The transmitted unit vectors are found as:
:<math> \mathbf{ \hat{k}'' = \hat{n} \times a - \sqrt{1-a \cdot a} \; \hat{n} } </math> :<math> \mathbf{ \hat{u}_{\perp}'' = \hat{u}_{\perp} } </math>Â :<math> \mathbf{ \hat{u}_{\|}'' = \hat{u}_{\perp}'' \times \hat{k}'' } </math><!--[[File:frml3.png]]-->
where
:<math> \mathbf{a} = (k_1/k_2) \mathbf{\hat{k} \times \hat{n}}</math>
Â
:<math> k_1 = k_0 \sqrt{\varepsilon_1 \mu_1} </math>
Â
:<math> k_2 = k_0 \sqrt{\varepsilon_2 \mu_2} </math>
Â
:<math> \eta_1 = Z_0 \sqrt{\mu_1 / \varepsilon_1} </math>
Â
:<math> \eta_2 = Z_0 \sqrt{\mu_2 / \varepsilon_2} </math>
Â
Â
:<math> \sin\theta'' = \frac{k_1}{k_2}\sin\theta \text{ if } \sin\theta \le k_2/k_1</math>
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The reflection coefficients at the interface are calculated for the two parallel and perpendicular polarizations as:
:<math>R_{\|} = \frac{ \eta_2(\mathbf{ \hat{k}'' \cdot \hat{n} }) - \eta_1(\mathbf{ \hat{k} \cdot \hat{n} }) } { \eta_2(\mathbf{ \hat{k}'' \cdot \hat{n} }) + \eta_1(\mathbf{ \hat{k} \cdot \hat{n} }) }Â = \frac{\eta_2 \cos\theta'' - \eta_1 \cos\theta} {\eta_2 \cos\theta'' + \eta_1 \cos\theta}Â = \frac{Z_{2\|} - Z_{1\|}} {Z_{2\|} + Z_{1\|}}</math>Â Â :<math>R_{\perp} = \frac{ \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) - \eta_1(\mathbf{ \hat{k}'' \cdot \hat{n} }) } { \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) + \eta_1(\mathbf{ \hat{k}'' \cdot \hat{n} }) }Â = \frac{\eta_2 / \cos\theta'' - \eta_1 / \cos\theta} {\eta_2 / \cos\theta'' + \eta_1 / \cos\theta}Â = \frac{Z_{2\perp} - Z_{1\perp}} {Z_{2\perp} + Z_{1\perp}}</math><!--[[File:frml6.png]]-->
=== Penetration Through Thin Walls Or Surfaces ===