Acta Physica Sinica, Volume. 69, Issue 15, 154206-1(2020)
Fig. 1. Symmetry relations between the forward and backward propagating modes: (a) Chiral symmetry; (b) time reversal symmetry; (c) parity symmetry.
Fig. 2. Real part of effective mode indices
versus
: (a)
in core layer 1 and
in core layer 2; (b)
in core layer 1 and
in core layer 2.
Fig. 3. Real part and imaginary part of effective mode indices
versus
: (a), (c) Real part of effective mode indices
; (b), (d) imaginary part of effective mode indices
.
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
; (c) real part of effective mode indices
; (d) imaginary part of effective mode indices
; (e)−(h) the real/imaginary part of
obtained from fullwave simulation is shown for the modes
Table 1. Symmetric relation of original field and adjoint field in the reciprocal waveguides with
View tableView in ArticleTable 1. Symmetric relation of original field and adjoint field in the reciprocal waveguides with
$\beta_i$ $-\beta_i$ $({\bar{{L}}}, {\bar{{B}}})$ $\left[\beta_i, {{\phi}}_i\right]$ $\left[-\beta_i, {{\psi}}_i\right]$ $({\bar{{L}}}^{\rm{a}}, {\bar{{B}}}^{\rm{a}})$ $\left[\beta_i, {{\psi}}_i\right]$ $\left[-\beta_i, {{\phi}}_i\right]$
Table 2.
Symmetry relations of original field and adjoint field in the reciprocal waveguides.
互易波导中原始场和伴随场之间的对称关系
View tableView in ArticleTable 2.
Symmetry relations of original field and adjoint field in the reciprocal waveguides.
互易波导中原始场和伴随场之间的对称关系
对称关系 算符 对称性关系 约束条件 手征对称 ${\sigma}$ ${{{\psi}}}_i({{r}}) = {\bar{\sigma}}{{{\phi}}}_i({{r}})$ ${{{\varepsilon}}}_{\rm r}^{zt} = {{{\varepsilon}}}_{\rm r}^{tz} = 0$ ${{{\mu}}}_{\rm r}^{zt} = {{{\mu}}}_{\rm r}^{tz} = 0$ ${\bar{ \chi}} = 0$ 时间反演对称 ${\cal{T}}$ ${{{\psi}}}_i({{r}}) = {\bar{\sigma}}({{{\phi}}}_i({{r}}))^*$ ${\bar{{{\varepsilon}}}}_{\rm r}$ ${\bar{{{\mu}}}}_{\rm r}$ ${\bar{ \chi}}$ 宇称对称 ${\cal{P}}$ ${{{\psi}}}_i({{r}}) = {\bar{\sigma}}{{{\phi}}}_i(-{{r}})$ ${\bar{{{\varepsilon}}}}_{\rm r}({{r}}) = {\bar{{{\varepsilon}}}}_{\rm r}(-{{r}})$ ${\bar{{{\mu}}}}_{\rm r}({{r}}) = {\bar{{{\mu}}}}_{\rm r}(-{{r}})$ ${\bar{ \chi}}({{r}}) = -{\bar{ \chi}}(-{{r}})$
Table 3. Comparison between CCMT, GCMT and GCMF.
View tableView in ArticleTable 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 $\displaystyle\iiint \phi _j ^{\ast} [{\bar{{L} } }{{\phi} }_i-\beta_i {\bar{{B} } }{{\phi} }_i]{\rm{d} }v \!=\! 0$ $\displaystyle\iiint \sigma \phi _j [{\bar{{L} } }{{\phi} }_i-\beta_i {\bar{{B} } }{{\phi} }_i]{\rm{d} }v \!=\! 0$ $\displaystyle\iiint \psi _j \cdot [{\bar{{L} } }{{\phi} }_i \!-\! \beta_i {\bar{{B} } }{{\phi} }_i]{\rm{d} }v \!=\! 0$
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Yun-Tian Chen, Jing-Wei Wang, Wei-Jin Chen, Jing Xu.
Received: Feb. 9, 2020
Accepted: --
Published Online: Dec. 30, 2020
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