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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)

enter image description here

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.

enter image description here

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.

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