:<math> \mathbf{ \hat{u}_{\|} = \hat{u}_{\perp} \times \hat{k} } </math>
<!--[[File:frml1.png]]-->
The reflected unit vectors are found as:
:<math> \mathbf{ \hat{u}_{\|}' = \hat{u}_{\perp}' \times \hat{k}' } </math>
<!--[[File:frml2.png]]-->
The transmitted unit vectors are found as:
:<math> \mathbf{ \hat{u}_{\|}^{\prime\prime} = \hat{u}_{\perp}^{\prime\prime} \times \hat{k}^{\prime\prime} } </math>
<!--[[File:frml3.png]]-->
where
:<math> \sin\theta^{\prime\prime} = \frac{k_1}{k_2}\sin\theta \text{ if } \sin\theta \le k_2/k_1</math>
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[[File:frml4.png]]
[[File:frml5.png]]
-->
The reflection coefficients at the interface are calculated for the two parallel and perpendicular polarizations as:
:<math> R_{\perp} = \frac { \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) - \eta_1(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) } { \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) + \eta_1(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) } = \frac{\eta_2 / \cos\theta^{\prime\prime} - \eta_1 / \cos\theta} {\eta_2 / \cos\theta^{\prime\prime} + \eta_1 / \cos\theta} = \frac{Z_{2\perp} - Z_{1\perp}} {Z_{2\perp} + Z_{1\perp}} </math>
<!--[[File:frml6.png]]-->
== Penetration through Thin Walls or Surfaces ==