PX284 - S1 - snell's law

\vec{E} = {\vec{E}}_{a} \exp (i ({\vec{k}}_{a} \cdot \vec{x} - ω_{a} t))

where, $a = {i, r, t}$

boundary conditions:
- matching phases: $ω_{i} = ω_{r} = ω_{t}$
- for all vectors, ${\vec{x}}_{b}$ in the boundary: ${\vec{k}}_{i} \cdot {\vec{x}}_{b} = {\vec{k}}_{r} \cdot {\vec{x}}_{b} = {\vec{k}}_{t} \cdot {\vec{x}}_{b}$ $⟹ ({\vec{k}}_{r} - {\vec{k}}_{i}) \cdot {\vec{x}}_{b} = 0 = ({\vec{k}}_{t} - {\vec{k}}_{i}) \cdot {\vec{x}}_{b}$
- components of ${\vec{k}}_{a}$ in the phase boundary must be equal, therefore the differences must be perpendicular in the boundary, ie: $\hat{n} \times {\vec{k}}_{i} = \hat{n} \times {\vec{k}}_{r} = \hat{n} \times {\vec{k}}_{t} ⟹ k_{i} \sin i = k_{r} \sin r = k_{t} \sin t$
the speed:

v = \frac{ω}{k} = \frac{c}{n}

where, $n$ is the refractive index

since $ω$ is same for $i, r$ and $t$ , $k \propto n ⟹ k_{i} = k_{r}$ , as they are both in $n_{1}$ , and therefore, $i = r$ (angles)
snell's law:

n_{1} \sin i = n_{2} \sin t

if $n_{1} > n_{2}$ , when $\sin i > n_{2} / n_{1}$ , total internal reflection is obtained