PX284 - S2c - total internal reflection

considering $n_{1} > n_{2}$
using snell's law:

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

if $\sin i > n_{2} / n_{1} ⟹ \sin t > 1 :$ there is no solution for $t$
there is no transmitted light, and there is total internal reflection for $i \geq i_{c}$ , where, $\sin i_{c} = n_{2} / n_{1}$
eg: for glass-to-air, $i_{c} \approx 41.8 °$
the transmitted wave:

{\vec{E}}_{t} \propto \exp ({\vec{k}}_{t} \cdot \vec{r} - ω t)

considering the components of the wavevector:

{\vec{k}}_{t} = k_{t x} \hat{x} + k_{t z} \hat{z}

$ω$ fixes $| \vec{k} |$ via:

k = \frac{ω n}{c}

$k_{∥}$ must be equal on both sides of the boundary:

k_{t x} = k_{i x} = k_{i} \sin i

\begin{matrix} | {\vec{k}}_{t} |^{2} = k_{t x}^{2} + k_{t z}^{2} \\ k_{t z}^{2} = | {\vec{k}}_{t} |^{2} - k_{i}^{2} \sin^{2} i \end{matrix}

this is less than $0$ if $k_{i} \sin i > | {\vec{k}}_{t} |$

k_{t z} = \pm i \sqrt{k_{i}^{2} \sin^{2} i - | {\vec{k}}_{t} |^{2}}

note: $i$ represents both $\sqrt{- 1}$ and the incidence angle, but incidence angle is always an argument of $\sin$ , or a subscript
to avoid confusion, defining $κ :$

\begin{aligned} κ^{2} & = k_{i}^{2} \sin^{2} i - | {\vec{k}}_{t} |^{2} \\ = \frac{ω^{2}}{c^{2}} (n_{1}^{2} \sin^{2} i - n_{2}^{2}) \end{aligned}

∴ {\vec{E}}_{t} \propto \exp (- κ z) \exp (i (k_{t x} x - ω t))

this is called an 'evanescent' wave, which means it is decaying, $\exp (- κ z)$
eg: for glass/air, $n_{1} = 1.5$ and $n_{2} = 1$
the critical angle, $i_{c} \approx 41.8 °$
considering a $45 °$ reflector:

κ = \frac{2 π}{λ_{0}} \sqrt{{1.5}^{2} \sin^{2} 45 ° - 1^{2}} \approx \frac{2.2}{λ_{0}}

the z-dependence:

\exp (- κ z) \approx \exp (\frac{- z}{10.45 λ_{0}})