+$, $\psi _j ^-$
    本征方程${\bar{{L}}}{{\phi}}_i = \beta_i {\bar{{B}}}{{\phi}}_i $${\bar{{L}}}{{\phi}}_i = \beta_i {\bar{{B}}}{{\phi}}_i $${\bar{{L}}}{{\phi}}_i = \beta_i {\bar{{B}}}{{\phi}}_i$
    测试函数进行测试$\displaystyle\iiint \phi _j ^{\ast} [{\bar{{L} } }{{\phi} }_i-\beta_i {\bar{{B} } }{{\phi} }_i]{\rm{d} }v \!=\! 0$$\displaystyle\iiint \sigma \phi _j [{\bar{{L} } }{{\phi} }_i-\beta_i {\bar{{B} } }{{\phi} }_i]{\rm{d} }v \!=\! 0$$\displaystyle\iiint \psi _j \cdot [{\bar{{L} } }{{\phi} }_i \!-\! \beta_i {\bar{{B} } }{{\phi} }_i]{\rm{d} }v \!=\! 0$
    下载: 导出CSV
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    Pierce J R 1954 J. Appl. Phys. 25 179Google Scholar

    [2]

    Gould R W 1955 IRE Trans. Electron Devices PGED-2 37Google Scholar

    [3]

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    Alaeian H, Dionne J A 2014 Phys. Rev. A 89 033829Google Scholar

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    Shen Y, Deng X H, Chen L 2014 Opt. Express 22 19440Google Scholar

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    Liu Z Z, Zhang Q, Liu X L, Yao Y, Xiao J J 2016 Sci. Rep. 6 22711Google Scholar

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    Xu J, Chen Y 2015 Opt. Express 23 22619Google Scholar

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出版历程
  • 收稿日期:  2020-02-09
  • 修回日期:  2020-04-22
  • 上网日期:  2020-05-15
  • 刊出日期:  2020-08-05

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