# Deriving Fresnel Equations for parallel polarization using Maxwell's boundary conditions

I'd first like to preface this post with the "right answer" per wikipedia (I've seen the same answer elsewhere on more reputable websites) The thing I find trouble some is the cross terms such as $$n_2 \cos \theta_i$$ where indices of refraction are "mixed with the other angle".

I have meticulously applied Maxwell's boundary conditions namely that the tangential components of $$\vec{E}$$ must be continuous across the boundary. I'm positing that $$\frac{B_r}{B_i} = \frac{E_r}{E_i}$$ since to my knowledge their magnitudes are always related by constants. the aforementioned Boundary Condition can be expressed as the following in terms of $$\vec{B}$$ and $$\vec{k}$$;

$$\sqrt{\mu_i \epsilon_i} \left[ \frac{1}{k_i} \left(\vec{B_i} \times \vec{k_i} \right) + \frac{1}{k_r} \left(\vec{B_r} \times \vec{k_r} \right) \right] \times \hat{n} = \sqrt{\mu_t \epsilon_t} \left[ \frac{1}{k_t} \left( \vec{B_t} \times \vec{k_t} \right)\right] \times \hat{n}$$

Using the vector identity

$$\left( \vec{B} \times \vec{k} \right) \times \hat{n} = - \left( \vec{k} \times \vec{B} \right) \times \hat{n} = -\left( \hat{n} \cdot \vec{k} \right) \vec{B} + \left( \hat{n} \cdot \vec{B} \right)\vec{k} = -\left( \hat{n} \cdot \vec{k} \right) \vec{B}$$

(since the magnetic field is orthogonal to the normal)

I'm going to do this term by term while paying close attention to the geometry

$$\frac{\sqrt{\mu_i \epsilon_i}}{k_i}\left( \vec{B_i} \times \vec{k_i} \right) \times \hat{n} = - \frac{\sqrt{\mu_i \epsilon_i}}{k_i} \left( \hat{n} \cdot \vec{k_i} \right) \vec{B_i} = - \frac{\sqrt{\mu_i \epsilon_i}}{k_i} k_i \cos \left( \pi - \theta_i\right) \vec{B_i} = \sqrt{\mu_i \epsilon_i} \cos \theta_i \vec{B_i}$$

$$\frac{\sqrt{\mu_i \epsilon_i}}{k_r}\left( \vec{B_r} \times \vec{k_r} \right) \times \hat{n} = - \frac{\sqrt{\mu_i \epsilon_i}}{k_r} \left( \hat{n} \cdot \vec{k_r} \right) \vec{B_r} = - \sqrt{\mu_i \epsilon_i} \cos \theta_r \vec{B_r}$$

$$\frac{\sqrt{\mu_t \epsilon_t}}{k_t}\left( \vec{B_t} \times \vec{k_t} \right) \times \hat{n} = - \frac{\sqrt{\mu_t \epsilon_t}}{k_t} \left( \hat{n} \cdot \vec{k_t} \right) \vec{B_t} = - \frac{\sqrt{\mu_t \epsilon_t}}{k_t} k_t \cos \left( \pi - \theta_t\right) \vec{B_t} = \sqrt{\mu_t \epsilon_t} \cos \theta_t \vec{B_t}$$

After invoking $$\theta_i = \theta_r$$ and $$\vec{B_i} + \vec{B_r} = \vec{B_t}$$ we have

$$\sqrt{\mu_i \epsilon_i} \cos \theta_i \vec{B_i} - \sqrt{\mu_i \epsilon_i} \cos \theta_i \vec{B_r} = \sqrt{\mu_t \epsilon_t} \cos \theta_t \vec{B_i} + \sqrt{\mu_t \epsilon_t} \cos \theta_t \vec{B_r}$$

Collecting like terms of $$\vec{B_i}$$ on one side and $$\vec{B_r}$$ on the other we have

$$- \left[ \sqrt{\mu_i \epsilon_i} \cos \theta_i + \sqrt{\mu_t \epsilon_t} \cos \theta_t \right] \vec{B_r} = \left[ \sqrt{\mu_t \epsilon_t} \cos \theta_t - \sqrt{\mu_i \epsilon_i} \cos \theta_i \right] \vec{B_i}$$

Taking the dot product of each side with itself

$$\left( \sqrt{\mu_i \epsilon_i} \cos \theta_i + \sqrt{\mu_t \epsilon_t} \cos \theta_t\right)^2 B_r^2 = \left( \sqrt{\mu_t \epsilon_t} \cos \theta_t - \sqrt{\mu_i \epsilon_i} \cos \theta_i \right)^2 B_i^2$$

After taking

(1) Dividing and taking the square root of the previous line

(2) Invoking the assumption that $$\frac{B_r}{B_i} = \frac{E_r}{E_i}$$

(3) Using $$\eta = \sqrt{\mu \epsilon}$$

(4) Using $$\frac{E_t}{E_r} = 1 + \frac{E_r}{E_i}$$

We find the Fresnel equations for parallel polarization are

$$\left(\frac{E_r}{E_i} \right)_{parallel} = \frac{\eta_t \cos \theta_t - \eta_i \cos \theta_i}{ \eta_i \cos \theta_i + \eta_t \cos \theta_t}$$

$$\left( \frac{E_t}{E_i} \right)_{parallel} = \frac{2 \eta_t \cos \theta_t}{\eta_i \cos \theta_i + \eta_t \cos \theta_t}$$

My answers conflict with wikipedia which corroborates lecture notes from both Brown University and MIT, but I wasn't able to understand\follow the derivations in those notes. The most startling concern with my answer is the lack of "cross terms" such as $$\eta_i \cos \theta_t$$ and $$\eta_t \cos \theta_i$$ and the like which are present in the first picture I posted. At the same time I can't see anything wrong with my math and I can't fathom how these "cross terms" appear.

If anyone could provide helpful hints or point me in the right direction I would be greatly appreciative.