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I Confusion With Derivation of Fresnel Equations

  1. Apr 29, 2016 #1
    Screen Shot 2016-04-29 at 22.05.38.png
    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!
     
  2. jcsd
  3. Apr 29, 2016 #2

    Henryk

    User Avatar
    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.
     
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