= Evanescent penetration depth
{title2=$\delta=1/\kappa$}
Define the amplitude <evanescent penetration depth> as the distance at which the <electric field> <wave amplitude> falls to $1/e$ of its surface value:
$$
\delta=\frac1\kappa=\frac{\lambda_0}{2\pi\sqrt{n_1^2\sin^2\theta-n_2^2}},\qquad
\frac{|\mathbf E(z)|}{|\mathbf E(0)|}=e^{-z/\delta}.
$$
The squared-field envelope decays twice as fast:
$$
\frac{|\mathbf E(z)|^2}{|\mathbf E(0)|^2}=e^{-2z/\delta},\qquad d_{1/e}=\frac\delta2.
$$
The same normalized <exponential decay> describes the tangential average <Poynting vector> for the <TE polarization> above. An intensity penetration depth convention often means $d_{1/e}$; it must not be confused with amplitude depth $\delta$ or interpreted as outgoing normal <light intensity>.
For the stipulated $n_1=1.5$, $n_2=1$, $\lambda_0=633\,\mathrm{nm}$ and $\theta=50^\circ$,
$$
\begin{aligned}
k_0&\approx0.00992604314\,\mathrm{nm}^{-1},&
\kappa&\approx0.00561812463\,\mathrm{nm}^{-1},\\
\delta&\approx177.99534\,\mathrm{nm},&
d_{1/e}&\approx88.99767\,\mathrm{nm}.
\end{aligned}
$$
At $z=200\,\mathrm{nm}$ the normalized <wave amplitude> is approximately $0.325099$, and its squared envelope is approximately $0.105689$. At $z=\delta$, these become $e^{-1}$ and $e^{-2}$, respectively. The quantity inside the <square root> is dimensionless, so $\kappa$ has inverse-length units and both depths have length units.
\Image[/evanescent-field-decay-and-penetration-depth.png]
{title=Constructed evanescent-field model at 633 nm: amplitude and squared-field decay at 50 degrees, and their penetration depths as incidence increases above the critical angle}
The sharp angular dependence has a simple explanation. For $\theta=\theta_c+\varepsilon$ with $\varepsilon>0$ small and measured in <radians>, a first-order <Taylor expansion> gives
$$
\delta\sim\frac{\lambda_0}{2\pi\sqrt{2n_1^2\sin\theta_c\cos\theta_c}}\,\varepsilon^{-1/2}.
$$
Thus $\delta\to\infty$ as $\theta\downarrow\theta_c$: exponential confinement disappears continuously. With fixed <refractive indices>, $\delta$ decreases as the <angle of incidence> increases and approaches $\lambda_0/(2\pi\sqrt{n_1^2-n_2^2})\approx90.1091\,\mathrm{nm}$ as $\theta\to90^\circ$. This is a <limit> of the normalized depth; exactly grazing incidence supplies zero incident normal <power>. Real beam widths, angular spreads and <optical dispersion> matter when applying these ideal <plane electromagnetic wave> predictions near the threshold.
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