# Confusion With Derivation of Fresnel Equations

## Main Question or Discussion Point 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 paralell 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!

Related Other Physics Topics News on Phys.org
Henryk
Gold Member
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.