円形導波管モードの遮断波数

　円形導波管（半径 $a$ ）のTE $_{m n}$ ，TM $_{m n}$ モードの遮断波長 $λ_{c}$ を低次モードから順に示すと次のようになる．

\begin{aligned} No. & Mode & χ_{m n}^{'} or χ_{m n} & λ_{c} \\ 1 & {TE}_{11} & χ_{11}^{'} = 1.841184 & 3.4126 a \\ 2 & {TM}_{01} & χ_{01} = 2.404826 & 2.6127 a \\ 3 & {TE}_{21} & χ_{21}^{'} = 3.054237 & 2.0572 a \\ 4, 5 & {TM}_{11} & χ_{11} = 3.831706 & 1.6398 a \\ 4, 5 & {TE}_{01} & χ_{01}^{'} = 3.831706 & 1.6398 a \\ 6 & {TE}_{31} & χ_{31}^{'} = 4.201189 & 1.4956 a \\ 7 & {TM}_{21} & χ_{21} = 5.135622 & 1.2235 a \\ 8 & {TE}_{41} & χ_{41}^{'} = 5.317553 & 1.1816 a \\ 9 & {TE}_{12} & χ_{12}^{'} = 5.331443 & 1.1785 a \\ 10 & {TM}_{02} & χ_{02} = 5.520078 & 1.1382 a \\ 11 & {TM}_{31} & χ_{31} = 6.380162 & 0.9848 a \\ 12 & {TE}_{51} & χ_{51}^{'} = 6.415616 & 0.9794 a \\ 13 & {TE}_{22} & χ_{22}^{'} = 6.706133 & 0.9369 a \\ 14, 15 & {TM}_{12} & χ_{12} = 7.015587 & 0.8956 a \\ 14, 15 & {TE}_{02} & χ_{02}^{'} = 7.015587 & 0.8956 a \\ 16 & {TE}_{61} & χ_{61}^{'} = 7.501266 & 0.8376 a \\ 17 & {TM}_{41} & χ_{41} = 7.588342 & 0.8280 a \\ 18 & {TE}_{32} & χ_{32}^{'} = 8.015237 & 0.7839 a \\ 19 & {TM}_{22} & χ_{22} = 8.417244 & 0.7465 a \\ 20 & {TE}_{13} & χ_{13}^{'} = 8.536316 & 0.7361 a \\ 21 & {TE}_{71} & χ_{71}^{'} = 8.577836 & 0.7325 a \\ 22 & {TM}_{03} & χ_{03} = 8.653728 & 0.7261 a \\ 23 & {TM}_{51} & χ_{51} = 8.771484 & 0.7163 a \\ 24 & {TE}_{42} & χ_{42}^{'} = 9.282396 & 0.6769 a \\ 25 & {TE}_{81} & χ_{81}^{'} = 9.647422 & 0.6513 a \\ 26 & {TM}_{32} & χ_{32} = 9.761023 & 0.6437 a \\ 27 & {TM}_{61} & χ_{61} = 9.936110 & 0.6324 a \\ 28 & {TE}_{23} & χ_{23}^{'} = 9.969468 & 0.6302 a \\ 29, 30 & {TM}_{13} & χ_{13} = 10.173468 & 0.6176 a \\ 29, 30 & {TE}_{03} & χ_{03}^{'} = 10.173468 & 0.6176 a \end{aligned}

$m = 1$ のモードを $n \leq 5$ についてまとめると次のようになる．軸対称のテーパ導波管に基本TE $_{11}$ モードが入射した場合，このような高次モードが発生するが， $m \neq 1$ のモードは発生しない．

\begin{aligned} No. & Mode & χ_{m n}^{'} & No. & Mode & χ_{m n} \\ 1 & {TE}_{11} & χ_{11}^{'} = 1.841184 & 4, 5 & {TM}_{11} & χ_{11} = 3.831706 \\ 9 & {TE}_{12} & χ_{12}^{'} = 5.331443 & 14, 15 & {TM}_{12} & χ_{12} = 7.015587 \\ 20 & {TE}_{13} & χ_{13}^{'} = 8.536316 & 29, 30 & {TM}_{13} & χ_{13} = 10.173468 \\ 37 & {TE}_{14} & χ_{14}^{'} = 11.706005 & 48, 49 & {TM}_{14} & χ_{14} = 13.323692 \\ 60 & {TE}_{15} & χ_{15}^{'} = 14.863589 & 73, 74 & {TM}_{15} & χ_{15} = 16.470630 \end{aligned}