Confusion With Derivation of Fresnel Equations

[bananabandana]

Okay, so I'm working with the diagrams above. $i$ denotes incident, $r$ reflected, and $t$ transmitted.

-We're working in two HIL dielectrics. Incoming and outgoing waves are of form $Aexp[i(\vec{k}\cdot\vec{r}- \omega t)$. As I understand it, Maxwell's equations give four boundary conditions for this - Let:

• $\vec{\hat{n}}$ be a unit normal vector to the interface,
• $\rho{SC}$ is the surface free charge
• Subscript 1,2 refers to medium 1,2

Let's just look at the p-case:

• The parallel component of the $\vec{E}$ field must be continuous over the boundary.
• But the perpendicular component is also continuous over the boundary, since we know:

(1) $$ \vec{\hat{n}} \cdot(\vec{D_{1}} -\vec{D_{2}}) = \rho_{SC} $$
and in a dielectric, $\rho_{SC}=0$, and since the dielectric is HIL - $|\vec{D}| = \epsilon|\vec{E}|$ , i.e:

(2) $$ \vec{\hat{n}} \cdot \epsilon ( \vec{E_{1}} -\vec{E_{2}} ) =0 $$

- For the wave to be continuous parallel to the boundary, we must have $\theta_{incident} = \theta_{reflected}$, as shown, and also that $n_{1}sin(\theta^{i}) = n_{2} sin(\theta^{r})$. [Since the exponential terms must be equal]. So we can just work in terms of the amplitudes:

Then, for parallel continuity:

(3) $$ (E_{0}^{i}+E_{0}^{r})cos \theta^{i} = E_{0}^{t}cos \theta^{t} $$

And for perpendicular continuity:

(4) $$ (E_{0}^{r}-E_{0}^{i}) sin \theta^{i} = -E_{0}^{t} sin \theta^{t} \implies E_{0}^{t} sin \theta^{t} = (E_{0}^{i} -E_{0}^{r}) sin \theta^{i} $$

Solving these simultaneously, we arrive at the result:

(5) $$ r = \frac{E_{0}^{r}}{E_{0}^{i}} = \frac{ cotan \theta^{t} - cotan \theta^{i} }{ cotan \theta^{i} + cotan \theta^{2} } = \frac{ cos\theta^{t} sin \theta^{i} -sin \theta^{t} cos \theta^{i} }{ cos \theta^{t} sin \theta^{i} + sin \theta^{i} cos \theta^{t}} $$

Except this is wrong, since if $n_{1} sin \theta^{i} = n_{2} sin \theta^{t}$, then we can rewrite (5) as:

(6) $$ r= \frac{n_{2}cos \theta^{t} - n_{1} cos \theta^{i} }{ n^{2}cos\theta^{t}+n_{1}cos \theta^{i}} $$

Which is not the result my lecturer gets! Can someone explain where I made my mistake? Would be very grateful!

 

[Henryk]

The mistake is in equation 2. Different media have different dielectric constants,
Rewrite the equation 2 as $\vec n⋅ (\epsilon_1 \vec E_1 −\epsilon_2 \vec E_2)=0$
Refractive index is related to dielectric constant by $n_1 = \sqrt{\epsilon_1 \mu_1}$ and $n_2 = \sqrt{\epsilon_2 \mu_2}$
Assuming non-magnetic materials, $\mu_1 = \mu_2 = 1$.
Then correct equations 4 and 5.