TEとTMモードのモード関数の面積分は先に示したように周回積分に変換できる$^\dagger$． \begin{eqnarray} \iint_S \VEC{e}_n^{\TE} \cdot \VEC{e}_l^{\TM} \ dS &=& -\iint_S \big( \nabla_t \Psi_n^{\TE} \times \nabla_t \Psi_l^{\TM} \big) \cdot \VEC{a}_z dS \nonumber \\ &=& -\oint_C \Psi_n^{\TE} \frac{\partial \Psi_l^{\TM}}{\partial \sigma} d\sigma \nonumber \\ &=& \oint_C \frac{\partial \Psi_n^{\TE}}{\partial \sigma} \Psi_l^{\TM} d\sigma \end{eqnarray}
TEモードの電界モード関数の微分とTMモードの電界モード関数の面積分の場合，上式において \(\VEC{e}_n^{\TE} \to \frac{\partial \VEC{e}_n^{\TE}}{\partial z}\) とすると， \(\Psi_n^{\TE} \to \frac{\partial \Psi_n^{\TE}}{\partial z}\)， さらにTMモードの境界条件\(\Psi_l^{\TM}=0\) (on C)ゆえ， \begin{eqnarray} \iint_S \frac{\partial \VEC{e}_n^{\TE}}{\partial z} \cdot \VEC{e}_l^{\TM} \ dS &=& -\iint_S \big( \nabla_t \frac{\partial \Psi_n^{\TE}}{\partial z} \times \nabla_t \Psi_l^{\TM} \big) \cdot \VEC{a}_z dS \nonumber \\ && \left( = -\oint_C \frac{\partial \Psi_n^{\TE}}{\partial z} \frac{\partial \Psi_l^{\TM}}{\partial \sigma} d\sigma \right) \nonumber \\ &=& \oint_C \frac{\partial}{\partial \sigma} \left( \frac{\partial \Psi_n^{\TE}}{\partial z} \right)\Psi_l^{\TM} d\sigma = 0 \end{eqnarray} よって， \begin{eqnarray} T_{I,ln}^{\mathrm{TM:TE}} &=& \iint \frac{\partial \VEC{e}^{\TE}_n}{\partial z} \cdot \VEC{e}^{\TM}_{l} dS = \oint_C \frac{\partial}{\partial \sigma} \left( \frac{\partial \Psi_n^{\TE}}{\partial z} \right)\Psi_l^{\TM} d\sigma = 0 \\ T^{\mathrm{TE:TM}}_{V,ln} &=& -\iint \VEC{e}^{\TM}_{n} \cdot \frac{\partial \VEC{e}^{\TE}_l}{\partial z} dS = -\oint_C \frac{\partial}{\partial \sigma} \left( \frac{\partial \Psi_l^{\TE}}{\partial z} \right)\Psi_n^{\TM} d\sigma = 0 \end{eqnarray}
一方，TMモードの電界モード関数の微分とTEモードの場合， \(\VEC{e}_l^{\TM} \to \frac{\partial \VEC{e}_l^{\TM}}{\partial z}\) とすると， \(\Psi_l^{\TM} \to \frac{\partial \Psi_l^{\TM}}{\partial z}\) より， \begin{eqnarray} \iint_S \VEC{e}_n^{\TE} \cdot \frac{\partial \VEC{e}_l^{\TM}}{\partial z} \ dS &=& -\iint_S \left( \nabla_t \Psi_n^{\TE} \times \nabla_t \frac{\partial \Psi_l^{\TM}}{\partial z} \right) \cdot \VEC{a}_z dS \nonumber \\ &=& -\oint_C \Psi_n^{\TE} \frac{\partial}{\partial \sigma} \left( \frac{\partial \Psi_l^{\TM}}{\partial z} \right) d\sigma \nonumber \\ &=& \oint_C \frac{\partial \Psi_n^{\TE}}{\partial \sigma} \frac{\partial \Psi_l^{\TM}}{\partial z} d\sigma \end{eqnarray} この場合はゼロにはならない． よって， \begin{align} T^{\mathrm{TE:TM}}_{V,ln} &= -\iint \VEC{e}^{\TE}_{n} \cdot \frac{\partial \VEC{e}^{\TM}_l}{\partial z} dS = -\oint_C \frac{\partial \Psi_n^{\TE}}{\partial \sigma} \frac{\partial \Psi_l^{\TM}}{\partial z} d\sigma = -T^{\mathrm{TE:TM}}_{I,nl} \end{align} $n$と$l$を入れ替えて， \begin{align} T^{\mathrm{TE:TM}}_{V,nl} &= -\iint \VEC{e}^{\TE}_{l} \cdot \frac{\partial \VEC{e}^{\TM}_n}{\partial z} dS = -\oint_C \frac{\partial \Psi_l^{\TE}}{\partial \sigma} \frac{\partial \Psi_n^{\TM}}{\partial z} d\sigma = -T^{\mathrm{TE:TM}}_{I,ln} \end{align} ここで，\(a=z(z)\)より， \begin{gather} \frac{\partial \VEC{e}_l}{\partial z} = \frac{\partial a}{\partial z} \cdot \frac{\partial \VEC{e}_l}{\partial a}, \ \ \ \ \ \frac{\partial \VEC{e}_n}{\partial z} = \frac{\partial a}{\partial z} \cdot \frac{\partial \VEC{e}_n}{\partial a} \end{gather}
$\dagger$ G. Figlia and G. G. Gentili, "On the Line-Integral Formulation of Mode-Matching Technique," IEEE Trans. Microwave Theory Tech., vol.MTT-50, no.2, pp.578-579, 2002.