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    • 5.5: Decomposition of a Wave into TE and TM Components
      https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/05%3A_Wave_Reflection_and_Transmission/5.05%3A_Decomposition_of_a_Wave_into_TE_and_TM_Components
      \[\begin{align} \hat{\bf e}^i &= \left(\hat{\bf e}^i\cdot\hat{\bf e}_{\perp}\right)\hat{\bf e}_{\perp} \nonumber \\ &+ \left(\hat{\bf e}^i\cdot\hat{\bf e}^i_{\parallel}\right)\hat{\bf e}^i_{\parallel}...\begin{align} \hat{\bf e}^i &= \left(\hat{\bf e}^i\cdot\hat{\bf e}_{\perp}\right)\hat{\bf e}_{\perp} \nonumber \\ &+ \left(\hat{\bf e}^i\cdot\hat{\bf e}^i_{\parallel}\right)\hat{\bf e}^i_{\parallel} \nonumber \\ &+ \left(\hat{\bf e}^i\cdot\hat{\bf k}^i\right)\hat{\bf k}^i \end{align} \nonumber

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